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How can a spiral curriculum support you and your learners?

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How can a spiral curriculum support you and your learners?

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Where can you see the spiral curriculum in Maths - No Problem! textbooks?

The spiral curriculum is built into the Maths - No Problem! teaching approach. It's an intuitive, consistent lesson structure that saves time needed for lesson prep, and gives you peace of mind that concepts are being revisited and reinforced.

Here's how it works.
1. Topics are revisited to consolidate understanding
Throughout the Maths - No Problem! Primary Series, learners revise topics and ideas on a number of occasions throughout chapters and year groups.

The books ensure that children are guided, step by step, through a series of ideas in a structured and sequenced way. In this example we can see how the development of fractions progresses through the years.

In these Year 1 to Year 6 examples, you can see certain features that encourage children to access new learning. In the initial lessons of the chapter, all examples are using rectangles and squares. This allows learners to use easily available materials like paper to model these situations. You'll also see that if the prior learning is not secure, there is an opportunity to revisit these ideas without having to provide a completely separate lesson. There is also reinforcement of the language the children have been developing over the years.
2. Topics increase in complexity
A spiral curriculum needs to continually build on the new ideas children have learned. In order to do this, the level of difficulty and complexity needs to increase until that idea is mastered. You can see how this is developed by looking at the lesson titles from the Maths - No Problem! Textbook 4A, Chapter 2.

In Lesson 1, learners revisit the previous year's objective, giving the teacher an opportunity to assess the prerequisite knowledge for the chapter, but also so children can remind themselves of what they have already learned.

In Lesson 2, learners are introduced to adding numbers that are larger than what they've previously used, but without having to rename. This allows children to attend to the new idea being developed. You'll see that because it's a skill they've practiced many times, it's likely they will understand this idea in a relatively short time. That is why there is only a single lesson on this topic.

Lessons 3 to 5 build on the previous lessons, but now include renaming. This is a more complex skill, particularly as there are more numbers that may require renaming than learners have previously experienced.

In Lessons 6 and 7, the process moves to mental methods where learners need to visualise the problem and associated calculation. The complexity in Lessons 6 and 7 increases as there are not the same visual prompts we find in a written method. In order to manage the mental process, the number of aspects the learner needs to attend to to solve the problem must decrease. This, however, has been carefully managed and supported in the previous lessons.
3. New learning is related to previous learning
As the children develop, practice and embed their understanding of a new idea, they no longer have to attend to that particular part of their work. They are ready to learn something new, now that they have a prerequisite knowledge to learn it. In the examples below, we can see two levels of variation that are needed to have an effective spiral curriculum.

In the first instance, we can see the variation between the questions supporting the children to use a 'making ten' strategy, with questions of increasing complexity. However, the step before this was crucial. The example used pots of flowers: children learn that they can only add the same thing, using the same noun. The ability to do this starts in the early years when they may see one boy and one girl but two children. This idea is further developed, as we can see in the next example. Lulu reminds us that the denominators are both ninths - they are the same noun, which allows a learner to carry out the addition calculation with the same simplicity as if they were adding one apple to four apples. The concept of adding parts is new and provides an increasing level of difficulty, but the ideas that were developed earlier allow children access to the new learning.
4. Learners' mathematical understanding increases
Another key aspect of a mastery approach is that everyone can learn. If a learner starts at the right place and is given the right amount of time, they are capable of learning anything. If we approach our lessons with this in mind, we need to be ready to respond to a child's changing understanding of new ideas. Their ability will change as new ideas are understood and practiced. This is true of everyone. We may have the ability to make a cake, but not the ability to make a three-tiered wedding cake. We're not saying that we lack the inherent ability to do so, we just haven't developed the ability to make such a cake yet. But if we choose to, there are many skills we need to learn and much practice we need to undertake, often under the guidance of an expert. And we need to be prepared to put the effort in to learn.

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How can a spiral curriculum support you and your learners? | MNP Blog

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The spiral curriculum is built into the Maths - No Problem! teaching approach. It's an intuitive, consistent lesson structure that saves time needed for lesson prep, and gives you peace of mind that concepts are being revisited and reinforced. Here's how it works. 1. Topics are revisited to consolidate understanding Throughout the Maths - No Problem!

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Applying Cognitive Psychology to Enhance Educational Practice

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Applying Cognitive Psychology to Enhance Educational Practice

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Applying Cognitive Psychology to Enhance Educational Practice

The primary goal of this research, which is funded by the James S. McDonnell foundation, is to promote learning and memory performance within educational contexts through the investigation of principles in cognitive psychology. Studies address issues of transfer-appropriate and material-appropriate processing between encoding and retrieval. Applying tests in order to enhance learning and determining the desirable amount and timing of feedback regarding an individual's memory performance are methods that are currently under investigation.

The overlying theme of "desirable difficulties," first introduced by Robert Bjork (1994), is also explored through manipulations in the spacing of learning events and the study schedule produced by interleaving various to-be-learned items, such as English-Swahili translated word pairs or prose materials. Interleaving occurs when a to-be-learned target item is initially presented and followed by different to-be-learned items prior to the target's subsequent presentations. An interleaving schedule used during the presentation of a painting matched with the name of the artist has been shown to lead to better performance on later recognition tests when compared with a massing presentation schedule in which each painting and artist name was presented back-to-back, with one presentation immediately followed by the next presentation of a different painting by the same artist (Kornell & Bjork, 2008).

Studies have also looked at the effectiveness of similar choices used in multiple choice tests for future test performance as well as the act of generating items when they are presented with missing letters. Additional research is targeted towards understanding the role of an individual preparing to teach on that individual's subsequent learning and test performance.

This line of work is also directed toward understanding the mechanisms behind metacognitive awareness of learning. Most people are inaccurate in measuring their own knowledge, through judgments of learning, because they mistakenly rely on the immediate access to knowledge in order to determine the long-term memory retention and the transfer of such knowledge to different contexts. The goal of these studies is to determine the type of instructions and study conditions that will foster accurate judgments of learning, which can lead to better predictions of future performance and optimal self-initiated study practices.

Table of Contents

I. Retrieval as a Memory Modifier

We draw upon our memories for a variety of reasons: We may recall something in order to relay a story to a friend. We may take a test to double-check that we know important material. Or, we may recall information in order to help us remember other details or to solve a problem. However, a much underappreciated fact is that retrieval itself can affect our memories.

Testing effect

Taking a test often does more than assess knowledge; tests can also provide opportunities for learning. When information is successfully retrieved from memory, its representation in memory is changed such that it becomes more recallable in the future (e.g., R. A. Bjork, 1975); and this improvement is often greater than the benefit resulting from additional study (Roediger & Karpicke, 2006). Interestingly, taking a test can modify memory for information that was not explicitly tested initially (provided that the untested information is related to the tested information in certain ways; Anderson, R. A. Bjork, & E. L. Bjork, 1994; Chan, McDermott, & Roediger, 2006; Hamaker, 1986). Sometimes later recall of this untested information is improved (see, e.g., Hamaker, 1986), but sometimes it is impaired (see, e.g., Anderson et al., 1994), often dependent upon the type of relationship existing between tested and untested information (e.g., Little, Storm, & E. L. Bjork, 2011).

Currently, we are exploring both the benefits and the costs associated with this type of selective testing. This section will focus on situations in which testing some information improves the later recall of untested related information (Note: for information regarding the impairment of related information as a consequence of retrieval, see Retrieval-induced Forgetting).

Recently, we have investigated the potential use of multiple-choice tests as a tool to improve the later recall of untested related information. For example, the recall of initially untested information (e.g., Titan, given the cued-recall question, What is the largest moon of Saturn?) is impaired on a final test as a consequence of trying to recall the answer to a competitive question (e.g., What is the second largest moon of Saturn?, Answer: Rhea) on an earlier cued-recall test; but, the later recall of this competitive information (Titan) can be improved if it is used as a competitive incorrect answer choice on an initial multiple-choice test (e.g., What is the second largest moon of Saturn? A. Titan, B. Rhea, C. Enceladas, D. Mimas). Little and E. L. Bjork (2010) argue that when students do not know the answer to a multiple-choice question, they may try to retrieve information pertaining to why the other answers are incorrect in order to reject them and choose the correct answer. It is this type of processing leads to the spontaneous recall of information pertaining to those incorrect alternatives, thus leading the multiple-choice test to serve as a learning event for both the tested and untested information.

In addition to investigating situations in which tests are used after studying some to-be-learned information, we are investigating situations in which tests are used prior to studying (i.e., pretests). Although pretest performance is poor (because students have not been exposed to the relevant information prior to testing), pretests appear to be beneficial for subsequent learning (e.g., Kornell, Hays, & R. A. Bjork, 2009). We have also investigated the effect of multiple-choice pretests on learning for both pretested and untested related information (Little & E. L. Bjork, 2011). We believe that multiple-choice pretesting is more beneficial than is cued-recall pretesting because the multiple-choice pretest directs attention more broadly during subsequent study-not just to information pertaining to the question, but also to information pertaining to the alternatives.
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Retrieval-induced forgetting

Memory cues, whether categories, positions in space, scents, or the name of a place, are often linked to many items in memory. For example, the category FRUIT is linked to dozens of exemplars, such as ORANGE, BANANA, MANGO, KIWI, and so on. When forced to select from memory a single item associated to a cue (e.g., FRUIT: OR____), what happens to other items associated to that general, organizing cue? Using the retrieval-practice paradigm, we and other researchers have demonstrated that access to those associates is reduced. Retrieval-induced forgetting, or the impaired access to non-retrieved items that share a cue with retrieved items, occurs only when those associates compete during the retrieval attempt (e.g., access to BANANA is reduced because it interferes with retrieval of ORANGE, but MANGO is unaffected because it is too weak of an exemplar to interfere; Anderson, R.A. Bjork, & E. L. Bjork, 1994, Experiment 3). We argue for retrieval-induced forgetting as an example of goal-directed forgetting because it is thought to be the result of inhibitory processes that help facilitate the retrieval of the target by reducing access to competitors. In this way, retrieval induced forgetting is an adaptive aspect of a functional memory system.

In recent years, we have explored this phenomenon in a variety of ways. For example, we found that items that suffer from retrieval-induced forgetting benefit more from relearning than control items (Storm, E. L. Bjork, & R. A. Bjork, 2008). We have also demonstrated that retrieval success is not a necessary condition for retrieval induced forgetting to occur. That is, when participants are prompted to retrieve with cues that have no possible answer (FRUIT: WO____, rather than the standard, FRUIT: OR_____), access to competing items (BANANA) is impaired, as demonstrated on a final recall test. Furthermore, we are currently exploring the impact of variations of the type of cue support provided for retrieval attempts (FRUIT: OR_____; FISH: ____ORE; WEAPONS: DAGG_____). Our efforts in this domain currently rest on testing various assumptions of theoretical accounts of retrieval induced forgetting.
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New Theory of Disuse

Sometimes people cannot access information that was well learned earlier (e.g., the address of the house where they grew up). And students find that although they can recall information over and over again the day before a test, they cannot always recall it at the time of examination. Finally, sometimes people cannot recall information at one point in time, but can recall it later. In looking at these situations, it seems that our memories work in strange and unpredictable ways. The function of our memories, however, may be predictable. The New Theory of Disuse (R. A. Bjork & E. L. Bjork, 1992) posits that there are two indices of memory strength: storage strength (SS) and retrieval strength (RS). Storage strength is how well learned something is; retrieval strength is how accessible (or retrievable) something is. To illustrate, imagine four possible situations. If something is well learned (e.g., the address where you have lived for several years), it has both high SS and high RS: You know it well and can retrieve it readily. The address of a friend that you visited for the first time this afternoon, however, may only have high RS (and low SS) because the address, although practiced recently, was not well learned. Thus, although you know the address now, you will be unlikely to be able to recall it in a few days because RS will decrease over time, especially for information with low SS. Sometimes information has high SS (due to it having been well learned), but cannot be retrieved (e.g., the address where you lived as a child). If you were provided with this address again, however, you would have the feeling that that information was somewhere in the recesses of your memory, and in fact, you would be likely to relearn it very quickly. Finally, information can have both low RS and low SS. This information would include things that you heard in class earlier today, but did not learn well and cannot recall now.

The New Theory of Disuse postulates that RS and SS interact in interesting ways. For example, the more SS information has, the bigger the boost in RS it will receive as a consequence of restudy (e.g., you can relearn a childhood address much more quickly than a new address, even if you can't recall either of them initially). The more RS that information has, the smaller the boost in SS as a consequence of restudy (e.g., simply repeating to-be-learned information over and over again is not very helpful for really learning that information because you are probably practicing rote rehearsal, without forming the deeper connections necessary to improve long-term retention).

The theory has implications for a variety of research projects in the Bjork Learning and Forgetting Lab (e.g., spacing effects, optimizing testing events). Recently, we have investigated situations in which the theory would make unintuitive predictions (based upon previous interpretations of empirical results). For example, Kornell, R. A. Bjork, and Cheung (2009) investigated whether spacing, per se, improves learning or whether the benefit in retention occurs as a consequence of reduced RS at the time of the second learning event. They had participants study passages about various regions of the world (containing information that would be competitive across regions) and then re-study a subset of the studied information (i.e., the earlier learned countries), either immediately afterwards or after a delay. The hypothesis was that the earlier learned information would actually benefit more from a shorter spacing interval than from a longer spacing interval because information at the shorter spacing interval would have lower RS than information at the longer spacing interval (due to retroactive interference). Indeed, their hypothesis was supported, providing evidence for the benefit of re-learning information that is lower in RS-not spacing per se-on later retention. Additionally, Storm, E. L. Bjork, and R. A. Bjork (2008) and Little, Storm, and E. L. Bjork (2011) showed that relearning information that was impaired as a consequence of retrieving other competitive information (i.e., retrieval-induced forgetting) was relearned better after a subsequent restudy event than was information that was also restudied, but not first impaired as a consequence of retrieval-induced forgetting. Finally, John Nestojko and Ben Storm are currently investigating how RS and SS are affected as a consequence of testing versus re-study.
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II. How We Learn versus How We Think We Learn: Desirable Difficulties in Theory and Practice.

Introduction to Desirable Difficulties

Imagine a scenario in which a teacher has students practice different examples of a single type of math problem for an hour in class. By the end of the hour, it may seem-both to the teacher and to the students-that this type of math problem has been mastered. On a test two weeks later, however, the benefit may not be evident. In fact, much to the dismay of the teacher and the students, performance during training is not always representative of long-term learning.

In contrast to the story told above, in which an easy training method was followed by poor performance later, imagine that the teacher had interleaved many different types of problems during in-class training drills. Recent research reveals that difficult training of this type produces higher scores on the test than the easier version described above (Rohrer & Taylor, 2007), and this is the kind of training that the Bjork Learning and Forgetting Lab believes enhances long-term learning.

There are, in fact, certain training conditions that are difficult and appear to impede performance during training but that yield greater long-term benefits than their easier training counterparts. R. A. Bjork (1994) dubbed these difficult but effective training conditions desirable difficulties. Other examples of desirable difficulties (explained in greater detail in later portions of this webpage) include spacing rather than massing repetitions of to-be-learned information (R. A. Bjork & Allen, 1970; for review, see Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006), testing rather than re-studying information (Halamish & R. A. Bjork, 2011; for review, see Roediger & Karpicke, 2006), and varying the conditions of practice instead of keeping them constant. Along with investigating these and other desirable difficulties in isolation, we are currently working to understand the complex interactions amongst these variables (see Appleton-Knapp, R. A. Bjork, & Wickens, 2005). For instance, could it be the case that spacing inexact repetitions (i.e., combining spacing and encoding variability) enhances learning more so than either training condition alone, or would spacing and encoding variability lead to a difficulty that is undesirable?
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Spacing

It is common sense that when we want to learn information, we study that information multiple times. The schedules by which we space repetitions can make a huge difference, however, in how well we learn and retain information we study. The spacing effect is the finding that information that is presented repeatedly over spaced intervals is learned much better than information that is repeated without intervals (i.e., massed presentation). This effect is one of the most robust results in all of cognitive psychology and has been shown to be effective over a large range of stimuli and retention intervals from nonsense syllables (Ebbinghaus, 1885) to foreign language learning across many months (Bahrick, Bahrick, Bahrick & Bahrick, 1993).

R. A. Bjork, Kornell and Cheung (2009) argued that the spacing effect, one of the most robust and general findings from the history of experimental psychology, reflects a more fundamental accessibility principle-namely, that reducing the accessibility of information in memory fosters additional learning of that information (see New Theory of Disuse). In this view, increasing the spacing between learning trials enhances learning because it decreases accessibility of the to-be-learned information. Indeed, R. A. Bjork and Allen (1970) found that increasing the difficulty of an intervening distractor task without changing the duration of the spacing interval leads to improved learning, much like a spacing effect. Using a procedure designed to eliminate the normal confounding of spacing and accessibility by inducing absolute recovery-so that accessibility increased, rather than decreased, with delay- R. A. Bjork, Kornell and Cheung (2009) reversed the spacing effect, a result that supports the accessibility principle and runs counter to other explanations of the spacing effect. The accessibility principle may be a kind of law of memory, even if the effect of spacing per se is not.

The study-phase retrieval theory (Thios & D'Agostino, 1975; R. A. Bjork, 1975) of spacing proposed that the benefits of spacing arise from the retrieval of the first presentation upon repeated presentations. Accessibility of the first presentation is diminished at longer intervals and given successful retrieval of the first presentation, longer intervals create more potent learning events than shorter intervals. However, when spacing intervals are too long and repetitions do not retrieve previous presentations, then the study-phase retrieval theory predicts no benefit of spacing. On the other hand, the encoding variability theory of spacing (Estes, 1955) predicts that spacing should be increasingly effective with longer intervals. Appleton-Knapp, R. A. Bjork and Wickens (2005) examined this with an A-B/A-D paradigm and showed that the first presentation (B) is indeed strengthened, providing support for study-phase retrieval.

This theory of spacing would predict that an optimal learning schedule is one with expanding retrieval practice, rather than equally spaced practices. With successive practices, information is better learned and becomes inaccessible more slowly. As the greatest learning occurs when information accessibility is low (but not impossible), increasingly longer lags between retrieval practices should lead to better long-term learning. Indeed, Landauer and R. A. Bjork (1978) showed that expanding retrieval is better than uniform retrieval in two well-controlled experiments.

We are also investigating the impact of spacing on inductive learning. In a first step, Kornell and R. A. Bjork (2008) demonstrated that spacing led to superior inductive learning of artists' painting styles compared to massing. However, new research from the Bjork Learning and Forgetting Lab shows that it is not necessarily spacing, per se, but rather interleaving that improves inductive learning (Kornell, Birnbaum, Bjork, & Bjork, in preparation; see Inductive Learning).
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Generation

One robust and longstanding finding is that generating words, rather than simply reading them, makes them more memorable (Slamecka & Graf, 1978). As an example, this effect is often achieved for single words through the use of a letter-stem cue (ex. "fl____" for "flower") or by unscrambling an anagram (ex. "rolwfe" for "flower"). The effects of generation on memory are being investigated from many different angles in the lab, from its basic role as a memory modifier (see Desirable Difficulties), to people's awareness of this role and subsequent use of generation as a strategy (see Metacognition), to the extended effects of generation on related material (see Retrieval-Induced Forgetting).

In one study that looked at the effects of generation with typical text materials (DeWinstanley & E. L. Bjork, 2004), it was found that participants who were required to generate certain words (hereafter, target words) in a paragraph had better memory for those words than for target words that were simply read. What's more surprising, however, is that when these participants subsequently read and were tested on a similar paragraph, again containing target words that were either generated or read, both types of words elicited the same, higher, level of recall. This suggests that the earlier experience of generating words (and specifically, the contrast in memorability between words that were generated and those that were read) made the participants more effective at remembering the paragraphs overall. Subsequent research (Little, Storm & E. L. Bjork, 2011) has indicated that this benefit is due, at least in part, to better memory for the context of the target words.
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Interleaving

Spacing is one of the most robust, effective ways of improving learning. However, spacing calls for intervals of time in between repetitions, and this may not be the most efficient use of time. Imagine you have three final exams to study for. If you were to space out study of three whole courses, you might as well start your course review before the quarter even begins! Particularly when one has several different things to learn, an effective strategy is to interleave one's study: Study a little bit of history, then a little bit of psychology followed by a chapter of statistics and go back again to history. Repeat (best if in a blocked-randomized order).

The benefit of interleaving is found over a diverse set of stimuli ranging from word pairs (Battig, 1979) to motor movements (Shea & Morgan, 1979) to mathematics problems (Rohrer & Taylor, 2007) and word translations (Richland, R. A. Bjork, & Finley, 2004). Interleaving benefits not only memory for what is studied, but also leads to benefits in the transfer of learned skills (e.g. Carson & Wiegand, 1979). The theory is that interleaving requires learners to constantly "reload" motor programs (in the case of motor skills) or retrieve strategies/information (in the case of cognitive skills) and allows learners to extract more general rules that aid transfer.

Benefits of interleaving have also been demonstrated for inductive learning. Although inductive learning occurs in natural settings and everyday life, it can also take place in educational settings. In such cases, it becomes important to direct study in order to optimize learning. One's intuition may guide him or her to focus on one category at a time before moving on to another (blocking). Kornell and R. A. Bjork (2008) investigated the effects of blocking and interleaving on category induction. Interleaving is to shuffle the exemplars of a given category between exemplars of other similar categories. Interleaving however has the effect of both creating temporal spacing between exemplars of a given category and creating temporal juxtaposition of exemplars of different categories. Kornell, Birnbaum, Bjork, and Bjork (in preparation) investigated these separate effects and found that temporal juxtaposition, which allows for discrimination processes, is more critical to inductive learning than simple temporal spacing is. Additionally, although spacing alone and interleaving alone offer benefits to inductive learning performance (both desirable difficulties, combining the two manipulations does not provide an additive benefit to inductive learning performance (see Interactions between Desirable Difficulties).
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Perceptual desirable difficulties

Fluency, or the subjective ease of processing information, can provide learners with a useful basis for judging how well information has been understood. Perceptual variations are among the most obvious-and, sometimes, the most misleading-cues to the fluency of information. For example, when you encounter fonts that are difficult to read or words in very small print, you may experience a sense of disfluency-that is, you may have a feeling that the unusual or small typefaces are more difficult to process than more common typefaces.

These types of perceptual cues often cause people to think that disfluent information will be harder to remember than fluent information. Some research, however, indicates that perceptual disfluency can be a desirable difficulty (Diemand-Yauman, Oppenheimer, & Vaughan, 2010). The subjective difficulty of processing disfluent information can actually lead people to engage in deeper processing strategies, which then results in higher recall for those items (Alter, Oppenheimer, Epley, & Eyre, 2007). the Bjork Learning and Forgetting Lab is collaborating with Alan Castel to study how the fluent or disfluent appearance of text (e.g., clear vs. blurred text; upright vs. inverted text) impacts people's estimations of how well they will remember information as well as their actual memory for that information (Sungkhasettee, Friedman, & Castel, 2011).
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Interactions between desirable difficulties

It is not enough to simply provide educators with a list of desirable difficulties and claim that our work in optimizing learning has been completed. It may be that certain combinations of desirable difficulties interact to yield super-additive or sub-additive effects, if the processes by which the desirable difficulties work enhance or interfere with each other. Research into these interactions is therefore important on both practical ("What is best for learning?") and theoretical ("How do these desirable difficulties work to influence memory?") levels. For instance, research investigating the combination of spacing and variation in advertisements (Appleton-Knapp, R. A. Bjork, & Wickens, 2005) has shown that whilst spacing and variation are each independently 'desirable', there is a sub-additive interaction between the two manipulations such that variation is beneficial only when spacing intervals are short. At longer spacing intervals, variation actually hurts memory performance. Furthermore, the results of this study supported the study-phase retrieval theory of spacing over the encoding variability theory of spacing (see Spacing). Current studies in this line of research include examining the interactions between spacing and generation and spacing and variation with educationally relevant materials, such as text passages and glossary-style definitions. Interactions are also being explored with inductive learning, and Kornell, Birnbaum, Bjork & Bjork (in preparation) have found a subadditive effect of spacing and interleaving in the induction of butterfly species (see Interleaving).
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III. Learning concepts and categories (Inductive learning).

We are constantly faced with the challenge of categorizing and organizing the world into meaningful units of information. For example, when we meet our friend's pet, we are immediately able to recognize it as a dog rather than a large rat with a collar. This knowledge is acquired through a process called inductive learning: although we have not seen this particular dog before, we have seen many examples of dogs and other animals, and from those experiences we have induced the category of "dog." This same process takes place in educational settings. Forms of inductive learning include differentiating a novel Monet painting from a Picasso painting, or a cancerous cell mass from a benign one. The Bjork Learning and Forgetting Laboratory investigates the cognitive mechanisms that mediate inductive learning and ways to optimize this type of learning. This endeavor is especially important because our findings indicate that the learning methods people believe are most beneficial are often not (see Metacognition section). Kornell and R. A. Bjork (2008) found that while many students believe that studying examples of a category all at once leads to more effective inductive learning than studying them intermixed with examples of other to-be-learned categories, the opposite is in fact true (see Interleaving).
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IV. Goal-Directed Forgetting

People often view forgetting as an error in an otherwise functional memory system; that is, forgetting appears to be a nuisance in our daily activities. Yet forgetting is adaptive in many circumstances. For example, if you park your car in the same lot at work each day, you must inhibit the memory of where you parked yesterday (and every day before that!) to find your car today.

Much work conducted in the Bjork Learning and Forgetting Lab has focused on goal-directed forgetting, that is, situations in which forgetting serves some implicit or explicit personal need (E. L. Bjork, R. A. Bjork, & Anderson, 1998). In recent years our research has supported the notion that mechanisms of inhibition-analogous to those proposed in many areas of lower-level cognition, such as vision (explain, perhaps parenthetically)-play an important role in goal-directed forgetting. We have developed and utilized a variety of experimental paradigms to investigate phenomena that exemplify goal-directed forgetting, including directed forgetting (R. A. Bjork, 1979) and retrieval-induced forgetting (Anderson, R. A. Bjork, & E. L. Bjork, 1994).

Retrieval-induced forgetting Please see the Retrieval-induced forgetting section above.
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Directed forgetting

Forgetting is often viewed as an uncontrollable, undesirable failure of memory. Yet it is possible to experimentally induce forgetting in an individual that can lead to unexpected benefits. One such paradigm is known as "directed forgetting." In the typical list-based directed forgetting paradigm (E. L. Bjork & R. A. Bjork, 1996), a participant will study two lists of words, and is notified after each list whether or not it will be tested later on. If a list is tested after the learner was notified that it would not be tested, the learner will show weaker recall for that list, compared to a baseline condition in which all lists are expected to be tested, demonstrating the costs of directed forgetting. Interestingly, it is commonly found that recall of any list that was expected to be tested will be greater than that of the baseline condition, demonstrating the unexpected benefits of directed forgetting.

Another common paradigm for directed forgetting is the item-based method, in which participants are told after each word whether or not it will be tested. A similar pattern of results is observed, in which recall rates for the to-be-forgotten words are depressed, while recall rates for the to-be-remembered words are increased. However, the mechanisms by which item-method directed forgetting occurs are purported to be different than the mechanisms by which list-method directed forgetting operates.

In addition to studying the basic phenomenon of directed forgetting, efforts in the lab are currently underway to further investigate the effects of list-based directed forgetting using different materials and different paradigms. For example, does the pattern of results extend beyond simple word lists to more educationally relevant materials, such as text passages or videos? What happens to the pattern of results when information between the two lists is related? In addition, we are investigating whether directed forgetting applies to other learning paradigms, such as induction learning.
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V. Metacognition

Metacognition refers to the subjective awareness of one's own knowledge. In the Bjork Learning and Forgetting Lab, we focus on how this relates to learning and study behaviors, or metamemory. Metamemory consists of both monitoring the state of one's memory (or lack thereof) as well as using this information to control study decisions. Accurate metamemory can be crucial for a student in determining the success of his or her own study program.

Fluency and biases

Metamemory can be remarkably accurate in certain contexts, but this is not always the case. For instance, metacognitive monitoring seems to rely on the fluency of an item at encoding (i.e., when it is first learned; for more on this topic, see Perceptual Desirable Difficulties). This is often an accurate basis on which to make judgments, but the Bjork Learning and Forgetting Lab has shown that people still rely on this type of fluency even when it is misleading (e.g., Benjamin, R. A. Bjork, & Schwartz, 1998; Koriat & R. A. Bjork, 2006). Learners also fail to take into account forgetting that occurs over time (Koriat, R. A. Bjork, Sheffer, & Bar, 2004) and learning that occurs with repetition (i.e., the "stability bias", Kornell & R. A. Bjork, 2009); they instead seem to assume that what they know at the time the judgment is made is an accurate reflection of what they will know at a later time point.
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Awareness of study strategies

Another important issue in metamemory is people's awareness of the effectiveness of different study strategies. Zechmeister and Shaughnessy (1980) showed that people tend to think massed repetitions are more effective than spaced repetitions. Massed repetitions lead to greater short-term performance, but impair long-term performance (e.g., Simon & Bjork, 2001); this dissociation could explain why people think massed repetitions are more effective. Kornell and R. A. Bjork (2008) found that in the case of inductive learning, the belief that massed presentation is better than spaced presentation holds true even after people have taken a test and have done better with spacing! At the same time, other recent work (e.g., Benjamin & Bird, 2006; Toppino & Cohen, 2010) has found evidence that people do prefer later re-study to immediate re-study when allowed to choose between the two. Current work in the Bjork Learning and Forgetting Lab is examining further whether and in what contexts people are aware of the benefits of spaced practice when they are allowed to choose how to space repeated study opportunities.
Back to Table of Contents Visit gocognitive.net to see a full series of Dr. Robert Bjork's interview

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Research | Bjork Learning and Forgetting Lab

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The primary goal of this research, which is funded by the James S. McDonnell foundation, is to promote learning and memory performance within educational contexts through the investigation of principles in cognitive psychology. Studies address issues of transfer-appropriate and material-appropriate processing between encoding and retrieval.

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Designing Curriculum That Treats Students as Sense-Makers

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Designing Curriculum That Treats Students as Sense-Makers

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How do we design high-quality curriculum that supports students and their teachers seeing themselves as sense-makers, not mistake-makers? David will share effective teaching practices embedded in curriculum that allow all students to make mathematical connections. We will also discuss as a group how to embed features that are educative for teachers, so that the use of curriculum can lead to both student and teacher learning. Another critical goal of our curriculum work is to develop curriculum that explicitly aims to be responsive to student ideas and which includes instructional supports that give all students access to the mathematics through supporting students and teachers in having authentic mathematical discussions. In the presentation, participants will learn how those supports have been embedded within our curriculum through the use of instructional routines. Our assumption in the design of our curriculum work: It is possible for all students to learn, given appropriate support. Presented by David Wees

Hosted by: Paula Torres

Watch the full presentation at: www.bigmarker.com/GlobalMathDept/Designing-Curriculum-That-Treats-Students-as-Sense-Makers

Sign up for the Global Math Department Newsletter at: globalmathdepartment.org

Presented on August 21, 2018

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Designing Curriculum That Treats Students as Sense-Makers

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How do we design high-quality curriculum that supports students and their teachers seeing themselves as sense-makers, not mistake-makers? David will share effective teaching practices embedded in curriculum that allow all students to make mathematical connections. We will also discuss as a group how to embed features that are educative for teachers, so that the use of curriculum can lead to both student and teacher learning.

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Visible Maths, planning lessons and running a department - Podcast

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Visible Maths, planning lessons and running a department - Podcast

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On this episode of the Mr Barton Maths Podcast, I spoke to Peter Mattock.

Pete is Director of Maths & Numeracy in a secondary school in the Midlands. He is a Secondary Mastery Lead for the EMS Maths Hub, and Maths SLE & PD Lead - he has most of the alphabet covered there. He is also the author of a brand-new book, called Visible Maths.

I have had the pleasure of hearing Pete speak at several maths conferences over the years, and always found his sessions thought-provoking. And the same is very much true for his book. As we discuss in the interview, this is not an approach to teaching mathematics that I am experienced in, or entirely comfortable with, but it has certainly got me thinking.

So, in what turned out to be a wide-ranging conversation, we chatted about the following things and plenty more besides:

Pete does the earliest ever plug for a book, managing to mention it in his first sentence!

What is Pete's favourite failure, and what did you learn from the experience?

Then Pete dives into a fascinating discussion about how he planned out a sequence of lessons on fractions for a mixed-attainment Year 7 group, and we go off on lots of tangents along the way, covering how before why and the issue of differentiation

Then I quiz Pete on his approach to running a maths department, including the mistakes he has made, his marking policy, what his departmental meetings look like, joint approaches to topics, and his tip for any aspiring head of department listening.

And then it is time to take a deep dive into Visible Maths, where we focus on Pete's visual approach to 3 concepts: order of operations, division of fractions, and completing the square. The images Pete uses are available to look at in the show notes, and I strongly recommend you check them out

Finally Pete reflects on an important book, something he has changed his mind about, and what he wishes he'd known when he first started teaching that he knows now.

Once again, I loved this conversation. It left me with plenty to think about and reflect on, including how before why, mixed-attainment, sense making and more. Stick around at the end of the conversation where I mull over this in my Takeaway.

Two quick plugs before we crack on:

There are 20 free Diagnostic maths revision quizzes for Key Stage 2 SATs, GCSE Foundation and GCSE Higher available at diagnosticquestions.com/Revision2019 to check out the questions and quizzes.

And if you are interested in spreading the word about your product, service or event to 1000s of intelligent, engaged, and quite simply incredible listeners, then I am now offering the opportunity to sponsor episodes of this podcast. Just drop me an email at [email protected].

The images we talked about in the second half of the show are:
1. Order of operations: 2. Fraction division 3. Completing the square
On Twitter Pete is @MrMattock
His blog is: educatingmrmattock.blogspot.com
Pete's book, Visible Maths, is available Pete's Brockington College Maths Homework books are available from TES here

Pete Mattock's Big 3
1. twitter.com - seriously, it is so useful and one place every maths teacher should be using regularly.
2. atm.org.uk So much good stuff here, so much thinking that has already been done that we can learn from. Much of it is only open to members, but at £8 a month it isn't much for the level of knowledge available.
3. mathsbot.com - my go to site for most things - virtual manipulatives, quick question sets, loads of stuff.

Thanks so much for listening, and I really hope you enjoy the show!
Craig Barton

I am a maths teacher, currently teaching at Thornleigh Salesian College, Bolton, UK. Here are links to some of my work:
mrbartonnmaths.com
Mr Barton Maths Blog
Twitter: @mrbartonmaths
Diagnostic Questions
Mr Barton Maths Podcast
My book: How I wish I'd taught maths

Podcast: Play in new window | Download | Embed

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Peter Mattock: Visible Maths, planning lessons and running a department - Mr Barton Maths Blog

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On this episode of the Mr Barton Maths Podcast, I spoke to Peter Mattock. Pete is Director of Maths & Numeracy in a secondary school in the Midlands. He is a Secondary Mastery Lead for the EMS Maths Hub, and Maths SLE & PD Lead - he has most of the alphabet covered there.

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Planning, memorable teaching and teacher expertise - Podcast

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Planning, memorable teaching and teacher expertise - Podcast

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On this episode of the Mr Barton Maths Podcast, I spoke to Peps Mccrea.

Peps is a former Fasttrack maths teacher and Senior Lecturer in mathematics education. He has been a National Curriculum Advisor for the DfE, External Examiner at the OU, and is the author of Lean Lesson Planning and Memorable Teaching. Peps has three Masters degrees (in Engineering Design, Educational Leadership, and Educational Research) and holds Fellowship Awards from the University of Brighton and the Young Academy. He now leads on the Institute for Teaching's Masters in Expert Teaching course.

In short - I was once again well out of my depth.

I absolutely love Peps' books on lesson planning and memorable teaching and I am fascinated by the challenges of developing expertise in teaching. Hence, I had been wanted to get Peps on the show for ages, and he did not disappoint.

In a wide-ranging interview we covered the following, and more:

Does Peps believe you can teach problem solving, and if so how?

What are Peps' 7 habits of highly effective lesson plans, and what are the practical implications of implementing these?

How can teachers work together to joint-plan effectively?

What are the key principles of memorable teaching, and how can we design our teaching to lead to more memorable experiences?

What does Peps think of my ideas about keeping still, teaching in silent, and getting rid of all classroom displays in terms of his understanding of memory?

How would Peps present a worked example?

Is there a consensus as to what expert teaching looks like?

Can you judge how good a teacher is from a lesson?

Can the principles of deliberate practice be applied to teacher education?

What would Peps consider to be three essential research findings or principles that all teachers should know?

I don't want to build this up too much, but I am going to have to. This is one of those special episodes. I reckon I need to listen to it about 48 times to take in all the nuggets. It is also one to share with your non-maths colleagues as the vast majority of Peps' points about planning, memory and the development of expertise are applicable to all teachers and all subjects. So please spread the word. And if you enjoy this episode, or any of the other 30-odd, then a quick review on iTunes would be hugely appreciated.

I'd also like to take this opportunity to officially announce that I have written a book. It is called "How I wish I'd taught maths", with the subtitle "reflections on research, interviews with experts, and 12 years of mistakes", and is being published by John Catt Education, the home of such wonderful books as What every teacher needs to know about psychology and the recent What does it look like in the classroom?. My book is essentially a collection of all the lessons I have learned from speaking to my wonderful podcast guests, reading the books and research they pointed me to, and trying out ideas in the classroom. It is very much a tale of regret when I think back to all the things I used to do, but hopefully it will be of interests to you, my dear listeners. It is currently being read by Kris Boutlon, and assuming I survive that experience, I hope it will be available before Christmas.

On Twitter, Peps is @pepsmccrea
His website ispepsmccrea.com
His first book isLean Lesson Planning
His second book isMemorable Teaching
Links to the books Peps recommends teachers read can befound here, along with the recommendations of all my other guests.

Peps Mccrea's Big 3
1.John Sweller TES piece
2.David Wees stuff around Instructional Activities
3.Keep an eye on the IFT website > going to be publishing some stuff

Thanks so much for listening, and I really hope you enjoy the show!
Craig Barton

I am a maths teacher, currently teaching at Thornleigh Salesian College, Bolton, UK. Here are links to some of my work:
Twitter: Mr Barton Maths Podcast
Just the Job Podcast

Podcast: Play in new window | Download | Embed

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Peps Mccrea: Planning, Memorable Teaching and Teacher Expertise - Mr Barton Maths Blog

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On this episode of the Mr Barton Maths Podcast , I spoke to Peps Mccrea. Peps is a former Fasttrack maths teacher and Senior Lecturer in mathematics education. He has been a National Curriculum Advisor for the DfE, External Examiner at the OU, and is the author of Lean Lesson Planning and Memorable Teaching .

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Critical ingredients for a successful mathematics lesson

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Critical ingredients for a successful mathematics lesson

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What are the ingredients for an effective mathematics lesson? Teachers are continually faced with a range of advice or ideas to improve their mathematics lessons and often this just creates confusion. It's a little bit like being a cook. New recipes appear online and in cookbooks on bookstore shelves, but often they're just adaptations of classic recipes that have been around before, their foundation ingredients are tried and tested, and often evidence based. There are always the staple ingredients and methods that are required for the meal to be successful.

The following is a list of what I consider to be important ingredients when planning and teaching an effective mathematics lesson. The list (or recipe) is split into two parts: lesson planning and lesson structure.

Lesson planning:

Be clear about your goal. What exactly do you want your students to learn in this lesson? How are you going to integrate mathematical content with mathematical processes? (The proficiencies or Working Mathematically components) Will you consider the General Capabilities in your planning?

Know the mathematics. If you don't have a deep understanding of the mathematics or how students learn that aspect of mathematics, how can you teach it effectively? Where does the mathematics link across the various strands within the mathematics curriculum?

Choose good resources. Whether they are digital or concrete materials, make sure they are the right ones for the job. Are they going to enhance students' learning, or will they cause confusion? Be very critical about the resources you use, and don't use them just because you have them available to you!

Select appropriate and purposeful tasks. Is it better to have one or two rich tasks or problems, or pages of worksheets that involve lots of repetition? Hopefully you've selected the first option - it is better to have fewer, high quality tasks rather than the traditional worksheet or text book page. You also need to select tasks that are going to promote lots of thinking and discussion.

Less is more. We often overestimate what students will be able to do in the length one lesson. We need to make sure students have time to think, so don't cram in too many activities.

You don't have to start and finish a task in one lesson. Don't feel that every lesson needs to be self-contained. Children (and adults) often need time to work on complex problems and tasks - asking students to begin and end a task within a short period of time often doesn't give them time to become deeply engaged in the mathematics. Mathematics is not a race!

Lesson Structure:

Begin with a hook. How are you going to engage your students to ensure their brains are switched on and ready to think mathematically from the start of each lesson? There are lots of ways to get students hooked into the lesson, and it's a good idea to change the type of hook you use to avoid boredom. Things like mathematically interesting photographs, YouTube clips, problems, newspaper articles or even a strategy such as number busting are all good strategies.

Introduction: Make links to prior learning. Ensure you make some links to mathematics content or processes from prior learning - this will make the lesson more meaningful for students and will reassure anxious students. Use this time to find out what students recall about the particular topic - avoid being the focus of attention and share the lesson with students. Talk about why the topic of the lesson is important - where else does it link within the curriculum, and beyond, into real life?

Make your intentions clear. Let students know what they're doing why they're doing it. How and where is knowing this mathematics going to help them?

Body: This is a good time for some collaboration, problem solving and mathematical investigation. It's a time to get students to apply what they know, and make links to prior learning and across the mathematics curriculum. This is also a time to be providing differentiation to ensure all student needs are addressed.

Closure: This is probably the most important time in any mathematics lesson. You must always include reflection. This provides an opportunity for students to think deeply about what they have learned, to make connections, and to pose questions. It's also a powerful way for you, the teacher, to collect important evidence of learning. Reflection can be individual, in groups, and can be oral or written. It doesn't matter, as long as it happens every single lesson.

There are many variables to the ingredients for a good mathematics lesson, but most importantly, know what and how you are teaching, provide opportunities for all students to achieve success, and be enthusiastic and passionate about mathematics!

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Tips for Teachers: Critical ingredients for a successful mathematics lesson

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What are the ingredients for an effective mathematics lesson? Teachers are continually faced with a range of advice or ideas to improve their mathematics lessons and often this just creates confusion. It's a little bit like being a cook. New recipes appear online and in cookbooks on bookstore shelves, but often they're just adaptations of...

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Kyle Pearce: Hey hey Math Moment Makers, it's Kyle here, I've got Jon right next to me if you're watching on YouTube.

Jon Orr: Woo, right here.

Kyle Pearce: Some of you are listening on your favorite podcast app, today we are so geeked to dive into a deeper discussion about spiraling our math curriculum. A few episodes back, we were chatting about problem-based lessons and the topic of spiraling came up and we promise that we would dive a little bit deeper on YouTube. Some of you are listening on your favorite podcast app. Today, we are so geeked to dive into a deeper discussion about spiraling our math curriculum. A few episodes back, we were chatting about problem-based lessons and the topic of spiraling came up and we promise that we would dive a little bit deeper. And for those of you who have actually kind of followed us on our spiraling journey, there's a lot for you here. We're not just going to give you the same old, same old, we've got a lot of things that we want to do, including some of the things that we're doing a little differently than when we began this journey. So Jon, what are we aiming to do here? What are we hoping people will hold us accountable to when they listen to this episode?

Jon Orr: For sure. So we're going to get right into it, but we're hoping, and we're going to share with you how we began the journey so that if you've listened to podcasts episodes before on spiraling, that might be a little bit of review, but it's good to kind of talk about that because it's going to lead into what we've learned along the way. So some of the new stuff is that we are still both practicing teachers and we are learning things, modifying things, changing things, and then how we've changed that approach to spiraling. And then more importantly, how you can get started and kind of skip the pitfalls that we fell into and you can speed things up. So that's what we're going to be doing. Pal, let's get to it.

Kyle Pearce: Welcome to the Making Math Moments That Matter Podcast. I'm Kyle Pearce.

Jon Orr: And I'm John Orr, we are two math teachers from Make Math Moments and together...

Kyle Pearce: With you, the community of Math Moment Makers worldwide, we all want to build and deliver problem-based math lessons that spark curiosity.

Jon Orr: Fuel sense-making.

Kyle Pearce: And ignite your teacher moves. So as folks, all you Math Moment Makers heard in the intro, we're really eager to dive into kind of a quick summary of our own spiraling or interleaving journey. Talk about some of the things that worked for us as well as what inspired us to get on this journey. Some of you might be wondering, "Why would I want to do this in the first place?" And also what we've learned along the way and some of the things that we're sort of tweaking along the way as we always do, pivoting our plan and trying to continually improve our practice. So today, we're going to dive into that and we can't wait to dive deep into spiraling here.

Jon Orr: Yeah, let's do that, Kyle. So let's start off with how we began but also, in this part, we should talk about what spiraling really is for the folks who are, maybe you just tuned into our podcast and is your first episode and you're like, "What are you guys talking about? You're going to talk about like what you've changed along the way, but I don't really know what you're referring to in terms of spiraling." So it's also possible you've called it something different. Maybe you're calling it interleaving lessons or interleaving practice. We've been calling it spiraling after our good buddy, Alex Overwijk from Ottawa and that's where our journey began right Kyle?

Kyle Pearce: Yeah, absolutely. I feel like Alex and both you, myself and actually a bunch of educators from Ontario, many of which who have joined us on the podcast, we were all in this place where we're trying to figure out, how do we improve retention? So we came across and it seems like everybody came across maybe different instances of it. You can kind of think of spiraling as just mixing it up, in general, mixing it up. And there's so many different ways we could do that right Jon?

Jon Orr: There's a couple of things, and I think one main thing that I think got us into this pathway was that kind of feeling at the end of the course, we were teaching semesters in high school and when we were looking at the end of the chorus, we were having these kids who couldn't, you were like, "Oh, we have to stretch back to September and think about these concepts again." And it's like, they're gone. You teach something and then don't touch on it again, and by the time you get to the end, it's like we have to almost relearn it. And you were thinking, we're almost cramming for this test because we actually forgot a lot of the material when we have to kind of re bring it back in. And it was like, "How can we fix that? How can I get actual, deeper learning so that it's not just, I now need to mass remember this so that I can just do well on a test and then forget it again."

Kyle Pearce: Yeah, in something I remember vividly myself is like, so back to this idea that at the end of a course, when you really pause and think about it, if you supposedly taught something, students did this evaluation and I'll call it an evaluation because that's how we used to assess and evaluate our students, it was like, "At the end of the unit, we're going to evaluate where you are with this thing called a test and that's going to be the mark you get there and that is who you are as a mathematician for that concept this year." And really, the only thing that could influence that would be how they did on this thing called the exam at the end of the year.
But as many teachers who are listening, anyone who's ever had to do a standardized tests, which many of our American friends would know. Anyone in Ontario who teaches grade three or grade six or grade nine, we have standardized testing in those grades. And then any high school teacher who has exams, they're dealing with this sort of end of the year crunch time. And I guess what we eventually did was we sort of paused and we thought to ourselves, if students supposedly, and let's just pick a random student in your class who received an 80% on unit two, they got an 80%. But at the end of the course, it appears as though they know nothing about that idea, right." They come and they sort of go, "I don't remember any of this." It really should cause us to pause and wonder how useful is that form of evaluation being used in our course?
And now, Jon and I, we do more assessment than evaluation throughout the experience of a course. We assess for growth, we don't just evaluate an peg a number anymore. We still have grades, but we use them as a means to determine where we go next. And at the end of the day, we sort of went, "Okay, if I'm reteaching this, how long after this test is that going to be useful for?" So I would take like, I remember in some classes two weeks at the end of the year and I would just regurgitate, I would stand up there and I would lecture my students, and this is in high school, and I would basically reteach everything. And I remember it being one of the worst times of the year. And I feel like a lot of teachers feel that way because you feel like, "What the heck did we do this year if I'm up here, sort of reiterating everything we did in order to get these students to supposedly achieve?"
And then the very next year, I remember I had a class of students in the spring and it was a grade nine course, and then I taught the grade 10 course in the fall, and it was essentially the same group of students. And when they come back and you sort of think to yourself, "How "hard" we all worked in grade nine and how little was sort of there at the beginning of this grade 10 course," I really paused and I thought, "We're going to have to dig deeper here and figure out how can we help with this retention thing?" So for us, it was all about retention, and Jon, we learned about all kinds of things. What was one of the books that we sort of picked up and it kind of opened our mind to how the brain works and why some of these things were actually happening in the first place?

Kyle Pearce: Yeah, exactly. And you might see, depending on how big the chunks are, the way I had taught for many years was that's what I'm after, I'm after you to be able to do problems that you had seen before or very familiar problems. And once in a while, I would throw in more of a challenging problem. We would always get slack from students that we had never taught them that so that was unfair to put that problem on there because we didn't teach them. But that was sort of what we had trained our students to believe. But even if I'm teaching in that way, if I do space out my practice, if I do kind of come back to ideas a little bit later, you're probably going to see some retention improvement.
But again, at the end of the day, it's still a bunch of memorized ideas, which will eventually fade if we're not using them and we're not making connections to them. So what we had realized over time, at the same time, Jon and I were experiencing with these things called three-act math tasks, we've talked on the podcast many times of how we fell on our face many times with those types of lessons. We usually use them as sort of like end of the unit summaries or something to get kids interested. But then we would kind of go back to that, "Okay, now here's the steps you need to memorize."
And I think the big aha moment we've had over these years is by all of the different math educators who have influenced us, many of which, who have been on this podcast. So you've got Dan Meyer on there, you've got Marian Small on there, you have Cathy Fosnot on there. There's so many educators who have come and we've taken different ideas and we've started to make sense of where this fits. So this spiraling idea and this problem-based lesson idea and making connections and building conceptual understanding has really come together and sort of dovetailed nicely to create an up on the screen for those who are watching the YouTube video, you can see these little spirals, we now kind of are back to calling them units, we do these many units of study, but we mix up when they might happen and we try to keep it developmentally connected because there's something that I think we lose if we take that t-shirt cannon of concepts and we just sort of fire them everywhere.
It can be really hard unless we're very intentional for students to make connections even if the teacher's working really hard to make those connections and help students make them, it can be really hard for students to kind of make sense of it. Why is this happening? Does this connect to this lesson or does it connect to that lesson? So we're now in this place where we try to keep our little spirals or our units are more like four, sometimes all the way up to eight day chunks of ideas that emerge over a few days. But then after that work we'll maybe move on to a different area of the curriculum, as long as it's developmentally appropriate, right? We don't want to just pop around to any random unit, we want there to be some logic and rhyme and reason to why we're doing that, and then we might come back.
So if let's say I was working with part of division, I might be working with that idea and then maybe in the next unit, maybe I go into a fractions unit. Then I come back and we start looking at quoted of division and start referencing, "Wait a second. How is this different from partitive division?" And making connections that way. So you can see how there's still a spiral happening, but we're trying to keep the chunks more recognizable, we want to make sure that students are making sense and given an opportunity for some purposeful practice throughout. And then something that I think is okay to do a little bit more randomly, like the t-shirt cannon, would be some of that purposeful practice or some of the assessment pieces that we give students.
So I could give them something that we haven't looked at in four or five weeks, just to see where they are with that idea. Are students develop? Has that idea crystallized? Have they conceptualized it? Or is this something that, oh, we might need to pivot and come back and continue going a little bit deeper with that idea?

Jon Orr: Yeah, and another great thing that I think we realized as we kind of got deeper and deeper into spiraling is the benefit of those tasks and those kind of mini-units and trying to find those mini-units that have multiple standards that you can touch on in that same progression. And they're sometimes hard to create or hard to find but one that you can go, "Okay, well, you know what, in these group of three lessons, we use the same context, but we touched on these three different strands, we brought them to the forefront." And I think that's one of the big benefits of spiraling where you can save a lot of time is that you get to kind of expose or merge those ideas and then they're out in the open and you talked about different strands all in the next in those couple of days, but then when you come back to it, after you went somewhere else, you come back to it, you can start to strengthen some of those things up.
And that's where that retention, things start to build is that we spaced those ideas out, we brought these ideas up to the forefront. Maybe the learning goal for that day was on this one thing but we also touched on these other two things. And when we come back to it, we can go, "Okay. Our learning goal now is on this thing, and we touch on these things." When you have multiple content strands where a content standard's kind of emerging in the same lesson or progression of lessons, I think we saved a lot of time on our end because you can talk about those things in that day, but that's where we strengthen a lot of ideas up.

Kyle Pearce: I think one piece as well, that I think is really important for us to mention here before we start talking about maybe some of the next steps, some of the getting started ideas that some of our listeners might want to explore is just the idea of how the problem-based lesson approach really helped us with getting to what we were after, which was students being able to solve problems, being resilient problem solvers, persevering through problem solving and relying less on this idea. I used to always think if there was a problem that we hadn't "covered" yet, I was always concerned that I was like, "Well, we can't do that because we haven't done blank yet." We haven't explored fractions yet, so that's not something we can do.
But through a problem-based model, where the task is so low floor, that gives students an opportunity to enter where they are, and that's what gives us an opportunity to assess where they are and determine where do we need to go next? Where should we be focusing? And that to me, I think, is really important because if I am teaching procedures first or maybe procedures only where students are memorizing things and they're not really conceptualizing them, then it is a siloed, it is an isolated idea that they won't be able to necessarily apply to another area of mathematics. So for us, I think that piece is so key. And I would say that that is one of the most important lessons that we've learned along the way is that spiraling can be really helpful, especially interleaving in spacing practice.
But the reality is, is changing how we introduce math concepts so that when students come into this, their first response isn't, "I don't know where to start." When students say, "I don't know where to start," that usually means one of two things. It means either, "I don't understand what the context of this problem is, right? So I don't understand the problem itself, or I'm trying to think of a formula or a step or a procedure that will do this work for me and I just don't know which one it is," which again, I think is kind of related to the first problem, "I don't understand what's happening here."
Through a problem-based lesson, students are understanding the context and they're allowing the context to sort of dictate what strategies and models they're going to leverage. And as they do this work, that's what's going to trigger some of the previous mathematics they've explored. As they start modeling it, that's where they might go, "Oh, you know what? I'm seeing a relationship between these two quantities. I think this is a ratio that we're working with and I know these things about ratios, for example." So those I think are the big, big pieces.
Again, no matter where you are, I think getting started can look very different for one person to the next one. Thing we would advocate you not do is sort of taking what you're doing now and just sort of trying to go all the way to the extreme and doing like the exact opposite, because you're probably not going to see much success there. Yeah, you'll get burnout. You might almost feel like, wow, this is like, things are going to happen and the results are going to be less impactful than the method that you've been using because you've been working with that method for a long time and making tweaks and changes along the way.
So whatever it is that you choose to do, we encourage you to do things slowly and methodically. We want you to feel like the change you're about to make is going to have a difference because blank what's that hypothesis you have as to why what you're about to do is actually going to improve learning for your students. And if you're not sure what that is, then maybe you need to do a little bit more reflection on that or maybe a little more reading or a little more thinking on that. And I think that's what we want to share with you now, before we wrap up, right Jon? Where could people start if let's say they've heard, they've listened to our story, maybe they see themselves in their students in our story a little bit? And they're wondering, "Okay, I want to get started, but I also don't want to completely overwhelm myself or my students along the way. What could we do?"

Jon Orr: Yeah, and so we're going to share our spiraling guide link in the show notes page, we'll toss it out here in just a moment after we kind of share these kind of getting started tips. But I think there's a couple of ways we can get started if you want to go, I want to go in, like Kyle said, you don't want to slowly do that, but it could be taking your units and consider switching the orders of some of those units. And that gets you into this place where you're doing a little bit here, you're doing a little bit there, and you're doing a little bit later on. Keeping in mind that I think what's really important though is to do that with teaching this developmentally, thinking about those progressions and also with this idea of problems first approach.
This is why all of the tasks that we've created on our Make Math Moments website makemathmoments.com/tasks are problem-based to start, and a lot of them are now full five-day units because there's that progression. And then consider taking one of those, doing the five days, I think the five days Kyle is also built on almost if you have 60 minute learning periods. And then maybe switch to another unit and then try that and then come back, you'll see some overlap, right, with strategies with models, but then you're covering different areas, so you might want to consider that. That's probably how we would definitely recommend getting started is switch those units up, but then make sure that you're doing developmentally because you don't want to go back to being like, "we didn't see any benefit, we saw them actually worst results because we were still teaching procedurally." So I think if you're going to make a switch that deep, you have to do both at the same time.
A couple other ones that we recommended that are really easy switches to get into, to kind of dip your toes in, which is seeing some benefit of the retention side of interleaving, which is partly also how we kind of maybe even started before we did our problem-based lessons, which is using some warm-ups, like small things at the beginning of your lesson, math talks on different ideas not related to today or those learning goal that you're planning on that day, maybe you're going to do a math talk here on this idea, then next day, you're going to do a little warmup idea on something that's coming up. A lot of preview is really nice too because you're like a touch on idea and then you go and do the learning goal for that day. And then it comes back later on. So those are some easy kind of not technically, I would say, spiraling ideas, but interleaving ideas.
And I think another one we've heard from many teachers who have kind of ventured in this as this idea of lagged practice. If you are say making the switch into this idea, but maybe you're like, I don't want to go full in yet, you might want to consider any practice ideas, purposeful practice ideas you give to your students. You might want to mix some of those up, which means like today, you're going to do some practice on today's learning goal, but also you're going to do some on yesterday's learning goal or a month's ago is learning goal, and then that kind of keeps things rolling. So there's some interleaving, there's some space practice versus mass practice happening.
Those would be some kind of getting ready, getting started ideas. Hey, and if you've started already and you've got different ideas, we want to hear about it. You can get onto the show notes page, put a comment in there of how you got started and maybe recommend for other people. Kyle what's our mini course link and I know that you've been sharing some resources here up on the screen as I was talking. So if you're watching this over on YouTube, you can kind of go and watch some of the video, the visual support here that we're providing over on YouTube. But Kyle, where can people go learn more deeper than this particular episode on spiraling?

Jon Orr: We are wrapping things up. We also touched on that we are going to throw in the show notes page a couple of episodes in the past from podcasts, we talked about episode 40, which is how do I know when to proceed or pivot? And then episode 42, how do I spiral math standards without spiraling out of control? Which is part of that case study that we mentioned that you can listen to those episodes there. But Kyle, let's wrap things up because as always, our episodes come out every Monday morning and we want you to make sure that you subscribe. If you have not subscribed to this podcast, hit that button now. Maybe this is the first time you listened, maybe this is the hundredth time you've listened. Maybe it's the seventh time, but hey, subscribe so that you don't miss out our episodes. We release them on Monday mornings so do that on your favorite podcast platform.

Kyle Pearce: Awesome stuff my friends. Head on over to the show notes page at makemathmoments.com/episode145 or episode 145, makemathmoments.com/episode145. Until next time my friends, I'm Kyle Pearce.

Jon Orr: And I'm John Orr. Hi fives for us, and hi fives for you.

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Episode 145: What We've Learned About Spiralling

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Jon & Kyle share insights on their journey with spiralling their math lessons over the last 8 years. They dive into what spiralling is, why you should consider this structural change in your classes, and how they changed to this model from their mostly traditional approach to math lessons.

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Formative assessment is the practice of gathering and interpreting information about student learning as it is happening in your classroom. It can involve a variety of methods. Using formative assessment helps you know where students are at in their learning so that you can adapt your teaching to meet their needs.

Formative assessment has a strong evidence base that meets our highest standards of evidence. For more information about the evidence base, s ee below:

To be effective, formative assessment needs to be implemented well. See below some 'things to know' when using the practice.

Formative assessment may look different in different contexts. See below examples of formative assessment in a variety of classrooms and settings.

Here you will find tools to help you implement formative assessment in your classroom or setting.

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Formative assessment is the practice of gathering and interpreting information about student learning as it is happening in your classroom. It can involve a variety of methods. Using formative assessment helps you know where students are at in their learning so that you can adapt your teaching to meet their needs.

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Mastery learning is a way of designing units of work so that each set of tasks focuses on a particular learning objective and students must master a task to move onto the next one. By monitoring student progress through informal and formal assessments, teachers can provide additional support or enrichment to students depending on their level of mastery.

Mastery learning has a strong evidence base that meets AERO's highest standards of evidence. For more information about the evidence base, see below:

To be effective, mastery learning needs to be implemented well. See below some 'things to know' when using the practice.

Mastery learning may look different in different contexts. See below examples of Mastery learning in a variety of classrooms and settings:

Here you will find tools to help you implement mastery learning in your classroom or setting.

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Mastery learning is a way of designing units of work so that each set of tasks focuses on a particular learning objective and students must master a task to move onto the next one. By monitoring student progress through informal and formal assessments, teachers can provide additional support or enrichment to students depending on their level of mastery.

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Overview

Explicit instruction involves breaking down what students need to learn into smaller learning outcomes and modelling each step so that students can see what is expected of them. Providing explicit instruction limits the mental effort for students allowing them to process new information more effectively.

Research

Explicit instruction has a strong evidence base that meets our highest standards of evidence. For more information about the evidence base, s ee below.

Using the practice

To be effective, explicit instruction needs to be implemented well. See below for more information on the 'things to know' when using the practice.

Planning for explicit instruction

Archer and Hughes (2011) provide a useful planning guide with their 6 teaching functions of explicit instruction. By cycling through this planning process, you can deliberately embed explicit instruction in your lesson processes.

To plan explicit instruction activities and supports as referenced by Archer and Hughes, it may also be helpful to work backwards from where you want your students to be at the end of the learning session to see how you can slowly remove scaffolds and supports to enable student mastery.

Snapshots of practice

Explicit instruction may look different in different contexts. See below examples of explicit instruction in a variety of classrooms and settings.

Implementation tools

Here you will find tools to help you implement explicit instruction in your classroom or setting.

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Explicit instruction involves breaking down what students need to learn into smaller learning outcomes and modelling each step so that students can see what is expected of them. Providing explicit instruction limits the mental effort for students allowing them to process new information more effectively.

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The I Do - We Do - You Do model involves you in following a series of steps starting with you leading instruction and finishing with students working independently. In the I Do stage, you explain what students need to understand or model how to do a process. Then, in the We Do stage, you help your students by providing scaffolds such as prompts or partially completed procedures. Finally, in the You Do stage, your students do the procedure or show their understanding on their own.

Why You Should Use the l Do - We Do - You Do Model

When used well, the I Do - We Do - You Do model:

Suits novice learners (which most students are)

Nurtures self-efficacy

Reduces task anxiety

Students feel less anxious due to the clear modelling and the scaffolded opportunities to practice. These same factors help students to experience independent success, which in turn builds their self-efficacy ¹.

The I Do We Do You Do model is based upon the gradual release of responsibility from teacher to student.

Next, let's explore each stage in more detail.

This is true whether the procedure involves a younger student writing the letter m in cursive or a senior student keeping an historical research journal. Modelling is a powerful tool that draws on Albert Bandura's Social Cognitive Theory. Modelling involves: If you want to increase students' knowledge and understanding of a topic, you need to explain it. Research on the modality and multimedia effects² shows that you should explain things both:
Example

I Do Conceptual Learning

Visually

Example

The examples are faded as you start by completing many of the steps yourself and progressively get your students to more steps with each example.

Examples

We Do Conceptual Learning

The key to conceptual learning is getting your students to think about the material they need to understand. This involves helping them encode and retrieve the material in an organised way. Within the We Do stage of the I Do - We Do - You Do model you provide support through prompting comparative and organised thinking. You can make use of partially completed information and have your students fill in the gaps.

You can help your students encode information by getting them to:

You can help your students retrieve information using 2 strategies:

You can also do this:

Example

You Do Procedural Learning

With You Do activities as the procedural level, you simply have your students complete all of the steps involved in the task themselves.

Some people ask if you should get your students to apply what they have learned to solve problems at this stage. The answer depends on whether you showed them how to so and then did so together in earlier stages.

You Do Conceptual Learning

At the conceptual level, You Do activities involve recalling information without prompts. This can involve:

You Do Conceptual Example

More Than Just A Lesson Structure

Many people think of the I Do We Do You Do model as a way of structuring a single lesson, and it can be. Yet, it can also be much more. The structure can also occur:

When you believe your students will find lesson material to be quite challenging, you can break it up into more manageable chunks. You follow the the I Do - We Do - You Do structure for each chunk and may get a few chunks covered within a single lesson.

By contrast, you can also use the structure over several lessons. Especially as students get older, learning doesn't occur over a single lesson, so you follow the structure over a small series of lessons - with each lesson picking up from where the previous lesson finished.

Beyond the I Do We Do Model

The essence of the I Do We Do You Do model is scaffolding tasks as your gradually release responsibility to your students. Yet, many of the strategies listed above are grounded in research that fits nicely with, but is separate from, the I Do We Do You Do model, particularly:

However, there is one potent strategy that wasn't mentioned. It is a simple but powerful strategy - spaced practice (also known as distributed practice). Below, this is shown as a new stage Do Again (and again, and again).

Beware

You can adapt the model but some adaptations fly in the face of strong research. Be particularly wary of:

The rapid release of responsibility model turns the I Do We Do You Do model on its head and starts with the You Do stage. It is a euphemism for inquiry learning (and its cousins discovery learning and problem-based learning). The research is clear that while experts learn well from such approaches, novice learners do not. Most school students are novices in what they are learning, although they build expertise as they progress. As a result, you should not use inquiry learning with novice learners. Experts are competent in both the material and the processes or procedures involved.

You should also be wary of the we do stage being confused with collaborative learning. Collaboration can extend the model in useful ways, but the essence of the we do stage is teacher + student,not student + student. That said, some productive ways to use collaboration within the I Do We Do You Do model are:

Retrieval Pairs

One example of retrieval pairs would be flash card buddies, where one student holds up a card (or paper) with an explanation of a concept on the side facing their buddy and the name of the concept on the side facing them. Next, their student gives their buddy the first sound of the answer or of each word (prompts). The buddy answers and gets told if they were right or wrong, as well as the correct answer. The process repeats with different concepts and with students reversing roles. You could use a similar process with other retrieval exercises, such as prompted concept mapping. The teacher involvement in this stage is largely about preparing cards or other prompts.

Brain Dump 1-2-4

Students write down everything they can remember from a lesson or topic (brain dump 1). Then students pair up (brain dump 2), discuss each others' brain dumps, then make corrections and additions based on that discussion. They agree on a single, merged brain dump. Pairs then join with another pair (brain dump 4) and repeat the discuss and amend process. Note although there are 4 people in the group they are only discussing 2 merged brain dumps from the previous stage.

1 Person per Step

If your students are learning a multi-step procedure such as 2 digit by 2 digit multiplication you can use the one person per step strategy. 2 digit x 2 digit multiplication has 3 steps:

Multiply the top number by the ones

Multiply the top number by the tens

Add the answers from steps 1 and 2 together

Place you students into groups of 3, then have each person complete one step. Give your students at least 3 problems to solve and have them rotate the step they do each time. For uneven groups, one person may do 2 steps (groups of 2 are better than groups of 4). Note, you should use this strategy after your students have done independent practice as it reduces social loafing.

Footnotes

SHAUN KILLIAN
(MEd., MLead.)

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The I Do WE Do YOU Do Model Explained - Evidence-Based Teaching

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The I Do, You Do, We Do model is a simple yet potent way to use explicit instruction to improve your students learning. Discover how you

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In the hustle and bustle of everyday life, it's so easy to let our lesson plans move down our priority list. And when that happens, our classes can lack the impact they might have otherwise.

Thankfully, there's hope. I've found several strategies, tips and tricks to make lesson planning more meaningful and effective. If we can write more effective lesson plans - improve our pedagogy, kick up our use of best practices - we can have an enormous impact on our students' learning.

But that's probably complicated and time-consuming, right?
Nope. Not a bit.
In fact, I'm going to give you four tools you can add to your lesson planning tool belt that can help you write more effective lesson plans right away. You can refer to these four frameworks any time for ideas and lesson inspiration. They're research-based and highly effective. Don't think of them as yet another thing you have to add to your already full plate of teacher responsibilities. Instead, think of them as tools you can add to your tool belt that you can pull out and use whenever you need to.

1: Depth Of Knowledge

Let's start with the first one, a favorite of mine - Webb's Depth of Knowledge. There's a good chance you were exposed to Webb's Depth of Knowledge, or DOK, in your studies to become a teacher, but if you haven't checked it out since then, you're missing out on a really useful lesson planning tool. DOK is all about the complexity or the kind of thinking your students are doing. It's applicable to any grade level, from the littles to the bigs.

There are four DOK levels.

The first is Recall and Reproduction. Basic definitions. Identify steps. List examples. Name the character in the story.

The second is Skills and Concepts. This level builds upon the first level. This is where students analyze cause and effect. Compare and contrast. Summarize. They're taking the information they learned in level 1 and they're acting on it, trying to understand relationships between different parts.

The third is Strategic Thinking. In this level, students can use their reasoning skills to make a hypothesis and justify it. If level 2 is applying information in routine, expected situations, this third level is about applying it in non-routine situations. The exceptions and the anomalies. They may refer to prior knowledge and have to think through their justification.

The fourth is Extended Thinking. This is the level where students put everything together. They can apply their new learning to a project or a real-world scenario. They can connect it to other subjects. They see how it fits in their view of the world and in light of what they've learned in other classes and in life.

I love the examples that Karin Hess shares in her Depth of Knowledge videos on YouTube. Here's one of them. In a unit on temperatures and global warming, students may progress through the DOK levels like this:

Having Webb's Depth of Knowledge handy is really useful for lesson planning. It can help you generate new teaching ideas to help students investigate and understand new ideas in different ways.

2: Revised Bloom's Taxonomy

Our second framework is the Revised Bloom's Taxonomy. The standard Bloom's Taxonomy, created by Benjamin Bloom in the 1950's and 1960's, is a staple of teacher prep programs. It focuses on student tasks, what the student is doing. (Depth of Knowledge, on the other hand, focuses on how the student is thinking.)

In 2001, a group of experts in psychology, curriculum and instruction published the Revised Bloom's Taxonomy. The new version includes a verb to describe the thinking process and an object to describe the knowledge.

The verbs are familiar from the original Bloom's Taxonomy and include remember, understand, apply, analyze, evaluate and create. The objects are categorized as factual, conceptual, procedural and metacognitive. They actually match up with what we learned about Depth of Knowledge in some ways.

By matching these two categories together in a grid, the Revised Bloom's Taxonomy gives dozens of ideas to base learning activities. When you're stuck, just refer to the grid and read the explanations of the verbs and the objects. Apply them to what your students are learning, and there's a good chance that new ideas will come to mind.

Another way to approach this is to have students look over Revised Bloom's and its suggested activities. They may give you some new ideas for activities that help them get deeper into new content and ideas.

Before we move on, let me stop for a second and encourage you. Remember, I'm not sharing these frameworks as a set of expectations that you must hold yourself to. I'm sharing them as a source of ideas. Think of them like tools in a toolbox. You aren't required to use every tool in the toolbox for every project. They're resources. They're supposed to serve you. You aren't required to serve them! OK, with that little bit of perspective, let's move on to the next one.

3: Universal Design for Learning (UDL)

The third framework is Universal Design for Learning. This framework helps us optimize teaching and learning for everyone based on science on how the brain learns. Thankfully, they've done all the scientific analysis and have given us suggestions that can help us start to implement it right away.

Universal Design for Learning, or UDL, is organized into three areas:

So getting students interested, getting them information, and helping them demonstrate it. Pretty basic, right?

Each of those three areas has its own set of guidelines. These UDL guidelines give us even more great ideas on how to improve our lessons so they really work with our students such as:

giving students choice, setting the right amount of challenge and helping students self-assess to increase engagement.

making learning auditory and visual, making language clear, and helping students transfer learning to life to present content in different ways.

using multiple media to communicate, setting appropriate goals and helping students manage information and resources allow for more action and expression.

Each of these guidelines has specific suggestions. On the CAST website, where the guidelines are kept, you can click through to any part of any of the guidelines for help and ideas to implement them. There's lots of information, so start small and grow little by little. Remember - I'm sharing UDL here as a source of ideas, for it to serve you and not for you to serve it.

4. The ISTE Standards

The fourth and final framework is the ISTE standards. ISTE, or the International Society for Technology in Education, has published standards for students, educators, education leaders, and technology coaches. By looking over the student standards, you can get new ideas for broadening a lesson or a unit of study. These are the ones that can help your lesson plans to flourish. The other standards will encourage you to grow as a professional, but the student standards can be a really helpful resource for generating ideas. They include empowered learner, digital citizen, knowledge constructor, innovative designer, computational thinker, creative communicator, and global collaborator.

So, there are your four frameworks for creating powerful lessons - Webb's Depth of Knowledge, Revised Bloom's Taxonomy, Universal Design for Learning and the ISTE Standards.

I hope your head isn't spinning with all of these frameworks, but if it is, I can totally understand. They are definitely not four sets of checklists that you have to check off every time. Just think about it. There are TONS of educators around the world that don't know about these - or just don't use them. By at least being aware of them and using them for whatever they're worth - just to get ideas and plan better lessons - you're leveling up your teacher game.

No pressure. No guilt.

So, what do you do with all of these frameworks? My suggestion is to find the one that seems like the best fit for you. Bookmark it in your web browser. Print a copy of it off. When you sit down to create lesson plans and you struggle, pull it out. Look it over and see if it gives you any inspiration. If you get comfortable with it and are ready to move up another level, check out another framework and learn a little more about it. Don't worry. These frameworks aren't going anywhere.

They'll be here waiting for you whenever you're ready for them.

I'd love to hear from YOU. Which of these frameworks seems the most useful? What do you like best about it? If you use any of these frameworks already, how do you use them - so that they serve you instead of you serving them?

Please leave a comment below. I can't wait to see it!

Ready to take your teaching skills -- and student learning -- to another level?
Check out our online courses. ALL under $50!

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4 frameworks to level up your lesson planning

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In the hustle and bustle of everyday life, it's so easy to let our lesson plans move down our priority list. And when that happens, our classes can lack the impact they might have otherwise. Thankfully, there's hope. I've found several strategies, tips and tricks to make lesson planning more meaningful and effective.

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Watch: Introduction to the Primary Connections 5E model

Are you a teacher in Australia interested in learning more about using the 5E model in your classroom?
Enrol now in our online self-paced course Unpack the 5E Model!
Primary Connections resources and professional learning is based on the 5E teaching and learning model. This evidenced-based approach is effective in guiding teaching and learning of science because it supports active, constructivist learning; students draw on their prior knowledge, pose questions, participate in hands-on experiences, and conduct exploratory and formal investigations, to develop their own explanations about scientific phenomena. Students are given opportunities to represent and re-represent their developing understanding using literacy skills. They are actively engaged in the learning process. Students develop science inquiry skills and an understanding of the nature of science.

Teaching and learning progresses through five phases: Engage, Explore, Explain, Elaborate and Evaluate. The phases of the Primary Connections 5E teaching and learning model are based on the 5Es instructional model (Bybee, 1997).

The 5Es

Engage

The purpose of the Engage phase is to elicit students' prior knowledge, stimulate interest and gather diagnostic data to inform teaching and learning. Each unit begins with a lesson that mentally engages students with an activity or question. It captures student interest, provides an opportunity for them to express what they know about the concept or skill being developed, and helps them to begin to make connections between what they know and new ideas.

Explore

Students carry out hands-on investigations in which they can explore the concept or skill. They grapple with the problem or phenomenon and describe it in their own words. This phase allows students to acquire a shared set of experiences that they can refer to to help each other make sense of the new concept or skill. The Explore phase is characterised by multiple opportunities for students to experience hands-on learning and represent their thinking.

Explain

The purpose of the Explain phase is to support students to develop scientific explanations, drawing from experiences and observations, using representations. Students continue to develop knowledge of concepts and demonstrate their developing understanding.

Elaborate

This phase provides opportunities for students to apply what they have learned to new situations and develop a deeper understanding of the concept or greater use of their science inquiry skills. It is important for students to discuss and compare their ideas with each other during this phase. Investigations in the Elaborate phase build student capability for science inquiry skills in a meaningful context.

Evaluate
Learn more about the 5E model (PDF, 72.4KB)
The final phase provides an opportunity for students to review and reflect on their own learning, and on their new understanding and skills. Students represent changes to their understanding, beliefs and skills.

TWLH charts
Download the at-a-glance guide
A TWLH chart is a learning tool used to elicit students' prior knowledge by asking what students Think they know, determine questions students Want to know answers to, document what has been Learned, and How students know they've learned.

Questioning

Effective questioning techniques are an essential part of a teacher's toolkit. This guide provides suggested question stems that can be used during each phase of the 5E model.

Download the at-a-glance guide

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The 5E model: a framework for guided-inquiry

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Are you a teacher in Australia interested in learning more about using the 5E model in your classroom? Enrol now in our online self-paced course Unpack the 5E Model! Primary Connections resources and professional learning is based on the 5E teaching and learning model.

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SOLO ( Structure of Observed Learning Outcomes) offers a structured outline for the learners to use to build their learning and thinking. It motivates students to ponder where they are presently in terms of their level of understanding, and what they must do to progress.

SOLO Taxonomy was developed by John Biggs and Kevin Collis, two educational researchers who were interested in creating a framework that could help teachers design more effective learning experiences. The framework is based on the idea that there are different levels of understanding, and that students can move through these levels by engaging with increasingly complex tasks and ideas. By using SOLO Taxonomy, teachers can create learning experiences that are tailored to each student's current level of understanding, and that help them progress towards more sophisticated levels of knowledge.

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SOLO Taxonomy is often used in conjunction with the concept of constructive alignment, which is the idea that learning outcomes, teaching activities, and assessment tasks should all be aligned with one another. By aligning these three elements, teachers can ensure that their students are learning in a way that is both meaningful and effective.

With SOLO Taxonomy, teachers can design learning experiences that are aligned with the specific level of understanding that each student has already achieved, and that help them progress towards more advanced levels of understanding. This approach allows students to build on their existing knowledge and skills, and to develop a deeper understanding of the subject matter over time.

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SOLO Taxonomy for enhancing student learning

Solo Taxonomy is a systematic way that describes how learners' understanding build from easy to difficult while learning different tasks or subjects. The Solo Taxonomy can be used to enhance the quality of learning within the classroom teaching and provide a systematic way of developing deep understanding (Damopolii, 2020). Student learning can be guided in ways that promote deep learning.

SOLO Taxonomy is a valuable tool for assessing the depth of knowledge that students have achieved in a particular subject or task. It allows teachers to identify where students are in their learning journey and determine what steps need to be taken to move them to a deeper level of understanding.

By using SOLO Taxonomy, teachers can design learning experiences that are appropriate for each student's level of understanding and encourage them to move towards deeper levels of knowledge. This can lead to a more effective and engaging learning experience for students, and ultimately, better academic performance.

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What are the SOLO Levels?

The Structure of Observed Learning Outcome, presents a compelling way to structure the complexity and quality of students' thinking into distinct levels. It's a versatile tool that allows educators to gauge attainment levels and foster quality learning. This taxonomy consists of five levels, each representing a different depth of knowledge and ability level.

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1. Prestructural Level: Here, students exhibit a lack of understanding, often missing the point entirely. The level of thinking is minimal, with a focus more on lower-order verbs such as identify, memorize, and recall. It's the first stepping stone, a difficulty level that needs overcoming before progressing.

2. Unistructural Level: At this stage, students can identify singular aspects of knowledge, and their understanding is limited to isolated disciplinary knowledge. For instance, a student might identify that water boils at 100 degrees Celsius but may not understand why this happens.

3. Multistructural Level: The quantity of knowledge increases at this level. Students begin to gather multiple pieces of information, but they struggle to relate them coherently. For example, a student in this stage might know the boiling point of water and that heat energy is involved, yet fail to link these facts.

4. Relational Level: This is where the magic of student-led learning starts to manifest. Students begin to connect the multistructural elements into a coherent whole. Their level of thinking becomes more complex, and they start to understand the relationships between facts. For instance, a student at this level would understand that water boils at 100 degrees Celsius due to the increased kinetic energy of water molecules.

5. Extended Abstract Level: The zenith of the SOLO taxonomy. Here, students not only connect facts but extrapolate and hypothesize beyond the given context. The attainment level is high, and students exhibit the ability to apply their coherent knowledge to new, abstract scenarios. For example, a student at this stage might predict the behavior of other fluids based on their understanding of water's boiling point.

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A research survey of 500 high school science teachers revealed that less than 20% were aware of the SOLO taxonomy, indicating the untapped potential this model holds for shaping pedagogy. By incorporating the SOLO taxonomy into teaching, educators can gradually raise students' level of thinking from unistructural to the abstract level, thereby encouraging deeper, more conceptual understandings. It's a robust approach to foster students' transition from just acquiring facts to connecting and applying these facts creatively and thoughtfully.

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Comparing SOLO Taxonomy to Bloom's Taxonomy

Understanding the paradigms of SOLO Taxonomy and Bloom's Taxonomy can enrich teaching methods and learning programs. Both taxonomies serve as frameworks for constructing learning objectives, but they differ fundamentally in their structure and focus.

Biggs & Collis and Biggs & Tang, the architects of the SOLO Taxonomy, designed it as a measure of the quality of learning, while Bloom's Taxonomy was developed as a hierarchy of learning objectives.

SOLO taxonomy elucidates the learner's depth of understanding, from the basic concept to the broader concept. It is more learner-focused and aims to measure the learner's quality of understanding along a continuum: from pre-structural, unistructural, multistructural, to relational and extended abstract levels.

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Here, the focus is on the learner's progression in understanding classroom concepts and their ability to connect them. A well-structured assessment question in the SOLO taxonomy can enable students to demonstrate their understanding at the conceptual level.

On the other hand, Bloom's Taxonomy is more content-oriented, focusing on the definition of science and the classification of learning objectives within the cognitive domain. It provides a framework for the understanding of science from lower-order cognitive skills (knowledge, comprehension, and application) to higher-order skills (analysis, synthesis, and evaluation).

However, it falls short of assessing whether students can integrate and extend their knowledge in the same way the SOLO taxonomy does.

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As Biggs and Tang put it, "The SOLO taxonomy not only suggests an item writing methodology, but the same taxonomy can be used to score the items." This points to the versatility of the SOLO model in both constructing and evaluating connected questions, a feature less prominent in Bloom's taxonomy.

In essence, both taxonomies offer valuable insights, but their application depends on the educational context and the desired learning outcomes. For a more holistic understanding of student learning, educators might consider integrating the strengths of both taxonomies in their pedagogical approach.

1. The Bloom's taxonomy implies that there is an essential relationship between the questions asked and their responses; whereas, both the question and answer can have a different level in SOLO taxonomy.

2. Bloom's Taxonomy differentiates 'knowledge' from the intellectual processes or abilities that function on this 'knowledge' whereas the SOLO taxonomy is at its core based upon the processes of understanding used by the learners when responding to the prompts. Hence, knowledge pervades across each level of the SOLO taxonomy.

3. Bloom argues that his taxonomy does not only relate complexity but also relates a sequence of difficulty; whereas, there is no such requirement of increasing difficulty in case of the SOLO method.

4. Bloom's taxonomy does not accompany criteria for guessing the results of any activity; whereas, SOLO taxonomy is precisely useful for guessing the outcomes (Crompton, 2019).

5. Both taxonomies along with the Universal Thinking Framework, enable children to talk about their learning more effectively.

1. University of Sydney - SOLO Taxonomy

2. University of Washington - Creating Exams

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Why is The SOLO Taxonomy a good fit for Hattie's ideas?

The intersection of SOLO Taxonomy and John Hattie's ideas about education presents a compelling synergy. Hattie, renowned for his work on Visible Learning, emphasizes the impact of effective feedback and the importance of students understanding their own learning process.

SOLO Taxonomy aligns perfectly with these principles. It offers a structured framework for students to assess their own learning and for teachers to provide meaningful feedback. The taxonomy's stages - from prestructural to extended abstract - provide a clear ladder of progression. This enables students to visualize their current understanding and what the next level of comprehension looks like, promoting a sense of self-regulated learning.

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Moreover, Hattie advocates for teaching strategies that move students towards higher-order thinking skills. SOLO Taxonomy serves as a roadmap for this journey, providing clear signposts for moving from surface to deep, and ultimately, to conceptual understanding.

In essence, the combination of Hattie's emphasis on visible, self-regulated learning and the structured progression inherent in the SOLO Taxonomy creates a powerful tool for enhancing student learning outcomes. Together, they provide a robust framework for guiding students towards deeper, more insightful learning experiences.

Teachers can use:

1. No Idea - (like the pre-structural level)

2. One Idea - (like uni-structural level)

3. Many Ideas - (like multi-structural level)

4. Relate - (like relational level)

5. Extend - (like extended abstract)

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Why is Solo Taxonomy useful for designing instructional tasks?

SOLO Taxonomy is based on the idea of cognitive processes, which are the different ways in which we process information and understand concepts. These cognitive processes include surface learning, which is focused on memorization and recall, and deep learning, which involves understanding concepts and applying them in new situations.

By using SOLO Taxonomy, teachers can help students progress from surface learning to deep learning, and ultimately to the highest level of cognitive processing, which is extended abstract thinking. This approach not only helps students develop a deeper understanding of the subject matter, but also prepares them for real-world problem solving and critical thinking.

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Solo Taxonomy is useful because:

1. It helps learners to reflect meaningfullyon what the next levels in their learning are.

2. It helps instructors to thoughtfully design learning experiences and learning intentions.

3. It is used by the teachers and students together.

4. It makes it convenient to identify and implement effective success criteria.

5. It offers feedforward and feedback for the learning outcomes.

6. Students understand the reasons for everything they do and realise improvements are due to their own strategies

7. It shows the difference between deep and surface understanding, helping learners understand where they are on that spectrum, and what they must do to progress.

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Classroom Examples of SOLO Taxonomy

As we have seen, the SOLO taxonomy is a powerful tool that teachers can use to design learning activities that progressively deepen a student's understanding of a subject. Here are eight fictional examples of how SOLO taxonomy can be applied across various subjects in primary education:

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1. Mathematics: In a lesson on fractions, a teacher could start with the unistructural level by asking students to identify fractions in a group of shapes. Moving to the multistructural level, students could be asked to compare and order different fractions. At the relational level, students could be tasked with finding equivalent fractions, and finally, at the extended abstract level, students could apply their understanding of fractions to solve real-world problems, such as dividing a pizza or a bar of chocolate into equal parts.

2. English: In a lesson on narrative writing, students at the unistructural level could identify key elements of a story (characters, setting, plot). At the multistructural level, they could describe these elements in detail. At the relational level, they could analyze how these elements interact to create a cohesive story. Finally, at the extended abstract level, students could create their own original narrative incorporating these elements.

3. Geography: In a lesson on climate zones, students could start at the unistructural level by identifying different climate zones. At the multistructural level, they could describe the characteristics of each zone. At the relational level, they could compare and contrast different zones. At the extended abstract level, they could discuss the impact of these climate zones on human life and culture.

4. History: In a lesson on the Roman Empire, students at the unistructural level could identify key events or figures. At the multistructural level, they could describe these events or figures in detail. At the relational level, they could explain the cause and effect relationships between these events. At the extended abstract level, they could evaluate the impact of the Roman Empire on modern society.

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5. Science: In a lesson on the water cycle, students at the unistructural level could identify different stages of the water cycle. At the multistructural level, they could describe these stages in detail. At the relational level, they could explain how these stages are interconnected. At the extended abstract level, they could discuss the importance of the water cycle for life on Earth.

6. Art: In a lesson on color theory, students at the unistructural level could identify primary colors. At the multistructural level, they could mix primary colors to create secondary colors. At the relational level, they could create a color wheel showing the relationship between primary, secondary, and tertiary colors. At the extended abstract level, they could create an original artwork using complementary colors to evoke specific emotions.

7. Physical Education: In a lesson on basketball, students at the unistructural level could learn to dribble the ball. At the multistructural level, they could learn to pass and shoot. At the relational level, they could play a game, applying these skills in a dynamic environment. At the extended abstract level, they could reflect on their performance and devise strategies for improvement.

8. Music: In a lesson on rhythm, students at the unistructural level could clap a simple beat. At the multistructural level, they could clap a complex rhythm. At the relational level, they could perform a rhythm in a group, listening to and synchronizing with others. At the extended abstract level, they could compose their own rhythm and perform it for the class.

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These examples demonstrate how the SOLO taxonomy can guide the design of learning activities that progressively deepen students' understanding, moving from surface to deep learning. The SOLO taxonomy not only provides a clear structure for these progressions, but also allows teachers to easily identify the current level of a student's understanding and to design appropriate learning activities to move them to the next level.

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Key Insights:

SOLO taxonomy provides a structured framework to guide the design of learning activities.

It helps in moving students from surface to deep learning by progressively increasing the complexity of tasks.

It is applicable across various subjects and aids in the holistic development of students.

"The function of education is to teach one to think intensively and to think critically. Intelligence plus character - that is the goal of true education." - Martin Luther King, Jr.

According to a study by the Journal of Educational Psychology, students who were taught using strategies that promoted higher-order thinking performed 20% better on academic assessments.

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How does the Universal Thinking Framework compliment the SOLO Taxonomy?

At its core, the Universal Thinking Framework from Structural Learning Theory is built upon the seminal cognitive development theories of the 20th century, including the works of Piaget and Vygotsky, and even integrates principles from Bruner's theories. It posits that true learning isn't merely a process of rote memorization or assimilation of facts, but an intricate dance of assimilation and accommodation, constructing and reconstructing cognitive schemas. This echoes Piaget's theory of cognitive development, which emphasizes the dynamic, constructive nature of learning.

This framework guides students through a structured journey of identifying and connecting key concepts, effectively facilitating the creation of mental models that reflect their understanding. The Universal Thinking Framework is not a static process; it's iterative, fostering a continuous cycle of refinement and expansion of these mental models as new information is assimilated, thereby promoting generative learning.

A significant advantage of this approach is its alignment with the Cognitive Load Theory, reducing the strain on working memory by helping learners organize and structure complex information. This process naturally enhances metacognitive awareness, empowering learners to monitor and regulate their own cognitive processes more effectively.

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Moreover, the Universal Thinking Framework aligns with Vygotsky's concept of the Zone of Proximal Development, as it scaffolds the learning process, guiding learners from their current understanding towards higher levels of comprehension. It's a tool for gradual release, enabling students to eventually become independent learners.

Its universal applicability is a defining feature. Whether it's the humanities or the sciences, the framework can be employed across disciplines, reinforcing the assertion by Bruner that "any subject can be taught effectively in some intellectually honest form to any child at any stage of development."

In essence, the Universal Thinking Framework embodies the essence of active, structured, and deep learning. It's not just about absorbing knowledge, but generating it. It's a powerful methodology for educators aiming to nurture learners who can engage with content at a profound level, creating rich, interconnected networks of understanding.

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References to SOLO Taxonomy

Biggs, J., & Collis, K. (1989). Towards a model of school-based curriculum development and assessment using the SOLO taxonomy. Australian journal of education, 33(2), 151-163.

Biggs, J. B. (2011). Teaching for quality learning at university: What the student does. McGraw-hill education (UK).

Crompton, H., Burke, D., & Lin, Y. C. (2019). Mobile learning and student cognition: A systematic review of PK‐12 research using Bloom's Taxonomy. British Journal of Educational Technology, 50(2), 684-701.

Damopolii, I., Nunaki, J. H., Nusantari, E., & Kandowangko, N. Y. (2020, June). The effectiveness of Inquiry-based learning to train students' thinking skill based on SOLO taxonomy. In Journal of Physics: Conference Series (Vol. 1567, No. 4, p. 042025). IOP Publishing.

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Other Links

Want more? Here is a link on Problems with Bloom's Taxonomy (Invalid, unreliable, impractical) Want to dive into SOLO model? Check out Pam Hook's Website. Start with these two introductory books: pamhook.com

To review the SOLO taxonomy you can view three minutes of the video Understanding, from Teaching Teaching & Understanding Understanding, section 3 (3:15 - 6:18). (youtube.com)

SOLO Taxonomy: A Guide for Schools Bk 1: a Common Language by Julie Mills and Pam Hook SOLO Taxonomy: A Guide for Schools Bk 2 by Pam Hook Julie Mills Leave a reply © 2019 Eductechalogy. All rights reserved. (amazon.com)

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What is the SOLO Taxonomy and how can it be used to promote deeper learning outcomes in the classroom?

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How Do I Spiral My Math Standards Without Spiralling Out Of Control!

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How Do I Spiral My Math Standards Without Spiralling Out Of Control!

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Michael Rubin: If I'm going to take this task, so I'm going to try to consolidate it. I don't imagine myself sitting there and saying, "Hey why don't we try this?" I imagine myself posting a solution that somebody else did and say, "This person or this student or whatever, did this. What did they do? Why did it work? Compare and contrast it to what you did, and have them think of those things, so that even in that consolidation I'm not just talking at them." So I've got that idea of this purposeful practice [crosstalk 00:00:25]-

Kyle Pearce: You're listening to a snippet of our conversation with Michael Rubin, a Math Moment Maker who is back to join us for a continuation of Episode #40, because the conversation was so rich, we couldn't leave it at just one episode.

Jon Orr: Yeah. So if you haven't listened to Episode #40, be sure to stop and go listen right now. If you already have listened, then let's pick up the conversation where we left off and pivot the conversation into a deep dive around spiraling his math course for the first time. You'll grab some great value bombs from this episode.

Kyle Pearce: Let's get ready to talk spiraling your math curriculum. Here we go. Welcome to the Making Math Moments That Matter Podcast. I'm Kyle Pearce from TapIntoTeamMinds.com.

Jon Orr: And I'm Jon Orr from MrOrr-IsAGeek.com. We are two math teachers who together-

Kyle Pearce: With you the community of educators worldwide who want to build and deliver math lessons that spark engagement, [inaudible 00:01:37] learning and ignite teacher action. Welcome to Episode #42. How do I spiral my curriculum for the very first time? This is part two of Math Mentoring Moment episode with Michael Rubin from Northern California. Let's do this.

Jon Orr: Throughout this conversation with Michael, you'll be hearing about this dedicated math teacher who is a clearly life-long learner. Not only does Michael continually reflect on his own practice, in order to focus his mathematics professional development on his own educator-learning needs, but he is also extremely intentional in how he organizes his long-range course plans.

Kyle Pearce: One of the ways Michael has continued to fuel his own professional learning, was by participating in our six-module, online workshop. Michael continues to stretch his content knowledge and pedagogical practice by participating in our ongoing Make Math Moments Academy, so he can continue to access the online workshop content, as well as engage in new learning modules, math tasks and the Math Moment Maker Community area.

Jon Orr: Are you ready to do a deep dive with us in the online workshop like Michael did last fall?

Kyle Pearce: Well, you're in luck because we run this online workshop twice a year, once in the spring and once in the fall. So fall 2019 registration is open now. It just opened on Friday September 13, but it will close on Friday September 27, 2019. So be sure to learn more at MakeMathMoments.com/OnlineWorkshop.

Jon Orr: And if taking a deep dive alone isn't enough, the workshop is accredited for two to four PD credit hours to Brandman University. If you're interested in learning more about registering, be sure to check out MakeMathMoments.com/OnlineWorkshop. If you're listening after the fall 2019 registration, you can still head over to MakeMathMoments.com/OnlineWorkshop to join the waiting list, in order to get notified of your next opportunity to participate.

Kyle Pearce: That is MakeMathMoments.com/OnlineWorkshop.

Jon Orr: And now here is the second half of our chat with Michael.

Kyle Pearce: Welcome back on the podcast Michael. We had you on just a few episodes ago in Episode 40, so welcome back. How have you been seen we last chatted?

Michael Rubin: I've been great and thank you for having me back. I've been super busy. My wife runs a summer camp, so is taking care of a couple of 10-year-old girls for a few weeks, and got everybody at the camp mad at me because I taught them Nim. And they went and they just kept beating everybody, once they figured it out, so that was a lot of fun. And I've been working on spiraling my curriculum so [crosstalk 00:04:28]-

Kyle Pearce: Oh beautiful. Beautiful. Do you mind just quick reminder for those who may have either not been with us on Episode 40 or they haven't heard it yet, who are you? Where are you coming from? What's your coaching role or teaching role, I should say? And then we'll dive right into the meat and potatoes here.

Michael Rubin: Yeah. So my name's Michael Rubin. I've been teacher on special assignment in Northern California in The Bay Area. I'm going back to the classroom next year. I'll mostly be teaching Course 1 which is an integrated freshman math to English language learners, and either a Course 3 or precalc.

Kyle Pearce: Nice. Nice.

Jon Orr: So when we had you back Michael, or had you back on Episode 40, you were really concerned and really diving into this idea of spiraling your math curriculum. And you had questions and struggles, and we realized that there was so much about spiraling curriculum, that we had to extend it to two episodes. So that's why we have you back and I think we just want to dive right into questions you have about spiraling. So if you're listening to this right now, you haven't listened to Episode 40, you might want to start there because we talk about what spiraling is and then we move into talking about some struggles on spiraling.

Kyle Pearce: Yeah. And actually as we get and dive in here, John and I are actually on a video call with Michael on this episode. So what we're going to be doing for our academy members, is on and off we're going to actually be doing some screen sharing. I'll do it from probably my end, probably easiest.

Kyle Pearce: But just so those who are listening at home, we are going to be referencing some of the work that Michael's been doing around spiraling and really just providing one perspective of what this might look like for this particular course that Michael's working on. So Michael, do you mind framing that out for us a little bit? What got you on spiraling? What's your intent here of actually trying to spiral a curriculum and then we'll start diving in and figuring out where you started and where you're heading to with that plan?

Michael Rubin: I was first introduced to the idea of spiraling in your grassroots workshop. I'd never heard of it before. It had never been in my frame of reference, so I was eagerly anticipating it as the modules came out every week. I'm like, "What's this thing called spiraling? Is this a task? Are they going to be drawing constructions?" I didn't know what it was.

Michael Rubin: And when I saw it I really liked it because one of the struggles that we deal with as a school and as a district, is that we get a lot of incoming students throughout the year. And what's very difficult with that is let's say you teach Unit 1 on solving equations in Course 1. And then you're done with solving equations and then the kids come in March, so they don't get any of that.

Michael Rubin: And so the main reason that I wanted to do it other than the other benefits that you talked about, just to give kids an opportunity to engage with these concepts, even if they do come in sometime in the middle of the year.

Kyle Pearce: So that's great to hear. There's some intentionality behind that. This idea of helping students with retention is key. I know for me, one of the big pieces and I don't know if I knew it was going to happen the first time I did try to spiral, just even this idea of really being able to start at that surface level with a bunch of ideas, rather than going really deep, really fast and losing someone along the way. That for me was really, really big.

Kyle Pearce: I'm wondering, so you're framing out this course and did you have a starting point? I'm going to start sharing on the screen, for those in the academy who are watching this little mini course here but did you have a starting point? What got you thinking along these lines, because I'm picturing myself sitting at home listening to this or watching this and saying, "Okay, he's going to spiral like I have no idea where to begin." So can you help someone get into your mind as to where you started and how that's progressing along.

Michael Rubin: Yeah. So I had no idea where to really start. You guys had given a lot of good advice. So I'd figured I'd start with the scope and sequence documents that our district had come up with, and really to go through those because they were written in terms of units, to go through them, find the big ideas and come up with the main topics, which I found were algebra, geometry and statistics, the main subject areas, and then break those up into a limited number of primary objectives.

Michael Rubin: And one of the reasons that I want to do that is, and this is beyond the scope of spiraling I think, but I think it works with it as well, is that I do plan on switching to standards base grading, which we don't need to get into. But I was like, "I don't want to have 90 different objectives or whatever, so I'd like to do 13 that we can work on with those little sub-objectives over time."

Michael Rubin: So once I came up with those 13 primary objectives, I broke them up into sub-objectives and then I listened to one of your podcasts and you introduced me to this No, Understand, Do Form, so I went through and fixed them that way. That actually helped a lot with the spiraling because I was able to start thinking about well, when I spiral and I'm spiraling by what kids can do like the skills, am I spiraling based on the understandings like how am I going to do that? So that helped me figure that out a little bit more.

Michael Rubin: And so once I did that, I didn't really know what to do so I was like, "You know, I'm just going to start searching for tasks." And I had a bunch of websites, yours, TapIntoTeamMinds.com, Jon's, mrorr-isageek.com. I had the spreadsheets from Andrew Stabell, Dan Myer. I also had some websites I'd collected from, I'm not sure how to say it, Fawn Newman I think is how you say it?

Kyle Pearce: [Fawn Nguyen 00:10:03].

Michael Rubin: Fawn Nguyen. Yeah. All the Twitter peeps and if listeners aren't on Twitter, I highly recommend it. Even this morning I was struggling with thinking about tasks for geometry which I've had trouble finding. And I think it was John Roe who posted a whole list of geometry tasks and activities and inquiry-based stuff, so it's really helpful.

Michael Rubin: And so I just started going through tasks and that's where I left off. I was just about going into tasks, and my general idea the time was that once I got the task I was going to start spiraling by task. Things have changed since then, but that's an update from last time and where I was and when I left you guys last.

Kyle Pearce: Amazing.

Jon Orr: Okay. So we started that we too. We talked about that before, that we started with tasks and teaching through activity-based tasks instead of just saying, "I'm going to grab 1.1 from the chapter and then I'm going to do 1.2, but then I'm going to switch it up and I'm going to do 2.1 and 2.2 and then I'm going to go 3."

Jon Orr: We talked last time on the podcast, the benefit is to teach through activities and tasks, rather than just chunk it up, for so many reasons on connecting ideas together and also for retention. And I guess I'm interested when, because we did it through tasks like that to begin with and then you said your ideas have changed since then. What do you mean by that?

Michael Rubin: I've never taught through task, so I spent many, many hours going through tasks and trying to think about how I'd teach them. Think about the main concepts that I could get out of them. I know that I was going through a lot of your tasks and you'd purpose them for one topic and I was like, "Oh, but I think I could kind of do this with that." And I realized if I sat down and tried to plan out a lesson for each one of these, I'd get through at most, 15 by the end of the summer and I'd be no closer to my goal.

Michael Rubin: So I pivoted instead of proceeding as planned, and decided once I'd found a bunch of tasks that really and I almost feel like I think was strategy number seven where you organize by units, instead of looking at the tasks so much and thinking about how I'm going to spiral out the tasks, I decided instead that I would turn my KUD statements into learning progressions and I used your format.

Michael Rubin: You had one a while back, which was all the way from, I think, counting on or something, to linear relations. And I used that to come up with a general progression, like what are the ideas that students should be comfortable with first, as we go through the progression. And then when I spiraled, I spiraled based on the progressions figuring that what I'll do is I'll insert the task later.

Michael Rubin: So I don't think I'm doing exactly what the two of you did, but not being as familiar with the task and what concepts come out, I feel like it's going to be a lot more sanity inducing for me.

Kyle Pearce: Yeah. I think if I could it again and again, I always look at and think and we can dwell like hindsight's 20/20, all those things. But knowing what I now know, I think that might be a great way to go about it because now you know you have the intentionality at front and center. And now you can focus on we'll call, I know backfilling sounds like a sloppy way to say it, but you're taking now content and tasks and activities and all of these things and you're starting to place them in.

Kyle Pearce: And some of those as well, it might be really convenient to toss in hey a Dan Meyer task or one of John's tasks or anything John Roe or any of those things fit here and there, but then also that there's going to be some little gaps. And then now it's okay, now that I see this gap it's like, I can focus my attention on how can I address that piece and ensure that I'm pulling out the learning that I'm hoping that know, understand and do.

Kyle Pearce: And for those in the academy who are watching here, I've just been scrolling through this particular sheet, what is your I can statements. At the top they were I can statements, and then you've taken those I can statements and it looks like for each concept, I'm just going to flip right to the top here, really quickly here. So you have your overall objectives that you've included, so you have a collaboration as an objective that you have. You have reasoning quantitatively, you have reasoning structure, reasoning repetition, algebra, single variable, algebra inequality.

Kyle Pearce: I really like how you've started with these big ideas and you've tried to figure out, okay, what do I want the kids to walk away with? And then you've taken these and you have one I can for each of these, which I think is great because you've essentially focused on okay, how am I going to hit that for this one concept?

Kyle Pearce: So for statistics I can summarize, represent and interpret data on a single count or measurement variable. And then now you go, "All right, what does that mean? Now what do I need? What do kids needs to know, understand and do in order for that I can statement to become a reality? So I really like how you started there, and you're now fleshing those things out. And I think oftentimes what we do is the opposite because of the way our curriculum is written and designed, you're given all these standards in the US or all these expectations if you're here in Ontario like Jon and I, and then you're looking to address every single one of these little tiny pieces, and you're not even sure how they all interconnect.

Kyle Pearce: And if you go back that Episode #10 with Jo Boaler, she says something along those lines about this idea of when we create curriculum, we're taking these big ideas, or the curriculum writers are taking these big ideas and they're chopping them up into a huge number of little, tiny pieces, and then they get scattered and the connections are lost.

Kyle Pearce: So I just want to make sure that we don't overlook the idea of starting with these big ideas and really going, "Okay, what do I want these kids to walk away from?" Because a lot of the little, tiny pieces may get lost along the way, especially if I'm introducing them in little siloes. I'm introducing this little concept because I have to but not explicitly connecting it to one of these bigger items, so I love that.

Kyle Pearce: I'm wondering, where did these overall big ideas come from for you? What inspired them? Did they come from somewhere or were a mix from different places? Just for those at home so that they know where they might want to start digging in.

Michael Rubin: Yeah. So they were definitely a mix. So I've got these 13 objectives, the last 9 are the content objectives. And so I got these big ideas from our district's scope and sequence document, and so for each unit that the district created, there's a topic and then it's got guiding questions, the main objectives and then essential skills. And I went through all of those to figure out what are these big ideas, and then I rearranged them.

Michael Rubin: So some of the units I thought there were more connections between two of them, then I know that the district did use the textbook a lot and trying to figure out how to break things down because a lot of our teachers do use, just teach out of the textbook, and so I wanted to make them a little more connected in some ways.I also didn't want 30 of them since I'm going to be assessing students on these, just for ease of communication with the kids.

Michael Rubin: And then the collaboration objective I came up with, again referring back to Jo Boaler. She talks a lot about, and I think you guys have talked about this too, assess what you value. That if you're just assessing the content then you're not showing them that you value the collaboration or you value the reasoning skills, so I decided to put those in there.

Michael Rubin: The collaboration one I worked on myself, but the Routines For Reasoning, I actually have the book with me this time, by Grace Kelemanik and Amy Lucenta and Susan Janssen Creighton, I pretty much took those reasoning objectives right out of that book. I went through and figured out what are the main ideas in there. They refer specifically to the mathematical practices, the ones that are more thinking oriented like how do mathematicians think, And my plan is use some of their routines and reflect that in some way on the assessments. So I might have kids draw a picture on an assessment to reason quantitatively, so I included those as well. I highly recommend that book [crosstalk 00:18:29].

Jon Orr: Can you say the title again?

Michael Rubin: It's Routines For Reasoning. It's called Fostering the Mathematical Practices in All Students. So it's got a great introduction and it teaches the mathematical practices through routines, specifically to support English language learners and students with learning disabilities.

Jon Orr: Michael, I want to go back. When you were talking with your big ideas on your sheet and you listed them, and then you said you changed the orders of them around based on some connections, I wonder if you can share which ones you decided to group together and why those ones?

Michael Rubin: Yeah. So for example, one of the units in our scope and sequence, is just an entire unit on arithmetic sequences. We just deal with the arithmetic sequences separate from solving linear equations and linear relations, whereas I felt that those were inherently linked. So instead of treating arithmetic sequences as its own thing, I felt like the arithmetic sequences could really drive a lot of our investigations into linear equations and even just solving equations in one variable, and they were super visual, so I think that they'll help a lot with my students in particular.

Michael Rubin: So that's an example of where even though I know my colleagues will be spending two weeks or three weeks just on arithmetic sequences, I'm just wrapping that in to everything else. So I won't specifically be teaching that as a unit, I'm just bringing it up as we go through single variable and algebra and the linear relations.

Jon Orr: That's a great example of how textbooks or other documents, have chunked things up to keep them separate, but you can save time by linking other things together.

Kyle Pearce: Yeah. I'm looking here. I just scrolled down to the, so that's your fifth objective. "So algebra can write linear expressions and equations in one variable and use them to solve problems." And I see you have four knowledge chunks or objectives here, so you have A sub N as the Nth term of a sequence A. The order of operations is something you want them to know. The absolute value of a number is the distance from ... And so on and so forth, and then, A linear expression or equation contains a single variable with no exponents or radicals."

Kyle Pearce: So you've got some key pieces for things you want kids to know when they walk out of there, and then you also have some key things for understanding as well. And you can see that you've taken these ideas and traditionally they're separate but you're bringing them together and it's you're sequencing it in a slightly different way than a textbook would. And I feel like you're providing students with a rich opportunity to make those connections, so that they don't just see it as some couple day thing, and then you move on and you never come back to it. So super cool. I love it.

Michael Rubin: Yeah. And you might also notice, I put some little, I guess symbols in there, C squared and P squared. I put in some little indicators.

Kyle Pearce: Yeah. I love it, so your C squared it looks like is the common concept throughout.

Michael Rubin: Yeah. Throughout.

Kyle Pearce: P squared use in purposeful practice, so I really like how you thought those things through. What does that mean for you as you laid them out?

Michael Rubin: Yeah. So this was a process I went through before I actually started spiraling the ask, and I went through and I counted all the what can students do, what do I want them to do. And the nine objectives, I had 92 statements. It was like, "Well, I can't. I've got 180 days. I don't want to break them down and be, 'Okay, we're working on that one.'" And I realized that there were bigger ideas or concepts or things you might bring in task, and that there were a lot of others that just either were an extension of something.

Michael Rubin: So I can solve equations for which there is one solution, multiple solutions, no solutions and infinitely many solutions. That might not be something I'm going to find a task for, but that would be a great, I'm thinking of maybe an open middle problem or something, where you give kids and say, "Okay, here's some sort of prefab equation with some blanks in it. I want you now to fill in the numbers, so that there's one solution or no solutions or infinitely many solutions."

Michael Rubin: And it can be something to get kids thinking about it, where it's not necessarily a task, but it's a great extension, purposeful practice of students that are understanding, solving equations really well. It's something to bite their teeth into and practice, but it's not something I need to go out and try to find a task for.

Kyle Pearce: And I think too, sometimes you can even take that idea of what is a task and maybe for some people, I know I used to always think it has to be a three act math task for everything, but an open middle, I would call that an open-middle task as well. It just might be serving or it might be coming from a different angle, so from the angle of a challenge or whatever it might be, but it could still come under that umbrella of task. So just things for people to think about at home, that we don't have to have this really flashy task for everything.

Kyle Pearce: Sometimes it's the challenge and as we build that culture of not answer getting but for actually doing the math and actually challenging ourselves and building productive struggle, or at least this willingness to engage in productive struggle. I think those opportunities for grabbing tasks, as long as there's something there like that element.

Kyle Pearce: It's for me, curiosity is always great but challenge is another reason, or just when it's a puzzle. You give them something that's more like a puzzle and it's like you really have to do some deep thinking about it as well. It might not necessarily have the super-flashing context, but don't feel bad if you want to label that activity as a purposeful task in your classroom.

Michael Rubin: Yeah. You just summarized it. I was taking notes about how I was going to respond as you were saying, and you just addressed everything I was say. So I think the only thing I'll add to that, just so we're clear because I haven't talked about my general plan and I think it might help because I started formulating my general plan as I was going through and doing these. I worked backwards from how much time we have in the school year, and thinking about them not as individual chunks but as a complete spiral.

Michael Rubin: What's a complete spiral going to look like? And as I was thinking about that I was like, "Okay, let's say a complete spiral is like two weeks. I can do between 13 and 16 complete spirals depending on time, and that will give me a lot of flexibility. Then as I break those down, what am I going to be doing in those two-week chunks?" And the way I was thinking about tasks was like, those are the introductory ideas.

Michael Rubin: So I'm going to use a task, and a task is something that I'm going to spend maybe a couple days on maybe doing a follow-up task, but they might not be things where it's like, "Okay, now I'm going to get into the nitty-gritty of the consolidation phase." It's like, "Let's get the big ideas out." And then as I follow up to that, I'll have these consolidation days, where we then talk about some of the ideas.

Michael Rubin: I don't have a great idea of how I'm going to do that yet, but I was thinking a lot of it is still going to be that curiosity, deeper thinking task base. Well let's look at this. I was thinking a lot of ... Are you familiar with Sara VanDerWerf?

Kyle Pearce: Yes. Yeah. We just brought here up on the last episode we recorded.

Michael Rubin: Oh perfect. Perfect.

Kyle Pearce: Jon referenced one of her sessions, so-

Michael Rubin: Yeah. I saw one of her sessions about the same time that I became familiar with you guys. She's got this great line. Let's see if I can not butcher it. It's like, "Students will see it before I show them and they'll say it before I tell them.

Kyle Pearce: That's the line, huh. That's the line Jon used.

Jon Orr: You actually said it better than I did, than I tried to reference it on the last episode.

Michael Rubin: And I actually have that at the top of, I've got my own little HyperDoc. I have that at the top of my HyperDoc, as the idea of being if I'm going to take this task and I'm going to try to consolidate it, I don't imagine myself sitting there and saying, "Hey, why don't we try this?" I imagine myself posting a solution that somebody else did and say, "Hey, this person or this or student did this. What did they do? Why did it work?"

Michael Rubin: Compare and contrast it to what you did, and have them think about those things, so that even in that consolidation, I'm not just talking at them. And so I've got that idea of this purposeful practice, and so when I look and say, "Oh, this is like a great purposeful question." It's like I'm not going to spend a bunch of time revisiting this concept. I'm not going to spend a bunch time hashing this out. This is something that we can extend with our thinking, maybe bring it up once or twice, but it's within the main idea is an extension but it's not something that I need to vote three days in a row to.

Michael Rubin: And then the other one was a concept throughout, where I realized that some of these ideas would just naturally come out in the task in the purposeful practice of everything else. So I didn't really need to focus on those as things to find activities or tasks for either.

Jon Orr: What's an example of one like that?

Michael Rubin: I can create linear expressions and equations in one variable that describe quantities in their relationships. So in my mind, students are going to be doing that when they're doing a task. Students are going to be doing that when they're working with sequences. Students are going to be doing that when they're working with linear relations and systems of linear relations. That's just something that's going to carry throughout, so as I come up with the tasks and I'm trying to be intentional.

Michael Rubin: And it's like okay, I might go down to something that I actually care about like a key idea. I can solve linear equations in one variable using the distributive property. I'm going to sit there and say, "Okay, well I don't just want to give them an expression with the distributive property. I want them to create a linear expression using the distributive property in one variable, and then use what they know about that to solve it."

Michael Rubin: So I'll be referring back and that guides me to say, "Okay, how am I going to teach this task? What am I going to come up with? How am I going to present this, and it's something that I will be referring back to." My general plan is probably to print the list out and as I hit them, do a little check mark, so I can keep track of what I've covered when, to make sure I'm gathering all the ideas and they're all coming out. That's my general plan with this.

Kyle Pearce: Yeah. Very cool. I'm seeing too, as I'm looking through your list of dos, a lot of these are really awesome and I feel like over time, they may as you go through this, it might lead to more understands because I'm looking at for example, the do. For people watching in the academy, the 5.D.5.D which I believe is your fifth objective, it's a do and it's the fifth do and it's A, B, C, D.

Kyle Pearce: So you've got this structured and organized really well, probably way easier for you to navigate than maybe someone else who comes in for the first time. I look at that one and I see and it says, "I can use the structure of an equation to help me solve the equation. For example, 2X plus 1 all over 3 is equal to 5/3rd means 2X plus 1 equals 5.

Kyle Pearce: Even just this idea of I'm picturing a student who understands what that's all about, would be a student who could actually model both sides of the equipment into three groups or it's 1/3rd the size on either side. And this other equation is three times bigger, so that tripling and thirding idea exists there that we can pull out number talks.

Kyle Pearce: So there's so many things here that I think because you've organized this so well, that over time you're going to probably see some of these ideas maybe flip-flop or maybe mirror in some ways in different areas of the no, understand, do part. Because I always found for me, I was big on the know part for a really long time, and then after some time I started shifting some of those into the dos, whether I realized I was doing this or not, and now I'm really heavily into the understands part. And the know and dos come out through the understanding in a lot of ways.

Kyle Pearce: So just something that I'm noticing, I have my own little epiphany here as I'm going through this document. And I feel like you're the type of person Michael, that as you go through and you have these little epiphanies as you go along, I can picture you having no problem with pivoting from that plan and maybe revising or creating this new version of the document and doing the revisions over there and changing it for next time.

Kyle Pearce: So I want people who are watching and listening, to be getting the sense that you don't have to have a finalized document or plan that you're going to do for the rest of time, because that will really, really limit you in how effective you can be. I've worked with some educators, amazing educators who I would call them leaders, math-content leaders in my own district.

Kyle Pearce: And they've done something similar, not from a spiraled approach but from a scope and sequence approach, and they've put so much effort into it, but now I feel like they've gotten to a place where they're stuck because they don't want to change anything. They put so much sweat, blood, tears, not so much blood but the tears and the sweat, into this document that now they feel like if they make any changes they'll ruin it.

Kyle Pearce: And I think that's something that we want to make sure people at home know is you don't have to start this process and feel like it's got to be complete or in its final draft the first time you use it. It's a process I think, of learning. Do you feel like this has helped you better learn your course content from just engaging in this process, Michael?

Michael Rubin: Yeah. Absolutely. And I'd also like to echo what you were saying with that, and you talked about this on the last episode I was with you too well, about flexibility. And as I was creating these, I had this idea that as I go through the year, I bet more things are going to come out and I'll just add them and that will be totally fine because I'm one person and I don't have a buddy I'm working with on this. So it's like I know there are things I'm going to miss, especially as I'm going through and doing these from my other courses as well, where I'm not as familiar with the content. I realize okay, I'm going to be able to add things to it.

Michael Rubin: You also just answered one of the questions I think, that I was going to ask you later, which is do you spiral by those essential understanding or do you spiral by the skills? And so it sounds like you were saying that you try to spiral by those key understandings and I've gone the other direction. So I thought it would be a lot easier in terms of organizing by task or organizing on how I'm going to teach to do it by skills, and just keep in my that as I'm doing that, I need to focus on those essential understanding, so I thought that was interesting.

Michael Rubin: From a beginner's perspective it seems like going through and saying, "Okay well, we're going to solve simple linear equations first and then we might bring up the distributive property and then multiple terms on a side, combining terms and then variables on both sides." That would be like a natural progression where these equations get a little bit more complex. And that as I'm spiraling I can start with just the simple ones and the next spiral through I'll introduce something with maybe multiple terms, and then after that variables on both sides and then after that the distributive property.

Michael Rubin: And seemed for a novice to be easier than say, "Well, I'm going to focus on expressing relationships and then I'm going to focus on equations as a process of reasoning and explaining that reasoning. I don't see how to do that yet."

Jon Orr: I think what Kyle is saying is that we've said that already too, it's like there's no really one way to do this, and like you said that you are ready to modify this whenever you need to. And I think I have revamped my plan for my grade nine class every semester. I have an outline of the way I think it might run but then it doesn't flow that way because of the pivoting in the moment. So we've talked about different ways to spiral in the last one. Do you spiral by Kyle suggesting or you can spiral by a big idea and then think about always those big ideas.

Jon Orr: Some people spiral by the conceptual and then move to more specifics afterwards. Think about one teacher was spiraling everything by doing graphs and pictures. Everything was represented by graphs, pictures, concrete materials and then the next time was like, "I'm going to slowly integrate abstract representations eventually, and by the end we're doing those same ideas over and over again but we're getting more specific." That would be big idea down to more specific skill sets, and starting with skill sets and then going big ideas, also can work and there's nothing wrong with that idea.

Jon Orr: I think the biggest piece of why spiraling is so great is doing what you're doing right now. As Kyle said, you are becoming a master of the curriculum right now. You are understanding so much more about the connections between math concepts as a teacher, and all that is doing, you making this plan is preparing you to be flexible on the fly. You're going to be able to know the connection.

Jon Orr: You've done all the hard work to see these connections in the curriculum that you have to teach, rather the standards that you have to teach, so that when you're in the classroom, and see a student doing a solution here, you know where that fits in the plan. And you're going to be ready to modify whatever you need to do on that. I think that is the biggest, in my opinion, takeaway of trying to spiral your course is you're becoming a master of the math and that's going to be able to help you in the classroom.

Kyle Pearce: Just to echo what Jon's saying as well, I think you just nailed it. I picture eighth overall objective here, this algebra objective and it involves systems of linear equations, inequalities and two variables. My apologies if you're an early elementary teacher and you're thinking, what the heck are we talking about? But the reason I pick this one was because it is like you've got it chunked down into the least number of parts here.

Kyle Pearce: I think Jon just mentioned this idea that how you choose to frame or organize things, the one thing that we just don't want to forget is that the end goal is that overall objective at the top there, right? So that's a big thing, and then when I go one step removed from that I go, when I say solve systems of linear equations, what are kids thinking about? What are they conceptualizing and what is their conception of what that is all about?

Kyle Pearce: In their mind, could they tell me that, "Oh okay, so you're talking about a scenario where there's like two taxis. One taxi starts with this flat rate and then it increases by this amount. And this other taxi doesn't have a flat rate and it increases by this amount. And we want to know essentially, when one's the better deal over the other. To me that's like a conception about solving linear systems of equations where that student's going, "I get what we're trying to accomplish here."

Kyle Pearce: And then I think when you go down a step deeper than that, so we take one more shovel out. Then we start to go, "Okay, so how can I actually do those things?" And I guess my only caution for folks is to avoid what I did for a very long time, which was trying to check off those individual skills, and at the end I tried to bow them together. Tie them together under this big clump to go like, "There is your overall objective."

Kyle Pearce: Whereas I think by doing so, it was harder for kids to take these separate pieces and glue them together. Rather than starting with big idea piece, which we just described what that big idea is. So we're like, "That's it. Hey guys, gals. This is all we got. This is the big thing that we want to do."

Kyle Pearce: Here's the hard part. The hard part is there's so many different ways to attack it. It's like look at our conversation about spiraling. Spiraling is this. We dig deeper and deeper and deeper, we don't want to forget the overall goal of why we wanted to spiral in the first place. It's so easy for me to dig deep enough, and then all of a sudden I find myself trying to mix as many concepts together as possible because that's what the word spiraling reminds me of is just this blender.

Kyle Pearce: Hit the blender and now it's spiraled but what a second, what am I doing this for again? Oh yeah. I'm doing it because I want to make sure that the concepts are accessible by all students in the class and for as long as possible in my class so we don't dig too deep, too fast. I want to make sure kids retain it, meaning that I'm now giving them an opportunity to come to ideas.

Kyle Pearce: And I also look at skill piece as whenever I come back to an idea, regardless of the choice of how to spiral, I come back to an idea, but then I start a step below where students are ready. Meaning, I'm going to start a little bit behind where we left off last time, and it's not going to consume a lot of time. It's going to actually allow it make sure everybody's on that bus with me, or everyone's on that elevator with me and we're going to go up a level together.

Kyle Pearce: Instead of saying, "Well, this is where we got off the elevator last time, let's everyone get on." When in reality we're like, "Wait a second, a bunch of kids are not on that floor. They either didn't retain it, they didn't get it the first time. Whatever it was, now I'm giving them that second opportunity to practice some of those skills, and then those skills come along the way.

Kyle Pearce: At the end of the day if a student doesn't know how to solve by graphing, substitution and elimination, those are three big ideas, its like can they do one of those? That's way better to me, than doing all three of those and not realizing why they did any of it, which was my whole beginning of my teaching career. They could do those three things but they didn't realize that you could have used any of those three in all those scenarios, or that one of those was way easier to do in this scenario than in that scenario. So I know that was a mouthful but just some things to think about anyway.

Jon Orr: I was going to bring that example up because that skill about solving systems of equations, I remember teaching it exactly that. It would be like it would go over your head. We're going to solve where two lines cross by graphing, and then we're going to solve by substitution, and sometimes kids see the connection between those two and don't, and now I'm going to solve by elimination. Then you're going to go, "Hey, by the way, that was three different ways to solve any problem where you had two equations and two unknowns. And now on our last before the test, we're going to throw them all in. Now we're going to say, "Which way would you do it now?"

Jon Orr: It makes a huge amount of sense like you said Kyle, to start with the whole idea of I want to figure this out the way I've modified it this last few years I think was based on some of the work from Alex Overwick and the first lesson is like you said also really low so that everyone can do it. It's like, "Hey, there's four jujubes and three Smarties at this store and they cost this much cents. And the same store there's three more jujubes and five Smarties that cost this much. Here's a whole bunch of pennies. Here's the pennies themselves. Here's something that could represent jujubes. Here's something that could represent Smarties. Take the pennies that those combination is supposed to add to for the one situation. Here's the pennies for the other situation. Put the pennies on the counters. Put the pennies on those shapes and make it work."

Jon Orr: What are kids really doing there? Very concrete. Very hands-on and then slowly start to build up into those other things. Start with that big idea about we're trying to figure out what these values are in these situations. Then by the end you don't have go like, "Oh okay, now we're going to do the problem-solving day where we have to decide." You've been deciding the whole way through to begin with.

Kyle Pearce: Yeah. I'm just playing in the background here. Jon I don't know if you see it but just sharing my screen showing Accounting Candy Sequel. And that first video doesn't show a whole lot, but it just gives you a quick glimpse of three groups or three colors of candies, and then we zoom in on it so you can't see it. And then basically the big question that kids are really having to figure out is listen, if I know there's 57 chocolates in total, they're just colored different colored candies. I know there's 44 blue and pink, there's 35 blue and purple and then I've just got a summary of all three of those pieces of information by how many of each color are there?

Kyle Pearce: And I usually use this task way before we even introduce this idea of solving systems of linear equations at all. And what happens is rather than me teaching it, what ends up happening is kids do all kinds of really cool solutions strategies. They just use basic arithmetic to figure this out. Students take the total candies. They subtract 44 blue and pink, and they know that's how many purple there must be.

Kyle Pearce: So right away, they're using this idea of elimination without even realizing it. They're even using some substitution in some cases. And meanwhile for me, I would have done it this big, long way. When I show on the task side I'll include it in the show notes, but you see this big, long solution where I'm taking stuff and substituting this into that and this into this. And students have done this in 13 seconds flat because they have that understanding of what it is that they're trying to achieve.

Kyle Pearce: So that just popped into my mind as Jon was sharing there, but I'm wondering we could pause for a moment and get your thoughts there Michael, on this idea of starting with the idea of spiraling and how you want to be thinking about organizing, and then also just in terms of the intentionality of the content that you're going to be delivering.

Michael Rubin: Yeah. Perfect and that might a great segue too. So first of all I've just took a page of notes on what you just said and wrote "takeaway" because you always ask what your key takeaway and that's a stressful moment. So I'm writing this down. I really appreciate that idea of always going back. That was an idea I had, not something that was forefront in my mind that whatever skill I'm working on, make sure it's going back to this main concept and that students understand how it's all connected. That is my intention, but I also see how it could be really easy to say, "Oh well, now we're solving these linear equations by graphing or linear systems by graphing or whatever, and now we're going to solve them with substitution and have them unrelated."

Michael Rubin: I had this idea as you were speaking, one really good way of doing that is to maybe use the same context. I can see you've got something that we're graphing as a nice way of solving it, and then just take the same context, modify the numbers where they just don't work out where the students try to graph it.

Michael Rubin: I don't even know. I might give students this problem and none of them might try. I might give them an introductory test and none of them might graph, so that might a place where I pivot and it's like, "Okay, so now we're talking about these types of problems."

Michael Rubin: My plan too, now that I'm thinking of it, I really like Jon's idea of these mastery days. And part of my plan with the mastery days is that's going to be the day when we go through and I formally introduce the content objectives where it's like we were working on solving systems of linear equations and we can talk about what does that mean, and what were some of the different types of problems we solved and how were they connected? How were they similar? How were they different? It seems like those might be really great days as well, to make some of those connections. So I have those two ideas.

Kyle Pearce: I just wanted to make sure before we move on, just the idea of really taking advantage of context and not trying to overdo new context all the time. I did that forever and even before pre-transformation for me, when I would go and I would try to make question for kids to do in my class, I would try to make 13 different contexts every day. It was like this problem and this problem that's completely unrelated and this problem.

Kyle Pearce: It's like I spent so much of my own planning time, and then also teaching time, helping kids to wrap their heads around a new context, rather than taking that one context, spending that little extra time up front to get them to really understand it, and to feel like they owned the context, and then building ideas off of that. So you can juice them, just totally wring them of all that they're worth, and not worry so much about ...

Kyle Pearce: Curiosity to me is really important at the beginning of an idea, beginning of a concept especially when you're culture building your classroom. But as you go and as trust builds and kids are comfortable in your non-threatening classroom environment, I feel like sometimes you don't have to rely on that so much because now it's just like they know what they're here for. It's like there's no fear. We're here to dive into some really, really important concepts and it's not just about answer getting, it's about math and discovering how math works and fueling that sense-making. Jon, what were you going to say? Hopefully I didn't just steal your entire mouthful there.

Jon Orr: Well, I wanted to talk about the mastery days and just when you said about the making consolidations on those days. That's a great idea but I just want to remind you, don't forget to, it's so important that on the day you tasks or the day you do a learning objective or a learning goal, to make sure by the end of that, we're going to talk exactly about what we did then too.

Jon Orr: So it's like when you said it, I got the image that you're waiting until a mastery day to talk about, "Hey, remember back on Monday when we did this thing? Oh, what we were doing was this thing over here." I think I would want you to do that both days. It's like do that on the day you're doing it, so that's like we're consolidating on what we're really doing here. What is the point of today? Dan Myer, I remember seeing him and he said that one of the things you could do at the end of a lesson was he had kids title the lesson.

Jon Orr: People always want to put up front what the learning goal should be at the very beginning of a lesson. And we talked many times I think, on the podcast about withholding that a little bit until later, and at the end is a great time to do that. Like, "What is the learning goal here today? If you had to title this lesson, what would you title it?" And then you can help shape what that title might be with attaching the learning goal to it.

Jon Orr: It's like asking someone their big takeaway like, "What did you get out of this lesson today?" And making sure everybody walks away with, "Oh that was related to yesterday's work." Or "That was related to the task we did a month ago." So but it's also a great idea to do it on those mastery days where you spend consolidating some stuff.

Kyle Pearce: That's a great point Jon. I didn't even think about that piece but now that you're mentioning it, I know for a fact that there was many years I was teaching where I thought that kids were getting it through the inquiry process, but oftentimes that's not the case. There's some kids that completely miss the point. It's like when you're watching a movie and you're totally in this movie and you totally get what's going on, and then the next person nudges you and says, "Wait, who is that person on the video?" And you're like, "It's the bad guy. How obvious to me was that but to you it wasn't?"

Kyle Pearce: I think in math class it's even more so. It's so much easier for me to drift. I'm sitting in class and I just drift off for just even a moment, and I miss that key piece. So I think having a starting and an ending each day, each lesson with some consolidation, to make sure that there was very explicit as to like, "Hey friends, this is why we did what we did today."

Kyle Pearce: It's great when a student comes up with it and they can share it, but then you're still reiterating it. So if you can use student work and use those five practices to select and sequence the work and share. Letting three students share, not everybody share but then you taking that and going, "Okay, based on what this student said, I'm going to rephrase what they said to make sure that everybody's clear because maybe they weren't clear on how they said."

Kyle Pearce: That's the other thing. I hear a student say something and I connect the dots because I already know what they're trying to say. What about the kids who don't know what they're trying to say? So I have to sometimes restate what the kids are saying, and then I go to this group and I let them share and I say, "Hey, can you just share about this part over here because we already know about this part. I really liked your strategy over here." Maybe it's that graphing idea that came out. They share that and then you can share something you liked about it but then maybe extended a little bit further.

Kyle Pearce: And then you go to that third group and then finally you have to think about okay, now given what they're sharing, what can I do to help and actually deliver some value here? I definitely don't want people to hear Jon and I speaking, thinking that Kyle and Jon only sit on the side or meddle in the middle like Jennifer Waddle said on the recent episode there Jon. I'd love that but I think the consolidation though, is one of those times where we can't be fearful to say what needs to be said, clarify what needs to be clarified.

Kyle Pearce: Otherwise what I think can happen and what happened in my classroom for many years was some kids got it but a lot of kids didn't. And then the result of that is that people call the type of learning in that environment, discovery-based math curriculum. And people will say, "Discovery math is a poor way to go." And I'm going to agree. Discovery math with no intentionality and no clear, explicit explaining of the concepts is aimless for a lot of students.

Kyle Pearce: And again like I said, I did it for years. I know there were kids who left and I wasn't clear, concise enough, explicit enough and partially was because of my fear of no, I'm not allowed to do this in this style of more progressive teaching. That was one fear, which I think is a misconception, but then the other part was I don't think I really understood the mathematics well enough, and that's what I was referencing earlier about a lot of those dos.

Kyle Pearce: As you start to continually dive deeper into the concepts, a lot of those will start to become concepts and things you want kids to be able to understand and really own. And as I started to own more of the math learning, I was able to help students explicitly see those concepts and pull at them. So any thoughts there? I saw you writing a few things down, so we'll let you share your thinking and Jon and I will settle down for a moment.

Michael Rubin: Yeah. I think I just wrote about four pages. So first of all I'm a math nerd and more specifically a teaching nerd, so when I say this I hope people don't judge me, but literally my adrenaline's pumping right now with these ideas. I'm so excited because that's something I've really struggled with and I tend to be an all or nothing person. So this idea that yeah, no it's okay to consolidate and consolidate and specifically tie to the big ideas.

Michael Rubin: It reminds me of a colleague who just this last year I was working with their, she was doing an inquiry task with drawing right triangles with the same angle to explore some trig ratios. And so she asked her students at the end of the class like, "What was the purpose of this activity?" And a kid raised their hand and they were very proud of themselves and they said, "We drew triangles and measured the side lengths." It's like they completely didn't understand what they were doing or why they were doing it.

Michael Rubin: So I actually just as you were doing that, I've got this overall template for each day and I added in what the concept introduction day is the mastery day, it's consolidate days. I wrote in, "Consolidate, tie to big idea. Why are we doing this? What are the concepts?" That this isn't something where you have the ... I was thinking about having students write a reflection. If I click there Kyle, on my day-to-day in that green-

Kyle Pearce: This one right here?

Michael Rubin: ... you can see my ... Yeah. So here's a link to my year long day-by-day plan. Yeah. There you go.

Kyle Pearce: Beautiful.

Michael Rubin: Initially when I was going to spiral by the way, I was just going to do it like this and I realized I had no flexibility in it. So I pivoted once again but I've got these, this is where I'm going to link my lessons, put all the links and stuff so I have it all organized and I've got these big ideas. I'm going to have an engaging start every day. I'm going to teach through some sort of task, maybe some purposeful practice or formative assessment, assessment for learning, reflections, exit tickets. On some days a gut community building and mindfulness tapped in there as well, and so I just added that consolidate tie to that big idea.

Michael Rubin: That's really exciting for me because I think that's something that I was and I probably still will struggle with a little bit, especially with having a lot of English learners. Again, I got really excited about that idea of juicing that context. One that I was thinking of was the stacking paper because you can take that and disproportional reasoning all the way through systems of equations. You just have a different height table and different stack sizes of paper and have them figure out at what number of stacks will they be the same height? And you can use that context and I think tying that context will also help them realize, oh these are the big ideas. This is how they connect. This is the purpose of what we're doing. How is this one different from the last one?

Michael Rubin: I love that idea of starting a little bit behind. It also answers another one of my questions like, "What do you do for the kids who missed a day or missed that task day or came in part way through the year that's starting that little bit behind get to them caught up as well?" And then it makes those connections explicit like what was the strategy here? What was the strategy there and what were the main ideas? That was incredibly helpful.

Jon Orr: It brings up a lot of ideas for me, specifically lately about what notes look like or the end. If you're going to consolidate at the end, what does that format look like as a takeaway for students? You had mentioned a reflection but what I've been thinking about lately is especially for my applied level students, what does that note look like are probably terrible at taking notes themselves, not dedicated enough to write notes down. They're disorganized.

Jon Orr: I was spending a lot of time lately on what should that takeaway look like, so that students have something to look back on if they need to, because for a long time I was like, "If they do a lot of work, I still believe this, if they do a lot of math thinking and math doing in my classroom for my hour and 15 minutes with them, do they need to take any notes away to reflect on and to look back on?"

Jon Orr: In the higher levels I get I'm thinking, yes, I think they do need something to not necessarily look back on but the act of writing it down I think is important. And they can reflect on it later but it's like what is that takeaway at the end, so when they write it down. We have big idea written down. It's connected to the big idea we looked at before. There might be an example written in there. I'm really concerned or interested in what people's notes look like now. I used to tell my kids, "Oh, you should take pictures of the work that they did." And that acts as their note but I still think that there's so much value in consolidating the learning they did, that they can take away.

Jon Orr: I'm really interested in the interactive notebooks that I've seen floating around the Internet and resources. Some of those are the foldables and the flip books, some of those can act as a great resource. I'm really interested in that. I'm still exploring what is the best way that a consolidation piece at the end of a lesson looks like, so that there's something to take away.

Kyle Pearce: What do you think Jon?

Michael Rubin: Yeah. I don't know if this is something that you'll care about at all, you might just want to cut this from the thing as it's not really related to spiraling but it's something I've worried about. And as I've been trying to incorporate all these new things, that reflection piece was the big idea, so I just in the chat notes I linked two documents.

Michael Rubin: We have, our district hired a company, I think it's called E.L. Achieve and they do this thing called constructing meaning. And so I went to this workshop, which is all about providing academic language supports, so they talk about the bricks and the mortar, meaning that you've got your brick language which is all the vocabulary that you do in class like the math vocabulary. But then you've got all this connecting language that kids don't necessarily know how to use, especially English-language learners. So do you see the links in the chat notes?

Kyle Pearce: Mm-hmm (affirmative).

Jon Orr: Yeah.

Michael Rubin: So if you click those and open those, one of them I use, I don't know if I'm allowed to talk about their stuff but I adapted it from what they did. So if you click on the other one first, you can see how I'm planning on doing this.

Kyle Pearce: We'll include a link. I just pulled up elachieve.org, so I mean you're properly citing and we'll throw it in the show notes, and then the academy as a link as well as a resource people can look at, so I'm sure they'll appreciate that.

Michael Rubin: Okay. So what they do is in that paragraph writing, they don't know how to do that. And so I've come up with this and it gets to the idea, and the question I took specifically out of Jo Boaler's Mathematical Mindsets book where she had her reflection question, which I initially was going to just throw at the kids. And as Jon was saying, I realized kids aren't going to necessarily know what to do there. They're not going to know what to write, and so you come up with this idea of the paragraph and the kids can literally, they don't need to use it but if they don't know how to start or what to say, they can use this framework where it's like the main idea or concept of in the check box is lesson, problem, activity, discussion or other thing.

Michael Rubin: And then you've got the verb, examined, addressed, introduced, supported, explored and then the next box. The idea of, the relationship between, the importance of, the reasons for, the difference between. And then they can fill that in and explain with the content language and it goes all the way through. They introduce. They explain and describe. Support with key ideas. So blank is a key detail or is an example or is an essential information that illustrates or that explains that suggests blank because blank.

Michael Rubin: I've actually gone through and I just took a topic and wrote a couple of examples of what those paragraphs might look like. This is something I planned on explicitly teaching with my students. So the first time we do this we're probably going to do it together, and it might even be the first 10 times we do it together. And I was thinking this would be a way that students could even if they feel like they have an exemplary one, they can share it with me and I can share it to the class through Google Classroom. So it's a way for kids to consolidate that information.

Michael Rubin: As I was doing this though, I realized Jon was saying as well, is that even if you have this great reflection, it might not include everything. So in the reflection and notes document, which was the one you had linked to before Kyle, I've got the topic and that's the title of the lesson. What's the topic? I don't fill that out for them. It's like what would you title this lesson? You've got the space for the question, their response and then any other notes that they feel would be relevant. So they could put in an illustration, an example problem. Maybe if they didn't write it out, an efficient method. Maybe a side comment that a kid made. An alternate strategy that they've got a lot of flexibility in doing that.

Michael Rubin: And this might be a place whereas we work on this, either the day of the task if the task is short enough, I only have 57 minutes, or the next day during a consolidation phase we trade these things. We have kids write feedback on them. We discuss something that maybe was helpful that they found in the students' notes and we used that to get to the main, big ideas as well. I mean I don't know if this is going to work, I've never done it but I felt like I needed a plan.

Kyle Pearce: No, that's good. All right so Michael, we've been doing a lot of chatting here. This has been super, super helpful. I'm in this tab here with it looks like your objectives that are summarized a little bit here. You've got a list of tasks and objectives. What did you do next with those, in order to start framing out what this is going to look like, as the new school year quickly approaches for you in California?

Michael Rubin: Yeah. So where we left off at the last discussion that we had, I'd started looking for tasks and I figured I needed a way to organize them. And so as I was going through tasks just searching websites, I created a list of all my main objectives and started just linking them in, in no particular order, and I realized to spiral I was going to need a little bit more organization than that. So I took the progression idea, and on that second tab I broke down every topic like algebra, geometry and statistics and I broke them down into the main progressions and the main ideas.

Michael Rubin: And so once I did that, I was able to sit down, write out any prerequisites I thought kids might need in order to deal with those. And then go back to that first tab where I listed tasks and figure out, well where do they fit in here? It's not that I'm going to necessarily use them all, especially as we've talked now about juicing those context, but I have a menu of tasks to use and I'm sure I'll be adding to this list as we go along.

Michael Rubin: And then came the task of actually spiraling it and this is where I had some concerns. I've learned a lot in our discussion today, but on that third tab one of my biggest concerns has been our benchmark assessments because the way that we teach it even though it's integrated, we really spend the first semester and we do algebra and we do the second semester geometry and statistics. And so I know that all my colleagues are going to be getting to all these, going in depth into all the algebra concepts right away. And so I had to figure out how was I going to spiral these out, while at the same time preparing my students for these benchmark assessments?

Michael Rubin: So I work on a compromise, so instead of saying, "Okay, we'll I'm just going to do every topic." All the basic concepts of algebra, geometry and stats and then just move down equally, I decided to try them or whatever. So I highlighted the ones that I'm focusing on and that's going to roughly correlate to what the rest of the school's doing at that time.

Michael Rubin: So if you look at, you're looking at the screen here. I'll give a quick description of the spreadsheet for those that are listening. I've got five columns and the first one's just a cycle number, so that's going to refer to maybe a two, two and a half week cycle. I'll be flexible with that. And then I've got an algebra concept, a geometry concept, a statistics concept and any other topics that I think should be revisited or reviewed either through purposeful practice or maybe some homework problems or warmups. And I figured these are going to be the main ideas I'm going to take out it.

Michael Rubin: And the process I went through, is I went through the progression and I figured out how do these progressions build? And I started with the simpler ones and just started building them in until they became more and more complex. Then I went through, I did the same thing with geometry, realizing that we're going to want to hit some of those bigger ideas more in that second semester when we're going to have those benchmark assessments. So I saved some of those big ideas of congruence and triangle, angle relationships and proof.

Michael Rubin: Until then I like the idea of introducing those concepts earlier, but because I'm going to have to spend a little bit more time on algebra, so that my students are ready for the final and the benchmark assessments, I had to make some choices about what I wanted to do. So that's how I organized my spirals.

Kyle Pearce: So clearly the amount of planning you have put into this is so key. So if someone is listening at home and they're like, "Well, I want to try doing this." And again when we say, "Spiraling" it's not like a this or that, it's like it might be dipping your toes in and you might just make certain modifications. But I think all of this preplanning you've done is huge and I think it probably makes this process of actually trying to organize things, make so much more sense.

Kyle Pearce: I really want to also reference how you've taken things and you've really tried to show them on a bit of a progression, which is great, or a trajectory as some people might call it. And you had also mentioned earlier that Jon and I are big on really thinking about the mathematics of where does it come from and where is it going, and trying to figure out how am I going to organize this along a trajectory? It doesn't necessarily all have to happen one after the other, but we just want to make sure that it's in at least a reasonable order as things are happening. And then how do those trajectories or those progression interconnect if any?

Kyle Pearce: So super cool. I love that and then I love this spiraling plan as well, where you've got things, you're focusing on different things as you go. And I've got to assume and again it's an assumption that you'll find out after this year as you do this, but by introducing things and not doing all the algebra at the beginning and then all the geometry. I think that's going to have some positive results on those benchmarks later because it's not like it's all gone and then you don't see it for X number of weeks while you're doing geometry.

Kyle Pearce: So I think that constant mix is really going to serve students well, despite the fact that you might not get as deep into certain things as maybe you might have in the past. So it will be really cool to see how that goes. Jon, any reflections on that?

Jon Orr: Yeah. No, I'm impressed with the level of organization, for sure. I'm thinking back and comparing the way we did it when we outlined it in the last episode. When we just did it, we didn't have anyone to talk to about it, we just shot from the hip. So it's very, very nice to see the level of understanding of the concepts going into your planning. I think as I said before, it's so key to think about that and I think if anyone gets anything out of this conversation, I think it is about understanding the connections amongst the concepts, and it's necessary to think about that before you start say, making a planning document yourself.

Jon Orr: So great job here. We are impressed and as Kyle said, we'd love to just talk to you after you've taught this and see what things you'd change, what you learned along the way. So it will be great to have you back again on another episode next year, once you've put this into place. I think that would be great for people to listen to.

Michael Rubin: Yeah. I'd also like to reiterate on that planning. For somebody that just did it, I could not have done this without the planning. I also felt that the planning was so important for me as I was making decisions. In the past we've had these geometry units and it's like, "Oh, well we're doing constructions for two weeks." And there was no reason for the constructions. It's like, "Okay, well it's just part of the unit." And now it's like, "Oh well, really we're going to be doing these constructions."

Michael Rubin: It's great for problem solving, especially if you problematize them. It's like, "Can you construct this and have the kids work collaboratively to do so?" That's great but then really getting that idea that you can use this concept of constructions to establish the triangle congruence postulates and that we're setting them up for that.

Michael Rubin: So the reason that we're spending all this time on the constructions is so that they're ready to engage in maybe some more meaty concepts of proof and justification. And these are understanding that I'd never had before, so being able to go through and ask myself, "What are the key concepts? What are the big questions? Why am I doing this?" This has been super helpful in this plan.

Kyle Pearce: Michael again, I think you've done so much work in knowing and being, I don't want to say a part of the whole journey because we definitely weren't, but you've done a lot of this work on your own. But the one thing that I want to make sure that people are aware is that you didn't try to do it completely independently.

Kyle Pearce: So right from the beginning when you were in the workshop last fall, there was lots of talk about it in the discussion area. Currently in the academy, you're still right in there and talking about it and really doing a lot of co-learning with people out there. And I know that you're going to be a huge asset to the rest of that Math Moment Maker Community.

Kyle Pearce: So a huge, huge thank you to you for being vulnerable enough to come on here, not just once but twice to share on the podcast. And as we had mentioned beforehand, we would love to be able to take the video we shot while we were having this discussion, so that those who are in the academy can really dive in a little bit deeper. And then hopefully others in the academy who are doing something like what you're doing, and maybe they're doing it in a different way, we'll try to get them on for some episodes as well. Try to get different perspectives in here. We can all learn from one another.

Kyle Pearce: I know for me a big takeaway was starting with those big ideas like you had done in your planning process. You had spent, I'm going to say, 80% of the work was in the preplanning and the small amount of work is actually now where you place everything and the little adjustments and even the tasks. I know that's a big too, but as more and more people are doing the work like you're doing here and sharing that out, people will have at least a great place to know that okay, these tasks are great for these ideas. And here's some really cool big ideas that I can start with, and then I'm just going to modify to suit my own context, my own class, my own course.

Kyle Pearce: So thank you to that or for that. Jon, any last minute comments here or are we ready to wrap this thing up?

Jon Orr: Man, I think we are ready. Before we go, we would love for you, Michael, I know that you were ready for this, do you want to chat about what you say your big takeaway was for today?

Michael Rubin: Yeah. I mean I feel like there were so many, but I think the one that really resonated with me, was especially in this idea of spiraling where you're not just dealing with one big idea at a time. It's how important it is to make those connections to the big main big ideas and the concept's explicit so that students really understand that hey, we're solving this problem today but this is similar to the concept that we studied two weeks ago. We're just using a different strategy or a different method and these are the reasons we're using that different strategy or method but we're really solving the same types of problems. We're thinking about the same ideas.

Michael Rubin: That was really helpful, that idea that we should consolidate every day, maybe not at the beginning of the lesson because you don't want to ruin the surprise of the task but definitely by the end that you make those connections, so kids walk away understanding that. That was really helpful.

Kyle Pearce: Amazing.

Jon Orr: Great. That's good.

Kyle Pearce: That's super helpful. Well thanks so much again, Michael. We really appreciate your time and as mentioned we're learning from one another. I know Jon and I have picked up a huge amount of ideas. I love the way you've structured your planning, so we're definitely going to be keeping a copy of that for our own explorations as we move forward. And hopefully we will continue to see you around the community area of the Math Moment Maker Community inside the academy. And we will be up-to-date I'm sure, throughout the year as things go and as you pivot and continue planning and pivoting along your spiraling journey.

Michael Rubin: Absolutely. Sounds great.

Kyle Pearce: All right.

Michael Rubin: Take care.

Kyle Pearce: Have a great one. We'll talk to you soon.

Michael Rubin: All right. You too.

Kyle Pearce: As always, both Jon and I learned so much from these Math Mentoring Moment episodes, but in order to ensure we hang onto this new learning, let's be sure to reflect on the big ideas and big takeaways that we've learned. An excellent way to do this and help learning stick, is to reflect and create a plan for yourself to take action on something that resonated with you in this episode. A great way to hold yourself accountable is to write it down and even better, share it with someone, your partner, colleague or with the Math Moment Maker Community by commenting on the show notes page, tagging at #MakeMathMoments on social media or in our free, private Facebook group, Math Moment Makers K-12.

Jon Orr: Another great way you can take action is to take the action that Michael did and do a deep dive with us in our six-module, online workshop each spring and fall, where we coach you step by step on how to create and modify your lessons to engage students, build deeper understanding of math and promote resilience in problem-solving. With over 250 educators already through the program, we're confident you too can create resilient, engaged problem-solvers just like Michael has learned how to do.

Kyle Pearce: And if taking a deep dive isn't enough, the workshop is accredited for two to four PD credit hours through Brandman University. If you are interested in learning more about registering, be sure to check out MakeMathMoments.com/OnlineWorkshop. Remember registration closes soon, so head over to MakeMathMoments.com/OnlineWorkshop.

Jon Orr: Are you interested in joining us on the podcast for an upcoming Math Mentoring Moment episode where you too can share a big math class struggle? You can apply over at MakeMathMoments.com/Mentor. That's MakeMathMoments.com/Mentor.

Kyle Pearce: In order to ensure you don't miss out on any episodes, be sure to tap the subscribe button in your favorite podcasting platform. And if you like what you're hearing, do us a huge solid and be sure to share the podcast with a colleague and help us reach a wider audience by leaving us a review on iTunes, Spotify, Google Play or whatever platform you are listening to this on. Every bit helps.

Jon Orr: Show notes and links to resources from this episode can be found at MakeMathMoments.com/Episode42. Again that's MakeMathMoments.com/Episode42.

Kyle Pearce: Well until next time, I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us.

Jon Orr: And high fives for you.

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Episode #42: How Do I Spiral My Math Standards Without Spiralling Out Of Control! [Part 2 of A Math Mentoring Moment] - Make Math Moments

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This week Michael Rubin from our Math Mentoring Moment Episode #40 is back to continue the conversation from proceeding vs. pivoting to organizing his course content. Yes, this episode packs in a great conversation about Spiralling your math standards WITHOUT spiralling out of control.If you haven't yet listened to episode #40, be sure to STOP - and GO LISTEN NOW!

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Both primary (65%) and secondary (78%) school teachers think general administrative work is the non-teaching task they are spending too much time on

Equally primary (52%) and secondary (41%) school teachers place organising resources and premises, setting up displays, setting up/tidying classrooms as the support/management activity they are spending too much time on

Interestingly, 53% of primary and 61% of second school teachers believe they are spending too many hours on marking and correcting pupils' work

In the backdrop of constant budget cuts, teachers are under huge pressure to provide students the best possible educational experience with the resources available to them.

This challenge can be signified by recent research by UCL's Institute of Education (IOE), who found that one in four teachers worryingly work more than 60 hours a week.

Interested in teachers' workloads, TheKnowledgeAcademy.com analysed the latest findings from Gov.uk, who surveyed 4,329 primary and secondary school teachers across the UK to discover the non-teaching tasks as well as the support/management activities they think they are currently spending too much time on.

The Knowledge Academy found that both primary (65%) and secondary (78%) school teachers think they are spending too much time on general administrative work.

Thereafter, primary school teachers believe they are spending too much time on individual planning/preparation of lessons (56%).

Whilst, secondary school teachers rank marking/correcting pupils' work (61%) as the second non-teaching task they are spending too much time on.

Contrastingly, only 11% of primary school teachers think they are spending too much time on pupil counselling. On the other hand, just 9% of secondary school teachers equally state they are spending too much time on engaging in extracurricular activities as well as on teamwork/dialogue with fellow colleagues.

Both primary (52%) and secondary (41%) school teachers place organising resources and premises, setting up displays, setting up/tidying classrooms as the support/management activity they primarily perceive to be spending more time on than they would like to.

Although 41% of secondary school teachers also feel they are spending too much time on staff meetings. Interestingly, 33% of primary school teachers say the same thing about staff meetings.

Opposingly, just 10% of primary school teachers believe they are spending too much time on appraising, monitoring, coaching, mentoring and training other teaching staff.

Whilst only 9% of secondary school's teachers think they are spending too much time contacting people or organisations outside of school other than parents.

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Ranked: The school tasks teachers think they are spending too much time on

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Both primary (65%) and secondary (78%) school teachers think general administrative work is the non-teaching task they are spending too much time on Equally primary (52%) and secondary (41%) school teachers place organising resources and premises, setting up displays, setting up/tidying classrooms as the support/management activity they are spending too

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Teachers don't have enough time to prepare well for class

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Teachers don't have enough time to prepare well for class

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Almost all teachers (92%) in our study out today said they don't have enough time to prepare effectively for classroom teaching - the core of their job.

The Grattan Institute surveyed 5,442 Australian teachers and school leaders across all states and territories, primary and secondary schools, and government and non-government schools. The survey was about teachers' use of time.

Teachers told us they are too stretched to do everything we ask of them. When teachers aren't supported to do their jobs well, teaching quality suffers and students lose out.

Beyond preparation for effective teaching, 86% of teachers reported they didn't have time for high-quality lesson planning.
Teachers say they don't have time to prepare well for lessons
Our findings consistently show many teachers feel overwhelmed by everything they are expected to achieve. One teacher told us:

[There is] not enough planning time to allow for how responsive we need to be to students' needs.

Many teachers, especially at disadvantaged schools, said they get too little support to help struggling students.

And many teachers point to heavy requirements relating to writing reports, communicating with parents and supporting student welfare. One teacher said:

Administration time takes up most of planning time - such as communication to parents, newsletters, displays, notes, permission slips, phone calls and talking to students about well-being issues.

Teachers' struggle with workload is not caused by a lack of effort - Australian teachers work hard. Census data show teachers in Australia work about 44 hours a week on average - much more than the 40 hours of general professionals.

Australian teachers' working hours are high by international standards, too. OECD data show Australian secondary teachers work an average of 45 hours a week, compared to the international average of 40 hours.

Read more: 'Exhausted beyond measure': what teachers are saying about COVID-19 and the disruption to education

Our survey also found school leaders are aware of the pressure on teachers' time, but feel powerless to do much about it.

We cannot expect each of the 9,500 schools around Australia to solve these challenges on their own. Governments must step up. Our report, Making time for great teaching, recommends governments adopt three reform directions.

1. Find ways other school staff can take on non-teaching work

Governments need to better match teachers' work to their expertise. To do this, they should find better ways to use the wider schools workforce, including support and specialist staff, to help teachers focus on effective teaching.

Significant numbers of support and specialist staff have been added to schools over the past few decades. But governments have not tracked the best way to deploy and use them well.

In our survey about 68% of teachers agreed support staff could cover their extra-curricular activities. We estimate this would free up an additional two hours a week for teachers.

2. Help teachers reduce unnecessary tasks

Teachers consistently say they feel overburdened by administration. Streamlining administration where possible is important. But there are also significant opportunities to improve how teacher time is spent on core teaching-related work.

According to the OECD's Teaching and Learning International Survey (TALIS), Australian teachers spend about one-third of their time each week on core teaching activities such as correcting student work, preparing for lessons, teamwork and professional development.

This is four times as much as the time spent on general administration (8%), so any improvements in core teaching work could potentially free up large amounts of time.
Teachers don't spend enough time actually teaching
Lesson preparation is one of the key activities teachers do weekly. Yet more than half of teachers say this involves a lot of searching for and creating their own curriculum unit and lesson plans, assessments and classroom resources. This work eats up a lot of time and is a significant barrier to teachers feeling prepared for the classroom.

Almost 90% of teachers agreed that having high-quality common resources for curriculum and lesson planning would help reduce their planning burden. This would free up an extra three hours a week.

3. Rethink how teachers' work is organised

Policy decisions and industrial agreements shape the fundamental ways teachers' work is organised in schools. For example, they set the number of face-to-face teaching hours required each week, the number of students in each class and expectations about the work teachers do during term breaks.

Governments should ensure school leaders have the flexibility to rethink teachers' work in ways that open up more preparation time. For example, school leaders should have the flexibility to make small increases in class sizes (say by two to three students). Most teachers said they were in favour of slight increases in class sizes in exchange for two hours of extra preparation time each week.

Read more: Making better use of Australia's top teachers will improve student outcomes: here's how to do it

School leaders should also have flexibility to schedule more structured preparation and planning activities in non-term time. In our survey 58% of teachers agreed working together for two or three extra days before each term could reduce their workloads during term.
Teachers agreed with several reforms
What next

Our report calls on Australian governments to commit to a $60 million program to investigate and pilot the concrete options, including those tested in our report, that create more time for great teaching. That investment would be a tiny fraction (less than 0.1%) of the $65 billion Australia's governments spend each year on school. This is a small price to pay to improve the way our schools operate and ease the workload burden on teachers.

Making sure teachers have enough time for great teaching should be a national imperative.

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Teachers don't have enough time to prepare well for class. We have a solution

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Almost all teachers (92%) in our study out today said they don't have enough time to prepare effectively for classroom teaching - the core of their job. The Grattan Institute surveyed 5,442 Australian teachers and school leaders across all states and territories, primary and secondary schools, and government and non-government schools.

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UDL lesson planning can help you teach all students. Learn how to lesson plan with UDL and download a UDL lesson plan template.

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Stage 3: Important Questions to Answer

What activities, experiences, and lessons will lead to achievement of the desired results and success in the assessments?

How will the learning plan help students acquisition, evaluation, and transfer?

How will the unit be sequenced and differentiated to optimize achievement for all learners?

Stage 3: W.H.E.R.E.T.O.

WHERETO is an acronym to help execute Stage 3 and is explained below. (Another acronym to help with the process of writing Stage 3 is GRASPS) When considering your unit you must recognize what is expected of your students throughout the process. This table includes how will they be evaluated, judged, and graded:

E

EQUIP students, help them EXPERIENCE the key ideas and EXPLORE the issue.

R

Provide opportunities to RETHINK and REVISE their understandings and work.

E

Allow students to EVALUATE their work and its implications.

T

TAILOR the unit to the different needs, interests, and abilities of learners.

O

ORGANIZE to maximize engagement, as well as effective learning.

Stage 3 Worksheet

Stage 3 Completed Worksheet

Learning Activities and Planned Lesson Instruction:

Sources

Front Loading

Activities - Worksheets and Handouts

Lesson 1 - What is an Essay?

Lesson 2 - What are Different Styles of Essays?

Lesson 3 - Taking a Peek at an Essay

Lesson 4 - The Bing, Bang, Bongo Method: 5 Paragraph Essay

Lesson 5 - Check Out My Body...Type

Lesson 6 - MCAS/PARCC

Lesson 7 - Audience Importance

Lesson 8 - Introducing the Introduction

Lesson 9 - "Girl, Look at the Body...I Work Out": Writing SEXI Body Paragraphs

Lesson 10 - Conclusion Confusion

Lesson 11 - IYOW (Write Your Own)

Lesson 12 - Self Editing

Lesson 13 - Peer Editing

Lesson 14 - Final Draft

Lesson 15 - Reflection

Works Cited

Ray, Rebecca L. CURRICULUM UNIT Composition Writing 10th Grade English Language Arts. 14 July 2012. Lesson Plan. Fitchburg State University, Fitchburg, Massachusetts.

Wiggins, Grant P., and Jay McTighe. The Understanding by Design Guide to Creating High-Quality Units. Alexandria, VA: ASCD, 2011. Print

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What is UBD? | Understanding By Design in the Classroom

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What activities, experiences, and lessons will lead to achievement of the desired results and success in the assessments? How will the learning plan help students acquisition, evaluation, and transfer? How will the unit be sequenced and differentiated to optimize achievement for all learners? WHERETO is an acronym to help execute Stage 3 and is explained below.

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Learn about CAST’s Universal Design for Learning framework, fostering inclusive educational experiences and improving learning outcomes for all learners.

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Sponsored by PowerSchool and ISTE U

This post contains Amazon Affiliate links. When you make a purchase through these links, Cult of Pedagogy gets a small percentage of the sale at no extra cost to you.

When I taught seventh grade language arts, one of my favorite things to teach was S.E. Hinton's book The Outsiders. Every year, we began the unit with a discussion about the cliques that formed in students' lives, how these groups interacted, the unwritten rules that governed their behavior, and what happened when groups clashed or people formed relationships across group lines. After we did some reflecting, writing, and talking, we were ready to start the book.

The reading went fine, more or less. Some chapters we did in class (I would read to them, then they would read silently), and others at home. Some students became as absorbed in the novel as I'd hoped they would; others, not so much. Predictably, some fell behind in the book like they did with all assigned reading.

I checked students' progress with occasional quizzes, we did some work on plot and characters, setting and theme, and then, after a unit test over the whole book-containing mostly questions that asked students to identify characters, setting, and key plot points-we spent nearly three class periods watching Francis Ford Coppola's movie version of the book. And I got to drool over Matt Dillon in the movie's opening scene again and again and again. From start to finish, the whole unit took about three weeks.

In retrospect, I'm not sure why it was my favorite. Seriously. I mean, even though I loved the book, my students' response to it was mostly lukewarm. Maybe it was just the idea of teaching it that I loved. Maybe it was the connections I was able to make to the stuff students dealt with on a day-to-day basis. I don't know. I taught that book a few times, and even though I looked forward to it every time, I always finished the unit a little unsatisfied.

And it's only now, years later, that I'm starting to understand that dissatisfaction: I can't say with any confidence that my students actually learned something from that unit. Upon deeper reflection, I'm not confident that any of my students learned anything of lasting value from at least half the lessons I taught.

This is a hard pill to swallow, because I wasn't half bad as a teacher. I had decent relationships with my students and I believe most of them had good experiences in my classroom, but real, durable learning? I can't say how much of that actually happened.

I don't love admitting that.

But at least now I understand why my teaching didn't produce a lot of actual learning: I never set clear, measurable learning targets.

Things would have been so different if I'd known about backward design.

When teachers talk to each other about the stuff they're teaching, they often say things like this:
"What novels do you do in 8th grade?""Oh that'll be perfect. I can use this when I teach the American Revolution!"
"I don't think I can fit that in; we're doing moon phases next month."

We tend to talk about our teaching plans in terms of the broad topics we cover. This shorthand is practical; we're not going to drill down into specific skill and knowledge objectives while waiting our turn at the bagel table. But when I think about the lessons I gave my students, the ones I observed in my colleagues' classrooms, and the work I've seen my own children do, I think this shorthand might be a pretty fair representation of what many of us are still doing: churning out lessons that keep students busy with our content without ever getting clear about what we want them to learn.

Instead of starting with a topic, we'd do better if we start with an end goal, and that's where backward design comes in.

Traditional vs. Backward Planning

Traditional Lesson Design

For many years, teachers have been planning lessons and units of instruction like this:

Step 1: Identify a topic or chunk of content that needs to be covered.

Step 2: Plan a sequence of lessons to teach that content.

Step 3: Create an assessment to measure the learning that should have taken place in those lessons.

Notice that in this approach, the assessment is created after the lessons are planned. Sometimes it isn't created until most of those lessons have already taken place. The assessment is kind of an afterthought, a check to see if students were paying attention to the stuff we taught them.

For most of my teaching career, this is how I planned. It's presumably how most of my colleagues planned. I believe it's still how many teachers plan.

So what's wrong with it? Well, when we plan this way, we're more likely to include content and activities that have questionable value. When teaching the American Revolution, for example, if our goal is just to "teach about the American Revolution," we can throw in anything that has any relation to that topic: a coloring page of the Boston Tea Party, a colonial flag craft project, or a worksheet where students unscramble words like minuteman, independence, and Hancock.

This random approach creates two problems.

The first and most important problem is a lack of durable, transferable learning. One reason so many of us don't remember much of what we learned in school is that we learned it through this haphazard, topic-driven approach. These random activities are taking up precious time that could be spent on much more valuable stuff.

The other is poor student engagement. Our students know when they're being asked to do something pointless. If they don't see the relevance of what they're learning or a direct line between the content of your course and a desirable outcome, they'll tune it out. Sure, many students will do what you ask anyway, because they want good grades and the benefits that come from them. But they're not learning. If you don't believe me, ask them.

Backward Design

In their book Understanding by Design, which was originally published in 1998, Grant Wiggins and Jay McTighe introduced us to backward design, an approach to instructional planning that starts with the end goal, then works backward from there. The "full" version of Wiggins and McTighe's original approach is pretty complex and can be time-consuming to implement. If you're ready for that, I recommend digging into their book. For now, though, I'm just going to share the most basic version of backward design.

Here are the steps:

Step 1: Identify what students should know and be able to do by the end of the learning cycle.

Step 2: Create an assessment to measure that learning.

Step 3: Plan a sequence of lessons that will prepare students to successfully complete the assessment.

The difference in order is significant: Plan the assessment first, then plan only lessons that will contribute to student success on that assessment.

I was first introduced to this concept in my sixth year of teaching, and the genius of it completely blew me away. I used it when planning my next unit and experienced the biggest spike in student success I'd ever seen. On top of that, I was actually excited about teaching the lessons I had planned. For the first time, it felt like none of my class was wasted; everything actually mattered. There was something a lot more satisfying about doing things this way.

Let's take a look at an example to illustrate the difference between a unit planned the traditional, topic-driven way, and the same unit planned with backward design.

Before and After: The Lunar Cycle

The Before, Where the Final Product is a Test

At some point in their school careers, students study the phases of the moon. One pretty typical way to teach this is as follows:

A lecture or video about the phases of the moon, followed by a worksheet to label the phases.

An interactive activity like scraping the filling out of Oreos to represent the lunar phases.

Following a teacher's sample, creating a physical model of the moon phases using something like styrofoam balls.

A unit test that requires labelling the phases of the moon from memory and answering multiple-choice questions about the lunar cycle, eclipses, and seasons.

In many classrooms, teachers also have students track the appearance of the moon over the course of a month, so that might be added as well.

Following this plan, a teacher would feel pretty satisfied that they "covered" the topic of moon phases. But if we look at the Next Generation Science Standards (NGSS), the standard relating to moon phases says that grade 6-8 students will:

" Develop and use a model of the Earth-sun-moon system to describe the cyclic patterns of lunar phases, eclipses of the sun and moon, and seasons" ( MS-ESS1-1).

Note the language here: Students are meant to develop a model, then use it to describe these patterns. But in the plan above, students merely copied a model, and they didn't use it to describe anything; even if the model required some written captioning to explain what was going on, because the model was a copy, it can't be safely said that students were really the ones describing the system.

Then there's the test. If we assume that a large portion of a student's grade is based on the test, then students are not being measured on their achievement of that standard. The standard does not require students to memorize the phases of the moon. Nor does it ask them to "demonstrate knowledge" of how the whole system works. The standard wants students to develop a model and use it to describe the system.

It would be easy to blow off this distinction, to say Bah, same difference. The test asks students a lot of questions that would show an understanding of these concepts, so we're covered.

But not really. Asking a person to develop a model is a much higher-order task than asking them to copy a model. Describing systems and patterns is way more challenging than selecting the correct description.

Developing models and explaining things is the work of real scientists: They notice phenomena, study it, then figure out how to represent those phenomena in order to make it clear to other people. To say, "Look at this! It's interesting and it explains why things are the way they are!"

And that's exactly what the NGSS authors had in mind: "Any education that focuses predominantly on the detailed products of scientific labor-the facts of science-without developing an understanding of how those facts were established or that ignores the many important applications of science in the world misrepresents science and marginalizes the importance of engineering." (National Research Council, 2012, p. 43).

In other words, a superior education will teach students to think and practice like scientists. If we don't plan learning experiences that make that possible, we're giving them a sub-par education.

The After, Where the Final Product is a Model and a Presentation

So if we re-do this unit plan with backward design, we'll need to start by developing an assessment that would measure success with that standard. That means the assessment would not be a test where students merely label the moon phases, but a student-developed model of the moon phases along with some kind of presentation where students use that model to explain lunar phases, eclipses, and seasons.

When designing this final assessment, it's essential that the teacher crafts a rubric that clearly outlines specific, high standards for both the model and presentation. The rubric should list criteria for the accuracy and functionality of the model, plus the quality of the presentation itself. Here's an example of what that might look like:

With a good rubric in place, we then work backwards to determine what lessons students need to do excellent work on the final assessment.

Direct Instruction: Students still need to know the basics of the lunar system. Memorization is not the goal, so we're not grading for that, but students do need to know the information well enough to explain it and use the right vocabulary, so we can keep the lecture or video from the original plan. And having them work with the information a bit on some kind of worksheet or with some kind of online practice is fine, but I wouldn't give points for this work. If you need to motivate students to do it, require them to demonstrate proficiency (like scoring at least 80% on the match game with a set of Quizlet flashcards) before they can move on to the next step.

Active processing with models: Next, give students a chance to work with a model so that they experience the cycle in action. This will deepen their understanding of the earth-sun-moon system, which will better prepare them to give their own presentations. Something like this interactive model from CK-12 allows students to manipulate the time of day and the position of the moon to see how these variables change what we see in the sky. Working in groups, students could use the simulator to answer questions about the moon's appearance depending on what phase it's in and the time of day, then transition into making and checking predictions based on those variables.

Presentation practice: If students are going to do well on the final presentation, they'll need practice in explaining the lunar cycle in their own words. Using an existing model (such as the interactive mentioned in the previous step), have students explain the lunar phases, including information about the seasons and eclipses, to a partner or small group. Ask the listening partners to "coach" the presenter if their description has any holes or inaccuracies. Hearing their peers explain the system, along with attempting the explanation themselves, will help them use the language of the lunar cycle more fluently.

Model development: Now students begin developing their own models. The standard does not specify that this has to be a physical model, so you could offer students choice: The model could very well be made with styrofoam balls, but it could also be a hand-drawn diagram, a slideshow presentation, an animated video, a children's book, or even a short skit they present to the class or record on video. Many of these options, like the children's book, include the "presentation" piece right along with the model. Give students time in class to work on these models; this will ensure that they do their own work and will allow you to give feedback if a student is heading in the wrong direction. Remind them regularly that the explanation of the model is nearly half their grade, so they should consider scripting that out if their presentation is going to be "live."

Model presentations: Finally, students present their models. If you have a lot of students opt to give a "live" presentation this could take several class periods, which could be pretty mind-numbing since the subject matter is exactly the same for everyone. To cut down on that overall time, only have students present to the whole class if the class is actually participating in the model (like in some sort of skit or simulation). Otherwise, students can record their presentations on video or present to you one-on-one while their classmates work on something else independently.

With this "after" version, every lesson is designed to prepare students to give excellent presentations at the end. The whole time, they are using the lunar cycle vocabulary, correcting each other's misconceptions, and just like scientists, thinking about how to explain concepts to other people.

Some Questions About Your Lessons

Okay, so we've looked very closely at one small unit for a middle school science class. Now, take this same process and apply it to the things you teach.

1. What exactly do your standards require? Do they ask students to memorize and identify facts, or do they ask them to describe, explain, analyze, or create? (It's probably the latter.)

2. If it's the latter, how closely do your assessments measure those standards? Do they actually require students to do the describing, explaining, analyzing, or creating (which would likely require them to write, present, or create some product), or do they merely ask them to recognize when someone else does it in the form of an answer on a multiple-choice test?

3. Do you need to adjust your assessments so they more closely align with the standards?

4. If you do, the next step is to rework the lessons that lead up to that assessment. Does every lesson contribute to student success on that assessment? Could some of your lessons be omitted because they don't connect directly to the assessment? Are you missing anything? For example, if your assessment requires students to write in academic language and support their ideas with evidence, you should include some lessons that give students practice with that kind of writing.

5. Finally, will the assessment be weighted heavily in your gradebook? (It should be.)The lessons and activities leading up to the final assessment are there to give students exposure to the knowledge and practice with the skills necessary to perform on that final assessment; ideally, they should receive no grades at all on those activities. If you absolutely must assign some points, be sure the final assessment is worth a heck of a lot more than those smaller tasks.

Like I did, you probably also have some favorite lessons and activities. Some of these might turn out to be not just fun to teach, but also solid in terms of equipping students with knowledge and skills that will last.

If it turns out that those favorite lessons don't really align with any standards, you might be able to revise them so they do. Or you might keep them for other reasons-not every minute of class time has to be spent on standards-based instruction. Some activities have value because they help us get to know each other better, they help students develop social-emotional skills, or they simply offer a bit of fun. But if a lesson doesn't do any of these things, if it's disguised as learning but is doing little more than keeping students busy, it's time for it to go.

Using a process like backward design helps us get better at making these decisions. By making this approach part of our regular practice, we'll be able to look back on a day, a week, or a year of teaching and say with a lot more certainty that when they were under our care, our students learned.
Reference:
National Research Council (2012). A framework for K-12 science education: Practices, crosscutting concepts, and core ideas. The National Academies Press. https://doi.org/10.17226/13165.

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Are we planning with clear, measurable, meaningful learning goals in to guide us, or are we just keeping students busy?

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An illustration of the typical planning cycle

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18012019 Planning Cycle infographic v2_EDIT

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This video introduces the OneNote teacher planner for 2022. Below the video you will find links to download the planner.

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Retrieval practice and transfer of learning

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Retrieval practice and transfer of learning

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Transference of learning is difficult this guide discusses some ways to make it happen.

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Spacing and retrieval involves separating learning over multiple lessons and providing opportunities for students to recall what they have previously learned in different contexts. Science indicates that this helps to commit learning to long-term memory.

Spacing and retrieval practice has a strong evidence base that meets our highest standards of evidence. For more information about the evidence base, s ee below:

To be effective, spacing and retrieval practice need to be implemented well. See below some 'things to know' when using these practices.

Spacing and retrieval practice may look different in different contexts. See below examples of spacing and retrieval practice in a variety of classrooms and settings.

Here you will find tools to help you implement spacing and retrieval practice in your classroom or setting.

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Spacing and retrieval involves separating learning over multiple lessons and providing opportunities for students to recall what they have previously learned in different contexts. Science indicates that this helps to commit learning to long-term memory. Spacing and retrieval practice has a strong evidence base that meets our highest standards of evidence.

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How to use retrieval practice to improve student learning

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How to use retrieval practice to improve student learning

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A guide to retrieval practice

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How to use spaced retrieval practice to boost learning

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How to use spaced retrieval practice to boost learning

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A guide on how to use retrieval paractice.

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A guide to interleaving a maths curriculum

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On this page you will find links to courses, podcasts and more about how to interleave and spiral a curriculum.

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You searched for spiral - Make Math Moments

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The Make Math Moments 3-Part Framework Guidebook

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Planning resources for Mathematics from Victoria

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Back to Mathematics

Planning for Mathematics in the Victorian Curriculum F-10 supports whole-school curriculum planning and the progression of learning, assessment and reporting in the classroom.

The following resources have been developed to assist curriculum planning for Mathematics in the Victorian Curriculum F-10.

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Back to Mathematics Planning for Mathematics in the Victorian Curriculum F-10 supports whole-school curriculum planning and the progression of learning, assessment and reporting in the classroom. The following resources have been developed to assist curriculum planning for Mathematics in the Victorian Curriculum F-10.

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Time is short. This paper looks at how teachers can use time more efficiently.

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A great discussion on how to use innovative software to help students make more progress. So many students struggle with maths because of low literacy skills. Learn how new technology can reduce some of these difficulties.

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Happy New year and welcome to 2022!

In our first episode for the year, host Leanne McMahon talks to educator and digital guru Greg O'Connor from Texthelp. Greg poses some of the problems he sees in how our students are learning maths and suggests some solutions for overcoming these issues. Contact us at www.calculate.org.au or https://www.facebook.com/amsischools

Resources:

Contact Greg:

Email: g.oconnor@texthelp.com

Twitter @gregoconnor

Webpage: gregoryoconnor.com
Texthelp www.texthelp.com EquatIO
https://www.texthelp.com/en-au/products/equatio/
EquatIO - Free for Teachers http://text.help/freeforteachers
EquatIO For Schools/Universities - contact asiapac@texthelp.com
Other resources: Early math skills predict later academic success
Australia's Chief Scientist 2020 Australia's STEM Workforce Report

A couple of posts by Greg:
Literacy the real barrier to STEM uptake Can educational technologies help to reverse Australia's maths slump? How should schools tackle the maths slump?
Research has shown that mathematics textbooks contain more concepts per sentence and paragraph than any other type of text.
Joan M. Kenney (2005) Literacy Strategies for Improving Mathematics Instruction, ASCD, 2005
STEM texts are often written above the grade level for which they are intended

Barton, M. L., Heidema, C., & Jordan, D. (2002). Teaching Reading in Mathematics and Science. Educational leadership, 60(3), 24-28. PDF

Does reading *really* matter in mathematics?
https://www.theconfidentteacher.com/2021/04/does-reading-really-matter-in-mathematics/
Readability with Rewordify
https://rewordify.com/

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The Problem with Maths (and some digital ideas to fix it)

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Happy New year and welcome to 2022! In our first episode for the year, host Leanne McMahon talks to educator and digital guru Greg O'Connor from Texthelp. Greg poses some of the problems he sees in how our students are learning maths and suggests some solu...

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This link takes you to a list of papers on different evidence based pedagogies and how they can be used in the classroom to improve student outcomes.

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Our 'Tried and Tested' series covers education practices that have been proven to make a difference to learning outcomes for children and students.

The guides are written for teachers and educators, supporting ongoing professional development across key evidence-based practices. They can be read individually or together. Each guide is carefully sequenced, providing a step-by-step outline of how to effectively implement the practices.

How to use these guides

You can use these guides in a variety of ways.

If you are a new educator or teacher, you may use the guides to learn about how to implement certain practices that may be new to you, or to reflect and refine your practice.

If you are experienced, you may use these guides as a reminder or to gain new ideas for your own practice, or for instructional coaching and mentoring.

The guides could also be used in collaborative practice as part of informal or formal professional learning.

As part of AERO's ongoing work, we will be undertaking  research  into how teachers and educators use these guides in practice. We will use the findings to inform future directions, including how we support you to most effectively use evidence to improve practice.

Why are they important?

These guides are particularly important in light of the disruptions to learning as a result of COVID-19. Disadvantaged students and children are likely to have had the greatest learning losses during shutdowns associated with COVID-19. The practices in the guides have been proven to work quickly and effectively with this cohort, as well as others.

Can I trust these practices?

These guides only include practices that have been proven to work.

AERO has synthesised the most rigorous and relevant evidence-based practices from meta-analyses, systematic reviews and literature reviews. We have rated these sources of information against our Standards of evidence, focusing on evidence generated in an Australian context where possible.

Each individual guide has an annotated reference list outlining the evidence base. At times, some of the information in our resources draws on older seminal texts that inform later developments in the evidence base. We balance this information with more recent studies when presenting the evidence base.

Other resources

The Tried and Tested guides complement and support existing Australian resources such as the Australian Professional Standards for Teaching, the Early Years Learning Framework and guidance offered by schools and systems. Some specific guidance that may be relevant to you:

Teachers and leaders in ACT government schools are encouraged to consider these resources within the wider framework set out in Enabling Pedagogies, accessible with their system log-in.

Teachers and leaders in the Independent sector are encouraged to read these resources in conjunction with pedagogical guidance provided by their Association.

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These guides aim to support ongoing professional development by summarising key practices that can rapidly improve learning outcomes for students and children.

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Listen to my interview with Pooja Agarwal ( transcript):

Sponsored by Raymond Geddes and Kids Discover

We've talked about retrieval practice two other times on this site: First in 2015, when we read the book Make it Stick. Then in 2016, I interviewed cognitive psychologists Megan Smith and Yana Weinstein about the Six Powerful Learning Strategies You Must Share with Students.

In both cases, I noticed that although we were discussing collections of techniques, retrieval practice was at the core of both groups. Because it's such a powerful technique, I would think that by now it would have really taken off, that I'd hear teachers talking about it and see the term crop up in lots of education spaces, but I don't see that happening.

So I wanted to give it one more push by carving out a designated spot for it here. My goal is to help retrieval practice become a regular part of classroom practice and part of our common vocabulary as teachers. To help with this cause, I interviewed Pooja Agarwal, Ph.D., cognitive scientist and founder of RetrievalPractice.org.

What is Retrieval Practice?

Retrieval practice is the act of trying to recall information without having it in front of you. Suppose you're studying the systems of the human body-skeletal, muscular, circulatory, and so on. You could do retrieval practice by attempting to name those systems without looking at the list. Once you've listed all you can remember, you'd open up your book or notes and check to see if you got them right.

You might be thinking, This is nothing new. The whole concept of flashcards is built on retrieval practice, and flashcards have been around forever, right?

What's new is the research: In recent years, cognitive psychologists have been comparing retrieval practice with other methods of studying-strategies like review lectures, study guides, and re-reading texts. And what they're finding is that nothing cements long-term learning as powerfully as retrieval practice.

The Research

Agarwal and her colleagues studied the effects of retrieval practice with students in a middle school social studies course (McDaniel, Agarwal, Huelser, McDermott, & Roediger, 2011). Over the course of a year and a half, while the teacher continued teaching as normal, students were regularly quizzed on the material with no-stakes quizzes, meaning they wouldn't count against their grades. These quizzes only covered about one-third of what was being taught. The teacher left the room for every quiz, so she had no knowledge of what was included in the quizzes.

On end-of-unit exams, students scored a full grade level higher on the material from the quizzes than on any of the other material. The other concepts had been taught and reviewed by the teacher as they normally would; the only difference was that some things also appeared on the no-stakes quizzes, and those were the concepts students retained more fully when tested on the exam.

The very act of being quizzed actually helped students learn better.

Here's what this means for teachers: When we teach something once, then want to do something else to help students learn it better, instead of just reviewing the content, we're much better off giving something like a quiz instead. In other words, if we do more asking students to pull concepts out of their brains, rather than continually trying to put concepts in, students will actually learn those concepts better.

To read about other studies on retrieval practice, click here. For now, let's look at some ways you can incorporate retrieval practice into your instruction.

Ways to Use Retrieval Practice in the Classroom

Think-Pair-Share
This quick, low-maintenance strategy can be used at any time to have students recall information, then share it with a partner. You can use think-pair-shares with single-answer questions, or make them more open-ended, like "Think of one thing you learned yesterday about cells." Be sure to have students think on their own before turning to a partner. "It's important for students to retrieve individually as much as possible," Agarwal advises. "If you jump right into pairs, then we all know as educators that some students are retrieving and some may not be." One way to make sure all students retrieve is to have them jot responses down on paper before sharing them with a partner.

Low-Stakes Quizzes
These can be given on paper, in a Google Form, with an individual response system like clickers, Plickers, or Poll Everywhere; or by using a game like Kahoot or Quizziz. It's important to note that these quizzes are a learning strategy: Ideally, students wouldn't get scores on them at all, but if you must give some points, make them an almost negligible part of students' overall class grade.

Brain Dumps
Have students get out a sheet of paper and, within a certain length of time, write down everything they know about a topic of study. This can be done at the beginning of a unit (similar to how you'd use a K-W-L Chart), partway through as a way to reinforce learning, or near the end of a unit. Once students have completed theirs, they can exchange them in a think-pair-share or use them to compile a whole-class brain dump. Then they can return to their texts to see what they missed or what needs correction.

Flashcards
These can be a powerful retrieval tool in class or at home, but students need to be taught to use them correctly:

Once a card has been mastered, keep it in the deck a while.
Research has shown that students tend to "drop" cards out of their decks too soon after they have mastered them (Karpicke, 2009). Ideally, a fact should be successfully retrieved three times before a student moves on from it.

Actually retrieve.
Karpicke's research also found that when students see a familiar prompt on a flashcard, they have a tendency to tell themselves they know it, then flip it over to see the answer, rather than taking a few extra seconds to actually recall the answer-and ideally, say it out loud-before flipping the card over. The difference in timing is subtle, but important: Students will not get the same benefits from flashcards unless they actually retrieve the answer before seeing it.

Shuffle the deck.
Keeping the cards in the same order makes them predictable. Once a deck has been gone through a few times, it should be shuffled to make it more challenging.

For more tips, take a look at College Info Geek's excellent post on effectively using flashcards.

Tips and Cautions

Retrieval practice is not the same thing as assessment.
Although some retrieval activities might allow you to formatively assess student understanding, remember that retrieval practice is a learning activity. Again, if you must assign points for practices, make sure they will have a very low impact on student grades.

Space your practice.
Retrieval practice is even more effective if it's done in short bursts over time, rather than in a single long session. This spacing causes students to forget some of the material, and the struggle involved in trying to recall it strengthens their long-term learning.

Include feedback.
If students retrieve the wrong information, the practice won't be much good unless they find out the right information, so be sure to give them feedback as they go. This can be as simple as providing the correct answer or letting students check their texts after they have attempted to retrieve. "Part of that isn't just so students know if they got their answer correct or incorrect," Agarwal says, "but it adjusts students' metacognition," helping them get "better at estimating or judging what they know and what they don't know."

Match your practice to your assessment.
If your summative assessment for a chunk of content will be basic recall of facts, then the retrieval practice can be that simple. But it you're going to include some higher-order questions as well, then be sure to ask those same types of questions in retrieval practice.

Does it Fit with Inquiry Learning?

If your classroom is built around more hands-on, inquiry-driven approaches to learning that don't prioritize retaining specific content-structures like Genius Hour and project-based learning -you might wonder whether retrieval practice is relevant to your teaching.

Consider this: Within any of those frameworks, students still have to learn and retain discrete information. Take, for example, a student who has decided to learn Arabic as his focus for Genius Hour, or students who are studying soil pH as part of a gardening project. Both students will need to study and remember concrete information in order to make progress, so even if your instruction isn't set up in a traditional way, understanding how retrieval practice works can help you help your students learn better.

A Little Goes a Long Way

Agarwal's final piece of advice is to start small. "If a student just writes down two things or one thing they learned, and then you move on, that's okay," she says. "You'll still get benefits of retrieval without spending five minutes in a classroom discussion."

It can be a bell-ringer. An exit slip. A sponge activity. Something you do while students stand in line for lunch. Don't overthink it. Agarwal advises teachers to ask themselves this: "What is one way we can flip from reviewing information to students retrieving information? What can we do tomorrow?"

What will you do tomorrow? ♦

Learn More

Dr. Agarwal and her colleagues have put together this comprehensive guide to using retrieval practice. Visit RetrievalPractice.org/download to get a free copy for yourself. And be sure to follow Retrieval Practice on Twitter at @RetrieveLearn.
References:
Karpicke, J. D. (2009). Metacognitive control and strategy selection: deciding to practice retrieval during learning. Journal of Experimental Psychology: General, 138(4), 469.

McDaniel, M. A., Agarwal, P. K., Huelser, B. J., McDermott, K. B., & Roediger III, H. L. (2011). Test-enhanced learning in a middle school science classroom: The effects of quiz frequency and placement. Journal of Educational Psychology, 103(2), 399.

There's a lot more where this came from.
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Retrieval Practice: The Most Powerful Learning Strategy You're Not Using

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Nothing cements long-term learning as powerfully as retrieval practice. Learn how to incorporate it into your classroom.