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What are the features of 'teaching for mastery'?

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What are the features of 'teaching for mastery'?

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What are the features of ‘teaching for mastery’?

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Research - Summary of evidence for elements of teaching

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Research - Summary of evidence for elements of teaching

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related to mastery in mathematics

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Question: Just how much should students practice what they learn? On the one hand, it seems obvious that practice is important. After all, "practice makes perfect." On the other hand, it seems just as obvious that practicing the same material again and again would be boring for students. How much practice is the right amount?

Answer: It is difficult to overstate the value of practice. For a new skill to become automatic or for new knowledge to become long-lasting, sustained practice, beyond the point of mastery, is necessary. This column summarizes why practice is so important and reviews the different effects of intense short-term practice versus sustained, long-term practice.

That students would benefit from practice might be deemed unsurprising. After all, doesn't practice make perfect? The unexpected finding from cognitive science is that practice does not make perfect. Practice until you are perfect and you will be perfect only briefly. What's necessary is sustained practice. By sustained practice I mean regular, ongoing review or use of the target material (e.g., regularly using new calculating skills to solve increasingly more complex math problems, reflecting on recently-learned historical material as one studies a subsequent history unit, taking regular quizzes or tests that draw on material learned earlier in the year). This kind of practice past the point of mastery is necessary to meet any of these three important goals of instruction: acquiring facts and knowledge, learning skills, or becoming an expert.

Acquiring Facts and Knowledge

Intuition tells us that more practice leads to better memory. Research tells us something more precise: Memory in either the short- or long-term requires ongoing practice.

Let's first consider memory in the short-term, meaning days or weeks. Suppose I am trying to learn the procedures necessary for a bill to become a federal law. I might study these facts (using any number of techniques) and periodically test myself. Suppose further that I study until I perform perfectly on my self test. Do I know these facts? Yes, I know them now. But what about tomorrow? In order to protect this learning from the ravages of forgetting, I need to practice beyond one perfect recitation. Studying material that one already knows is called overlearning. Because memory is prone to forgetting, one cannot learn material to a criterion and then expect the memory to stay at that level very long.

Anticipating the effect of forgetting dictates that we continue our practice beyond the mastery we desire. In an illustrative experiment (Gilbert, 1957), subjects were read a brief paragraph about a fictional country and then asked 22 questions based on the paragraph. If the subject answered a question correctly, the question was discarded. Then the subject heard the paragraph again, and was asked those questions that he or she had missed. The procedure was repeated until the subject successfully answered all of the questions. Another group of subjects participated in a second condition that required overlearning. A question was not discarded until it had been answered correctly three times rather than once. All subjects received a surprise retest after a delay of either 15 minutes or two days. The overlearning group performed better at the short delay (22 questions correct versus 15) and also at the long delay (17 questions correct versus 13). Overlearning has been studied (although not extensively) for many years. These results are typical, but most of the experiments deal with short-term retention.

It may seem that the emphasis on short-term knowledge is peripheral to education. As teachers, we want long-lasting knowledge, not just knowledge for a few days. But, in fact, teachers may have goals that entail short-term knowledge. For example, a science teacher may want students to have a series of facts about certain species at their fingertips so that the teacher can introduce an important abstract concept concerning evolution on which those facts depend. Once the student has used the facts to gain a firm understanding of evolution, no great educational harm is done if the particular facts about particular species are forgotten. But without those facts well-lodged in memory for at least a short time, harm would be done to a student's ability to grasp the larger concept.

For other material, we most certainly do want longer-term retention. In this case again, practice past the point of mastery is essential. In the case of overlearning, the practice begins with active studying for the purpose of learning. Over time, practice will take the form of using old material in the course of studying some new material. For example, students will initially study the terms isthmus and delta to master their meanings, and will later practice these meanings as they use the terms in their continued study of geography.

Although practice takes on a different character for the longer-term, it is no less important. Studies show that if material is studied for one semester or one year, it will be retained adequately for perhaps a year after the last practice (Semb, Ellis, & Araujo, 1993), but most of it will be forgotten by the end of three or four years in the absence of further practice. If material is studied for three or four years, however, the learning may be retained for as long as 50 years after the last practice (Bahrick, 1984; Bahrick & Hall, 1991). There is some forgetting over the first five years, but after that, forgetting stops and the remainder will not be forgotten even if it is not practiced again. Researchers have examined a large number of variables that potentially could account for why research subjects forgot or failed to forget material, and they concluded that the key variable in very long-term memory was practice.* Exactly what knowledge will be retained over the long-term has not been examined in detail, but it is reasonable to suppose that it is the material that overlaps multiple courses of study: Students who study American history for four years will retain the facts and themes that came up again and again in their history courses.

Learning Skills

Acquiring factual knowledge is only part of what we want our students to gain from their schooling. We also want them to be skilled problem solvers, effective written and oral communicators, and creative thinkers. These skills-and indeed, all skills that involve thinking-rely on working memory capacity. Working memory is, to put it colloquially, the place in the mind where thought happens. It is often called the bottleneck of the mind because there is a limited amount of space in working memory. That is why it is difficult to mentally divide 34,516 by 87. It is hard to simultaneously maintain the numbers, employ the processes for long division, and update the answer as you derive it. This space limitation is relevant not just to mental arithmetic, but to most types of problems we would like our students to solve, such as writing a clear laboratory report, reading an essay with deep understanding, or seeing the links between historical events.

Our ability to think would be limited indeed if there were not ways to overcome the space constraint of working memory. One of the more important mechanisms is the development of automaticity. When cognitive processes (e.g., reading, writing grammatically, reading a map, identifying the dependent variable in a science experiment, using simple mathematical procedures) become automatic, they demand very little space in working memory, they occur rapidly, and they often occur without conscious effort.

For example, if you are reading this article, the process of reading is very likely automatic for you. You do not need to laboriously piece together the letters of each word to puzzle out its identity. Your mind seems to divine the meaning of prose immediately and without effort on your part. Try this classic demonstration of automaticity for advanced readers. In this task you are asked to name the ink color in which the words are printed, but ignore the word that the letters spell. Hence for the stimulus TURKEY, the proper response is "blue." First try this list:

Now try this list:

The second list is much harder to read than the first list because, for you, reading is automatic. Even though you try not to read the words that the letters form, you read them automatically and doing so conflicts with naming the ink color. For someone who cannot read, the second list is no harder than the first.

But most of the time automaticity is helpful, rather than disruptive. Picture a beginning reader slowly puzzling out the word "blue." Doing so consumes all of working memory, so it is difficult for the student to follow the plot of the story in which the word appears. Once reading is automatic, however, precious working memory resources can be devoted to considering the meaning of a text, the effectiveness of its argument, and so on.

Automaticity is important not only in reading, but in all mental life. Consider how difficult it would be to navigate an unfamiliar city by car if you had to focus on how hard to press the accelerator and brake, how far to turn the steering wheel, when to monitor your mirrors, and all of the other components of driving that have become automatized.

Automaticity is vital in education because it allows us to become more skillful in mental tasks. An effective writer knows the rules of grammar and usage to the point of automaticity-and knows automatically to begin a paragraph with a topic sentence, include relevant detail, etc. The effective mathematician invokes important math facts and procedures automatically. Readers who are able to visualize a map of the world will find various books and assignments easier to read (and learn more from them). In each field, certain procedures are used again and again. Those procedures must be learned to the point of automaticity so that they no longer consume working memory space. Only then will the student be able to bypass the bottleneck imposed by working memory and move on to higher levels of competence.

The development of automaticity for generalized skills depends on high levels of practice (e.g., Shiffrin & Schneider, 1984). There is no substitute. Ensuring consistent, sustained practice is the most reliable way to ensure that a student will become an effective reader, writer, or scientist. Following a complex written argument, writing a convincing essay, or engaging in scientific reasoning are all skills that are enabled by the automatization of each discipline's basics.

Becoming an Expert

What does it take to become an expert in a field? Consider a true expert, meaning one who is recognized not just as fully competent, but as a unique contributor to the discipline. In competitive arenas (e.g., athletics or chess), we would say that an expert competes at the national or international level. When asked how an expert gained such a high level of skill, non-experts usually attribute the success to innate talent. Experts themselves, however, tell a different story. They attribute their success to practice and to the ability to maintain concentration during long practice sessions (Ericsson, 1996). (The importance of practice doesn't mean that innate talent is meaningless, of course; practice is necessary for excellence, but it may not be sufficient to ensure it.)

Research studies indicate that experts are right, at least in that they do practice a great deal. Descriptive studies (Roe, 1953) of eminent scientists indicate that the most important factor predicting their success is not innate talent or intelligence, but the willingness to work hard for extended periods of time. This commitment to practice was reinforced by a large-scale study (Bloom, 1985) in which experts in athletics, science, and the arts were interviewed, along with their parents and teachers. Bloom proposed that the training of an expert typically involved four stages. The future expert was usually introduced to the domain under playful conditions as a child. His or her promise was noted, and in stage two, lessons were provided, usually with a teacher or coach who worked well with children, and regular practice habits were established. In the third stage, an internationally recognized teacher or coach was engaged, usually requiring a significant commitment of resources from the parents, as well as dedicated and likely exclusive study by the child. In the fourth stage, the student had absorbed all that he or she could from teachers and began to develop his or her personal contribution to the field.

Recent research that measures practice time more carefully paints a similar picture. The figure below depicts the estimated cumulative practice time of violinists separated by their ability levels. The best and good students were enrolled at a music academy that trains professional musicians; they were put into these categories, unbeknownst to them, by their professors for the purpose of this study. Subjects were asked to estimate the time they spent practicing each week. The graph below shows the total accumulated practice time at each age. Two conclusions may be drawn from the graph: Experts engage in a great deal of practice, and that even among very able performers, the best are those who have practiced more.

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The better violinists engaged in more practice during their training.
Figure adapted from K.A. Ericsson, R.T. Krampe, & C. Tesch-Romer (1993), p.379.

Some evidence that a great deal of practice, and not just talent, is a prerequisite for expertise is the "ten year rule," which states that individuals must practice intensively for at least 10 years before they are ready to make a substantive contribution to their field. What about prodigies like Mozart, who began composing at the age of six? Prodigies are very advanced for their age, but their contributions to their respective fields as children are widely considered to be ordinary. It is not until they are older (and have practiced more) that they achieve the works for which they are known.

How are such studies relevant to the average student? Few students will become a Mozart, Shakespeare, or Einstein, but if we want children to understand and appreciate excellence, we would do well to send the message that excellence requires sustained practice. The athletes and artists revered by many students excel not solely by virtue of their talent, but because of their hard work. Edison remarked that "genius is one percent inspiration and ninety-nine percent perspiration." The relative percentages of talent and practice are unclear, but the necessity of long periods of focused practice to exploit inborn talent is not.

What Material Merits Practice?

When we refer to "practice," it is important to be clear that it differs from play (which is done purely for one's own pleasure), performance (which is done for the pleasure of others), and work (which is done for compensation). Practice is done for the sake of improvement. Practice, therefore, requires concentration and requires feedback about whether or not progress is being made. Plainly put, practice is not easy. It requires a student's time and effort, and it is, therefore, worth considering when it is appropriate.

It was noted above that sustained practice over time is especially useful for developing automaticity in specific skills (which enables higher-level thinking) and in ensuring that a memory lasts as long as needed. Thus, the following types of material are worthy of practice:

1. The core skills and knowledge that will be used again and again. In this case, we give practice in order to ensure automaticity. The student who struggles to remember the rules of punctuation and usage (or must stop to look them up in a reference book) cannot devote sufficient working memory resources to building a compelling argument in his or her writing. The student who does not have simple math facts at his or her disposal will struggle with higher math.

2. The type of knowledge that students need to know well in the short term to enable long-term retention of key concepts. In this case, short-term overlearning is merited. For example, as noted earlier, a science teacher may want students to know a set of facts about certain species so that she can introduce an important abstract concept concerning evolution that depends on these facts. Or, a high school history teacher may want students to master the facts of several Supreme Court cases in order to build long-term understanding of a particular constitutional principle.

3. The type of knowledge we believe is important enough that students should remember it later in life. In this case, one might consider certain material so vital to an education that it is worthy of sustained practice over many years to assure that students remember it all of their life. A science teacher might spend the better part of a year emphasizing basic principles of evolution in the belief that the material is essential to consider oneself conversant in biology. Further, the curriculum might address and require practice in evolution in multiple years to assure that such knowledge will last a lifetime. Do we expect that a 40-year-old will have retained everything learned through the 12th grade? No, but do we expect that she will retain anything? Should she be able to grasp the basics of evolution or describe the different responsibilities of the three branches of the federal government or calculate the area of a circle? Exactly what sorts of knowledge merit the focus required to create long-lasting memory will be controversial, but that practice is required to create such memories is not.

* * *

How should practice be structured-should a teacher strive for overlearning in the short term or repeated learning over the long term? The answer will depend on whether the goal is automaticity in skills, short-term knowledge, or long-term knowledge-and what the teacher knows about the future curriculum students will encounter. For example, an English teacher might deem it very important that students understand the use of metaphor in poetry, but extensive, focused practice may not be practical or necessary. This knowledge will likely be developed over a number of years, and there will be opportunities for practice in the future. In other cases there will be future opportunities for practice, but the timeliness of the learning is important. For example, one teacher might provide just a cursory introduction to first-graders on how to tell time, figuring that the students will have ample opportunities for practice in the future. But another teacher might also reason that first-graders need to know how to tell time (so that, for example, they can monitor their activities during the day and be more self-directed) and so focus practice on this skill. Similarly, a French teacher may realize that students will have plenty of practice conjugating the verb être (to be) over the long term, but may justly believe that students must know this material early in their training or their ability to read, write, and understand French will be badly hampered.

Exactly when to engage students in practice, through what method, and for what duration are educational decisions that teachers will need to make on a regular basis. But, that students will only remember what they have extensively practiced-and that they will only remember for the long term that which they have practiced in a sustained way over many years-are realities that can't be bypassed.

Daniel T. Willingham is associate professor of cognitive psychology and neuroscience at the University of Virginia and author of Cognition: The Thinking Animal. His research focuses on the role of consciousness in learning.

*It is likely relevant that there is not only more practice in this case, but that the practice is distributed across time rather than concentrated in a few months (see " Allocating Student Study Time: "Massed" versus "Distributed" Practice," from the Summer 2002 issue of American Educator) ( back to article)

References

Bahrick, H. P. & Hall, L. K. (1991). Lifetime maintenance of high school mathematics content. Journal of Experimental Psychology: General, 120, 20-33.

Bahrick, H. P. (1984). Semantic memory content in permastore: Fifty years of memory for Spanish learned in school. Journal of Experimental Psychology: General, 113, 1-29.

Bloom, B. S. (1985). Generalizations about talent development. In B. S. Bloom (Ed.), Developing talent in young people (pp. 507-549). N.Y.: Ballantine.

Ericsson, K. A. (1996). The acquisition of expert performance: An introduction to some of the issues. In K. A. Ericsson (Ed.) The Road to Excellence (p. 1-50). Hillsdale, N.J.: Erlbaum.

Ericsson, K. A., Krampe, R. T., & Tesch-Romer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100, 363-406.

Gilbert, T. (1957). Overlearning and the retention of meaningful prose. Journal of General Psychology, 56, 281-289.

Roe, A. (1953). The Making of a Scientist. N.Y.: Dodd Mead.

Semb, G. B., Ellis, J. A., & Araujo, J. (1993). Long-term memory for knowledge learned in school. Journal of Educational Psychology, 85, 305-316.

Shiffrin, R. M. & Schneider, W. (1984). Controlled and automatic human information processing: II. Perceptual learning, automatic attending, and a general theory. Psychological Review, 84, 127-190.

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Ask the Cognitive Scientist: Practice Makes Perfect-But Only If You Practice beyond the Point of Perfection

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Question: Just how much should students practice what they learn? On the one hand, it seems obvious that practice is important. After all, "practice makes perfect." On the other hand, it seems just as obvious that practicing the same material again and again would be boring for students.

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Revisit - What are the features of 'teaching for mastery'?

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Revisit - What are the features of 'teaching for mastery'?

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What are the features of ‘teaching for mastery’?

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What Is Variation Theory? A Guide For Primary Maths Teachers

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What Is Variation Theory? A Guide For Primary Maths Teachers

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This blog will look at the concept of Variation Theory and attempt to demystify the confusing language surrounding it. It is hoped that this will make Variation Theory an accessible and exciting tool that you can begin to use in your maths lessons, or give you an understanding of what is happening if you recognise the principles, despite never having heard of Variation Theory before!
If you are a premium member of Third Space Learning, you can access a whole-school CPD presentation that covers these points in our Maths Hub.

What is Variation Theory in primary maths?

Variation Theory is a general theory of learning that is used predominantly in mathematics teaching and pedagogy. It emphasises a concept's essential features by focusing on what is kept the same and what changes, which offers the opportunity to make meaningful connections.

Origins of Variation Theory

The study of Variation Theory arises mainly from work across two groups of researchers who arrived at similar, but not exactly the same conclusions.

One was Swedish researcher Ference Marton and colleagues who worked on the concept of Variation Theory; the other was Gu Ling Yuan and his team in Shanghai who called it Bianchi (teaching with variation). As one would expect, the emergence of two similar theories has caused confusion in some areas of the sector. In the context of the United Kingdom, the distinction between them is rarely made and so aspects of Bianchi are used under Variation Theory.

They both agree that 'the central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features'. (Gu, Huang & Marton, 2004). This blog will mainly draw on the learning study of Gu and his team.

The application of Variation Theory is much wider than just the domain of mathematics teaching, but rather as a theory that underpins how students learn anything. The principle that underpins the theory is that new learning occurs when they understand how a critical aspect of a new concept changes (varies) against something that does not change (invariance).

By adjusting their perception of what has remained the same and what has changed, students are more likely to learn the object of learning and understand the underlying mathematical structures being studied. In other words, it is when an educator carefully constructs questions, considers the order in which they appear, and brings the thinking process between questions to the forefront for the students, so that meaningful connections to the mathematical structures can be made from the task.
It is unlikely that a worksheet on its own will bring about the required cognitive process in students needed to benefit from the advantages that Variation Theory offers students in their conceptualisation of mathematical structures.
Do not be deceived into purchasing paid resources that promise to bring about Variation Theory with just a worksheet, unless you can specifically identify how those questions enact the principles of the object of learning, and you feel confident that you can bring out these mathematical structures to the students through your questioning in a very deliberate way.

Indeed, as you will see through some examples, attempts at Variation Theory through only a worksheet approach and no dialogue risk students learning less than not trying to use Variation Theory.

What are the types of Variation Theory?

There are two types of Variation Theory; conceptual variation and procedural variation.

What is procedural variation in maths?

Procedural variation is a process for the formation of mathematical ideas stage by stage, in which students' experience in solving problems is manifested by the richness of varying problems and the variety of transferring strategies.

It is derived from three parts:

1. Varying the problem

2. Multiple methods for solving a problem

3. Multiple applications of a method to similar problems

As with conceptual variation, the key to success is to teach students to look for connections within the mathematical structures from an early age.

What is conceptual variation in maths?

Conceptual variation is a process that looks at providing students with multiple perspectives and experiences within a given concept or object of learning. It refers to the strategies a teacher may employ to bring to the forefront critical aspects of the desired learning outcome, and increase their understanding by demonstrating it from different perspectives.

Marton and his team of researchers concluded that students notice difference between different objects of learning before they notice sameness.

When employing conceptual variation pedagogy in classroom discourse, it is best to provide examples that are totally different before looking at examples that are similar. For example, if you wanted to demonstrate to students what a triangle was, it would be best to present a triangle along with other shapes that are not triangles to help students distinguish what it is that makes a triangle.

Is this an elephant?

A common non-mathematical approach to explain this is to look at the animal kingdom.

How is it that we are acutely aware of what an elephant is? We are able to distinguish it quite easily from the countless other members of the animal kingdom. If pressed to answer how we know the picture above is an elephant, the first two features that people describe to explain their certainty are the trunk and tusks.

To which my reply is this: is this therefore NOT an elephant?

Clearly, it is. But this is a tuskless elephant, and tusks are one of the distinguishing features of an elephant that many people claim. Perhaps we could go on to explain that the size of an elephant is a distinguishable feature, or the grey colour. An elephant must be grey, have a trunk, be big, and can have tusks, but this is not a requirement.

My reply is: if that is the case, is this an elephant?

Patently it is. Yet, this is hardly a big animal and so would not meet the conditions placed upon it to characterise it as an elephant from above. Therefore, an elephant can be large or small, have tusks or not have tusks, be grey and have a trunk. The necessary conditions of what constitute an elephant become ever clearer.

We also know, instinctively, that this picture is not an elephant despite it being small, grey and not having tusks. We know it cannot be because of the ears, the shape of the head and the lack of trunk. And here, through our visual perception, the idea of Variation Theory comes to life.

By seeing what it is to be something in as many different forms as possible and in forms that are related, be it closely or otherwise, we are able to see that concept from multiple perspectives and experiences to truly know it. At its most basic level, this is conceptual variation.

Examples and non-examples of Variation Theory

As mentioned above, we tend to recognise sameness before we recognise difference. It is from this principle that we can consider introducing critical aspects of mathematical structures through examples and non-examples that we share with students. In this instance, to remain true to Marton's findings, the examples that demonstrate the critical object of learning that we want students to learn will be invariant, but the background in which we teach will vary. Subsequently, when we show non-examples they act, as one might imagine, to define what the critical aspect is not.

For example, if teaching the meaning of the equals sign, you could have examples of different calculations using all four operations where the answer was correct and examples using all four operations where the answers were incorrect and so the equivalence sign could not be used.

Standard and non-standard in Variation Theory

Once students have developed a sense of what the critical aspect is and is not, it is important to then demonstrate this aspect in both standard and non-standard forms.

I remember the first time that students I was teaching came across a calculation where the equals sign came before any digits and other symbols '__ = 8 + 9'. There were looks of bewilderment as if they had been tasked with the impossible. The fault of this confusion was entirely my own as every question I had ever asked focused on the standard form when writing calculation '8 + 9 =__'. This meant that students had not fully grasped the concept of the symbol and its critical feature; that it is there to represent equivalence, not just as a symbol that is used to write down an answer afterwards.

In my teaching sequence for teaching equivalence and the equals sign, after looking at examples and non-examples, I should have then looked at standard and non-standard forms like those below:

__ = 8 + 9

10 + 7 = 8 + 9

This graphic might be useful in highlighting how these components fit together.

Procedural variation: what's the same, what's different and what do you notice?

We can use procedural variation to highlight patterns within a set of questions that bring about a critical learning point or object of learning. Say we wanted to draw students' attention to one of the field axioms of multiplication - distributivity - so that they can employ it at will when dealing with any multiplication. We may present this in the following way:

The incorrect way to go about this would be to treat this as a worksheet, in which you deliver an input and tell the students to crack on and complete it. There would be no method to that madness, and the opportunities for bringing out new and previous learnt relationships between the numbers would likely fall by the wayside for all students, or would only be found by those who are high attaining in mathematics.

Rather, a more controlled approach should be taken. You should ask students to complete the first two questions and then ask them, 'what is the same, what is different and what do you notice?' This will provide the opportunity to pick up on what the design of these minimally different questions is attempting to draw out.

Procedural variation steps

Firstly, you would want students to notice the relationship between the multiplicand from the given statement (12) and the combination of the multiplicands from the first two equations (11 + 1 and 10 + 2). Students will be drawn to the fact that they all total the same and that the product is also going to be 96.

From here, students can see if that holds true for the next two questions. The danger of letting students complete all 4 questions first without discussing it could result in students seeing that the product is the same, but not understanding the relationship between the multiplicands.

Notice that the 5 th question in the sequence does not maintain the same pattern exactly. The multiplicands total 12, but the multiplier is now half of the original. This is to stop students aimlessly writing 96 down without giving any thought.

Crucially though, the statement at the beginning can still be used to solve this as the answer will be half of 96. This relationship between the multipliers can also be brought out. Question 6 allows the commutative law of multiplication to come through, as well as the associative law, and the final question reinforces division as the inverse of multiplication.

At all points of getting students to answer these questions in a controlled environment, the questions, 'what is the same, what is different and what do you notice?' are the ones used most, so that students can attend to those differences and similarities. Indeed, completing the above exercise could take up the vast majority of the lesson and students would get more from completing it than a whole worksheet.

Problem solving in procedural variation

At the heart of procedural variation are three forms of problem solving:

1. Varying a problem - extending the original problem by varying the conditions, changing the results and generalisation;

2. The use of multiple methods of solving a problem by varying the different processes of solving a problem and associating different methods of solving a problem;

3. Multiple applications of a method by applying the same method to a group of similar problems.

1. Procedural variation: varying the problem

This form of variation aims to consolidate a concept by varying certain conditions to bring information, often information that is in conflict with students' current understanding of the concept, to light. Look at the example below:

The variation within these problems should be self-evident. The amount of apple juice remains a constant, with only the amount that each jar holds getting varied question by question. Students are often comfortable with division as sharing into equal groups (especially at primary school), where students may generalise that dividing results in making something smaller.

It is this misconception that the example above attempts to exploit through the last three questions. Indeed, we see that dividing by a whole number and decimal produces a quotient greater than the dividend.

2. Procedural variation: varying the method

This type of variation is one that many teachers do, but may not be aware of. Simply teaching and demonstrating different methods in solving a problem is the first step in this type of variation, but its success lies in being able to draw out what students notice about the similarities and differences between each representation. Here are a few worked examples to vary the method for solving 12 × 5.

Your school's calculation policy should set out what methods will be introduced and when. White Rose Maths have produced a calculation policy that looks at what different methods may be taught. That document is available from TES for free.

3. Procedural variation: multiple application of a method to similar problems

This final example looks at bringing about variation, not through changing the method, but with the application of a method or representation against a group of varied yet similar problems.

Variation Theory examples KS1

In KS1, this may look like the following:
1) James has 8 flowers and Julie has 5 flowers. How many flowers are there altogether?
2) James has 5 flowers and Julie has 8 flowers. How many flowers are there altogether?
3) There are 13 flowers, some are red and some are blue. 8 are red. How many are blue?
4) There are 13 flowers, some are red and some are blue. 8 are blue. How many are red?
Here the problems are varied, yet placed within a similar context, and the numbers used are the same, yet what each question is asking students to do differs slightly. For example, for question one, students may use Cuisenaire rods to visualise the problem. They would lay down the yellow rod and the brown rod to make a train and then the orange and the green above (or below) to help solve the total.

Question two enables students to experiment with the commutative law of addition. Notice how all the rods used are the same, but the order in which the bottom train has been constructed has changed. The underlying structure remains unchanged.

Question three tells us that there are thirteen flowers altogether and 8 of them are red. The next step would be to see how many more are needed to take 8 up to 13. Students may initially use white rods before exchanging it for a yellow rod of equal value. Again, the underlying structure of the problem remains unchanged.

The final question is merely the inverse of question 3. We know the total amount of flowers and that a certain subset of them equal 5. Therefore, to find the missing amount, we need to count up from 5 until we get to 13. Again, students may need to use 8 white rods first, but they will see that this can be exchanged for a brown rod.

By the end of this exercise, students will be able to see that the 4 questions have similar underlying structures despite the variation within the problems.

Variation Theory worked examples

As mentioned above, just providing students with questions that contain aspects of Variation Theory will not bring along the benefits of the approach. Therefore, if you do not understand the variation taking place, it would probably be best not to use those examples.

For each of these, we need lots of examples of what the type of variation you're referring to looks like 'in practice'. In other words, with reference to regular sorts of questions and learning objectives from the national curriculum, so teachers can read it and see how they could try implementing it in their own lesson.

If you can pull from each of the main national curriculum topics that would be great, but it's really just a case of showing the wide variety of ways people can think about including this practice in their teaching.

Number examples: number and place value

The examples below vary the representations to give access to the same idea. This approach can be easily transferred across all year groups to look at place value objectives. Using this in a KS1 class would involve looking at each representation and guiding the students through the process of asking 'what is the same, what is different, and what do you notice?' Introducing students to these questions early on will hopefully result in them becoming internalised automatically and so students will come to naturally ask themselves these questions.

The use of colour is particularly useful here as it supports connections between the place value counters, cards and base 10, but the presence of the Cuisenaire rod and the Numicon (whose colours do not conform to this) stops the association of 'tens' being green etc.

A crucial element to draw out of students is the unitization of 2 in some representations or where the 2 has been conserved as 2 ones. The number grid provides an opportunity to link these representations within the abstract number system.

Number examples: addition and subtraction

563 - 100 = ___ - 200

387 - 100 = ___ - 200

983 - 100 = ___ - 200

___ - 100 = 874 - 200

This question focuses on subtracting by multiples of 100 and students' understanding of the equals sign. It is the subtrahend that remains invariant while the minuend varies. This is done so that generalisations can be brought about in regard to the subtraction of multiples of 100. The minuends themselves play no major role in bringing about the effects of Variation Theory.

Here, we would want students to understand that 'equals' does not mean 'answer', but rather that two mathematical statements are balanced and produce a product that is the same as each other.

We can also bring about the relationship between the two minuends, one of which is unknown, when compared to the effect of subtracting the first one by one hundred. Namely, that the tens and ones do not change and that as the subtrahend has increased by 100 from 100 to 200. In order for the proportionality to remain constant between each statement, the minuend must also increase by 100.

It would be in the interest of students to quickly subtract a multiple of 100 from a 4-digit number where the difference is a 3-digit number in subsequent lessons so then they can see the impact this can have on the generalisations they are producing.

Number examples: multiplication and division

What is the same/different and what do you notice?

__ × 5 =__

5 × __ =__
Write what you notice below.
__ =__ × __

__ + __ + __ + __ + __ = __

__ × 7 =__

7 × __ = __

__ =__ × __

__ + __ + __ + __ + __ + __ + __ = __

This task links multiplication to the array/area model of multiplication, the understanding of the equals sign, the commutative law of multiplication and multiplication as repeated addition.

Pupils will recognise that the answer is 35 and that in the first model they can see 7 groups of 5 when looking vertically across from left to right. When looking at it horizontally from top to bottom, 5 groups of 7 can be seen. They can use these facts to write a multiplication calculation where the product is written first to bring about their understanding of the equals sign, and relate the multiplication of each vertical column to repeated addition of the number of columns in total.

The opposite is the case with the second representation where there are 5 groups of 7 on the horizontal and 7 groups of 5. Such activities will aid in pupils' understanding of the difference in visual representation when the same numbers are used as multiplicand and multiplier across two different calculations.

Number examples: fractions, ratio and proportion

This series of questions is designed to link all the above areas of fraction as well as ratio. This is important as the connections between percentages and ratio can be missed and so more time than needed is used to teach the concept of ratio.

Here the methods used vary but the question is exactly the same, albeit in the context of the concept that it is drawing out. As such, the answer will be exactly the same for each question (24), but the methods and mathematical thinking behind each approach will draw out the connection between the question, as well as the different representations that have been used, such as the numerator and denominator of a fraction. Notice how all of the above are either concrete or pictorial.

Measurement examples

The following task comes from @J0shMartin and his contribution to Craig Barton's Variation Theory website. Although this is predominantly a secondary-based website, there are some examples that are relevant to KS2. The following task requires students to notice patterns when converting between measurements (something students should have some understanding of doing BEFORE they embark on this task).

The answers are provided below:

As this is a context that many teachers are familiar with, here it would be best for you to try and consider how the questions have been carefully crafted to bring about variation, and how you might go about using this sequence within your own lesson. Consider the questions you would ask, the number of questions you would get the students to answer before stopping them to discuss the task so far, and how you would explain the patterns to the students.

Geometry examples: properties of shapes

Property of shape is a particularly useful curriculum area to deploy Variation Theory and can be used from year 1 onwards to great effect. Below is a sequence of slides used in a lesson on describing rectangles with some year 1 students.

The rotation, colour and the type of shape have all varied so that students can generalise about the properties of rectangles (many believed that the ones on slide 17 and 19 were not), so they can really see the distinguishing features of the shape and not get caught on elements that are not important.

This is the conceptual variation as outlined at the start of this blog in action. It uses examples and non-examples, standard and non-standard examples to draw out what it means for a shape to be a rectangle.

Variation Theory FAQs

What is variation theory?

Enjoyed this? Why not try our 'How I Wish I'd Taught Maths' series, inspired by Craig Barton's bestselling book:

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What Is Variation Theory? A Guide For Primary Maths Teachers

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Demystifying Variation Theory for teachers and offering examples of how you can bring this to life in the classroom.

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Teaching for Mastery - Variation - Handout

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Teaching for Mastery - Variation - Handout

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Research - Kullberg, Angelika, Ulla Runesson Kempe, and Ference Marton

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Research - Kullberg, Angelika, Ulla Runesson Kempe, and Ference Marton

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Teachers spend hours creating their own resources for their pupils to use after a concept has been taught. Over the years, I have made lots of resources which I change every year, simply because I think this exercise won't help this set of pupils grasp the concept being taught. I'm now at a point where I can make an exercise of questions which all pupils can use to learn the intended concept. What has changed for me? I have developed a better understanding and can now apply principles of variation theory when creating resources. This has allowed me to use the same practice exercise with all my classes.

Variation Theory is posited on the view that It has been previously pointed out that "when certain aspects of a phenomenon vary which its other aspects are kept constant, those aspects vary are discerned." (Lo, Chik & Pang 2006:3) "to see or experience a concept in a certain way requires the learner to be aware of certain features which are critical to be the intended way of seeing this concept." (Lo and Marton 2012) Here is an example, I am attempting to teach pupils how to factorise a quadratic expression. The critical aspect I want pupils to focus on is determining the factors of 'c' that sum to give the 'b' coefficient. I have kept the 'a' coefficient the same, and I have kept the 'c' constant the same too. I want pupils to work on the aspect that the 'b' coefficient is always the sum of two factors of the constant. Here is a similar example:

Here is another example, but here I am focusing pupils' attention on the critical aspect of determining the factors that multiply to give the 'c' constant. The factors will always sum to 10 which is the 'b' coefficient.

I have deliberately included the last example. Where pupils are taught that two numbers that add to give 10, but the product will be 0. I have included it at the end because it is a nuanced example of the set of expressions above. It is nuanced because its explicit features are not visible for pupils. It does not follow the structure of (a + __)(a + __) because the bracket of (a + 0) = a

What about introducing a practice exercise where you want pupils to focus on their negative mental arithmetic? The critical aspect for pupils to focus on is determining the factors of -24 that sum to give the 'b' coefficient stated.

What I really like about the above exercises is that pupils are experiencing different forms of variation from the previous set of expressions. However, in this practice exercise, I have changed only one aspect of each expression, which is the focal point of the intended concept being taught. If I change both the 'c' and 'b' coefficients of a quadratic expression then I am asking pupils to make two decisions, essentially to focus on two different things, two numbers that sum to 'b' and the same two numbers that multiply to 'c'. This above exercises constrains a pupil's thinking deliberately for two expressions at a time so the intended process of factorising a quadratic expression becomes clear. They can also see how the negative arithmetic changes the value of the 'b' coefficient.

I can now bring a practice exercise where pupils are now able to put their understanding of how to factorise a quadratic expression to the test:

A few key points about the design of this exercise set:

1. I have only changed one aspect of each expression from the previous expression.

2. I have focused on changing the surface of each expression. The procedural process is the same.

3. Between (b) and (c) I have deliberately changed the focus from finding the sum of the constant's factors to then finding the product of two factors of 12 that also give a sum of the 'b' coefficient.

4. I have structured the calculation for the coefficients of 'b' and 'c' because I want pupils to explicitly see how they are determining the numbers that go into the factorised form. I wanted pupils to see the depth underneath the surface of each quadratic expression. I also want pupils to see the procedural relationship between the factorised form and its quadratic expression.

To summarise, this is simply an attempt to widely implement variation theory when resourcing. I have seen that pupils are able to draw the intended mathematical understanding being taught sooner. A greater proportion of the class are able to apply their understanding of the intended concept correctly. The design of this exercise also aids pupils to deliberately think and apply their understanding at each expression they had to factorise. The above exercise is an improvement from previous exercises I have made in the past, and can be improved further after seeing how pupils respond to it.

Paper used: http://www.cimt.org.uk/journal/lai.pdf

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Resourcing: Applying Variation Theory - Conception of the good

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Teachers spend hours creating their own resources for their pupils to use after a concept has been taught. Over the years, I have made lots of resources which I change every year, simply because I think this exercise won't help this set of pupils grasp the concept being taught.

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Information about diagnostic maths activities to help teachers assess understanding and lay foundations for KS3