# data[].attributes.headline
The Levels of the Structure of Knowledge

# data[].attributes.subject
The Levels of the Structure of Knowledge

# data[].attributes.body
Has 3 main levels.

# data[].attributes.body
There are 5 main levels, 6 if including inquiring/investigating.

# data[].attributes.body
Macro concepts tend to the main different disciplines one may study in math. Such as Algebra, Geometry, Stats, Calculus, etc.

Meso concepts are those more specialized areas in those disciplines. They tend to be a bit more niche than the main disciplines. Such as trigonometry & discrete math.

Micro concepts tend to be the specific concepts of each subject. For Algebra some examples are system of equations, graphing quadratics, or finding the value of certain variables.

# data[].attributes.attachment_link.readable_text
Content is the curriculum. Therefore, curriculum planning needs to focus on content.

Rationale

Curriculum planning needs to start with the big picture and work its way in. At a school level, this means considering what the purpose of your curriculum is - what some people call a 'rationale'. What is it that your curriculum should be focusing on teaching your students, and why?

Because we are talking about the big picture, a school's curriculum rationale will be broad in scope. For example, it might state:

We want our students to:

Learn the knowledge they need to understand the world around them.

Develop the skills they need to continue to learn and contribute to society.

Develop attributes that reflect our school values.

Achieve a portfolio of qualifications that open the doors they need them to open and that reflect their very best

Love learning for the sake of learning, viewing it as interesting, exciting and empowering.

A school's curriculum rationale sets the direction for the curriculum in its entirety: subjects, assemblies, clubs, committees, sports, shows, trips, volunteering and more besides. However, it is in subjects that students spend most of their time and in which most learning happens. Therefore, it is subjects that we are going to focus on here.

Levels of organisation

Thought about in terms of subjects, the curriculum exists at different levels of organisation: macro, meso and micro. Let's explore each of these.

Macro-curriculum

The macro-curriculum relates to the specific subjects taught, the time allocated to each, and their purpose.

The subjects taught and the time allocated to each are the timetable. Most school leaders have a good handle on this. However, too few have as good a handle on the purpose of subjects. They haven't really considered why particular subjects are included in their school's curriculum. What this means is that they don't have a good enough handle on the real substance of the curriculum: content.

Why are we including this?

Why is it that art, history and science are included in your curriculum? The answers 'because these seem like important subjects for students to study', 'because other schools teach them', or 'because we've always taught these' aren't really good reasons. They reflect a very superficial way to think about the curriculum. School leaders need to go much deeper than this. They need to be clear about what is being taught and why.

Clarity in what and why

For example, if a specified purpose of art in the curriculum includes to "develop students' cultural literacy", school leaders should be clear that there needs to be a focus on the history of art, including timelines and the specific work of particular artists. If this purpose isn't clear, then it isn't clear that this content needs to be taught.

Clarity in what students should be doing with knowledge

Clarity in the purpose of subjects also helps to bring about clarity in what it is that we want students to do with the knowledge they are taught. A school's curriculum should be knowledge-based, skills-orientated. Being able 'to do' things with knowledge is the whole point of learning it.

For example, the purpose of a science curriculum might include that students:

Describe and explain the big ideas and key concepts of science.

Discuss contemporary scientific issues.

Plan and carry out experiments safely.

Present information in reports and graphical formats.

Make predictions and generalisations.

Draw valid conclusions supported by evidence.

At a macro level, we need to be clear about what students are expected to do with knowledge so we can be clear about the sorts of learning activities we should be planning and what we should be assessing.

Meso-curriculum

The meso-curriculum relates to the topics and sub-topics taught withina subject. For example:

In science, a topic might be 'chemical changes'. Sub-topics might include 'rates of reaction' and 'acids and bases'.

In history, a topic might be 'democracy'. Sub-topics might include 'Ancient Greek democracy' and 'democracy in the UK'.

In effect, the meso-curriculum is a thematic content organiser.

In setting this out, it can be useful to provide a broad overview of what students will learn in topics or sub-topics. For example, the overview of an 'introduction to business' topic might read:

This topic focuses on helping students develop knowledge about the basics of business. In it, they will learn how businesses satisfy needs and create consumer wants. They will explore the contribution of local entrepreneurs to the community.

Again, this is about starting with a bigger picture and working your way in.

Micro-curriculum

The micro-curriculum relates to the specific content within topics or sub-topics, set out as specific knowledge, skills and experiences.

Knowledge and skills

For example, in a science topic on chemical reactions, specific knowledge might include:

Key vocabulary: 'chemical reaction', 'reaction rate', 'concentration', 'particle size', 'temperature'

Everyday examples of chemical reactions, including: iron rusting, burning, milk turning sour

Factors affecting reaction rate, including: concentration, particle size and temperature

In a history topic on Ancient Roman Civilisation, this might be set out as follows:
Julius Caesar

Defeats Pompey in civil war, becomes dictator, 'Veni, vidi, vici' ('I came, I saw, I conquered')

Cleopatra of Egypt

Caesar assassinated in the Senate, Brutus Augustus Caesar

Life in the Roman Empire

The Forum: temples, marketplaces

The Colosseum: circuses, gladiator combat, chariot races, roads, bridges, and aquaducts

Eruption of Mt Vesuvius, destruction of Pompeii

Persecution of Christians

Setting a curriculum out in this way should be infinitely more useful to teachers, students, parents and school leaders than setting a curriculum out like this:

I can use my knowledge of a historical period to interpret the evidence and present an informed view.

I can explain the similarities and differences between the lifestyles, values and attitudes of people in the past by comparing Scotland with a society in Europe or elsewhere.

By studying groups in past societies who experienced inequality, I can explain the reasons for the inequality and evaluate how groups or individuals addressed it.

Irreducible form

For it to be most useful in curriculum documentation, we should be trying to get the wording of content down to its most irreducible form.

For example, ' to describe factors affecting reaction rate, including: concentration, particle size and temperature' is reducible to: 'factors affecting reaction rate, including: concentration, particle size and temperature'. At the micro level, it is the content that matters most, not what students should be able to do with this. There might be all kinds of things we want students to be able to do with specified content: describe, explain, compare, contrast, predict... Where we don't need to, we shouldn't put limits on this in curriculum planning.

That said, there will be some areas of the curriculum where the relationship between knowledge and what students should be able to do with this - skills - are inextricably linked. This will tend to be when content relates to procedural knowledge rather than declarative.

For example:

in maths, 'Multiply decimal numbers by 10, 100 and 1000', for example, 0.1, 0.01 and 0.001'.

in music, 'Compose chord sequences on the keyboard or guitar in C major or A minor using mainly primary chords'.

In such cases, this is the irreducible form. You can't take the 'doing' away from the learning as you can with some other content. However, specificity remains important. Contrast the maths example with, 'Carry out calculations involving decimals.' A lack of specificity means that wouldn't be a particularly useful content statement.

Experiences

Some learning can only come about as a result of experiences. For example:

in science, observing the 'screaming jelly baby' experiment.

in music, listening to Mahler.

in English literature, watching a production of An Inspector Calls.

in history, going on a trip to a museum.

The learning that comes from such experiences can be difficult to articulate, but it is learning all the same. Experiences of this kind are subtly different from what we typically mean by 'pedagogy'. By pedagogy, we typically mean things like 'direct-interactive instruction', 'formative assessment', 'researching' and 'investigating'. There are good reasons why we shouldn't be specifying pedagogy in curriculum plans (which are discussed in The Teaching Delusion 2). However, by 'experiences', we mean something different.

The experiences we are talking about are those that have value in their own right. They are the types of experiences that:

1. enrich learning.

2. are enriched by learning - knowledge that students have already learned adds to the experience.

3. bring about learning in a way that no other approach can.

For example, students could learn about the screaming jelly baby experiment if a teacher describes this to them or they read about it themselves. However, neither approach can bring about the learning that comes from seeing it in action. If students have been taught relevant knowledge before watching the experiment, they will get more from the experience than if they haven't.

At their best, experiences of this kind complement and consolidate learning that has come in other ways. They also help create excitement and enjoyment, the importance of which we should never underestimate.

Taken from The Teaching Delusion 2: Teaching Strikes Back by Bruce Robertson, published by John Catt Educational.

Available at:

# data[].attributes.attachment_link.title
A 5-Minute Guide to: Curriculum Planning - Macro, Meso & Micro

# data[].attributes.attachment_link.description
Content is the curriculum. Therefore, curriculum planning needs to focus on content. Rationale Curriculum planning needs to start with the big picture and work its way in. At a school level, this means considering what the purpose of your curriculum is - what some people call a 'rationale'.

# data[].attributes.headline
The Structure of Knowledge & Process Hand in Hand

# data[].attributes.subject
The Structure of Knowledge & Process Hand in Hand

# data[].attributes.body
While these different structures may seem seperate, it's only when put together that students are truly able to understand math concepts. These two structures rely on each other. We would not be able to apply any of our processes if we didn't have any facts or theorems to base it off of. Nor would we have any plan of action with these facts & theorems if we didn't have any processes to carry out our intuitions.

# data[].attributes.body
This is the level most math students have traditionally performed at. This involves remembering key terms, formulas, & representations at the surface level. Things learned at the factual level may be easier to memorizing, but it doesn't inherently lead to a genuine understanding of what is being learned. Students learn the what, but not the why.

# data[].attributes.attachment_link.title
Math Fact Fluency Expectations by Grade Level - Rocket Math

# data[].attributes.attachment_link.description
In most school situations, students take tests on one-minute timings. fluency expectations by grade level of automaticity vary somewhat.

# data[].attributes.body
Topics include broad math concepts that can be broken down into further detail. These further details are also known as micro concepts.

# data[].attributes.body
Earlier we discussed how our theorems work as our principals for math. These theorems can help students make connections to math topics. Math is known to build off of itself. So if students are able to make a meaningful connection with a topic, they're more likely to make one with following topics. Generalizations are the statements that can be made based off of theorems.

# data[].attributes.body
Problem Solving is an essential skill our students should develop & use in their math experiences. Doing so will make them a more developed mathematician. Problem solving has four main principles/steps that we will briefly cover.

# data[].attributes.attachment_link.readable_text
WTAMU > Virtual Math Lab > Intermediate Algebra

Learning Objectives

Introduction

Tutorial

As mentioned above, I use Polya's four steps to problem solving to show students how to solve word problems. Just note that your math teacher or math book may word it a little differently, but you will see it all basically means the same thing.

If you follow these steps, it will help you become more successful in the world of problem solving.

Polya created his famous four-step process for problem solving, which is used all over to aid people in problem solving:

Step 1: Understand the problem.

Step 2: Devise a plan (translate).

Step 3: Carry out the plan (solve).

Step 4: Look back (check and interpret).

Make sure that you read the question carefully several times.

Since we are looking for a number, we will let
x = a number

*Remove ( ) by using dist. prop.

*Get all the x terms on one side

*Inv. of sub. 2 is add 2

If you take twice the difference of 6 and 1, that is the same as 4 more than 6, so this does check.

FINAL ANSWER: The number is 6.

Make sure that you read the question carefully several times.

We are looking for two numbers, and since we can write the one number in terms of another number, we will let
x = another number

ne number is 3 less than another number:
x - 3 = one number

*Inv. of sub 3 is add 3

*Inv. of mult. 2 is div. 2

If we add 90 and 87 (a number 3 less than 90) we do get 177.

FINAL ANSWER: One number is 90. Another number is 87.

Whenever you are working with a percent problem, you need to make sure you write your percent in decimal form. You do this by moving the decimal place of the percent two to the left. For example, 32% in decimal form is .32

When you are wanting to find the percentage of some number, remember that 'of ' represents multiplication - so you would multiply the percent (in decimal form) times the number you are taking the percent of.

Make sure that you read the question carefully several times.

We are looking for a number that is 45% of 125, we will let
x = the value we are looking for

Make sure that you read the question carefully several times.

We are looking for how many students passed the last math test, we will let
x = number of students

Make sure that you read the question carefully several times.

We are looking for the price of the tv before they added the tax, we will let
x = price of the tv before tax was added.

The following formula will come in handy for solving example 6:

Perimeter of a Rectangle = 2(length) + 2(width)

Make sure that you read the question carefully several times.

We are looking for the length and width of the rectangle. Since length can be written in terms of width, we will let
w = width

length is 1 inch more than 3 times the width:

1 + 3w = length

*Remove ( ) by using dist. prop.
*Combine like terms

*Inv. of add. 2 is sub. 2

*Inv. of mult. by 8 is div. by 8

Make sure that you read the question carefully several times.

We are already given in the figure that

x = one angle
5x = other angle

Consecutive integers are integers that follow one another in order.

For example, 5, 6, and 7 are three consecutive integers.

If we let x represent the first integer, how would we represent the second consecutive integer in terms of x ? Well if we look at 5, 6, and 7 - note that 6 is one more than 5, the first integer.

In general, we could represent the second consecutive integer by x + 1. And what about the third consecutive integer.

Well, note how 7 is 2 more than 5. In general, we could represent the third consecutive integer as x + 2.

Consecutive EVEN integers are even integers that follow one another in order.

For example, 4, 6, and 8 are three consecutive even integers.

If we let x represent the first EVEN integer, how would we represent the second consecutive even integer in terms of x ? Note that 6 is two more than 4, the first even integer.

In general, we could represent the second consecutive EVEN integer by x + 2.

And what about the third consecutive even integer? Well, note how 8 is 4 more than 4. In general, we could represent the third consecutive EVEN integer as x + 4.

Consecutive ODD integers are odd integers that follow one another in order.

For example, 5, 7, and 9 are three consecutive odd integers.

If we let x represent the first ODD integer, how would we represent the second consecutive odd integer in terms of x ? Note that 7 is two more than 5, the first odd integer.

In general, we could represent the second consecutive ODD integer by x + 2.

And what about the third consecutive odd integer? Well, note how 9 is 4 more than 5. In general, we could represent the third consecutive ODD integer as x + 4.

Note that a common misconception is that because we want an odd number that we should not be adding a 2 which is an even number. Keep in mind that x is representing an ODD number and that the next odd number is 2 away, just like 7 is 2 away form 5, so we need to add 2 to the first odd number to get to the second consecutive odd number.

Make sure that you read the question carefully several times.

We are looking for 3 consecutive integers, we will let

x = 1st consecutive integer

x + 1 = 2nd consecutive integer
x + 2 = 3rd consecutive integer

Make sure that you read the question carefully several times.

We are looking for 3 EVEN consecutive integers, we will let

x = 1st consecutive even integer

x + 2 = 2nd consecutive even integer
x + 4 = 3rd consecutive even integer

*Remove ( ) by using dist. prop.
*Combine like terms

*Inv. of add. 10 is sub. 10

*Inv. of mult. by 6 is div. by 6

In a business related problem, the cost equation, C is the cost of manufacturing a product.

In the revenue equation, R is the amount of money the manufacturer makes on a product.
If a manufacturer wants to know how many items must be sold to break even, that can be found by setting the cost equal to the revenue.

Make sure that you read the question carefully several times.

We are looking for the number of cd's needed to be sold to break even, we will let

Practice Problems

Practice Problems 1a - 1g: Solve the word problem.

1e. Complimentary angles sum up to be 90 degrees. Find the measure of each angle in the figure below. Note that since the angles make up a right angle, they are complementary to each other.

(answer/discussion to 1e)

Need Extra Help on these Topics?

Last revised on July 1, 2011 by Kim Seward.

All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.

# data[].attributes.attachment_link.description
WTAMU > Virtual Math Lab > Intermediate Algebra Learning Objectives Introduction Tutorial As mentioned above, I use Polya's four steps to problem solving to show students how to solve word problems. Just note that your math teacher or math book may word it a little differently, but you will see it all basically means the same thing.

# data[].attributes.body
This involves students making arguments while being backed up by coherent explanations. This involves heavy critical thinking & reflection. This may be challenging & mistakes will be made. It's important to remind your students that from those mistakes new insights are made. This allows them to grow & build their understanding of a topic. Mistakes are just as valuable as finding the correct solution in math.

# data[].attributes.body
In math communication involves sharing insights, strategies, & more with your peers. By communicating you can see things from different perspectives, this may help you make your own discoveries. This is beneficial to all involved & their is multiple methods when it comes to communicating in math. The main point is that everyone build off of each other to gain understanding of a topic. This helps students build connections with what they learn as they're more directly involved in the process.

# data[].attributes.body
Math is known for it's connectivity when it comes from topic to topic. Making connections in math often entails seeing how facts relate to one another. How certain math symbols lead to certain procedures. How what is being learned can apply to the real world. Lastly, how what is being learned is connected to what has been learned.

# data[].attributes.body
Here students represent their math process in visual form. A student who can represent their math in different ways may have a better conceptual understanding of what's going on in the problem. These representations can also make it much easier for someone else to understand what's going on in the problem without doing the exact same work.

# data[].attributes.attachment_link.readable_text
by Francis (Skip) Fennell, NCTM President 2006-2008
NCTM News Bulletin, September 2006 (PDF)

Representation is NCTM's Professional Development Focus of the Year (FOY) for 2006-2007. The theme we will use to promote this FOY is "Show Me the Math: Learning through Representation." Movie buffs will notice a play on the phrase "Show me the money," which was popularized by the film Jerry Maguire. We chose this phrase because it says exactly what representation does-it shows mathematics to students who are trying to learn and to teachers who are looking for signs that students are understanding what is being taught.

The process of representation includes using models to organize, record, and communicate mathematical ideas, as well as selecting, applying, and translating these models to solve problems and interpret mathematics. The models can be used to "show" math, through the use of manipulative materials, diagrams, graphical displays, and symbolic expressions. Representation also includes internalizing or taking in mathematical ideas and understanding them. Let's think about it from the perspective of the classroom.

Representation and Planning. Representation should be an important element of lesson planning. Teachers must ask themselves, "What models or materials (representations) will help convey the mathematical focus of today's lesson?"

When planning a lesson that involves place value and three-digit numbers, an elementary school teacher might consider the use of place value or base-ten blocks (or computer versions of these materials*). A place-value chart could also be used by students or the teacher to demonstrate that a number, such as 137, can be represented as 1 hundred, 3tens, and 7 ones, or as 1 hundred, 2 tens, and 17 ones, or in other groupings. A 200-chart could also be used to show that 137 is closer to 150 than it is to 100 or 200.

Representation and Teaching. What is the instructional role of representation? Will particular models and materials always work? How do we know they work? How should they be used? What do students learn from their use, and how will this be communicated? These questions related to teaching and learning help teachers think about what students might "see" in a drawing that they create and provide a glimpse of the internal representations students create as they interact with any representation.

Representation can be used by teachers to instruct and by students to solve problems and communicate their mathematical ideas to others. When planning instruction, practice, or reinforcement activities, teachers should consider how they and their students can use representation in today's mathematics lesson.

For example, students and teachers may consider using region, set, or number line representations for comparing 4/5, 6/8, 8/10, and 1/2. What if they decide to use the number line for this lesson? This representation may be fairly easy for many students to create and use, and it allows them to interpret whole numbers, negative numbers, and fractions, decimals, and related percents. Should the students create their own number lines for this lesson? Should software be used? Should the students' number lines be organized by tenths? Should the students determine the intervals on the number line? Should the students be able to use the fractions above to compare equivalent decimals and percents? Notice the attention to the representation and how it may be used.

Representation and Assessment. As teachers present lessons, they can use representation to assess and improve students' performance and to make decisions about future instruction. The observations they make, the questions they ask students, and the instructional activities they provide all have potential as formative assessments. They give teachers a glimpse into what students have, or have not, understood. Questions that may help improve performance and guide instruction are:

How are students using representation to model and interpret the mathematics being presented?

What do the representations that a student is using tell about that student's understanding of the mathematics?

What do students provide when asked to use diagrams, sketches, or equations to explain their solution to a problem or task?

As we all recognize the importance of representation, we also need to realize that there is much more to representation than plopping manipulative materials in front of students and directing the modeling of a particular problem. Interviews with students often confirm that some students understand concepts presented with one type of representation but are unable to comprehend the same concept presented with another type of representation (e.g., a student may be able to use fractional cutouts to show that 5/8 + 1/4 is equal to 7/8 but will give 6/12 as the sum when encountering the equation symbolically). In short, we cannot always rely on particular models or responses to a particular model. We must deliberately plan the use of representation in lessons and continually assess whether or not students are "seeing" and learning mathematics.

So, show your students the math! Consistently provide opportunities for representation and encourage them to use it regularly to record and communicate the mathematical ideas that they are learning.

* NCTM's Illuminations Web site recommends a computer version of base-ten blocks that can be accessed free of charge, by visiting
http://illuminations.nctm.org/WebResourceReview.aspx?ID=131.

# data[].attributes.attachment_link.title
Representation-Show Me the Math! - National Council of Teachers of Mathematics

# data[].attributes.attachment_link.description
by Francis (Skip) Fennell, NCTM President 2006-2008 NCTM News Bulletin, September 2006 (PDF) Representation is NCTM's Professional Development Focus of the Year (FOY) for 2006-2007. The theme we will use to promote this FOY is "Show Me the Math: Learning through Representation."

# data[].attributes.body
Investigating in math requires the students to be proactive. They have to go out of their way to learn more about what they may not know how to describe. This can be done by asking questions in class, but more impactfully researching it yourself. Look up & synthesize any relevant information you may find. Question what you find, doing all of this should lead the students to their own new discoveries.

# data[].attributes.body
Formulae are different equations represented by different numbers & variables. They're usually tied to certain scenarios & there use are factual.

Theorems are in statement form & work as proofs for math concepts. There are explanations for why things work & it's heavily tied to conceptually understanding the "why." These theorems work as principles to our math concepts.

# data[].attributes.body
This first step involves students thinking more critically about the questions. What is being asked for you to do, what are the key terms for this problem, & what is it not.

# data[].attributes.body
This is the step where students decide how they're going to solve the given problem. What strategies would work best in the given case. Maybe working on it in "chunks", working backwards, or any other strategy. What's important is that the student uses a strategy that makes the most sense to them.

# data[].attributes.body
In this step you take your strategy into practice. It's important that our students persevere & do their best to see their strategies through. Seeing what tweaks could be made. Of course recognizing when a strategy may not work & switching to another is just as important.

# data[].attributes.body
Here student's will review their solutions. They should ask themselves if their solution is reasonable given the context of the problem. What worked & didn't work for them. This is a perfect opportunity to share & discuss solutions with other students.

# data[].attributes.headline
What are Mathematical Processes, Strategies, Algorithms, & Skills.

# data[].attributes.subject
What are Mathematical Processes, Strategies, Algorithms, & Skills.

# data[].attributes.body
Mathematical Processes can vary from topic to topic as each topic has it's own unique concepts. Strategies, algorithms, & skills are the main components of our mathematical processes.

Strategies often involve use of many different math skills, these skills are used in ways that will help develop their understanding of a problem.

Algorithms also involve use of certain skills, but in a more structured manner. Properly using an algorithm often leads to a specific outcome.

We've mentioned them a lot, but what are skills? As we see skills are engrained into our algorithms/strategies. These skills are small actions that allow these algorithms/strategies to work properly. Such as knowing your order of operations, knowing how to plot a line for a linear equation, etc.

# data[].attributes.body
Generalizations are explanations that give people (students) an understanding of how two or more concepts relate. These generalizations are the key points that we hope a learner would digest after a unit. Since math build off of itself, these gained generalizations will have future applications that can help assist students in learning new concepts.

# data[].attributes.attachment_link.readable_text
Making generalizations is fundamental to mathematics. Developing the skill of making generalizations and making it part of the students' mental disposition or habits of mind in learning and dealing with mathematics is one of the important goals of mathematics education. Making generalizations is a skill, vital in the functioning of society. It is one of the reasons why mathematics is in the curriculum. Learning mathematics (if taught properly) is the best context for developing the skill of making generalizations.

What is generalization?

There are three meanings attached to generalization from the literature. The first is as a synonym for abstraction. That is, the process of generalization is the process of "finding and singling out [of properties] in a whole class of similar objects. In this sense it is a synonym for abstraction (click here to read my post about abstraction). The second meaning includes extension (empirical or mathematical) of existing concept or a mathematical invention. Perhaps the most famous example of the latter is the invention of non-euclidean geometry. The third meaning defines generalization in terms of its product. If the product of abstraction is a concept, the product of generalization is a statement relating the concepts, that is, a theorem.

For further discussion on these meanings, read Michael Mitchelmore paper The role of abstraction and generalization in the development of mathematical knowledge. For discussion about the importance of generalization and some example of giving emphasis to it in teaching algebra, the book Approaches to Algebra - Perspectives for Research and Teaching is highly recommended. There is a chapter about making generalizations and with sample tasks that help promote this attitude.

Sample lessons

Mathematical investigations and open-ended problem solving tasks are ways of engaging students in making generalizations. The following posts describes lessons of this type:

1. Sorting number expressions

2. Lesson study: Teaching subtraction of integers

3. Math investigation lesson: polygons and algebraic expressions

4. Polygons and teaching operations on algebraic expressions

Of course it is not just the type of tasks or the design of the lesson but also the classroom environment that will help promote making generalization and make it part of classroom culture. Students will need a classroom environment that allows them time for exploration and reinvention. They will need an environment where a questioning attitude is promoted: "Does that always work?" ,"How do I know it works"? They will need an environment that accords respect for their ideas, simple or differing they may be.

# data[].attributes.attachment_link.title
Making generalizations in mathematics - Mathematics for Teaching

# data[].attributes.attachment_link.description
Making generalizations is fundamental to mathematics. Developing this skill and making it part of the students' mental disposition or habits of mind...

# data[].attributes.body
Utilizing algebraic tools such as algebraic multiplication, subtraction, & division allow highly complex problems to be solved & displayed.

This example explains how the proper use of the order of operations can help someone solving a many different problems of varying complexity. These concepts can also be used to help convey how it may be used in real life applications.

# data[].attributes.body
Logarithm Laws give a means of changing multiplicative process into additive process, and this can provide the means to find inverses of exponential functions, which represent continuous compounded growth shows deep intellectual depth.

This example highlights the usage that logarithms can have instead of simply stating it as an operation. The key points in the generalization are that logs are an additive process as opposed to the inverses (exponential functions) multiplicative process.

# data[].attributes.body
Generalizations...

Can refer to beyond one specific concept in math (they're versatile).

Are truths supported by factual examples.

May not always be true, as they're some exceptions.

Note: Video discusses both generalizations below (yellow boxes)

# data[].attributes.attachment_link.title
130. Overarching and topical generalisations

# data[].attributes.headline
What are the Differences Between Generalizations & Principals in Math

# data[].attributes.subject
What are the Differences Between Generalizations & Principals in Math

# data[].attributes.body
Principals...

Define a particular concept in math.

Work as proofs for math concepts, meaning they should be true in every case.

Always have qualifiers in their statements.

# data[].attributes.body
Generalizations/Understandings that are specific to a certain topic in a unit. These are what teachers aim for students to learn first & foremost. Learning how certain concepts work where they're introduced. Then hopefully being able to apply it later on (overarching focused).

# data[].attributes.body
Generalizations/Understanding that go beyond a specific unit. These are the concepts we would want students to understand & apply in later topics when they'll be needed.

# data[].attributes.headline
What are the Differences Between Level 1, 2, & 3 Generalizations.

# data[].attributes.subject
What are the Differences Between Level 1, 2, & 3 Generalizations.

# data[].attributes.body
Level 1 generalizations give the basic idea of what students should understand from a concept. They're not too in depth & are very generic. Level 1 scaffold how to go past the usage of "no no verbs" at the very basic level. Level 1 doesn't fully inform students of what they're learning, more so what they're doing. The Level 1 generalizations could be used as a scaffold, ultimately we should be approaching more frequent usage of level 2 generalizations.

Level 2 generalizations are our teaching targets for our the instruction we give. These go into the how & whys for the concepts students learn. This level of generalizations should be where the students gain the most understanding (not entirely application) of the topics. They get an idea of the importance of the topic, at level 3 they get to a point of application.

Level 3 generalizations while not mandatory can help build students depth of what they learned. Showing them a bigger scope of the usage of a concept, highlighting its importance. How may it be used in the real world? After giving the level 3 generalization it's a great idea to follow with real life examples students may see in their day to day lives.

# data[].attributes.headline
How to Draw Conceptual Understanding From Students

# data[].attributes.subject
How to Draw Conceptual Understanding From Students

# data[].attributes.body
Part of drawing understanding from students is providing opportunities for them to show you what they have learned & understand. Not so much asking them to answer questions that you come up with. Let them express their thoughts, get a little confused, & bring up insightful questions. Their learning should come from themselves just as much as it comes from the teachers instruction.

Some ways to do this are letting them put their math thoughts into writing, graphic organizers, concept maps, & much more.

# data[].attributes.attachment_link.readable_text
Multiplication is repeated addition. Keep, switch, flip. The butterfly method.
These are all examples of math shortcuts, tips, or tricks that many students learn to rely on from an early age. I taught many students throughout my 16 years in the classroom who quickly pulled out these strategies!

But my students couldn't explain why these tips and tricks work and would often stumble and become frustrated when they encountered situations where the tricks didn't work or they forgot exactly what to do.

That's why math education has transitioned in recent years to focus on teaching deep conceptual understanding rather than encouraging students to rely on shortcuts. Educators know that teaching children to deeply understand math leads to the development of problem-solvers and critical thinkers.

But how can we ease away from teaching tips and tricks so our students have the opportunity to become true mathematicians?

Don't worry; we've got a few ideas for you! Check out these seven tips for getting rid of the shortcuts and teaching true conceptual understanding in math.

1. Spiral Practice Through a Well-Thought-Out Scope and Sequence

Mathematics is a body of conceptual knowledge made up of interrelated concepts-it isn't just a list of disconnected topics to check off a list as students move from grade to grade. Using a carefully considered scope and sequence to structure your school year is the first step in avoiding the pitfalls of math tips and tricks.

During my last few years of teaching, my district used the Carnegie Learning High School Math Solution for our Algebra 1 and Geometry classes. For the first time, I saw how much the scope and sequence really matter. Watching my Algebra 1 students pull from their Module 1 experiences in Module 5 to make sense of quadratic functions was a lightbulb moment-for all of us!

It's even better if your scope and sequence leaves room to interweave spiral reviews so students can solidify their base knowledge and see the connections between previous and future learning. Because my Algebra 1 students could pull from their foundational knowledge and saw the concepts revisited throughout the year, they didn't need to rely on shortcuts or tips and tricks.

A thoughtful scope and sequence incorporating spiral review is key to teaching deep conceptual understanding in math. If we rely on teaching the "easy" shortcuts instead of giving students the time and space to master grade-level skills and see the connections between concepts, they'll struggle to develop a body of conceptual knowledge that will help them understand more complex ideas in the future.

2. Use High-Order Tasks to Build Critical Thinking Skills

Although many students (and teachers!) love math shortcuts because they lead to quick "success," having a toolbox packed with critical thinking skills and problem-solving strategies for students to pull from is so much more valuable. These skills will serve your students in various situations, whether they're in advanced math classes or have to think critically about real-world problems.

One way to help students develop their critical thinking and problem-solving skills is to assign high-order math tasks in your classroom. When working on these rich tasks, they can think about what they already know and test out different ways to complete the task until they identify one that works. In the process, your students fill their toolbox with problem-solving strategies and critical thinking skills, eliminating the need for tips and tricks.

Some of my favorite high-order tasks to use with my Algebra 1 students were in a lesson titled, "Do You Mean: Recursion?" This lesson is filled with activities that encourage students to think critically about arithmetic and geometric sequences and how to develop and deeply understand explicit and recursive formulas. They're even asked to compare the pros and cons of using explicit or recursive formulas, using evidence developed over the last series of lessons!

The fact that there's no "plug and chug" in this series of high-order tasks meant that my students were constantly using and developing their critical thinking skills and problem-solving strategies.

I was always amazed at the deep conversations I heard around the room as they completed tables of cell divisions and eventually used those observations to understand why explicit and recursive formulas work.

3. Visual Representations for Better Retrieval

Visual aids are powerful tools for helping students to develop a deep, conceptual understanding of mathematical concepts. I loved supplementing as many lessons as possible with diagrams, graphs, anchor charts, manipulatives, and even high-quality math videos. In doing so, every learner had an entry point into even the most upper-level mathematic concepts.

When students visualize math concepts, they can more easily see patterns and make connections that might not be immediately apparent from written or verbal explanations. And when they have a visual cue stored in their brain, it makes retrieving information much more manageable.

For example, suppose a student can recall that a quadratic function looks like a parabola because they've interacted with graphs illustrating a pumpkin catapult or diving into a swimming pool. In that case, they're more likely to be able to interpret the formula of a quadratic function and apply that conceptual knowledge to different scenarios.

4. Manipulatives and Hands-On Learning

Another way to eliminate the need for tips and tricks ("A negative times a negative is a positive," anyone?) is with manipulatives such as algebra tiles, counting chips, and even interactive number lines.

And I promise those hands-on materials aren't just for the younger kids-your high schoolers won't mind abandoning the paper and pencil note-taking in favor of digging into algebra tiles occasionally!

I'll never forget using algebra tiles for various purposes with my high schoolers. From watching a student with complex special needs finally understand the meaning and applications of a zero pair to seeing upper-level students suddenly "get" factoring trinomials, each visual and hands-on learning experience was pure magic!

5. Connect Concepts Instead of Teaching Math Shortcuts

Teaching is all about making connections. And while, yes, connecting with your students is one of the best ways to increase engagement, here we're talking about making mathematical connections.

Teach your students to look for the interconnectedness of mathematical concepts, so they see how ideas fit together and build on one another, and watch as they develop a deeper understanding of the underlying concepts. Then, it's time to kiss the shortcuts goodbye!

For example, the scope and sequence I used encouraged my students to apply their foundational knowledge of concrete geometric investigations and reasoning with shapes to formalize their understanding. Circles were also integrated throughout the course, rather than treating them as isolated geometric figures (as many other curriculums do).

Watching my Geometry students make connections between circles and angle relationships and complete constructions using arcs was a game changer! They retained much more information when they saw the connections between concepts and were able to apply their knowledge and skills in new situations that I never expected.

6. Help Your Students Make Real-World Connections

Another vital connection that will lead to the elimination of shortcuts, tips, and tricks is between the mathematics your students learn in the classroom and the real-world applications of the concepts.

When you help your students discover these links to the real world, math suddenly loses its abstract nature. It becomes relevant, practical, and motivating.

Now your students will be more likely to remain engaged and acquire conceptual knowledge that can be generalized across various situations. Here are some examples using real-world scenarios to model integer subtraction that could be used in a 7th-grade class.

7. Don't Use Math Tips and Tricks-Collaborate!

Most kids love to work in groups, right? It enhances the social aspect of school that many students value, and when structured correctly, these collaborative learning experiences can be the perfect setting for developing deep mathematical understanding.

Collaborating to create their conceptual knowledge is a powerful experience for your students. They may productively struggle, disagree, and even argue a bit, but these experiences are where the magic happens.

"Allow students to experience and play and notice and wonder," writes Tina Cardone, author of Nix the Tricks: A Guide to Avoiding Shortcuts That Cut Out Math Concept Development. "They will surprise you! Being a mathematician is not limited to rote memorization...Being a mathematician is about critical thinking, justification, and using tools from past experiences to solve new problems."

And I can think of no better opportunity to notice, wonder, think critically, and justify those thoughts than when collaborating with peers. It may be hard to give up that "sage on the stage" lecture style (I definitely struggled!), but hearing your students engage in rich, mathematical conversations and watching them abandon the shortcuts in favor of deeply understanding the math is worth it. The feeling is second to none!

Don't Let Tips and Tricks Take Away the Beauty of Math

Math is a beautiful, creative, and thought-provoking subject that sets the perfect stage for your students to become critical thinkers, problem solvers, and leaders of tomorrow. Don't let a reliance on math shortcuts, tips, and tricks rob them of that experience!

I hope you're ready to ditch the tips and tricks in your classroom, but if you need more convincing, check out this case study from Muleshoe Independent School District in Texas. They were able to teach their students deep conceptual understanding in math and get rid of the shortcuts-with some great results to show for it!

# data[].attributes.attachment_link.title
7 Concrete Strategies to Teach Conceptual Understanding in Math

# data[].attributes.attachment_link.description
Please Excuse My Dear Aunt Sally. Multiplication is repeated addition. Keep, switch, flip. The butterfly method. These are all examples of math shortcuts, tips, or tricks that many students learn to rely on from an...

# data[].attributes.headline
The Importance of Students Understanding Generalizations & Principals

# data[].attributes.subject
The Importance of Students Understanding Generalizations & Principals

# data[].attributes.body
Both generalizations & principals helps students get further insights on the topics they're learning in their math class. It goes beyond step by step modeling, plug & chugging, & general explanation of what numbers/variables in formulas mean. It's more so about why & how certain math concepts work. For example we care less about what number is being repeatedly added in a pattern and more so why that number specifically.

Example:

How many sides (outward facing) are added each time you add a cube next to another (in a horizontal straight line)?

Four sides are added each time you add a cube next to another.

Why are four sides added each time (outwardly) you add a cube next to another (in a horizontal straight line)?

The reason four sides are added each time is because when cubes are put together, the "connected sides" between the cubes disappear. So the only sides that remain are the top, bottom, front, & back for each cube (besides the end cubes which have five sides).

# data[].attributes.attachment_link.title
Why Should Kids Learn to Generalize in Math?

# data[].attributes.attachment_link.description
What does it mean to generalize? Why does generalization matter? And why do my kids need to learn how to do it?

# data[].attributes.body
The main focus of the Unit Web, this is the main topic you are working towards with your students for them to learn. Hopefully by the end of it being able to apply the concepts they learned to other contexts.

# data[].attributes.body
The lens you choose is meant to help you focus your unit title & should help you connect different concepts in that unit.

# data[].attributes.body
The unit strands make up the main ideas for your unit topic. You may write as many unit strands as necessary, just remember one has to be of the mathematical processes. Unit strands can be seen as the focal point of a lesson plan one makes. What's the goal for the day?

# data[].attributes.headline
Unit Strand: Concept in Mathematical Processes

# data[].attributes.subject
Unit Strand: Concept in Mathematical Processes

# data[].attributes.body
This should state what a student needs to know in order to effectively work through what they're learning. For functions that would include rates of change, intercepts, what a linear function is, etc.

# data[].attributes.body
These are the sub topics to your main ideas in a unit. These finer points are key to helping students understand the concepts they're learning. Without them what is being learned holds little substance besides their title. Linear functions imply functions that are linear. The micro concepts help us understand what exactly makes a linear function, besides it being a function that looks like a straight line.

# data[].attributes.body
Generalizations are statements on concepts that can be supported by factual examples. These statements may have certain exceptions to them, they should be correct most of the time though (principals should always be true).

# data[].attributes.headline
How Will I Encourage Collaboration When Unit Planning?

# data[].attributes.subject
How Will I Encourage Collaboration When Unit Planning?

# data[].attributes.body
While working on a unit plan on your own is a valid strategy, it's much more effective to work on it as a team. As someone who wants to be a high school math teacher, I will likely be part of a math team for the grade level I'm teaching. In this math team we can discuss common needs that our students have & how to best assist them. This can involve effective teaching strategies, differentiation, etc. We can also share different ways we (the teachers) could make the lessons engaging. Maybe one teacher shares an activity they've been doing with the class & how they've taken a liking to it. Work as a team to make an engaging lesson that can meet the needs of the students. I can borrow ideas from my peers I may think will work in my classroom. They may do the same as well.

# data[].attributes.headline
How Does Unit Planning Empower Teachers?

# data[].attributes.subject
How Does Unit Planning Empower Teachers?

# data[].attributes.body
Teachers are able to take control of what they teach. Instead of teaching based on a textbook/provided lesson plans, they get to create their own unit plans. This allows them to meet the different needs that each classroom may have. They're also much more deeply connected with the content they're teaching as they planned it out themselves. This can help make pacing a bit easier compared to a lesson you're not familiar with on that intimate level. Simply put teachers are more likely to be engaged with the content they make themselves than the ones they're given.

# data[].attributes.attachment_link.description
As members of the Professional Development Team at Faria Education Group, we spend a tremendous amount of time with school leaders and teachers discussing curriculum and unit planning. We coach teachers to design standards-aligned curriculum that uses the unit plan as the major organizational structure.

# data[].attributes.attachment_link.title
Guiding Questions to Check for Math Understanding - Corwin Connect

# data[].attributes.attachment_link.description
When planning a unit of work I would recommend teachers design guiding questions to help students understand the relationship between factual content…

# data[].attributes.body
The simplest of the three question types. These questions often have one answer & bring out key points of information students will need during a unit. These can also be considered leading questions as they are meant to guide students toward discovering a specific concept. For example, "Which line on the graph is the y-axis?" Each Generalization should have 3-5 of these questions.

# data[].attributes.body
These questions are much more thought provoking & should lead to students answers being reflective of the concept. Students are expected to explain or apply the concepts they are learning in the unit. These questions are meant to guide students towards that more involved point. A conceptual question could be "What two concepts make the y-axis unique?" Each Generalization should have 3-5 of these questions.

# data[].attributes.body
These questions should elicit a variety of responses, often encouraging discussions in the class. Students should defend their answers & listen to others answers respectfully. Through these discussions students should become much more familiar with the concepts in a unit as they go through what may & may not work. An example of a debatable question is "Should Vertical/Horizontal line graphs be considered functions?" The unit should have 2-3 of these questions.

# data[].attributes.headline
Strategy 1: Create A Social Learning Environment

# data[].attributes.subject
Strategy 1: Create A Social Learning Environment

# data[].attributes.body
This strategy puts a focus on collaboration & discussion between the students. Math doesn't always have to be about independent practice, it can be done as a community (classroom level or beyond) as well. Students just like everybody else (in most cases) thrive when it comes to interacting with peers. There are multiple ways to get students talking & thinking about math. The text uses the placemat activity to help students organize their thoughts & facilitate discussions. I would use the Frayar Model for example. The Frayer Model involves splitting a concept/vocab word into 4 quadrant. The quadrants including the definition, characteristics, examples, & non-examples. Students could work on this independently, but it would be much more beneficial to share what they wrote to the class. Or even better making Frayer Models with a peer or in a group. There are a lot of ways to bring some engagement to the math class. The trick is finding & utlizing the right strategies.

# data[].attributes.attachment_link.title
Helping Youth Learn Math Vocabulary: The Frayer Model

# data[].attributes.headline
Strategy 2: Provide an Open, Secure Environment to Allow for Mistakes as Part of the Learning Process

# data[].attributes.subject
Strategy 2: Provide an Open, Secure Environment to Allow for Mistakes as Part of the Learning Process

# data[].attributes.body
With any class, your goal as a teacher should be to provide a safe, inclusive classroom environment. This is especially important if we want to have mistakes to be a part of the learning process publicly. There is a lot to learn from mistakes, both figuratively & literally. Your brains forms many more connections when you're challenged than when you already know the answer. These connections can solidify what you learn & make you an effective learner. Of course accepting mistakes can be a challenge for some students. This is why we encourage them (everyone really) to adopt a growth mindset. A growth mindset tells you that you may not know something "yet", but you will learn in time. It tells you that you learn from your mistakes & you can recover from the challenges you face. The ability to bounce back can make a world of difference. Both in math & beyond the classroom. Personally I would encourage students to share even if they're uncertain. Other students may have the same thoughts & by their sharing they can help ease everyone, themselves, & help everyone learn something new. From mistakes we grow, if mistakes were never made, we would never think beyond what we know.

# data[].attributes.attachment_link.title
"Students Share Mistakes" Clip | A GROWTH MINDSET Documentary

# data[].attributes.headline
Strategy 3: Use Appropriate Levels of Inquiry & Employ Inductive Approaches to Develop Conceptual Understanding

# data[].attributes.subject
Strategy 3: Use Appropriate Levels of Inquiry & Employ Inductive Approaches to Develop Conceptual Understanding

# data[].attributes.body
Conceptual understanding is very important for any students success in a class. While correct answers are appreciated it can very much be worth deviating from the main path if there is something to be gained. The non-example is very under appreciated in education. Be ready for it & build off of it in your lessons. Students should be given the opportunity in lessons to make discoveries, come to conclusions, & discuss it with their peers or to the class.

# data[].attributes.attachment_link.title
Engaging Students in Inquiry Based Tasks Geometry: 1st Grade

# data[].attributes.headline
Strategy 4: Reduce Whole Class Teacher Talk Time

# data[].attributes.subject
Strategy 4: Reduce Whole Class Teacher Talk Time

# data[].attributes.body
Putting it briefly don't lecture for too long, at most 15 minutes. Make sure to break up what students are doing every now & then in class to make the lesson more easily digestible for them. Students have to be doing something through most of the class & lecturing should really only be for introductions. Let's put it this way, you're a comic & you have your opening act (intro to lesson). Your main routine will focus making material off of the crowds responses in order to boost engagement.

Note: The intro was pretty uncanny for me haha.

# data[].attributes.attachment_link.title
Teaching Tips from AE - Reducing Teacher Talk Time

# data[].attributes.headline
Strategy 5: Cater to Everyone in Your Class; use Differentiation Strategies

# data[].attributes.subject
Strategy 5: Cater to Everyone in Your Class; use Differentiation Strategies

# data[].attributes.body
As teachers we would all love to differentiate our instruction, we may at first believe it to be challenging. After all it's a very involved process. Making sure it fits for each individuals needs (which may feel daunting). Or sometimes we may make it so simple that it isn't as beneficial to those involved. This doesn't have to be the case. Instead of providing more or less work, we could provide an equal amount of work in varying difficulties. Allow students to choose the challenge of their questions, they can increase/decrease the challenge based on their performance. Students should also be allowed to present their work in different ways. Such as processes that make sense to them or different mediums of showing their understanding (in words/drawings/orally/math symbols). Lastly building rapport with your students beyond the math class is important. Students are more likely to feel motivated & desire to perform better when they know they're genuinely cared for, believed in, & seen as their own unique individual.

# data[].attributes.attachment_link.title
Differentiated Instruction: Why, How, and Examples

# data[].attributes.body
Assessment Strategies are primarily used to see where everyone is at in their understanding, providing useful information. While this is important on it's own, it's absolutely vital that you use said information to adjust your teaching as needed. Do you need to reteach or quickly brush up on past concepts. Is it a minor misconception that will get students on track when cleared? If you don't use your assessments to adjust your instruction then there is little point to them & your students. Assessments when done with proper care can have students think beyond questions & answers. They can explore concepts in a new light.

# data[].attributes.attachment_link.title
3 Ways to Assess Math Understanding More Deeply

# data[].attributes.headline
Strategy 7: Be Purposeful When Asking Students to Answer Questions; There is Safety in Numbers

# data[].attributes.subject
Strategy 7: Be Purposeful When Asking Students to Answer Questions; There is Safety in Numbers

# data[].attributes.body
When picking questioning strategies one has to be aware of possible impacts, both negative & positive. Hand raising is traditional & comfortable, but often doesn't accurately represent where the whole classroom is at. Random calling while convenient in getting a response is more detrimental than any benefits that one may get. Anxiety goes through the roof, a students may not have a response, or may be caught unaware. Random name calling is very situational, working best when established for smaller activities in groups. Although the anxiety is still present if a bit smaller. There are strategies to help push students towards particpating in the class. It can be discrete like raising fingers 1-5 or thumbs up/down. You can call on students while letting them know you may call them in advance. A random wheel picker, & a lot more. The strategy you choose should be suited for your classroom environment.

# data[].attributes.attachment_link.readable_text
These intentional response strategies create entry points for students who may feel reticent to share in class discussions.

"The person doing the talking is the person doing the learning." I heard this saying many times, but it finally stuck with me one day as I looked over my students' math performance data and noticed that all of my students with low mastery had something in common: They rarely participated in class discussions.

Meanwhile, many students with high levels of math mastery were those whose hands were frequently raised. Was it possible that not joining class discussions could contribute to low levels of mastery?

Ever since this aha moment, I've shifted my teaching practice to emphasize intentional response protocols that ensure all students contribute to classroom discussions, and my students' mastery has shifted with it.

The Impact of Intentional Response Protocols

When my classroom depended on hand-raising, I found that discussions were biased toward students who knew that their answers were correct. This meant missed opportunities for the class to engage with interesting mistakes.

There is benefit in hearing from a wide range of student voices, and intentional response protocols-frameworks for intentional sharing-ensure that every student has a pathway to participating in class discussions.

These protocols don't have to completely eliminate hand-raising from the classroom, but if they become routines, they normalize the participation of all students and shift the dynamic of who is doing the talking and, in turn, the learning.

Getting Started

One of the biggest barriers to student participation in my math class is concern over getting an answer wrong. Before launching new routines for intentional response in my classroom, I consider how to make my students feel comfortable taking risks by normalizing mistakes as part of the learning process.

This means breaking down preconceived ideas about who is a "math person" as well as what types of answers are worth sharing. At the beginning of the school year, I give students sentence stems for agreeing and disagreeing (e.g., "I disagree with your idea because..."), offering students a positive way to engage with errors in the classroom.

It's important to be attentive to student responses when errors are discussed and to notice and address whether students use these sentence frames. One strategy that went a long way in shifting my classroom culture was consistently praising mistakes and specifically naming the ways in which they help everyone learn.

This could sound like "I'm so glad that you gave that answer, because that is a common confusion. Now we can all look out for this type of mistake!" The specific narration about the learning process not only validates taking an academic risk but also names the direct benefit to others.

Differentiating the Protocols

In addition to classroom culture, I work to make intentional responses accessible for all students in my classroom, particularly students with learning differences or anxiety.

One strategy that is particularly helpful for students who need more processing time is to give students independent work time followed by a partner turn-and-talk before calling on students. This provides an opportunity for them to conference with a peer and potentially ask questions.

During that time, I circulate to give quick prompts or checks for understanding to students who benefit from more support. When it is time to call on students, I preplan some "break it down" questions that can quickly assist a student in the moment if they are stuck.

Finally, I'm always aware of students who have 504 plans that specify "no cold calling" or students who check in with me to share that anxiety is a barrier. For these students, I utilize "warm calling" by checking in with the student before calling on them to make sure they are comfortable sharing their response.

Enhancing Engagement

Three techniques have proven most impactful in increasing student engagement and mastery in my class: randomized calling, strategic warm calling, and popcorn out.

In randomized calling, I use a random picker wheel or draw names to choose student responders. This protocol eliminates teacher bias in choosing which students to call on. Choosing students randomly ensures that diverse voices are heard, and students experience a sense of fairness within the classroom. Additionally, a random selection process increases variation in the types of solutions or mistakes that are shared.

It is, however, important for a protocol to match the academic task at hand. Following a complex group work task, I found that it made sense to use a different protocol- strategic warm calling -to sequence student responses in a way that elicited learning. Strategic warm calling is when the teacher selects student responses based on how they reveal important ideas in the learning and gives the students notice before calling on them.

You might warm-call students in a predetermined order or infuse joy by adding an element of spontaneity: "Choose the tallest group member to share," for example. While debriefing a series of quick review problems, I may choose a table group or row in my classroom and call on students to "popcorn out" the responses. This protocol involves calling on students in quick succession to give single-word or short-phrase answers to problems that they have already completed. It is particularly effective for debriefing the steps of a procedure that students are already familiar with.

By planning which protocol best matches the task in the lesson, I'm able to balance representation of student voices with the pedagogical needs of the task.

Embracing Change

Changing my teaching practice from relying on hand-raising to focusing on intentional response protocols did not happen instantaneously and required practice and patience. When I first made this shift, I focused exclusively on strategic warm calling for several years before working up to randomized calling.

The first time I used the random name picker, I was nervous about how my class would respond. I was surprised to find that they thought it was really fun. Now, I regularly have students who offer to spin the name picker wheel and even see an occasional student smile when the wheel calls on them. It made me realize that my own bias was one of the biggest barriers I had to overcome in shifting my practice.

# data[].attributes.attachment_link.title
Protocols That Get All Students Talking in Class Discussions

# data[].attributes.attachment_link.description
These intentional response strategies create entry points for students who may feel reticent to share in class discussions.

# data[].attributes.headline
Strategy 8: Flexible Fronts: Arranging Your Classroom

# data[].attributes.subject
Strategy 8: Flexible Fronts: Arranging Your Classroom

# data[].attributes.body
You can also mess around when the seating structure of your classroom. Certain formations bring more focus on the students discussions than the teacher themselves. Traditionally the teacher is in the front side of the room with all the desks facing that direction. To guide discussions one could make an oval like shape for desks, inside & outside circles, having students face each other, & much more. You can tailor the structure of your classroom to your lessons/students needs. Personally I've always enjoyed the groups of 4 tables. It makes it very easy for students to work independently, in twos, or as whole group. Discussion is easy since they're in close proximity & facing each other. Looking side to side is also less cumbersome than looking behind you when your desk is placed "poorly."

# data[].attributes.attachment_link.title
Classroom Arrangement Styles: Pros, Cons, & Analysis

# data[].attributes.body
Visible thinking routines involve students putting their thoughts onto paper. This helps them keep track of their thoughts as they advance in a lesson. Teachers are also able to see students depth of knowledge on what was learned. They're given the opportunity to make impactful changes to their instruction based on what was written.

Below are 3 of the visible thinking routines I enjoyed & would use in my class.

# data[].attributes.attachment_link.readable_text
His mohawk flopped side to side as he pointed rapidly at his scrawling mathematics work. My 6th grader was having a hard time explaining his thought process. "It's like, I started with this idea, and tried this formula with these numbers, but then the math didn't work out, and I wasn't sure what this number was for, so I just gave up."

Sound familiar? My students routinely struggle to verbalize their thinking in the mathematics classroom. As a result, one goal of mine is to assist students in revealing their thought process. I've grown in this practice by using thinking routines to elevate discussions, reflections, and formative assessments. Thinking routines normalize the practice of making thinking visible, growing student capacity to name "thinking moves" (Harvard Graduate School of Education, 2022a). Here are three thinking routines I regularly use in my mathematics classroom to elevate student thinking:

1. Notice? Wonder? Next?

2. I used to think... Now I think....

3. Parts, Purposes, Complexities

Notice? Wonder? Next?

The Notice? Wonder? Next? thinking routine asks students to share, "What do you notice? What do you wonder?" These questions provide students a starting point for exploring their curiosities about a picture, word problem, problem-solving task, graph, and so forth. The National Council of Teachers of Mathematics (NCTM, 2023) provides a resource page with articles, examples, and teaching aids to support the "Notice and Wonder" routine.

In my classroom, I often need one more step to help students fully utilize the Notice and Wonder routine, and that is by asking students, "What's next?" Now that we have noticed and wondered, what should we do mathematically? How do we solve the problem or question we have posed? This prompting of "What's next?" places students in the driver seat of their problem-solving capabilities.

I used to think... Now I think...

As we continue to emphasize growth mindset in the classroom, the I used to think... Now I think... thinking routine normalizes change in our understanding of topics (Harvard Graduate School of Education, 2022b). In this thinking routine, students spend time reflecting on their initial conceptions of a topic, naming how their understanding has shifted or grown with deeper exploration.

During a lesson exploring polygons, middle school students shared these reflections:

I used to think polygons had all equal angles. Now I think polygons can have angles of different measurements.

I used to think I needed to memorize formulas to find the area of every polygon, which I didn't think I could do. Now I think I can break a polygon into triangles and rectangles to find the area.

Having students examine how their thinking has changed creates flexibility and adaptability in learners, an important skill for any mathematician.

Parts, Purposes, Complexities

The Parts, Purposes, Complexities thinking routine encourages students to carefully observe an object or system to identify connections between the parts and the whole (Harvard Graduate School of Education, 2022c). This might include a detailed graph, data set, table, function, formula, or word problem. This routine asks students to examine each of the three questions:

What are its parts?

What are its purposes?

What are the complexities?

A high school Algebra I teacher used this routine when introducing the point-slope form of the equation of a line { y - y ₁ = m( x - x ₁)}. Students noticed "the parts" or the variables in this equation, examining the operations occurring between variables to determine the purpose of the equation. One student shared, "What's making this complex for me is the difference between ' y' and ' y ₁'. I don't understand how you know where to put the number and where to just leave the y alone. Does it matter?"

By exploring this thinking routine, students examined the why and how behind this formula, leading to deeper mastery of this topic versus surface-level memorization. This routine also teaches the general skill of analysis, which in general means to break a complex topic down into simpler parts to gain a sense of the whole.

Putting Thinking Routines to Work

Using thinking routines in the mathematics classroom promotes a deeper relationship with the content at hand.

Thinking routines center the learning on student construction of knowledge, developing a classroom culture where students drive their learning. Thinking routines serve as a scaffold to help all students achieve success. As you move toward more student-centered classrooms, consider how you might use a thinking routine as part of a problem-solving task, performance task, or project-based learning (PBL) experience (McHugh, 2023).
References
Harvard Graduate School of Education. (2022a). Project Zero's thinking routine toolbox. Project Zero. https://pz.harvard.edu/thinking-routines

Harvard Graduate School of Education. (2022b). Thinking routine: I used to think... Now I think... Project Zero. https://pz.harvard.edu/resources/i-used-to-think-now-i-think

Harvard Graduate School of Education. (2022c). Thinking routine: Parts, purposes, complexities. Project Zero. https://pz.harvard.edu/resources/parts-purposes-complexities

McHugh, Maggie. (2023). Bring Project-Based Learning to Life in Mathematics, K-12. Thousand Oaks, CA: Corwin Press.

National Council of Teachers of Mathematics (2023). What is Notice and Wonder? Classroom Resources: https://www.nctm.org/noticeandwonder/

# data[].attributes.attachment_link.authors[].name
author: Melissa Duclos and Ally Cottrell

# data[].attributes.attachment_link.title
Help Students Reveal Their Thought Processes in the Math Classroom - Corwin Connect

# data[].attributes.attachment_link.description
Learn how thinking routines normalize the practice of making thinking visible in mathematics classrooms.

# data[].attributes.body
This strategy has students think about what key characteristics/processes go with a concept. While at the same time thinking what misconceptions may come with what is being learned. Having 3 of each helps students understand that what they learn often has many moving parts & isn't one-dimensional.

Example of a Do & Don't for solving linear equations. DO: Balance the equation before you solve for a variable (especially when the equation is multi-step). DON'T: Subtract 3 when you should divide by 3 to isolate X from 3X.

# data[].attributes.body
Students summarize the key takeaway they got from the lesson. This can help the teacher know if they're on track or where a student may be confused. This also encourages students to write what they learned in a way that makes sense to them. While implementing some personal flair for a catchy headline.

# data[].attributes.body
This strategy has students pick a part of their learning that they are confident in, may need a point in the right direction, & are completely lost in. This allows students to privately communicate with the teacher on what they may need help with. This can save time for the teacher as they don't have to wait for a students responses 1 by 1. This is also useful because teacher's can give direct support to students who may need it. Or if a lot of students have similar issues, the teacher can focus on what may need to be retaught. Or what different approaches they may use to better help their students.

# data[].attributes.body
Self Assessments are just as important as any standard assessment that looks for students knowledge. Self Assessments asks students to look into what they may or may not understand in their learning. Reflection is important since it gives students an idea on what they need to do to move forward. It's much more impactful for a student think about their progress them having someone tell it for them.

# data[].attributes.attachment_link.readable_text
Studies suggest that student self-assessments can help students perform better. A study published in the online journal Assessment & Evaluation in Higher Education found that when students evaluate themselves and receive feedback, their grades improve. Feedback literacy means being able to understand and use feedback to improve learning.

Proper student self-assessment goes beyond simple self-rating or guesswork. It involves students evaluating their own work against specific criteria. Then, they can reflect on and judge their performance based on objective and select benchmarks.

The application of self-assessment can vary. Examples of student self-assessment range from using learning logs and reflective journals to rubrics self-assessments. Teachers need to understand how student self-assessment helps students learn so they can properly guide them and advance their educational development.

What Is Student Self-Assessment?

Student self-assessment is an approach that students can apply to gauge how they're doing in school. This can be accomplished by checking their own work, performance, behavior, as well as understanding what they've learned and what they still need to learn. This encourages students to take ownership of their learning and fosters accountability. Examples of student self-assessment activities can include establishing goals, identifying strengths and weaknesses, and setting plans for improvement.

Effective self-assessment means consistently showing students how to think about their own learning through the entire learning process. Insights gained from the self-assessment process can help improve their achievement. Teachers play a critical role in regularly making the process clear to students and removing obstacles for productive self-assessments.

Why Is Student Self-Assessment Important?

Student self-assessment is important for several reasons. It encourages independent learning, enhances learning outcomes, and can potentially reduce teachers' workloads. It also helps students understand what they are learning and the reasons why they are learning about a particular subject. This understanding is a key to educational growth and helps provide a clear path to follow in their ongoing growth journey.

Student-self assessment also promotes metacognition, which means understanding one's own thought processes. This helps improve cognitive functions, including problem solving, learning, and decision making.

What Are the Benefits of Student Self-Assessment?

Student self-assessment boosts motivation and engagement by empowering students to take control of their learning. Active involvement in assessing their progress increases their investment in education.

Numerous studies highlight the positive impact of student self-assessment. For instance, a study in the International Journal of Educational Research revealed marked enhancement in writing skills among students who use rubrics compared to those who did not.

Additional examples of student self-assessment benefits include the following.

Students can evaluate knowledge and learning processes.

Critical thinking and problem-solving skills can be improved.

It champions a growth mindset for viewing challenges as opportunities.

Student self-assessment not only benefits students themselves, it provides valuable feedback to educators. This helps teachers gain insights into students' understanding, strengths, and weaknesses. It also promotes a supportive and inclusive learning environment, as students feel that their perspectives and voices are valued.

7 Examples of Student Self-Assessments

For teachers, discovering effective self-assessment strategies, processes, and tools in collaboration with students is an ongoing journey that demands time and patience. Here are seven examples of student self-assessments.

1. Learning Log

A learning log serves as a personal journal for students to record their thoughts, questions, and experiences on their educational journey. It encourages deep self-awareness, helping students track their growth, identify strengths and weaknesses, and gain insights into their learning patterns. By updating their learning log regularly, students take ownership of their learning, fostering a deeper understanding of their academic development and promoting accountability.

2. Reflective Journals

Reflective journals prompt students to write about their learning experiences, challenges, and achievements, including personal reflections, emotions, and opinions. Such journals encourage deep reflection and enhance self-awareness. Reflective logs, however, differ from learning logs by being more open-ended and less focused on objective data. Journaling in an online format can also be beneficial. It can open versatile reflective tools for students in the form of documents, videos, or audio, fostering richer learning experiences and improved outcomes.

3. Goal Setting

Students can set specific goals that are measurable, achievable, relevant, and timebound (SMART) within their learning logs or reflective journals. Examples of student self-assessment SMART goals can include improving academic performance-more specifically, a student can say: "I will increase my grade from a B+ to an A next marking period."

Another example of goal setting can include working to increase class participation-for example, establishing a promise to participate in school at least 3 times every day. This approach provides students with clear roadmaps for their learning journey and fosters a sense of accountability.

4. Rubric Self-Assessment

Rubrics can aid in the self-assessment process. This helps students identify areas for improvement and take ownership of their learning process, providing a clear framework to evaluate their own work and gain an understanding of high-quality performance.

Commonly used rubric types in schools include holistic rubrics (an overall performance assessment) and analytic rubrics (a breakdown of performance into separate elements). Another type, developmental rubrics, measure growth along a proficiency scale from novice to expert. Assessment rubrics help students proactively self-assess their work against predefined criteria.

5. Questionnaires or Surveys

Students can regularly fill out self-assessment questionnaires or surveys to reflect on their learning experiences, as well as identify strengths, weaknesses, and improvement opportunities. Examples of student self-assessment questions can include: What did you learn today? Did you try your best on your assignment? Answers to these types of questions can help students set action plans for continuous improvement.

6. Self-Reflection Worksheets

Self-reflection worksheets enable students to reflect on the breadth of their learning experiences, identify areas for improvement, and develop action plans to address those areas. Including action plans in these worksheets empowers students to take tangible steps toward enhancing their abilities, turning self-awareness into a catalyst for growth and development.

7. Exit Tickets

Exit tickets are brief assessments or reflections completed by students at the end of a lesson or class session. They help educators gauge student comprehension, identify areas that need further attention, and tailor future lessons accordingly. Exit tickets also encourage students to reflect on their learning and provide valuable feedback to the teacher.

Become A Transformative Educator

Despite the noted benefits of student self-assessment, putting forward new strategies into practice can come with challenges. If you are looking to incorporate teaching strategies to better connect with students and help improve learning outcomes, an advanced degree in education can prepare you with essential skills to become a transformative educator.

Wilson College Online offers a Master of Education (MEd) degree , covering subjects such as differentiated instruction, best practices, and technology integration. This program hones teaching skills through both research and classroom practices. It also offers educators the flexibility to continue teaching in their current district, thanks to the program's asynchronous self-paced model.

Discover how the Wilson College Online Master of Education degree can empower you to further impact the lives of students.
Recommended Readings
Pros and Cons of Education Apps for Toddlers

Special Education Teacher Salary and Career Overview

Tips for Building an Engaging Special Education Classroom
Sources:
ASCD, "How to Provide Better Feedback Through Rubrics"

Assessment & Evaluation in Higher Education, "Self-assessment is About More Than Self: The Enabling Role of Feedback Literacy"

EducationWeek, "Rubric Do's & Don'ts"

Edutopia, "Teaching Students to Assess Their Learning"

E-Learning Heroes, "Using Learning Journals in E-Learning #344"

International Journal of Educational Research Open, "Examining the Impact of Self-assessment with the Use of Rubrics on Primary School Students' Performance"

Kami, "How to Use SMART Goals for Your Students"

NWEA, "2 Types of Student Goal Setting that Empower Early Learners"

NWEA, "Proof That Student Self-Assessment Moves Learning Forward"

ProProfs, "Self-Assessment for Students: The Ultimate Guide"

# data[].attributes.attachment_link.title
Benefits and Examples of Student Self-Assessments

# data[].attributes.attachment_link.description
What are examples of student self-assessments that help teachers better connect with students and improve learning? Explore student self-assessments benefits.

# data[].attributes.body
While more a more direct version of instruction is viable, it doesn't do much to directly engage the students. Any student could give & is expected to deliver the same result to a tee. Inquiry based learning calls for & encourages a variety of responses from our students. The same concepts are being assessed, but student personal input is required for it to work best. Students give their interpretations off of what they notice in the context of the concepts. As opposed to regular instruction that expects a single results while pointing you in that direction.

# data[].attributes.attachment_link.title
How do we use inquiry when teaching mathematics?

# data[].attributes.body
These tasks focus on checking students understanding, factual knowledge, & the skills they will use in a unit. These tasks often involve application of what is being learned to less familiar contexts (sometimes outside the general content area). The tasks should be engaging & thought provoking, not relying on "busywork." Their is a framework called RAFTS that helps teachers design these kinds of projects. RAFTS stands for

Roles: What the student will do in the project

Audience: Who will read/is affected by the project

Format: What medium students will use to represent their project

Topic: What is being learned & applied

Strong: Usage of words that will hook students .

These projects can have a huge range. Like finding the volume of school buildings, or using linear equations to represent how far railroad ties are from each other on a track.

# data[].attributes.attachment_link.description
Instead of doing math problems with no context, students at this school role-play real jobs.

# data[].attributes.body
Multiple Choice Questions: Often these questions are used for quizzes or to see if students get the correct final answer. Being more focused on product than concept. That doesn't have to be the case though. Multiple choice questions can be designed in a way where they test students on concepts instead of product. For example you can have an answer choice that is correct, one that is a misconception (conceptually), one that is true but for a separate concept, etc.

Zero, One, Two, or Three: This strategy is a quick, easy, & subtle way to check your student's understanding of what was learned. Students write a number from 0-3 on paper. Ranging from completely lost (0) to completely understood (4). You can use this to check in with your students mid lesson & see where to go from there. It can also be used on student's homework, which I personally would love to implement. Even though many students may complete the homework, there are many different ways they could have gotten to that point. Whether it was with a lot of struggle or relative ease. This would help me know who needs further assistance.

# data[].attributes.attachment_link.description
Formative assessment is a proven technique for improving student learning, and the strategies shared here by Jay McTighe work both in the classroom and remotely.

# data[].attributes.body
Graphic Organizers are clear visual ways for students to sort their thoughts. They can be used to learn concepts/key vocabulary. They're a vast array of graphic organizers that students can use across many different content areas. Many of those can be transferred to the math content area with proper care. Below are 3 forms of graphic organizers that can be used.

# data[].attributes.attachment_link.readable_text
The use of a graphic organizer can start as early as first or second grade and can continue to be useful for some learners all the way through high school. In subjects such as math, that grow increasingly complex as students get older, these tools can be especially helpful in maintaining organized work habits and enhancing problem-solving skills. If used correctly and consistently as students develop, the concepts of strategic thinking graphic organizers instill will likely have reached the point that many learners no longer need them by the time they reach high school.

How To Use Graphic Organizers in Math

Using graphic organizers has been a proven effective problem-solving strategy for helping young learners think and process information more efficiently by allowing them to both visualize and organize the information they need to solve problems. Creativity and careful attention to detail can be greatly enhanced via the use of visual maps-which is exactly what a graphic organizer is. A graphic organizer aids in organizing thought processes as well as creating a framework to collect and compare the information that's being gathered. That's why, in addition to structuring information, organizers can be used to improve students' abilities to comprehend and process that information by seeing it separated it into categories of what is more important and what less important.

​Over time, graphic organizers help learners become strategic problem solvers. Provided they're used effectively and consistently as an integral part of the problem-solving process, graphic organizers can also improve test scores.

How Graphic Organizers Work for Math

A typical graphic organizer has the problem printed on it. The paper is divided into four quadrants with the problem appearing at the top, although sometimes, it can be found in the middle of the page.

The first quadrant is used for the student to determine what the problem is actually trying to solve for. The second quadrant is used to determine what strategies are needed to solve the problem. The third quadrant is used to show the steps involved in order to solve the problem. The fourth quadrant is used to answer the question that is initially being asked and to indicate why the answer the reasoning behind how the answer was arrived at, and why the answer is correct.

Graphic Organizers: The Takeaway

Graphic organizers can be a parent or teacher's go-to problem-solving tool of choice for many reasons but the bottom line is, the better a student can visualize and organize the strategy that goes into arriving at their answers, the more likely young learners are not only to come up with the appropriate solutions but also, to understand how they arrived at those solutions and what makes their answers correct.

Ultimately, the learner:

determines what's being asked

considers and tries out strategies

determines and shows answer

looks back to ensure that all parts of the question have been answered

provides a final answer to the question

Some of the graphic organizers used for problem-solving in math are referred to as 4-Block, 4 Corners, 4 Square or the Frayer Model. Regardless of which template you choose, you'll find that when it's used effectively and consistently, enhanced problem solving will be the result.

# data[].attributes.attachment_link.title
How Do Graphic Organizers Help Students With Math Problems?

# data[].attributes.attachment_link.description
How to use graphic organizers in math. Graphic organizers are powerful tools that assist learners in solving math problems effectively.

# data[].attributes.body
The Frayer Model is one of my favorites. It's very versatile, it can be used in different content areas, cover concepts, or vocabulary if you'd like. The Frayer Model is made up of 4 main components (5 if you count the center which is the focus). Those components are the definition, characteristics, examples, & non-examples. Since the Frayer models are student oriented, you can see their thought process, their personal connections, & you're able to clear up any misconceptions the students may have. Since these models can be so personalized students are more likely to remember & understand what is written down on them.

# data[].attributes.body
As said in the title students will organize whether they agree, disagree, or if a prompt is situational. There are other versions of this, like the always true, never true, sometimes true. Essentially you split a prompt into three possible categories it can be in. Students decide on their own or with peers on where it would be sorted. This is fairly simple, but it's a great low intensity way of building student's comprehension skills.

# data[].attributes.body
This one works a lot like the Frayer model, focusing on examples & non-examples. Here you would give students prompts related or unrelated to what is being learned. Then it's up to them to decide whether it counts as an example or non-example. This can be done on a whiteboard/projector, passing out sets of prompts cards, & any other way you can imagine distributing the prompts.

# data[].attributes.body
Communication in math in my opinion is best taught by providing opportunities for students to write/say their thoughts while doing math. Such as using different visual thinking routines. Students often remember the procedures of what is learned but struggle more with comprehension. Procedures at a point can feel as simple as motor skills. Thinking isn't that simple, it relies on recalling & synthesizing information to make a response. That response may be correct the first time, or it may need reflection. I believe it's important my students learn beyond "doing", instead they should develop their "applying" skills.

Self management relies on students being able to pace themselves in their own learning & being motivated to do so. This one is a bit tricker but I think it starts in building students confidence in math. It's hard to get started when you don't know where you should be going. This is why scaffolding is important. You can provide easier problems that go up in difficulty or complexity. By starting this way you can help students they're prepared for what's coming next. You can also start with a hard question, but I must emphasize you have to be VERY careful if you do so. If done wrong you can damage student's self confidence. Show the questions as a way of representing an end point. Have them think about it & bring in thoughts, don't focus on their ability to solve the hard question. Assess where students are at & build from there. Reassure your students that you will work gradually to that end point & you'll be there for support every step of the way. When they get to the point where they're ready & do solve it there confidence will increase, knowing that they were able to overcome the once seen impossible.

Research skills in math are greatly helped by student's own curiosity. Provide questions that get them hooked, that leave them asking for more knowledge (this doesn't mean to not provide an answer). Admittedly this is much easier said then done & even I'm having difficulty imagining how I can get students to that point.