The Reflective Educator

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Mathematical Representations

There is evidence that students who have access to and understand how to use different mathematical representations of the same mathematical concepts are more successful learning mathematics than students who only have access to one representation type.

The issue is that mathematical representations are not intrinsically meaningful on their own. Some mathematical representations are completely arbitrary and for others it can be challenging to determine to what elements of the representation to pay attention.


Here is an example intended to highlight how some mathematical representations, even ones that are very familiar, are somewhat arbitrary. Check out the diagram below and ask yourself, “What is meant by each of these models for the less than, equals to, and greater than signs?”

The less than, equals to, and greater than signs are arbitrary. They are symbols to which we denote meaning and which otherwise do not contain any mathematical information without that meaning assigned.


Another issue is that students do not always attend to the critical features of a mathematical representation. For example, I have often seen a shape and a formula for calculating the area of that shape introduced together, possibly like it is shown below with a calculation of area alongside the visual.

But to what exactly in this representation do we expect students to attend? The most obvious features of the diagram of the rectangle that correspond to the area formula are the 5 and the 3. These refer to the quantities of length and width. But what is meant by the multiplication of those two quantities? How is this multiplication represented in the diagram? There is no special reason from diagrams like these that children will attend to the space occupied by the rectangle and match that to the area of the rectangle, so we need to find ways to draw their attention to this element of rectangles.


Mathematical representations have potential power to subtly introduce ideas to students as well. The number line is a good example of a representation that is often introduced early and may lead to some powerful questions by students.

What do those arrows on either side mean?
What does the space between the numbers represent?
What does going left on the number line mean?
When does the number line stop?

Each of these questions has a mathematical answer and the number line can again be used to represent this answer (warning: but not always very well).


I worked recently with a group of teachers, and we looked for shortcuts to solving the equation x + x = 116 – 84.
Here are some of their shortcuts.

Strategy 1 Strategy 2 Strategy 3
“I combined the x’s together and I subtracted the 84 from the 116, which gave me 32. I could do this quickly because I knew that 11 – 8 = 3 and 6 – 4 = 2. This gave me 2x = 32, so then I divided both sides by 2 to get x = 16.” “I saw the x + x and changed it to 2x. Then I decided to divide everything by 2 to make the calculation simpler, and got x = 58 – 42. Since 58 – 42 is 16, this means x = 16.” “I noticed that 116 and 84 are both 16 away from 100. So I can rewrite this as x + x = 16 + 16 and therefore x = 16.”

But what if we tried to represent Strategy 2 and 3 on a number line? Here are a couple of different visualizations of these strategies. What information is captured differently by the different visualizations?


In our math curriculum, when introducing an area model for factoring and completing the square, we first introduce the representation itself before we do any other mathematical work using it.

Try out this applet and ask yourself, “What relationships between the visual and the expression do you notice as you change the value of a?”

In a classroom setting, we could ask students to share their answers to this prompt with a partner and then we could ask some students to share their answers with the entire class. After this, if necessary, we could add an observation from another class, so that students know to what elements of this representation to attend.

Next, after students practice writing out expressions for unaltered diagrams, we could ask students to write expression for the following altered diagrams:

In my experience, this geometric approach to completing the square results in more students in having access to the algebraic approach, and makes the name of the algebraic strategy more obvious.


Mathematical representations can offer explicit ways for students to make connections across different mathematical topics. In our Algebra I curriculum, we do not have a unit on graphing. Instead interpreting and using graphs is part of all seven units, increasing the odds students make connections in and between those units and also that students remember key ideas from the course.


To summarize:

  • Don’t assume that mathematical understanding is transmitted by the representation.
  • Some mathematical ideas are easier to introduce using some representations rather than others.
  • Reusing a specific mathematical representation over and over again will both help students make mathematical connections and remember key concepts from the year.


Teaching to Big Ideas

On Big Mathematical Ideas, Cathy Fosnot writes:


Underlying these strategies are big ideas. Big ideas are “the central, organizing ideas of mathematics—principles that define mathematical order”(Schifter and Fosnot 1993, 35). Big ideas are deeply connected to the structures of mathematics. They are also characteristic of shifts in learners’ reasoning—shifts in perspective, in logic, in the mathematical relationships they set up. As such, they are connected to part-whole relations—to the structure of thought in general (Piaget 1977). In fact, that is why they are connected to the structures of mathematics. Through the centuries and across cultures as mathematical big ideas developed, the advances were often characterized by paradigmatic shifts in reasoning. That is because these structural shifts in thought characterize the learning process in general. Thus, these ideas are “big” because they are critical ideas in mathematics itself and because they are big leaps in the development of the structure of children’s reasoning. Some of the big ideas you will see children constructing as they work with the materials in this package are unitizing, compensation, and equivalence.


There are two pieces of evidence from cognitive science that support teaching to Big Ideas instead of teaching 180 discrete and perhaps haphazardly connected lessons.


1. It is much easier to remember ideas that are connected together in more complex schema.


This is why it takes many repetitions to remember meaningless information (like a string of randomly chosen numbers) and far fewer repetitions to remember meaningful information (like a poem).

Acronyms like SOHCAHTOA are far easier to remember than the three equations this acronym represents because the acronym provides some structure to the information to remember. The acronym allows us to chunk the information to remember into smaller, easier to manage pieces. The same principle applies to anything we want to remember.

I see Big Ideas as recursively being formed of many smaller ideas and that a Big Idea is a way of linking the smaller ideas together in the same way that the SOHCAHTOA acronym links together the words sine, cosine, tangent with opposite, adjacent, and hypotenuse.

Discrete ideas each day vs collections of Big Ideas


2. It is much easier to remember information that we keep coming back to and are asked to repeatedly recall.


If I teach to Big Ideas, which may last more than a single session, the odds are greatly increased that the smaller ideas from which the Big Ideas are formed will be repeated across different learning sessions. This is critical because our brains are designed to forget information we don’t re-use and to remember information that is repeatedly helpful.

Repeating small ideas within big ideas

Suppose I want kids to remember Big Idea A, which is formed of smaller ideas A1, A2, A3, etc… and I teach this Big Idea over the course of a week. As I teach, I might ask kids to use idea A1 when working on idea A2, and then use ideas A1 and A2 while working on idea A3, all while asking kids to periodically attend to the relationships between A1, A2, and A3 as they are part of the Big Idea A itself. This means that during the course of a week, students may need to study idea A1 once, and recall idea A1 many times as they make connections to the other ideas of the week.

As the image also indicates, when teaching to Big Ideas means we can deliberately and explicitly make links between different Big Ideas, which means that across different weeks of instruction, the small ideas that make up Big Ideas can be referenced and repeated many times during the year. In our Algebra I course in our curriculum project, we don’t have a unit on graphing functions as graphing functions comes up in all seven units of Algebra I.


Here are some consequences of choosing to teach to Big Ideas instead of discrete small ideas:

  1. You have to name both the Big Ideas and the small ideas from which they are formed.
  2. Your curriculum can no longer be a collection of your favourite individual tasks as task selection and sequencing is far more critical.
  3. Your lessons have to be designed to make the connections between big ideas explicit for children rather than implicit. A rich schema is unhelpful if it is woven invisibly into your curriculum.



A Task is Not a Lesson

Does this image represent a lesson or a task?



I’ve noticed that opinions are split on this question with some people calling the image above a task and others calling it a lesson. In my opinion, unless an image like this includes a description of how the teachers and students will interact with the image, it’s a task. (Aside: There’s too much variety in how lessons are described to have a very clear definition of lesson in this post, so I’ll have to save that for a future post, but this blog post by Annie Forest on different lesson structures is a great read).

One might be able to imagine how you would use this task with students and form a lesson plan based on a task, but without some insight into the intended use of a task there is enormous variation in how any particular task might be used.

This is not just pedantry. There is significant evidence that how one teaches matters and that there is far more to teaching than just putting tasks in front of kids. As a profession, if we are to have any hope of solving the problem of communicating nuance about teaching with each other, we should at least start with being clear about how we use some basic professional terms like task and lesson.


Why Inquiry Fails

Here are at least six problems that often make inquiry-based lessons fail.



Some Problems

  1. Students have too much information to process when attempting to solve a problem which can quickly overwhelm their working memory.
  2. When given a new problem type, students do not always have all of the prerequisite knowledge necessary to approach the new problem successfully, leading them to need to make leaps of logic larger than they are able.
  3. When students are sharing their ideas and strategies for solving a problem, other students are either not really listening or hearing what they want to hear rather than what is actually being said.
  4. Students sometimes focus on the short-term “How do we solve this type of problem?” rather than “What mathematical principle can I generalize to be able to solve other problems?”

    Alternatively, students attempt to generalize and run out of working memory since they often have to hold both their solution and the generalizations from their solution simultaneously in their heads in order to generalize.

  5. The goals and/or structure of inquiry-based lessons are often unclear. When students need to remember the goal of a lesson and the structure of an activity in order to be successful, they have less working memory available to actually be able to focus on the task at hand – the problem they are working on.
  6. Students often use “means-end analysis” when problem solving which means they tend to focus less on what process they are using (and improving that process) and more on what answer will get them out of needing to continue problem solving.


Some solutions
Note: The solutions described below assume that your lesson structure is still inquiry-based. It may be that the best solution to whatever goal you have for the day is not to use inquiry at all, but have students study worked examples instead.

There are a variety of solutions to problems 1, 2, 4, and 6 which are all based on problem-structure and task selection.

For example, in the instructional routine Connecting Representations, the goal of any given task is not to come up with answers to problems but to name connections between two representation types. Students are focused on a small amount of information at any given time, which is given students in a deliberately staggered way in order to reduce the amount of information to process all at once. Connections between representation types are also easier to generalize than solutions to problems.

A sample Connecting Representation task


To improve the odds that students can generalize from their problem-solving experience, make generalizations a focus of any whole group discussion following the problem-solving time and have students reflect (in writing, perhaps in a regular journal) on what they learned today that they think they may be able to use in the future. Sentence prompts like “Today I learned to pay attention to … because …” can scaffold these reflections for students.

Another suggestion is to select tasks for which students can use a lot of what they already know to solve the problem and only have to make small leaps. We want to balance our students’ use of their long-term memory to aid them in solving problems with their working memory to make the small new leaps or connections necessary. A good rule of thumb is that if students need to make 3 or more small discoveries or new connections in the course of solving any particular problem, they probably won’t.

We’ve also noticed in our work that it can be helpful to distribute problem solving tasks both over time and over a group of people.

Distributing a problem solving task over time means giving out portions of the problem to learners in smaller chunks rather than all at once. For example, it can be really helpful to give students some independent time to first consider a problem on their own before working with a partner. Or you can divide a more complex problem solving task into smaller pieces so that students can “chunk” their earlier work into their later, more complex solutions. The Math Forum’s “Notice and Wonder” protocol is one example of this principle being used in practice.

Distributing a problem solving task over a group of people means giving learners deliberate access to each other as resources while they are problem solving.

In Peter Liljedahl‘s “Building Thinking Classrooms” work, this looks like students working at vertical whiteboards in small groups or with a partner, during which I have often noticed groups borrowing ideas from each other.

An example of the use of vertical nonpermanent surfaces, shared with permission from Michael Pruner.


In our work with instructional routines (with lots of help from Grace Kelemanik and Amy Lucenta) this means making problem solving sessions short and/or interrupting an unsuccessful problem solving session with whole class opportunities for students to share observations and ideas. In order to improve the odds that students actually listen to and understand each other (and solve problem #3 listed above), we have students first share a strategy/idea while we or a student points, then another student restates a strategy/idea while we either continue to point, another student points, or we annotate the strategy using color/symbols/small amounts of text. We have also found it helpful to press students to provide complete explanations, especially when there are missing details/jumps in logic in their explanations.

To get a sense of how this supports students, try watching the following two videos to see how the use of restating and annotation makes a huge difference in your own clarity around a strategy being shared.


Student sharing a strategy: no gesturing or annotation

Student sharing a strategy: with gesturing and annotation


In terms of the 5th problem with inquiry, around the goals and structures of a lesson being unclear, I’ve written extensively here about how instructional routines support students (and their teachers) in minimizing unnecessary extraneous cognitive load focused on “what am i doing next?” If you don’t have time to read that other post, tl;dr: routines free up working memory by allowing students to delegate questions about what their role is, what they are doing next, and why they are doing it, to their long-term memory.



In many schools around the world, learning how to use inquiry-focused lessons in mathematics class is a focus of the school or mathematics department. However, inquiry-focused lessons come with their own set of challenges, raised above.

I have some proposed solutions to those challenges listed above, but I’d love to hear what other people are doing to tackle the same problems or what other problems people have noticed occur when they try to implement inquiry-focused lessons.



Teaching Two Courses on Instructional Routines

This spring I’m going to be co-teaching two courses with Kaitlin Ruggiero on instructional routines.


Learn more here

Learn more here

The first course will focus on using the instructional routine, Contemplate then Calculate, as a tool for learning how to use the 5 Practices for Orchestrating Productive Mathematical Discussions. The second course will focus on using another instructional routine, Connecting Representations, to better understand formative assessment strategies. These instructional routines were developed by Grace Kelemanik and Amy Lucenta. Connecting Representations is one of the routines described in their book, Routines for Reasoning.

Both courses are structured as a one-day workshop followed by 5 weeks of planning and reflecting, in an online discussion forum, on the use of the practices and strategies. The fee for the courses is minimal, set at $25 for non-New Visions’ teachers. If you are a teacher in New York City, you can also pay a $45 fee to the NYC DOE and receive 1 p-credit for each course.

I’m looking forward to teaching these courses and hope that some of you are able to participate!


Five traps of technology

“Mathematics is not done with a computer. Mathematics is not done with pencil and paper. Mathematics is done with the brain.”

~ An anonymous participant of the Computer Based Mathematics Summit, London, 2011


The heart of mathematics education is ensuring that students develop both knowledge of mathematics (here is a definition of mathematics) and productive dispositions towards mathematics. The minimum test for us to apply when considering the use of technology in mathematics education is this: “How does this use of technology help develop students’ knowledge of mathematics and/or their productive dispositions towards mathematics?”


An early form of technology for mathematics, source : Wikipedia


There are five traps to avoid when using technology. The first trap is that students end up not learning mathematics but instead only learning how to use a particular technological tool. The second trap is that someone who knows the mathematics already can see the mathematical principles illustrated by a particular technological tool but that a novice does not see or use the tool the same as the expert and therefore does not experience the mathematics the same. The third trap is using technology solely to focus on recall and repetition since students often lose opportunities to see patterns across problems (it doesn’t do much for most students’ productive dispositions towards mathematics either). The fourth trap of technology is that it can isolate learners, both from each other and sometimes even from their teacher. The final trap is that technology can sometimes make it harder to see (or hear) how students understand mathematical ideas.

Here are some questions we should ask ourselves when deciding to use any particular piece of technology with our students:

  • Does using this technology help my students learn mathematics that they can use without the use of this technology?
  • How will someone who does not yet know the mathematics embedded within this technological tool see the mathematics?
  • Does this technology focus solely on the acquisition of a limited set of mathematical knowledge or is it possible for students to use deliberate practice to identify patterns across different problems and acquire new mathematical ideas?
  • Does this technology make it harder for my students to interact with each other and with me?
  • How will I learn how my students understand the mathematical ideas that are the focus of this lesson?


In a follow-up post, when I have more time, I’m hoping to share examples of technologies that fall into these traps and how I might change the technologies to avoid the trap or how I might change might change my teaching to circumvent the trap.


Just Google It

In an age where fake news is beginning to dominate the media consumed by millions of people around the world and Google’s results are being gamed by racist organizations, claiming that students don’t need to know anything because “they can just Google it” is irresponsible at best and negligent at worst.

Students (and adults) are struggling to determine what’s real and what’s not. We need to do more as educators than just surface that fake news exists (like the Northwest Tree Octopus). We must ensure that our children leave school knowing enough history, geography, math, science, language, etc… so that they cannot be easily be fooled by fake news.



The best inoculation against misinformation is a rich base of knowledge and experience that contradicts that misinformation.


Is Teacher Marking Necessary?

Teachers do a lot of marking of student work. But is it necessary?



In this comprehensive review of the literature on feedback, corrective feedback (example shown below) without mechanisms for correcting that feedback were found, unsurprisingly, to have little impact on student learning in most cases.

Wrong answer

An example of minimal feedback


Unfortunately, there is also good evidence (see the same literature review) that taking even more time to add comments to student work does not lead, by itself, to improved student learning. So what can teachers do differently?



Here’s a simple strategy. Take a pile of student work and review it, looking for evidence of student performance, and find examples of feedback that you can meaningfully target to groups of students, and then design activities for the whole class to do that result in different groups of students getting feedback on their ideas. In other words, integrate the time you would spend marking with the time you spend planning but in response to what students did in your class.

One question that comes up when I suggest this strategy to teachers is “But what will I put in my grade-book?” Here I suggest that a grade-book can contain evidence of completion of tasks on a regular basis and that for a smaller number of assignments, more detailed information could be provided. Stopping grading everything doesn’t mean you can’t grade anything; just be more selective. A more radical suggestion is to work at the school-wide level and eliminate everything that isn’t absolutely necessary to improve student learning.


Moving beyond CUBES and keywords

It is well known that children often struggle to solve word problems in mathematics. One strategy that is used to support students with having access to word problems is called CUBES. Another is to have students identify all of the keywords in the problem. (Update: Margie Pearse wrote a longer response to these same two strategies here).


Circle the important quantities, Underline the question, Box any math words, Evaluate: What steps should I take?, Solve and check: Does my answer make sense?


In these strategies, students are encouraged to chunk the information given in the word problem in a variety of different ways. For the CUBES strategy, the word quantities is often defined as numbers including units and direction (if given).

Let’s try out the CUBES strategy with the following word problem from the Mathematics Assessment Project. Why don’t you try it yourself first?

Meal Out


Here is my attempt, as if I were a student, on this task for just the first three of the steps in CUBES.

CUBES with Meal Out


You’ll notice that I have circled a lot of unimportant quantities. I’ve also boxed some math words or expressions that are probably not helpful. These are reasonable things to expect many students to do. How does a child know that “three-course meal” is really a description of a kind of food and not a quantity in this context? We could easily imagine contexts in which the number of courses in a meal is important.

My point is that CUBES is an insufficient strategy to help students have entry to this problem. It might be helpful (sometimes) but it almost certainly not sufficient. There is a lot of thinking yet to be done before identifying the critical information from the problem and being able to solve the problem.


Here are some additional recommendations that you can combine as needed:

  • Make sure students have access to the context itself. In this case, if students do not know what a three-course meal or a two-course meal are, it might be helpful to have pictures or a story that describes these things. If the context is one that you think you will revisit more than once, it may be helpful to act out the story.
  • If you have students who are learning English as a new language, it may be helpful to work out, perhaps with other students in your class, a translation of the context into the language they know best.
  • Have students restate, in a variety of different ways, what they think the context of the situation is about. This will both help students hear different ways of describing the situation and give you information about how students are making sense of the context.
  • Let students ask questions about the context. While solving the problem for students may be counter-productive, answering questions students have about the context will both give you information about how they understand the context and give students helpful information that you may not have predicted they needed.
  • It might be helpful to have students describe the relationships between the quantities and other information given. This might be by drawing a diagram or a mindmap. For a pair of routines that may be helpful here, try Capturing Quantities or Three Reads, both described in this book.
  • After solving the problem as a group and ensuring that everyone understands the solution(s), come back and check which information from the problem was actually useful. Over time this may help students learn how to distinguish the relevant from the irrelevant information in the problem.


Writing Curriculum

An experiment


Let’s try a little experiment. Take a look at the following network graphs and think about what is different for each graph and what is the same for each graph.



Now look at this matrices associated with these network graphs.



Which network graph do you think goes with which matrix and why? (This might take you a few minutes. Be patient.)


This particular task is intended to be used with an instructional routine called Connecting Representations designed by Amy Lucenta and Grace Kelemanik. They describe this instructional routine and three other instructional routines in significant detail in the book they just published, Routines for Reasoning.

One goal of Connecting Representations is to support students in making connections between different mathematical representations which describe the same mathematics but on the surface look very different. Another goal of the routine is to support students in being able to see and describe the mathematical features (or structure) that are important to pay attention to in the individual representations in order to make connections between representations.

In the network graphs, as you no doubt noticed, the number of nodes, the labels for the nodes, the ways arrows are connected between nodes, and the direction of those arrows are all important features. You may have concluded as well that the actual positions of the nodes do not matter. Additionally, by looking at the network graphs and this paragraph, you may also know exactly which parts of the graphs I mean by the word ‘node’ (if not, I mean the circles with the numbers inside).

In the matrices, if you make connections between the graphs and the matrices, you almost certainly had to pay attention to the rows and columns of the matrices and the values of the entries in the matrices, most of which were zeros and some of which were ones.

At a meta-level, you focused on individual parts of each representation, you may have zoomed out to look across different representations, and you made connections between different representations.


Variation theory

I’ve recently been attempting to incorporate a critical idea from variation theory into the design of curricular resources; students learn from noticing differences across a background of sameness, rather than from seeing similar objects and discerning the important features by what is the same across each of the objects. Another way of saying this is that differences stand out much more than similarities do.

On Variation theory, Mun Ling Lo writes:


For the network graphs above, I deliberately varied the connections between nodes and the position of the nodes within the diagram. In the matrices, I represented the connections between nodes and did not represent the position (since I cannot represent the position in a matrix) which resulted in a deliberate variation across the rows and columns.


Tasks that are not really learning tasks

Not all tasks are tasks that students will make new connections from. Some tasks require students to demonstrate understanding of a concept they already know. While it is helpful for students to rethink about ideas periodically as there is significant evidence that this helps students remember ideas over time, we also need to use tasks that build new understanding.

Here’s an example of a task on the same content that assumes students know some mathematical ideas already.


Note that it is impossible for students to do this task without already knowing how to answer both questions. This is not a task that students are likely to learn something new from on its own.

Side note: This is by the most common kind of task I see when I have observed teachers over the past five years.


Instructional efficiency

One strategy for taking a task that serves both purposes of helping students remember things they have learned before and helping students build new connections is to have students practice solving problems but deliberately sequence the problems so that students see new connections between the problems they solve.

For example, try and solve the following mental arithmetic problems in your head, without a calculator, and without writing anything down. While solving these problems, deliberately try to use what you’ve done in an earlier problem to make the next problem easier to solve.


What did you notice yourself doing as you worked through the problems? What big idea might students get out of solving this series of problems?


Ideally you saw that 10 + 3 is the same as 9 + 4 and that may have helped you see 9 + 4 as 9 + 1 + 3 = 10 + 3 so that you could reuse your solution to the previous problem. For 19 + 4, you may have also regrouped to 19 + 1 + 3 = 20 + 3. 29 + 14 may have become 30 + 13 = 43 and 69 + 25 may have become 70 + 24. If not, then if I wanted you to see this, as a teacher I may have had a student who did regroup like this share their strategy with the class.

My point is that I have increased the odds that you saw this regrouping strategy by deliberately choosing the problems for you to try.



If you are designing curriculum or tasks for your students here are my two recommendations:

  1. Pay attention to what you are varying across different problems or representations you give students. What you vary across a group of related problems or representations is what students are more likely to notice.
  2. If possible, do this even when giving students problems to practice so that there is a chance students learn something new from that practice.