The Reflective Educator

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That’s Mathematics

Ed Southhall invited me to sing a line from the song, “That’s Mathematics” by Tom Lehrer and I happily obliged, along with a few other friends of Ed’s.



One thing I really like about this song is the variety of different things that Tom calls out as being mathematics. Mathematics is no one thing, it is a bunch of different things to different people.

Online Learning Recommendations

Given that many schools (and entire school districts) may be closed down during the coronavirus outbreak, I decided to write this post with recommendations for schools that may attempt to implement online learning during this time.

I read through this review of the research on online learning, which contains these high-level recommendations. There are some caveats with this research, especially given that most research on online learning has been done with older students and that the sample sizes with k to 12 students are relatively small. That being said, some evidence for effectiveness is better than no evidence at all. A further caveat: these recommendations are based on effect sizes, which I have not included since they are notoriously unreliable to compare.

  1. Instruction combining online and face-to-face elements had a larger advantage relative to purely face-to-face instruction than did purely online instruction.

    If your school is closed completely, then this may be impossible. That being said, online programs such as Zoom or Big Marker may allow for some “face to face” interaction to occur. These programs will also help with the next recommendation.

  2. Effect sizes were larger for studies in which the online instruction was collaborative or instructor-directed than in those studies where online learners worked independently.

    This basically means that you should design activities that are either led by a teacher or activities that have students work together in small groups. Resources like Google Docs and Skype will be helpful for students working together but given the high possibility that some students will engage in off-task and/or anti-social behaviour (such as teasing or bullying), having some moderation and oversight of these online spaces will be helpful.

  3. Elements such as video or online quizzes do not appear to influence the amount that students learn in online classes.

    Creating a bunch of video lessons of a talking head working through some math problems and then quizzing students on what they have learned afterwards is not supported by the existing evidence on online learning. Given that educator planning time is in short supply, it’s probably best to plan other types of activities.

  4. Online learning can be enhanced by giving learners control of their interactions with media and prompting learner reflection.

    A good example of a program that allows for this is Geogebra. See for example this set of constructions puzzles that require students to think and make decisions while they work through the problems.

    Another example is the DreamBox Learning math program, which also requires students to actively engage with mathematics. Disclaimer: I work for DreamBox Learning as a mathematician and senior curriculum designer.

  5. Providing guidance for learning for groups of students appears less successful than using such mechanisms with individual learners.

    This recommendation suggests that feedback and support for students should be individualized for online learning, rather than given to the entire group. This does not necessarily mean that one should avoid providing scaffolds (such as guiding questions) to the entire group or that teachers necessarily need to work with individual students, only that whatever guidance and feedback is provided, it should be directed where possible to individual students.

Based on my experience as a parent to my sons, who have both engaged with online learning and are in elementary school and high school right now, I have some further recommendations.

  1. Actively engage the learning guardians of students in the process of learning.

    My sons’ experiences have been far more productive when we have sat down with them while they work through the online course material. This does not mean that we do the work for our children, but rather than we are there to support, encourage, and nurture their development as learners.

    It will be helpful to offer explicit advice for how learning guardians can support their learners, especially given the range of knowledge and experience those learning guardians will bring to the task. You may even want to include videos of what class looks like and descriptions of instructional routines that learning guardians can use with their learners.

    Also, offer suggestions of activities to learning guardians that they can do with their children in their care that are not on a computer and do not require the learning guardians to be experts in any particular subject matter.

  2. Skip watch-the-video-then-fill-in-the-blank-spaces activities.

    I can say from experience that these types of activities are ubiquitous in online learning and result in nearly no learning. I watched my son listen to a video in one tab while dutifully recording the answers in his worksheet in another. I quizzed him 5 minutes later and he could remember literally nothing at all from the worksheet or the video.

    Given that completing these particular courses was a requirement at his school, I taught him a much more productive learning strategy. First, attempt the worksheet and fill in every blank, even if one has to guess. Next, watch the entire video without writing or doing anything else. Now go back to the worksheet and change as many of the answers as one can without going back to the video. Rewatch or listen to the video with the worksheet and change answers as necessary. This is still a terrible experience but it at least has the possibility to result in some learning.

  3. Provide devices for students to work if at all possible or at least ensure that any online learning activities can be completed with a smartphone.

    While access to computers and the Internet keeps increasing, there are still households that do not have access and so providing equitable access to resources to all families is a key responsibility of schools, particularly when expecting students to engage in online learning.

  4. Where possible, engage students in synchronous activities rather than asynchronous activities.

    One of the more successful online classes my son took was with the Art of Problem Solving. Each week my son met with the entire class in an online chat program where the teacher mostly posed questions and occasionally told the students information, while the students responded to the questions in the online platform. He also had a physical textbook, a bank of unlimited practice problems to work on, and challenging problems to complete each week. The chat program was nothing amazing, but it mostly kept my son engaged for the full 90-minute sessions.
  5. Use simple assignments that do not require students to navigate complex instructions.

    Even with assignments with simple instructions, there is a lot of potential for student learning. Given that your students will be working remotely and with limited direct support, you don’t want students spending too much of their time figuring out what they are trying to accomplish.


What other recommendations for teachers and schools who may be suddenly engaged in online learning do you have? What question do you have that I have not yet answered?

Why Is a Negative Times a Negative Positive?

There are different possible answers to this question, depending on the standard of proof one needs and the background knowledge one brings to the question.


Mathematical consistency and patterns

Try solving each of these problems, paying attention to the previous set of problems as you do so. Look for patterns to make solving the problems easier.

3 × 3 = ?
3 × 2 = ?
3 × 1 = ?
3 × 0 = ?
3 × -1 = ?
3 × -2 = ?
3 × -3 = ?
2 × -3 = ?
1 × -3 = ?
0 × -3 = ?
-1 × -3 = ?
-2 × -3 = ?
-3 × -3 = ?

The answers to these problems are below but I really do recommend taking the time to solve the problems above on your own first, so you get the sense of how students might think through this set of problems.

3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
3 × 0 = 0

At this stage, many people will notice the answers are 3 smaller each time and the number being multiplied by 3 is one smaller each time, so they continue that pattern to answer the following questions.

3 × -1 = -3
3 × -2 = -6
3 × -3 = -9

Now, we decrease the first number in the pattern by 3 and one has to make some deductions about what the answer should be.

2 × -3 = -6
1 × -3 = -3
0 × -3 = 0

One might now notice that the answers are going up by 3 each time as we increase the first number, and so it is reasonable to continue this pattern.

-1 × -3 = 3
-2 × -3 = 6
-3 × -3 = 9

While to some this pattern may seem obvious, when someone is still in the middle of learning this concept, they have less cognitive capacity available to accomplish the task at hand (multiplying numbers together) and accomplish the additional task of looking for patterns in their answers, so this is where someone else prompting them to stop and look for patterns in their work so far will be very useful.

Prerequisite knowledge: One has to know what these symbols mean, what is meant by finding one number times another, and how negative numbers work in terms of counting down and subtraction.


Mathematical consistency and mathematical properties

Let’s look at a problem that we can do in more than one way, borrowed from the Khan Academy.

5 × (3 + -3) = ?

If we add the numbers inside the parenthesis first, then this is 5 times 0 which is 0, since 3 + -3 = 0.

5 × (3 + -3) = 0

But what if we distribute 5 through both terms first?

5 × 3 + 5 × -3 = ?

Since distributing the 5 across the addition does not change the value of the expression, we know this is still equal to 0.

5 × 3 + 5 × -3 = 0

But this means that 5 × 3 and 5 × -3 are opposite signs, so since 5 × 3 = 15, then 5 × -3 is -15. Let’s look at another example.

-5 × (3 + -3) = ?

We know that this is the same as -5 times 0, so this has a value of 0.

-5 × (3 + -3) = 0

Similar to before, we distribute -5 through both terms.

-5 × 3 + -5 × -3 = ?

Again, the distribution of terms does not change the value of the expression on the left-hand side of the equation, so the result is still 0.

-5 × 3 + -5 × -3 = 0

We know from before that -5 × 3 is -15 so we can substitute that value for -5 × 3 in the left-hand side of the equation.

-15 + -5 × -3 = 0

Therefore -15 and -5 × -3 are opposites since they add to 0, so -5 × -3 must be positive.

Nothing in what we did for the two examples above is specific to the value of 5 × 3, so we can repeat this argument for every other multiplication fact we want to derive, so these two ideas can be generalized.

Prerequisite knowledge: One has to know what these symbols mean, what is meant by finding one number times another, how the distributive property works, and how negative numbers can be defined as the opposites of positive numbers.


Representation on a number line

Imagine we represent multiplication as jumps on a number line.

3 times 3 on the number line

For 3 × 3, we draw 3 groups of 3 moving to the right. Both the number of groups and the direction of each group are to the right.

But what about 3 × -3? Now we have 3 groups of the number still, but the number is negative.

3 times -3 on the number line

If we find -3 × 3, the size and direction of the number we multiply are the same, but now we are finding -3 groups of that number. One way to think of this is to think of taking 3 groups of the number away. Another is to think of -3 times a number as being a reflection of 3 times the same number.

-3 times 3 on the number line

So -3 × -3 is, therefore, a reflection of 3 × -3 across the number line.

-3 times -3 on the number line

In one sense though, this visual argument is just mathematical consistency represented using a number line. If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers as being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection.



Karen Lew has this analogy.

Multiplying by a negative is repeated subtraction. When we multiply a negative number times a negative number, we are getting less negative.

This analogy between multiplication and addition and subtraction helps students nicely connect the two concepts.

Joseph Rourke shared this context.

A gambler loses $10 per day. How much more money did they have 5 days ago?

Here, the loss per day is one negative and going backwards in time is another.

@M_Teacher_w_T shared this analogy:

“An enemy of my enemy is my friend.”

This aims not at the algebraic or arithmetic properties of numbers but more at the oppositeness of negative numbers.

Prerequisite knowledge: All contexts that build new understanding require students to understand the pieces of the context fairly well, so it is especially important to probe how students understand an idea when it is presented contextually.


Algebraic proof from first principles

From Dr. Alex Eustis, we have this algebraic proof that a negative times a negative is a positive.

First, he states a set of axioms that apply to any ring with unity. A ring is basically a number system with two operations. Each operation is closed, which means that using these operations (such as addition and multiplication on the real numbers) leads to another number within the number system. Each operation also has an identity element or an element that does not change another element in the system when applied to it. For example, under addition, 0 is the additive identity. Under multiplication, 1 is the multiplicative identity. The full set of axioms required is below.

Axiom 1: a + b = b + a(Additive commutivity)
Axiom 2: (a + b) + c = a + (b + c)(Additive associativity)
Axiom 3: 0 + a = a(Additive identity)
Axiom 4: There exists −a satisfying a + (−a) = 0(Additive inverse)
Axiom 5: 1 × a = a × 1 = a(Multiplicative identity)
Axiom 6: (a × b) × c = a × (b × c)(Multiplicative associativity)
Axiom 7: a × (b + c) = a × b + a × c(Left multiplicative distribution)
Axiom 8: (b + c) × a = b × a + c × a(Right multiplication distribution)

From these axioms, we can prove that a negative times a negative is a positive. I’ll reproduce Dr. Eustis’s proof below and include the reference to the axioms used. First, we prove that a = −(−a).

Corrolary 1
a = a + 0(Axiom 3 and Axiom 1)
a = a + (−a + −(−a))(Axiom 4 applied to −a )
a = (a + (−a)) + (−(−a))(Axiom 2 – the associative property)
a = 0 + (−(−a))(Axiom 4)
a = −(−a)(Axiom 3)

So now we know that if we introduce negative numbers a is equal to −(−?).

Corrolary 2
0 = a + (−a)(Axiom 4)
0 = (0 + 1) × a + (−a)(Axiom 3 and Axiom 5)
0 = 0 × a + 1 × a + (−a)(Axiom 8)
0 = 0 × a + (a + (−a))(Axiom 5 and Axiom 2)
0 = 0 × a + 0(Axiom 4)
0 = 0 × a(Axiom 3 and Axiom 1)

Proving that 0 = 0 × a is the kind of painfully obvious idea that hardly requires proof but it establishes a relationship between multiplication and the additive identity in the real numbers, which is not yet included in the axioms above.

Next, we prove that (−1) × a = −a.

Corrollary 3
a = −a + 0 × a(Corrolary 2 and Axiom 3)
a = −a + (1 + (−1)) × a(Axiom 4)
a = −a + 1 × a + (−1) × a(Axiom 8)
a = (−a + a) + (−1) × a(Axiom 5 and Axiom 2)
a = 0 + (−1) × a(Axiom 4)
a = 0 + (−1) × a(Axiom 3)

Now, finally, we can prove that (−a) × (−b) = ab.

(−a) × (−b) = (a × (−1)) × (−b)(Corrolary 3)
(−a) × (−b) = a × ((−1) × (−b))(Axiom 6)
(−a) × (−b) = a × (−(−b))(Corrolary 3)
(−a) × (−b) = a × b(Corrolary 1)

This last “proof” though is unlikely to justify that a negative times a negative is a positive for any students though. It’s the kind of thing which is a required level of justification for a mathematician interested in rigorous proof who would likely consider the other justifications “patterning” and not sufficient.

A critical idea of proof though is that the intended audience of a proof is left convinced that an idea is true, and so I posit that the algebraic “proof” presented here is no proof at all for almost everyone.

Prerequisite knowledge: While I went through and added the justification for each step of the proof that was missing, I needed a fair bit of fluency with the original set of axioms. I also needed to not lose sight of the overall goal and to be able to recognize the structure of each part of the argument and match that structure to the axioms.


A simpler algebraic proof

This algebraic proof from Benjamin Dickman is much simpler than going back to a proof based on the axioms of arithmetic.

a + (−a) = 0
a × b + (−a) × b = 0 × b
ab + (−ab) = 0

From this, we can show that ab and –ab have opposite signs and therefore that a positive times a negative is a negative. Using the fact multiplication is commutative, a negative times a positive is also negative.

Similarly, we can prove that a negative times a negative is a positive.

a + (−a) = 0
a × (−b) + (−a) × (−b) = 0 × (−b)
ab + (−a) × (−b) = 0

Since we know that −ab is negative, and the sum of these two terms is 0, therefore (−a) × (−b) is positive.

Prerequisite knowledge: The prerequisite knowledge for this proof is much less than the other one, but it does assume a fair bit of fluency with manipulation of algebraic structures.



Given that the goal of an argument that something is true is to leave the other person convinced of the truth of the argument, whenever anyone uses any justification, representation, or proof, it behooves one to check that one’s audience is left convinced.

Procedures or Concepts?

“What’s four times four?”, my son asks.

“Sixteen,” I respond and then add, “How do you know that’s true?”

“Hrmmm. I know four times four is the same as four plus four plus four plus four and that’s the same as eight plus eight, which is sixteen.”

During my career, I’ve found that mathematical ideas include procedures, concepts, habits of mind, and declarative knowledge. Even in the brief exchange above, I claim that all four of these types of mathematical ideas are being used.

I define procedures as a sequence of steps intended to be used to solve specific mathematical problems, concepts are ideas that can be used with different procedures, habits of mind are general problem-solving strategies, and declarative knowledge is that which is known to be true without reference to other ideas.

  1. Procedures: My son knows that if he wants to add 4 numbers, he first adds two numbers together, then the last two numbers together, and then these two results together.
  2. Concepts: My son knows that one definition of multiplication is repeated addition and uses this idea to transform 4 times 4 into 4 plus 4 plus 4 plus 4.
  3. Habits of mind: My son knows that if one is not sure how to solve a problem that one can often change it into a different problem that one can solve. In this case, my son decides to change the multiplication problem into an addition problem.
  4. Declarative knowledge: My son knows that four plus four is eight and eight plus eight is 16 without reference to other ideas. It is often the case that things that are currently declarative knowledge are based on procedures and conceptual knowledge learned earlier.

This is why I find arguments about whether we should teach children procedures or teach them conceptually confusing — it’s not possible to do one or the other, students are always learning some mixture of all four types of mathematical knowledge.

“Knowledge is not tiny bits that we can count and represent by numbers, but a network of logically interconnected ideas, beliefs, and generalizations structured so it can be searched and used to work out and evaluate new ideas.”
– Graham Nuthall, The Hidden Lives of Learners.

Observing Siegfried Engelmann

Here is a video of Siegfried Engelmann demonstrating the mathematical learning of some children he worked with for a period of 1 or 2 years. Engelmann is one of the co-developers of a program called Direct Instruction, not to be confused with its cousin, direct instruction.

At the beginning of this video, Engelmann is careful to frame the video as a demonstration of learning rather than a demonstration of teaching, but it actually ends up being both. Engelmann, being a caring educator, cannot help but do a bit of teaching in this video.



Lesson analysis:

First, Engelmann builds up student motivation by presenting problems as challenging. He sets up students with statements like, “That’s pretty good, but pretty easy! Wait til you see this one.” When a student responds with a correct answer, he responds with, “Yay!” and then asks everyone to repeat the question and the answer. This kind of interaction is an ongoing theme throughout the video.

The students do not raise their hands, nor does Engelmann call on any students individually. Instead students call out answers as soon as they have them, so that often many different students call out answers together.

Next, Engelmann chooses specific sequences of problems to ask or pairs of problems that make some idea more clear through their variation from each other. For example, here is one sequence of questions:

  1. What’s 8 plus 2?
  2. What’s 10 plus 2?
  3. 12 plus 2?

And here’s another sequence:

  1. 7 plus 2?
  2. 7 plus 3?
  3. 7 plus 4?
  4. 7 plus 5?

Choosing a sequence of problems to answer to elicit a particular strategy or idea is a reasonably common strategy in teaching, sometimes called a Number String or Problem String.

After nearly every question, the students respond by repeating the question and answer in unison, something they have clearly been taught to do. Underlying this strategy is presumably the belief that all children should be expected to participate in some way in class. Doug Lemov would call this, No Opt Out, and others have referred to it as holding high expectations for students.

Engelmann also circles back to previous problems students have done, increasing the connections students might make between problems. About 1 minute after students solved the problem “7 + 4 = ?” verbally, Engelmann writes the following on the board for students to solve.

7 + R = 11

While Engelmann does not make the connection between this problem and the one students solved earlier explicit, it’s not likely to be an accidental choice of problem, given how deliberately the problems appear to be selected through-out the video.

Students frequently can be seen to be counting to find the answers to the question, a practice Engelmann makes no effort discourage, in fact he prompts it as a strategy at one point during the video.

A child counting with their fingers

Throughout the video, the children are exuberant and excited to be participating in this lesson. I did not notice any times when children seemed “off task” or disruptive, except perhaps towards the very end of the session where some of the children appear a bit tired.

An excited group of children

At one point Engelmann makes a deliberate mistake, a common strategy used by teachers, and three of the children run up from their seats to correct the mistake for him, which ends up being part of the fun of the lesson. He also models how he responds to making a mistake and being corrected by the students — it’s no big deal.

Children correcting Engelmann

Engelmann pays careful attention to the language he uses. There are no tricks, no shortcuts, and no mathematical imprecisions. For example, in working through the example of 38 + 14, he is careful to refer to each digit by its place value meaning, so instead of saying 3 plus 1, he says 30 plus 10.

Engelmann asking a question

Engelmann asks many questions during the lesson. At one point, he asks, “Why can’t I erase the R here and put a 4?” which is likely intended to highlight the distinction between 54 and 5 × 4. Notice how he has both examples visible for contrast and that he is using an iconic gesture to emphasis what he is talking about with his question.

Engelmann asks, “How many squares would be in this shape?”

At one point, Engelmann draws a rectangle and marks it 3′ by 7′ and then asks students, “How many squares would be in this shape?” One of the student starts thinking out loud and says, “Okay, count by 3s. Count by 3s seven times.” The next problem he writes up he uses the same rectangle and relabels the sides 4 and c and marks the area as 24. He then asks students to figure out what c must be.

A “reverse” area problem from Engelmann

Two of the students call out 5, then another student calls out 6. Engelmann responds with, “Is that right?” as the students say 5 and then “6 is right!” when a student says 6. Engelmann then prompts students to “count by 4s how many times to get 24” as other students join him and repeat his instructions. As a group then, the whole class counts by 4. No students are singled out when they make mistakes, but no mistakes go without feedback either.

Near the end of the learning demonstration, Engelmann gives a fairly challenging problem.

The most challenging problem of the session

Engelmann waits 20 seconds before taking any action other than just watching the children think. Instead of telling students the answer or giving them a strategy, he changes the problem.

The comparison problem

Students figure out the revised problem, and then Engelmann puts the original problem back up on the blackboard. He then waits again, this time for 18 seconds. This amount of wait time is remarkable given that most of the evening, Engelmann usually never waits more than a second for students to respond. Finally a student responds, and Engelmann congratulations the student and shares their answer with the class.

The class ends by Engelmann thanking the students for their participation and shaking each students’ hands.


Additional observations:

  • Engelmann makes very little use of representations other than the symbolic although the students use their fingers as a tool many times.
  • During the video Engelmann does not tell students exactly how to solve any of the problems, although he is fairly leading in one instance. It may be that this prohibition exists to make it easier to demonstrate the learning of the students and it may also be that explicit explanations are actually rare in his teaching.
  • The pace is fast and variety of different arithmetic topics demonstrated through out the session is fairly large, but most of the numbers Engelmann uses are single or double digit numbers that students can still count with. The bulk of the mathematical work is on thinking about addition and subtraction in various ways with multiplicative ideas coming up as “counting by groups” or repeated addition.
  • Engelmann makes use of a number of strategies also recommended for teachers to use with inquiry-oriented instructional routines suggesting that these strategies may be lesson-type agnostic.


Watching this lesson was a very different experience for me than most of the direct instruction I have observed. In fact, the experience was so different for me as to leave me unsure what relationship, if any, exists between typical direct instruction and what was intended by Engelmann as direct instruction?


Here’s a video of my son practicing at the playground. Prior to this, my son had already spent about five minutes trying to accomplish the same task.


I have two questions now after seeing my son practicing at the playground:

  • How is this practice different than what practice typically looks like in math class?
  • How can we make practice in a math class more like practice at a playground?

Hands-on or minds on?

Children think with their minds, not with their hands. So when we assign activities because they are “hands on” it is more appropriate to think of these activities as being “minds on”. But of course, our minds are always on (to some degree) so now we want to specify, what exactly is it that is different about this task and what do I hope children will learn from this activity that they cannot learn without this activity?1 Or alternatively, what does this activity allow that another activity might not?

A child making a pattern using Unifix cubes.
Constructing patterns by hand is potentially faster and more efficient than drawing different patterns out so this allows children to more quickly iterate on their ideas.

For example, giving kids algebra tiles doesn’t mean that they will learn something important from using those algebra tiles. They might, but they might also learn the same thing from a structured activity using the area model.

A child playing with pattern blocks
What did this child learn during this free play activity?

My experience is that when I have vague hope that children will learn something from an activity that is related to the mathematics I want them to learn, they usually don’t.

Children using counters to keep track of quantities while working through a word problem.
These counters acted as a shared memory for the children while they solved a problem, allowing them to collectively keep track of their progress towards their goal.

To be clear, I see nothing wrong with students playing with manipulatives when free play is the goal. Sometimes having vague goals and experimenting with the use of manipulatives as a teacher is a good starting place for thinking about learning, but we should not conflate the ability of manipulatives to support learning with the actual learning students do as a result of using them.



Patterns and Algebra

Visual patterns are a way to introduce and extend students’ understanding of algebra and functions. As such, there are a number of principles for helping students better understand algebra by examining visual patterns that have the same structure as their algebraic counterparts.


Principle #1: We can use the visual pattern to give more meaning to the algebraic structure.

What do you see changing in the pattern below? What stays the same? How is this pattern related to algebra?

3 squares built from smaller squares, with the diagonals painted black
A typical visual pattern

In the pattern above, children are likely to describe the squares as growing from left to right. They’ll notice that the diagonals of the squares are shaded and that the squares are two longer each time.

Each of these observations can be described in terms of equations based on the term number. For example, a child might notice that each square has 4 arms and 1 center square shaded, so you might annotate the diagram to show the arms and the center square.

3 growing squares with the diagonals from the center circled in red and the center square circled in blue.
Annotating for structure

Another child notices that the length of each arm is one more than the term number and that the total number of black squares is 4 times this length plus one additional square. Algebraically, this is represented with S = 4(n + 1) + 1. We color the parts of this equation to correspond with the visual to make the connections more clear. By doing this, we assign meaning to each part of the expression based on the visual.


Principle #2: We can use visual patterns to justify algebraic relationships.

Visual patterns can also be used to give meaning and to justify that a particular algebraic relationship is true, beyond what is possible to do with pure algebraic reasoning alone or a single visual example.

One might start by giving students the following image and asking them what they notice about the image.

Growing squares where each square is larger than the square before it by an odd number.
What looks mathematically important in this visual?

Collectively, students will notice that there are five squares, each square is larger than the square before it, each square is composed of smaller squares, each square has the square before it embedded in the lower left-hand corner, the number of white squares added on each time is odd, and a whole of other mathematical and non-mathematical observations.

The observation that each square is embedded in the next square and that the number of white squares added each time is an odd number can be written as follows.

1² = 0 + 1 = 0² + 1 = 1
2² = 1 + 3 = 1² + 3 = 4
3² = 4 + 5 = 2² + 5 = 9
4² = 9 + 7 = 3² + 7 = 16
5² = 16 + 9 = 4² + 9 = 25

By starting with the visual, students can reason inductively that “each square is just the square before it plus an odd number” and then this reasoning can be represented algebraically as n² = (n – 1)² + (2n + 1).


Principle #3: Visual patterns can be used to help students understand some of the language used in algebra.

A growing pattern of algebra tiles, showing 2 missing single units in each square.
Each square needs 2 more units to “complete it”

I did not learn during high school why “Completing the Square” was called Completing the Square. It wasn’t until I started teaching the idea using a visual to represent the square1 that the language made sense.


Principle #4: Visual patterns can be used to distinguish between different algebraic functions2.

Look at the two patterns below. How is each pattern changing as it increasing? How are these changes different between the different kinds of visuals?

One pattern growing by 2 each time and one pattern multiplying by 2 each time.
How are the sequences similar? How are they different?

By using patterns we can more easily contrast the difference between y = 2x and y = 2x which in written form are far more similar the corresponding visual sequences.


Further resources and inspiration:


On Misconceptions

Misconception: a view or opinion that is incorrect because [it is] based on faulty thinking or understanding1; a wrong or inaccurate idea or conception2.

These two definitions for misconceptions vary slightly, but the gist of the definitions are the same — there are some ways of thinking which do not match the world as we know it.

When we examine children thinking closely, we find that thinking often differs from our own. But this makes sense given that children have different experiences of the world than we do and often have not experienced the parts of the world that we have.

What should a teacher do about misconceptions? Should teachers try to prevent kids from having misconceptions? Should teachers label children who have misconceptions as wrong? Is there any harm in labelling children’s ideas as misconceptions? 3

It’s clear that some ways we use language to talk about children cause harm. If I consistently use the words “low” and “high”4 to describe my students, then the odds are greater that I also associate low and high expectations for these groups of students, which is correlated with student learning5. Here the language is harmful because it over simplifies the relationship between children and their background knowledge and results in students learning less than they would otherwise be capable of learning.6

The most problematic nature of the idea of misconceptions is that it frames how we respond to children’s ideas.

  1. A child writes 2 × 3 = 5 when they meant to write 2 × 3 = 6. Why did the child do this? Maybe they were overwhelmed with the task or tasks they were working on and defaulted to a previous relationship they know. It’s not a misconception per se, it’s something the child could probably find for themselves if asked to look at their work again.
  2. A child looks at the two angles below and concludes that angle A is larger because the rays are longer. This definition of larger is likely to be entirely consistent with every other experience of smaller and larger for this child. This child is attending to different properties of the geometry than the one intended by the author of the question.
    Two angles, A and B, one with longer rays but smaller angle, one with smaller rays but a larger angle.
    Is this a misconception? Or is this an entirely consistent worldview based on a different world than their teacher? Do we say to the child, “No, that’s wrong,” or do we value the thinking this child did and consider how to increase the size of their world?

My preference when working with children is to avoid over-simplistic words and phrases to describe their thinking. By default, the word misconception assumes a deficit view of children’s thinking7 and ignores the great thinking children did to come up with their ideas. What I prefer to the word misconception is language that describes more precisely the varied ways of thinking that children have. While it is the role of teachers to expand the world view of children and we need language to talk to colleagues about our role, the language we adopt frames the conversations we have.


Humanizing The Mathematics Classroom

I can remember exactly where I had my first major mathematical discovery. We were driving up the highway toward the nearby town for a night out for dinner. I was sitting in the back of the car playing with sums of numbers in my head.

1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21

I noticed something strange about the sums with an even number of terms.

1 + 2 = 3 = 2 × 3/2
1 + 2 + 3 + 4 = 10 = 4 × 5/2
1 + 2 + 3 + 4 + 5 + 6 = 21 = 6 × 7/2

“That’s weird”, I thought. “Each of those answers is just the last number in the sequence times half of the number after it. I wonder if that works for the odd numbers?”

1 = 1 = 1 × 2/1
1 + 2 + 3 = 6 = 3 × 4/2
1 + 2 + 3 + 4 + 5 = 15 = 5 × 6/2

“Wow! It works! Let me try it for a bigger number.”

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 = 10 × 11/2

At that point, I cheered out loud and had to explain my discovery to my parents, who were now wondering what was going on in their back seat.

Two years later, I learned in my middle school math class that this specific relationship had been discovered by someone else before me. These sums I had been finding had a name, triangular numbers. I learned that an algebraic way of representing this relationship was Sn = n(n + 1)/2.

What I had not learned before this point was that real people invent mathematics. For all I knew, mathematics sprung up fully formed from the heads of my teachers. Mathematics was a god to me, my teachers were like Zeus giving birth to Athena, and I had no idea people were involved in the creation of mathematics at all.

There is an argument about what methods of teaching mathematics lead to students best knowing mathematics, but often lost in these arguments is what mathematics even is. Is mathematics a set of ideas? Is mathematics a process? Is it both? I spent most of my school career viewing mathematics as a set of concepts other people knew and would tell me. It was only in rare moments, like the one in the back of my parents’ car, that I got a glimpse of mathematics as something more.

It is possible to both teach mathematical ideas and mathematical processes. These two ways of knowing mathematics are complementary and not dichotomous. If one centers the ideas students have in class, then students learn both that their ideas matter and that they have mathematical agency.

But if one has never taught with the aim of centering student ideas, how does one get started?

Try watching this video of students engaging in an instructional routine called Choral Counting. The teacher starts with a quick discussion about pennies, and then the routine itself starts when the teacher gives students instructions on how they are going to count together. When you have time, watch the first six and a half minutes of the video.

You might be wondering how the counting-in-unison aspect of this routine centers students’ ideas, and I think that it’s a fair point to make. Note that the students do the counting and the teacher records what they say. While the thinking students do during this section of the routine is less obvious, that thinking enables students to have a count that they created together that they can analyze for patterns. At about 5 minutes and 45 seconds into the video, an individual student shares a conjecture and the class quickly tests this conjecture by continuing the count. By using annotation, gestures, and repeating what students have said, the teacher increases how much access students have to each other’s reasoning.

More importantly, these students get an opportunity at a fairly early stage in their mathematical careers to get a taste of the mathematical process of looking for patterns, coming up with conjectures and testing those conjectures while simultaneously building their understanding of place value in the decimal system.

It is unfortunate (and understandable, given privacy concerns) that this video shows only the actions the teacher takes and not the responses by students. There are some things I might do differently than this teacher but they and I agree on a critical point — students can behave as mathematicians and learn mathematics from the experience.

This next video example is truncated from a longer recording of a lesson from a middle school class and includes some thoughts from the teacher interspersed throughout the video. If you are interested, here is the link to the full-length video without the teacher commentary.

Contemplate Then Calculate: Grade 6 Telescoping Sums [14 minutes] from New Visions for Public Schools on Vimeo.


In this routine, Contemplate then Calculate, students are expected to delay any calculations until after they have stopped to look at the problem in some detail, a strategy that has a small amount of research to support it. A key difference between this routine and Choral Counting, besides the obvious differences in the structure and mathematical focus of the routine, is that Contemplate then Calculate has deliberate instructional strategies intended to support access embedded within the structure of the routine.

If efforts to humanize the mathematics classroom by surfacing the thinking of students do not include deliberate strategies to support the engagement of ALL students, then for some students this experience can be just as dehumanizing as many other math classrooms. If you do not feel included in the math classroom, then it cannot be a humanizing experience for you.

The most important idea here is that we can make the mathematics classroom a more inviting place and make mathematics itself more inviting by centering the ideas of students in the classroom. We can make this experience more humanizing for all students by using instructional routines that reduce typical barriers for entry for students and embed instructional supports within those routines that provide access to the mathematical thinking to all.

This post is part of the Virtual Conference on Humanizing Mathematics.