From Pedagogy to Mathematics, part 2

This post is the sequel to From Pedagogy to Mathematics, Part 1: Why we can’t get there from here. Below is a very brief summary of that post:

From Pedagogy to Mathematics, part 2: ACCESS & DEMOCRACY

(c) 2021, George Gadanidis, Western University

We have reasons and beliefs for not giving students better access to mathematics. But we need to put these aside not only because they are wrong but also because this enterprise is not about our reasons and beliefs.

This is not about us.

This is about students who spend over 10 years in mathematics classrooms. They deserve access to the structure and beauty of mathematics ideas.

It’s also about democratic ways of being (more to come after the next section).


What does access to mathematics look like?

Two examples were shared in the prequel to this post:

  1. Moving from “parallel lines never meet” to “assumptions about parallel lines that lead to different geometries”.
  2. Moving from “symmetry as an attribute” to “symmetry as a transformation”.

A third example is offered below (adapted from the book Mathematics Grade 9 by George Gadanidis).


What should we bring to the classroom about surface area (SA) and volume (V) of solids?

A focus on calculation?
  • Develop & use formulas for calculating V & SA.
  • Calculate V and SA of various solids, including composite solids.
  • Find missing dimensions in problems where SA or V is known and some other measurement is missing.
A focus on relationships?
  • Investigate relationships between V and SA.
  • Learn how to calculate V and SA to collect data about the SA & V of elephants of various sizes.
  • Represent the data in various ways (concrete, diagram, table, graph) and look for patterns.
  • Tell a story of how SA and V change as dimensions change.
  • Tell a story of the meaning of V/SA and SA/V and their graphical representations.
  • Tell a story about the meaning of linear and non-linear relationships in the context SA, V and their relationships.
  • Tell a mathematical story about elephants’ big ears.

why do elephants have big ears?

[Adapted from Gadanidis, G. (2021). Mathematics Grade 9.]

People give all sorts of answers when asked “Why do elephants have big ears?”

  • Big ears help elephants hear better.
  • They use them to fan themselves, to keep cool.
  • They cover their eyes when something scares them.
  • Veins bring blood to the ears to be cooled.

Large ears do help elephants keep cool.

  • The body (volume) of an elephant generates heat.
  • The surface area of an elephant dissipates heat.
  • Ears increase surface area without adding much volume.

Do all animals that live in hot climates need big ears? Or, is there something — perhaps something mathematical — that makes elephants different?  

model an elephant AS a cube

Let’s see what patterns we notice when we measure the volume and surface area of elephants of various sizes.

But we have a problem: How do we measure the volume (V) and surface area (SA) of an elephant?

In such cases, mathematicians develop and use models that simplify the problem. For example, instead of measuring V and SA of an elephant, they may measure V and SA of a cube.

So, for the next little while, think of an elephant as a cube.

What geometric solid would you use to model an elephant? Perhaps a rectangular prism, or a cylinder, or a sphere, or … ?

A small elephant:

  • The dimensions are 1 x 1 x 1.
  • V = 1 cubic unit.
  • It has 6 square sides.
  • SA = 6 square units.

A bigger elephant:

  • The dimensions are 2 x 2 x 2.
  • V = 2 × 2 × 2 = 8 cubic units.
  • SA = 6 × 4 = 24 square units.

An even bigger elephant:

  • The dimensions are 3 × 3 × 3.
  • V = 3 × 3 × 3 = 27 cubic units.
  • SA = 6 × 3 × 3 = 54 square units.

look for a pattern

Use the table below to record measurements for the first 10 elephants.

  • What is the relationship between SA and V?
  • Which grows faster: SA or V? Why?
  • Notice how the ratios of SA:V and V:SA change.
  • What is the meaning of each of these ratios?
  • What is the effect on heat dissipation as the elephant (cube) becomes larger?

Why do elephants have big ears?

PLOT On a graph

Comparing SA & V

The scatter plots on the right compare the growth patterns of side length, SA and V for cubes for side lengths 1-10.

  • Which scatter plot represents SA? Green, red or blue?
  • How can you tell?
  • What is the intersection point of the red and blue scatter plots?
  • What is the meaning of the intersection point?

The above graph is produced by the Python code shown on the right.

Go to to view and execute the code.

  • Which part of the code calculates SA and V?
  • Which part of the code plots the data?
  • How are the data stored before they are plotted?
  • What part of the code determines the colour of each scatter plot?
Comparing ratios SA : V & V : SA

The scatter plots on the right compare the ratios SA:V & V:SA, for side lengths 1-10.

  • Which scatter plot represents V:SA? Red or blue?
  • How can you tell?
  • Both scatter plots compare SA & V. Why are the shapes of the scatter plots different?
  • What is the meaning of each scatter plot in relation to “Why do elephants have big ears?”

The above graph is produced by the Python code shown on the right.

Go to to view and execute the code.

  • Which part of the code calculates the ratios SA:V and V:SA?
  • Which part of the code plots the data?
  • How are the data stored before they are plotted?
  • What part of the code determines the colour or each scatter plot?


 Which will evaporate first?

 The containers on the right have the same amount of water. Which one will evaporate first? Why?

Most people choose the container on the left. They say that the water surface exposed to the air is greater, so it should evaporate first.

A few people think this is a trick question, and they pick the least obvious answer: the container on the right.

What do you think?

 Which will dry first?

Both sponges on the right are fully soaked with water. Which one will dry first? Why?

Most people pick the small sponge, as it has less water.

Some people pick the larger sponge, since it has a larger surface. The greater the surface area, the faster the evaporation.

A few people think it will be a tie. The larger sponge has more water but also more surface area for evaporation. The smaller sponge has less water but also less surface area for evaporation. The two properties (amount of water and amount of surface area) work against each other and create a balance or a tie.

What do you think?

Puppies in cars

Did you know that it is unsafe to leave a puppy or other small dog in a car on a hot summer day? 

Because puppies are small, and their surface area is large compared to their volume, they dehydrate quickly.

When we are hot we perspire. As the perspiration evaporates we feel cool.

This situation is like the two sponges we looked at earlier. 

Like the elephants we modelled, the small sponge has a greater SA:V ratio. This is why it dehydrates more quickly.


Educative experience

Dewey (1938) said that although “all genuine education comes about through experience … not … all experiences are genuinely or equally educative” (p. 25).

Educational experiences offer continuity: they “live fruitfully and creatively in subsequent experiences” (Dewey, 1938, p. 28).

Continuity of experience

Educative experience provides a continuity among past, present and future experience.

There is little continuity between “parallel lines never meet” and mathematics itself. Rather than living fruitfully in further mathematics, “parallel lines never meet” is something to be unlearned in order to make conceptual progress.


The continuity of mathematics experience creates access.

Continuity connects mathematics ideas conceptually, adding depth and understanding and space for wonder.


Democracy is typically associated with the right to vote.

Although the right to vote is very important, we should also consider what other aspects of our society are democratic in nature.

For example, consider the following statement:

American Library Association (

Can you imagine a library where the access to books was restricted based on age or perceived ability or a narrow reading of a curriculum document?

Can you imagine a classroom where the mathematics that students experience is restricted?

How democratic are our mathematics classrooms?

Pape Library
ACCESS -- A personal anecdote

I arrived in Canada in grade 6 and my family settled near Danforth Avenue in Toronto. I came across the Pape Avenue Library early on. I was thrilled when I discovered that I could borrow any books that interested me. 
Canada was so amazing! 

The librarian gave me a temporary library card and off I went to the stacks. I came back to her desk with a pile of wonderful books. I was devastated when she told me that I couldn't take that many on a temporary library card. She consoled me that soon I would be able to.
I read and read and read.

In the summer after grade 8, I came across a book called College Algebra. Its content seemed intriguing. The librarian looked at the book, she looked at me, and then smiled. 
I lost myself in mathematics that summer.

Sitting at my desk in grade 9 I wondered, "Why is the teacher holding back? Why is he not sharing the interesting mathematics connections I see?"
School mathematics seemed so shallow.


The following is a draft of design principles I will use to engage teacher candidates in learning how to create educative / accessible / democratic mathematics classroom experiences, in a course I am teaching this year.

1. Pick a mathematics topic

Pick a mathematics topic from the curriculum.

  • Start with a broad topic, like a strand: number, algebra, measurement, probability, and so on.
  • Research that topic, with an eye on the subtopics from your curriculum.
  • Look for ideas that capture your imagination and offer a sense of wonder.
  • Pick the most interesting idea as your focus.
2. Understand its conceptual structure

Research in depth the conceptual structure of your mathematics idea of interest.

  • What is its historical development?
  • What other concepts are connected? How?
  • Draw a concept map to show mathematics landmarks and how they connect.
3. Script a story

Script the most interesting mathematics you learned as a story — as a sequence of experiences.

  • The core ideas in good stories are expressed in ways to be accessible to a wide audience (low floor).
  • The story sequence offers conceptual surprise.
  • Surprise leads to opportunities to think, wonder and gain deep conceptual insight (high ceiling).


Is the design process outlined above pedagogy or is it mathematics?

It is definitely mathematics.

It is about understanding the conceptual structure of mathematics ideas in ways that may be communicated to others who are not experts in the field.

If we can’t tell a story that offers conceptual surprise and insight about what we know — if we can’t communicate a sense of wonder to a broad audience — then we don’t understand deeply enough.


Three examples have been shared in these 2 posts:

  • on parallel lines
  • on symmetry
  • on surface area and volume

You can see more examples in videos 4-15 at

Start by trying one that best relates to your curriculum.

But don’t stop there:

  • Use / adapt the above design principles to create your own math stories.
  • Engage students in also researching / creating such math stories.



Dewey, J. (1938). Experience and education. New York, NY: Collier Books.

From Pedagogy to Mathematics, part 1

From PEDAGOGY to MATHEMATICS, part 1: Why we can’t get there from here

(c) 2021 George Gadanidis, Western University


Imagine this scenario: You visit a mathematics classroom and you are excited to see student math work displayed on classroom walls, students working collaboratively on math tasks using a variety of materials, with the teacher monitoring progress and offering help and encouragement. Pedagogically, it looks like an ideal classroom. But when you join a couple of the groups and more closely observe their work you notice that the mathematics they are engaged with is shallow and incomplete.

What is the problem? How might it be addressed?


Schwab (1983) identified four commonplaces of education we typically address when we talk about curriculum, teaching and learning:

  1. teacher
  2. student
  3. milieu
  4. subject matter

Pedagogy – how we teach – can significantly impact the roles of teachers and students, and it can affect the teaching and learning milieu. However, pedagogy has much less effect on subject matter.

In this post I draw a distinction between pedagogy and mathematics: 

Although pedagogy is extremely important, it cannot improve, or only superficially improves, shallow mathematics we bring to the classroom.

To illustrate this distinction between how and what we teach, I offer two examples of the geometry we typically bring to classrooms.


What mathematics do we teach?

What mathematics do we bring to the classroom when we teach about parallel lines? 

For over 15 years, our mathematics teacher education program has included activities to disrupt prospective teachers’ conceptions of mathematics, as a form of math therapy (Gadanidis & Namukasa, 2005, 2007).

When we ask, “Tell me everything you know about parallel lines” prospective teachers typically share that “parallel lines are straight and they never meet.” A few also remember “something about transversals” and the angles that they create.

How might we teach these ideas?

A reasonable pedagogical approach may involve the following:

  • Using a math scavenger hunt, students identify in the world around them lines that are parallel, as well as transversals.
  • Students work in pairs to construct parallel lines and transversals, using paper and pencil or dynamic geometry tools, and identify, share and discuss patterns in angle measurements.

But the mathematics is still shallow 

About 2,300 years ago, Euclid tried to prove his Parallel Lines Postulate but was unable to do so. Neither were mathematicians that followed. It turns out that the idea that “parallel lines never meet” is not a theorem to be proven. It is an assumption and different assumptions create different geometries. 

This is roughly how Euclid looked at the problem:

We have a straight line and a point near the line. How many straight lines can we draw through the point that will not cross the first line?

Euclid wanted to prove that the answer is 1.  

If the answer is “1” then we live on Flatland (a.k.a. Euclidean geometry). 

If the answer is “0” then we live on a sphere, where “parallel” lines “meet”. If we take a globe and create a “triangle” using the equator and 2 lines of longitude,  the sum of interior angles can have many answers.

If the answer is “infinite” then we live on a hyperbolic surface.

More on parallel lines

Consider this riddle:

Molly steps out of her tent. Walks south 1 km. Walks west 1 km. Sees a bear. Runs north 1km, arriving back at her tent. How can this be? And what colour is the bear?

Not sure of the answer? 

See an interview on the theme of parallel lines with Dr. Megumi Harada of  McMaster University at  

See also animated video #11, Parallel lines parallel universe, at

Read the story Parallel Lines.

The effect of what we teach

Here is a comment from preservice teachers in our teacher education program, reflecting on their school mathematics experience (Gadanidis & Namukasa, 2007):

I feel like I was misled, misguided, told the half-truth about parallel lines. It is the first time that I have realised and felt that math isn’t just BLACK & WHITE and can cause quite creative outcomes and discussions. 

How is it possible that students learn parallel lines never meet when they live on a sphere?

Why would we think that the solution to shallow mathematics is better pedagogy when what is really missing is a good conceptual structure of the mathematics of parallel lines?


For millennia, symmetry was primarily used to describe the properties of shapes (Stewart, 2013), which is what most elementary school children learn about symmetry today (Healy, 2003). For example, the letter A has 1 line of symmetry and the letter H has 2 lines of symmetry.

Symmetry as a transformation

Over the last couple of centuries mathematicians have come to view symmetry not as an attribute but as a transformation that leaves an object apparently unchanged (Gadanidis, Clements & Yiu, 2018).

Looking at the image below, what transformation may have been applied to the square?

Once we see symmetry as a transformation, more interesting mathematics emerges. 

When the ideas of symmetry we bring to the classroom are deeper, more mathematical, they create opportunities for surprise and insight – for wonder.

In GRADES 2-8 classroomS

I’ve had the pleasure of working with “symmetry as a transformation” in several grades 2-8 classrooms. When students combine the 4 rotation symmetries of the square, they notice that the result is always a rotation symmetry.

The rotation symmetries of the square are a tightly-knit group. They are happy spending time just with one another.

Students also notice that the 4 reflection symmetries of the square behave differently.

“When I put 2 reflections together I get a rotation.”

“Let’s try it with 2 different reflections.”

“It’s a rotation again!” 

The reflection symmetries of the square also like to meet new symmetries when they transform one another.

Isn’t this wonderful?

different mathematical structures

When we rotate a square, all the rotation symmetries “live” on the same side of the square. The square never flips over.

When we reflect a square, the reflection “lives” on the other side. Reflecting a square twice (or any even number of times) brings it back to its rotation side.

Rotation symmetries and reflection symmetries are different mathematical structures.

You can also see this difference by labelling the vertices of the square 1-4 and recording how their number order changes. The image below shows the number order for each of the 4 rotation symmetries of the square: 0, 1/4, 1/2 and 3/4 turn.

See more about symmetries at

In a grade 6 classroom 

In a grade 6 classroom (Gadanidis, Clements & Yiu, 2018), after we worked with the symmetries of squares and triangles, we posed this problem:  

Imagine we put the 4 rotation symmetries of the square in a rectangle and let them move and bump into one another, like bumper cars. Also imagine that when they bump they transform one another using their symmetries. For example, the 90o rotation symmetry would turn the 180o rotation symmetry another 90o and transform it to a 270o rotation symmetry. What might happen if we let them bump  and transform one another in this way for a while?

As serendipity would have it, the first three student predictions were all correct but expressed differently:

  1. They will all become red
  2. They will all become 0o
  3. They will all become 360o

Why is this the case?

Try it yourself at

Would the same thing happen if we used the 3 rotation symmetries of the equilateral triangle?


Read the story Symmetry.


Teaching mathematics is not just about how we teach and how students learn. It is also about what mathematics we bring to the classroom.

We have become very good at pedagogy. We can walk into a classroom, or reflect on our own teaching, and recognize what works well pedagogically and what does not. 

We also need to be able to recognize and be concerned about the quality of the mathematics we bring to the classroom.


Sequel to this post

In the sequel to this post I offer a model, with an example, for recognizing and designing better quality mathematics that we bring to classrooms.



Gadanidis, G, Clements, E. & Yiu, C. (2018). Group theory, computational thinking and young mathematicians. Mathematical Thinking and Learning 20(1), 32-53.

Gadanidis, G. & Namukasa, I. (2005). Math Therapy. The Fifteenth ICMI Study: The Professional Education and Development of Teachers of Mathematics, State University of Sao Paolo at Rio Claro, Brazil, 15-21 May 2005.

Gadanidis, G. & Namukasa, I. (2007). Mathematics-for-teachers (and students). Journal of Teaching and Learning, 5(1), 13-22.

Healy, L. (2003). From symmetry as a property to reflection as a geometrical transformation: Evolving meanings and computational tools. Paper presented at the Third Computer Algebra in Mathematics Education Symposium. Reims, France.

Schwab, J.J. (1983). The practical 4: Something for curriculum professors to do. Curriculum Inquiry, 13(3), 239-265.

Stewart, I. (2013). Symmetry. A very short introduction. Oxford, UK: Oxford University Press.

paper-and-pencil computation

In a previous post, I wrote: Children need practice with a wide variety of basic computation skills.

One of the basic skills listed was: Paper-and-pencil computation.

But which methods of paper-and-pencil computation?

Let’s look at multiplication, as an example.


Most parents I meet learned the multiplication method shown on the right.

Here’s the sequence of steps:

  • 6 times 4 is 24
  • write the 4
  • carry the 2
  • 6 times 2 is 12
  • add the 2 we carried to get 14
  • write the 14
This method has some advantages:
  • every multiplication we do involves only single digits
    • you can multiply very large numbers by only knowing your nines multiplication table
  • it always works the same way
    • regardless of how big the numbers are, the process is always the same
  • it’s concise and efficient
    • if you have a job where you have to do a lot of multiplications every day, and you don’t have access to a calculator, this method is an asset
It also has some disadvantages:
  • it un-teaches place value
    • before learning to multiply, children learn that 24 is 20 + 4 (or that 24 is 2 tens and 4 ones)
    • using this multiplication method, children practice calling numbers by the wrong names
      • for example, when we say “6 times 2 is 12”, the 2 is really 20 and the 12 is really 120
      • similarly, when we say “carry the 2”, the 2 is really 20
  • the method gives the correct answer even if you don’t understand
    • in other words, you don’t have to have good number sense to make it work
    • which also means that practicing this method does not help develop your number sense
    • for example, I once received a package where on it somebody had computed 10 x 12, as shown at right
    • this is why when you ask adults to explain this method, they often can’t: they learned it as a set of rules without conceptual understanding
  • it does not have a mathematical future
    • because the method breaks math rules (calling numbers by their wrong names), you can’t make connections to other math children have to learn (unless that math also breaks math rules the same way)

Because of the conceptual disadvantages, some education jurisdictions are hesitant to teach such paper-and-pencil computation methods to young children.

However, this does not mean that all paper-and-pencil computation methods have these disadvantages.

Let’s look at an example for multiplication:

Multiplication v.2

Let’s multiply 6 x 24 using a different computation method, as shown at right.

Here’s the sequence of steps:

  • 6 times 4 is 24
  • 6 times 20 is 120
  • 24 plus 120 is 144
How this method is different:
  • it reinforces place value understanding
  • it is similar to the way most people multiply in their head
  • it is easy to explain
  • it has a mathematical future, as it correctly uses place value, expanded notation, and the distributive property:
    • 6 x 24 = 6(20 + 4) = 120 + 24 = 144
  • it gets students ready for algebraic processes: 6(y + 4) = 6y + 24
  • it also requires students to know, and practice, how to multiply with multiples of 10 (as in 6 times 20 is 120)
What about bigger numbers?

Don’t we eventually have to carry?

Let’s take a look at 9 x 24.

Here’s the sequence of steps:

  • 9 times 4 is 36
  • 9 times 20 is 180
  • at this stage, if you can’t see that the sum is 216, you can use partial sums:
    • add the hundreds to get 100
    • add the tens to get 110
    • add the ones to get 6
    • altogether, that’s 216

Is there Multiplication v.3?

There are variations to the above method.

Whatever paper-and-pencil methods we decide children should learn, we should ensure that they are understandable and have a mathematical future.

That is:

  • paper-and-pencil computational methods should build on children’s previous knowledge
  • they should prepare children for more complex mathematics

kids need a wide variety of basic computation skills

In another life, I was a district math consultant. In my first month in the job, I received a call from an elementary school principal, inviting me to a parent council meeting for 10-15 minutes to talk about the new curriculum.

I asked, “Would it be possible to have an hour, or an hour and a half, so we can do some math together?” The principal checked with his parent council and we scheduled a math night for parents at the school library.

To make a long story short, word got around and I started receiving invitations for parent math nights across the district. Sometimes the turnout was 20 parents. Sometimes it was over 200, with tables and chairs filling a gym. It was one of my favourite things to do!

I quickly noticed 2 things:

  1. Many of parents feared or even hated math.
  2. Quite a few commented on their children’s basic computation skills.

To get parents thinking about what basic computation skills might be, I did 3 warm-up activities with them:

1. Counting money

I brought some bills and coins and I laid them out on a table: a collection of twenties, tens, fives and coins.

I invited a parent volunteer to count the money. I asked them to touch the bills and coins as they counted.

Then I invited a second parent volunteer to check the first answer.

How would you count the bills and coins?

If you are a typical person (like the parents at the Math Nights), you would first count the $20 bills, then the $10 bills, then the $5 bills, and so on … finally counting the coins, with the smallest last.

I said: “Great! We agree on the answer.”

And I asked: “How many of you would add money this way?” Most parents raised their hands.

“But we have a problem,” I added. “The math is backwards.”

Most parents (if not all) learned in school to add numbers from the right, starting with the smallest place value.

But when asked to add bills and coins, their minds naturally added from the left, starting with the greatest place value.

This is interesting, isn’t it?

Parents noticed:

  • The calculation methods they learned in school are not always (actually, not often) the ones they naturally use in everyday situations.

2. All you need is a calculator (?)

Next, I pointed out that calculators were abundant and inexpensive. (Today, every cellphone and tablet has a built-in calculator. Or just type the question in your browser’s address bar and it will find you the answer.)

So I asked parents: “What do you think about not teaching children how to add, subtract, multiply and divide and just let them use a calculator?”

Their answer was that this would not be a good idea.

They explained that kids need to understand the math. With the calculator, they just push buttons.

So, I handed out calculators and asked them to solve 12 x 25.

I added, “But there’s a catch: you’re not allowed to use the #2 key. No cheating by doing this in your head. You have to get the answer using the calculator.”

Here are 3 of the calculator methods parents used.

  • 3x4x5x5 (multiply the factors of the numbers)
  • 6×50 (half the fist number, double the second one)
  • 6x5x5 + 6x5x5 (a combination of the two methods above)

Can you think of a different method?

Parents noticed:

  • They wanted their children to understand.
  • A calculator can be a tool to think-with.

3. On paper & in your head

Last, I presented parents with the problem, 16 x 24.

I asked 2 parents at each table to solve this using pencil and paper and the rest to solve it in their heads.

Before you read further, try calculating 16 x 24 in your head.

I asked the parents who used paper and pencil to share the methods they used. They hesitated to volunteer. When a few did share, we noticed that they used the same method.

When asked to explain the paper-and-pencil method, they had a very difficult time. They could describe how, but could not explain why. For example, “Why did you indent the second row?” Some parents became frustrated and said: “It’s just a rule!”

The atmosphere changed when I asked parents to share the mental methods they used to solve the same problem. There was a palpable energy in the room. Parents were eager to share their personal solutions. They praised others who came up with different methods. They expressed delight at methods that surprised them.

Here are some of the mental methods parents shared:

  • They multiplied 10 x 24, 6 x 20 and 6 x 4, and then added the three products.
  • Some multiplied 20 x 16 and then 4 x 16.
  • Some multiplied 16 x 25 and then subtracted the extra 16.
  • Some multiplied 4 x 4 x 4 x 6.
  • In some rare cases, parents used algebraic structures:
    • 16 x 24 became (20–4)(20+4) = 400 + 80– 80 – 16
    • or, more simply, (20–4)(20+4) = 400 – 16.

Parents noticed:

  • When using a mental method, they understood what they were doing and could easily explain it to others.
  • The paper-and-pencil method was more like using a calculator: pushing buttons to get an answer without much understanding.

Basic computation skills?

I’m old enough to know that I don’t have all the answers.

I do spend numerous days each year in elementary school classrooms, working with students and teachers. Here’s my opinion:

[bctt tweet=”Children need practice with a wide variety of basic computation skills.” username=”georgegadanidis”]

Some basic computation skills:

  • Paper-and-pencil computation.
  • Mental computation, with ability to think flexibly, creatively and playfully about numbers and operations.
  • Solving number-based puzzles with and without tools (like calculators, concrete materials, games, coding, etc).
  • Making sense of numbers and operations in big idea contexts, such as the one below from Grades 1-2 classrooms.

The lyrics in this music video come from parent comments, after children shared their learning at home. From a project supported by KNAER.

In search of discovery learning

And now, as adults, we want the same for our children, which I think is really another issue (having to do with the “good old days” and the older generation underestimating the younger generation), but here are some of my thoughts on all the talk on “discovery learning”:

  1. Ontario education. Over the last 10 years I’ve spent on average 40-50 days each year in elementary classrooms, and I haven’t seen “discovery learning”. I’ve seen guided investigations, where the teachers prompt and scaffold student thinking to help them understand concepts in new ways. And I’ve seen a variety of other approaches too.
  2. Discovery learning. Years ago, Constance Kamii, who studied with Jean Piaget, did a lot of research on students discovering solutions rather than being told how to solve math problems. In one study, she compared Grade 3 children from three different classrooms, who learned how to add sums like 7 + 52 + 186, in three different ways: 1) by learning the addition algorithm you and I learned in school; 2) by not being told how to add and having to discover how to add by themselves; and 3) using a mix of methods 1) and 2) (for example, see this paper). Interestingly, on standard paper-and-pencil tests, the “discovery math” children did the best and the “let’s learn the algorithm” children did the worst. It was also interesting that when the “discovery math” children gave wrong answers, their answers were close, while the wrong answers of the “algorithm” children would sometimes be off by thousands (showing that they had not developed any number sense).
  3. Right vs left. You wouldn’t want to push analogies too far, but if we stereotype a bit, Kamii’s discovery learning approach can be seen as belonging to the right of the political spectrum. Like building a business from scratch, she asks children to construct their own, individual understanding of what addition is. The way I was taught math in school was more like a hand-out: my teacher gave us “the” way to add, ready-made.
  4. Some of this, and some of that. I think we need a balanced approach. Some of this, and some of that. Even if we think we know the “best” way (which I don’t). Children spend a lot of time in math class and need variety, across the spectrum of “tell” and “discover”. Most of all, we need to appreciate the wonderful minds of young children.

If you want to see some Ontario math classrooms in action, take a look at these lesson studies on Repeating Patterns in Grades 1-3.

To code, or to model, that is the question!


There is a growing momentum in education to engage K-12 students with computational thinking. At the same time, there is a gap between coding (as an end in itself) and authentic computational modelling practices of scientists and professionals to solve real-world problems and build knowledge – to learn – through computational “conversation” and “interaction” with their field (Barba, 2016), “with and across a variety of representational technologies” (Wilkerson-Jerde, Gravel and Macrander, 2015, p. 396).


Our societies are growing in complexity, in big part because of the intertwining connections afforded by new technologies. The use of computational tools to model phenomena, processes and relationships is becoming a prerequisite to scientific progress and economic success, as evidenced by the emergence of numerous computational modelling fields, such as computational biology, computational mathematics, computational finance, computational medicine, to name a few examples.


A focus on computational modelling in education, which is not isolated but integrated with curricular subjects, not only prepares students for future success: it also provides students a powerful learning tool with which to design, test and refine conceptual models and build powerful understandings of what they are studying .


The SSHRC Computational Thinking in Mathematics Education Research Partnership and the  Computational Thinking in Mathematics Education Community of Practice, which is part of KNAER‘s Mathematics Knowledge Network, have been working in the direction of computational modelling in collaboration with the Wellington Catholic District School Board, the Robertson Program at the Eric Jackman Institute of Child Study, and St Andrews Public School.


Barba, L.A. (2016). Computational Thinking: I do not think it means what you think it means. Blog post, retrieved 6 January 2018 from

Wilkerson-Jerde, M.H., Gravel, E.G. & Macrander, C.A. (2015). Exploring shifts in middle school learners’ modeling activity while generating drawings, animations, and computational solutions of molecular diffusion. Journal of Science Education and Technology 24, 396-415.

Educating young mathematicians (#4): Our greatest asset

Our greatest asset

Trigonometry in Grade 3

Related resources

Gadanidis, G. (2012). Trigonometry in grade 3? What Works: Research into Practice, Research Monograph #42.

Video Transcript

Our greatest asset, and our greatest opportunity in mathematics education, are the young mathematicians that we work with, and their amazing minds.

Chances are that we all carry some negative experiences with us from our mathematics learning in school. We have to be careful not to pass this on to young children.

Underestimating children

It’s quite common in our society for adults to underestimate young children. In fact, this is a historical pattern, dating back thousands of years.

The ancient Sumerians wrote on clay tablets to complain about their new generation: that young kids didn’t pay attention, they didn’t have a work ethic, they weren’t polite, and generally were a disappointment. Every generation that followed has said something similar, including our generation.

It’s not easy for us to create an education system worthy of children’s incredible minds, if we underestimate them.

Can young children think abstractly?

Jean Piaget brought to us the wonderful idea of constructivism: that young children develop an understanding of mathematics and other ideas from the inside out. They develop their own understanding as opposed to having it ready-made, transferred to them.

Jean Piaget also brought to us the idea of stages of cognitive development. And he said that young children are concrete thinkers and they develop their capability to abstract later on, maybe around age 12.

Seymour Papert who worked with Jean Piaget disagreed. He said these stages you’ve identified are not in children’s minds, they are in the learning cultures that we create in classrooms. They are a symptom of the way we educate them, and not their potential.

Kieran Egan said, young children can’t abstract? How could that be possible? Because if they can’t abstract, how would they ever develop language?

When we develop understandings of words like table, chair, dog, we look at a variety of these objects in the world around us, and extract the essential characteristics.

For example, there are dogs that are big, small, different colours, different dispositions, and we extract the essential characteristics, and with that abstraction that we create, we then look around the world and identify them, and distinguish them from other things that look like them.

So, young children naturally abstract at a young age, to be able to develop understanding of language.

A thirst for mathematics

Young mathematicians thirst for mathematical ideas, for big ideas, for surprises, and for mathematical insights.

And when they share these ideas with their parents, we get incredibly positive feedback.

For example, working in a classroom where the teacher wanted to cover the content of bar graphing, we developed an activity where students would measure the height of time, measuring the height of hours on a clock and graphing those as a bar graph, and noticing the patterns in them.

And a related activity is having a yellow dot on a car tire, and imagining what path that dot would travel as the car moves forward. What would that graph look like?

Children created comic strips of how they might share these ideas with parents. And then they shared those when they went home, along with some of the other artefacts.

We took parent comments, we put them into a Word document, we organized them into themes, we took out any repetition, and then we made that into a song. And then children sang that song back to their parents.

And we’ll end this series of videos with giving you an opportunity to listen to the children, singing to their parents.

The music video, Dots, Clocks & Waves, plays at the end of this video. Here are the song lyrics:

Dots, clocks and waves

my daughter explained
how to conduct experiments
and make bar graphs
plotting the results

she was amazed
by the wave pattern
excited to explain it
to her brothers at home

a dot on a car tire
makes a wave pattern
at first I thought
it would be a spiral

the wave pattern
is still there
even if the wheels
even if they are square

it’s great to see my son excited 
about school and about math
it’s great to see enthusiasm 
and interest in school math

my son enjoyed
testing his hypothesis
he was surprised
surprised by the result

he shared his comics
of what he learned
about math waves
on tires and clocks

the height of every hour
on a grandfather clock
plotted on a bar graph
makes a wave shape

like the height of a dot
on a rolling tire
or seasonal temperatures
or sunrise and sunset times

it’s great to see my daughter excited 
about school and about math
it’s great to see enthusiasm 
and interest in school math

Educating young mathematicians (#3): Five As for coding + math

Five As for coding + math


Computational Thinking in Math Education Community of Practice @ Math Knowledge Network

Gadanidis, G., Brodie, I., Minniti, L & Silver, B. (2017). Computer coding in the K-8 mathematics curriculum? What Works? Research into Practice: Research Monograph #69.

Gadanidis, G. (in press). Five affordances of computational thinking to support elementary mathematics education. Journal of Computers in Mathematics and Science Teaching 36(2), 143-151.

Video transcript

Today, there is a lot of pressure in our society to have young children to learn to code: from industry, from government, from academics in computer science, from non-profit and from for-profit organizations.

Coding is not new in education.

Forty, fifty years ago, Papert and his team developed Logo, which was an environment for young children to learn to code.

The difference between today and what we did in the past, is that we used to have a strong connection to mathematics. When Papert developed Logo, he said, learning math in Logo is like learning French by going to France.

Coding and math are complementary subjects, and they are easy to integrate.

Coding on its own, is one more thing to teach in an already crowded curriculum.

Coding and math are a natural fit. In fact, coding offers 5 important affordances that help us teach mathematics better.


The first affordance is agency.

When children are working in a coding environment, the world is open to them. They can explore the activity that the teacher gave to them, but they can also ask what-if questions and try other things.

For example, when I was working in a grade 3 classroom, and this was the first time that this classroom did coding and mathematics, one of the young students came up to me and said, “I’m not going to do what you did on the screen, I’m going to do something of my own, and then I’m going to come afterwards and say to you, watch this!” That’s agency.


The second affordance is access.

The beauty of the coding environments we have today, and the one that Papert developed with Logo, is that it has a low floor and a high ceiling.

You can engage with coding, with minimal prerequisite knowledge. At the same time, you have the whole coding environment available to you, so you can do very complex programming as well.

So you can explore mathematics ideas that are quite simple, or quite complex. This is what we mean by differentiated instruction. Students can enter at their level, and work to their potential.


The third affordance is abstraction.

When you want, for example, to teach the computer to draw a square on the screen, you need to know what a square is, you need to know its essential characteristics.

You can put the block in of walk forward 100 steps, turn right, and put a loop around it, do this 4 times, and that would create a square. So what you’ve done there, you created the essential characteristics of a square, and you can even take that and create a new block called draw square.

And then you can take this block and use it as an object of other things you write, so you can put it in another repeat loop, you can make it turn around, you can make it spin, you can make it do a variety of patterns. So what coding does is not only does it help you create abstractions, but those abstractions have a concrete feel, because they are objects on the screen that you can manipulate.

Making mathematical abstractions have a concrete feel, also makes them more accessible to young children.


The fourth affordance is automation.

Normally what you would do if you were working with pencil and paper, you would draw one sketch out and then you want another variation of that, well you have to draw it again.

But in a coding environment, you have your code, and you can say well I would to do this a little bit different, and change one little parameter, and then all you have to do is press on start and then it does it for you.

You’ve automated that process, you can try it over and over, you can show it to others, you can ask what-if questions, change this, see what happens, all very quickly. So it allows for that dynamic modeling, and that exploration, that normally you wouldn’t be able to do.


The last affordance is that of audience.

Audience is very important for learning for young students because they love to share ideas, they love to see the ideas of others, and you can do this very easily in a coding environment. It’s digital so you can share with parents at home, you can share with peers, and you can access the ideas of others and build on them.

Video ends with animated clip of Talk Math to your Computer.

10 affordances of computational modelling

Components of the above 5 affordances described above may be subcategorized to create a list of 10 affordances that come into play when computational tools are used to model mathematical concepts and relationships:

Educating young mathematicians (#2): What did you do in math today?


Math that feels good: A model for math education reform

Interview with Western applied mathematician Lindi Wahl

Video transcript

For over 10 years we’ve been working in classrooms in Ontario and in Brazil, collaborating with teachers to develop activities that will engage young mathematicians with mathematical ideas that will surprise them, that capture their imagination.

Our starting point has been to ask the question, How can we help?

So, we start with teacher needs and student learning needs, and then we wrap around those more complex ideas of mathematics.

For example, when teachers in grade 3 asked us for ideas for area representations of fractions, and how to teach that in different ways, we added around that content the concept that I described in the previous video, of infinity and limit.

So, students had lots of practice representing fractions as areas, but also they had opportunities to see that you can hold infinity in your hand.

We also take the time in classrooms to prepare students to communicate these ideas to others.

Our success is dependent on how well students can share these ideas in ways that will capture their parents’ imagination, that will offer them mathematical surprise and insight. We want to see that in the feedback that we get back.

Following this, is the animated video of Grades 1-2 students singing parents’ comments sent to their teacher, after students shared at home what they learned about growing patterns.

Educating young mathematicians (#1): Surprise me!

The first in a series on Educating Young Mathematicians.


“Infinity” interview with Western mathematician Graham Denham.

Gadanidis, G. (2012). Why can’t I be a mathematician? For the Learning of Mathematics 32(2), 20-26.

Video transcript


There’s this idea that good teachers make math easy to learn. Seems logical and common sense. Let’s consider an example:

Imagine sitting in a movie theatre, watching a movie. And, typically what you do is you try to predict; you try to guess what might happen next. Now, if the movie is easy-to-learn, your predictions will be correct. That might feel good the first or second time, but eventually that movie isn’t going to work for you.

We don’t pay for movies to experience our guesses, our predictions, to be correct. We pay for movies to be surprised, to flex our imaginations, to see new things that aren’t expected, and to think hard about things that happen.

When was the last time you were surprised mathematically? I don’t mean that your teacher dressed up as Pythagoras, and that surprised you. I mean mathematically, conceptually.


Surprise is not a frill. It’s not something we can ignore. Surprise is a biological necessity, actually. Humans enjoy creating, experiencing and learning from surprises. If that wasn’t the case, then our ancestors, a long time ago, would have been some wild animal’s lunch.

The word surprise would be a rare find in a math curriculum document. In fact, any curriculum document.

Let’s put it in there.

Although we don’t have a focus or a tradition in education of engaging young children with mathematical surprises, math is full of beautiful surprises.

Where odd numbers hide

For example, let’s take the first four odd numbers: 1, 3, 5 and 7. We can build them using block like these. There’s the first one, number 1. Number 3. Number 5. And number 7. Made out of 1, 3, 5 and 7 blocks.

If you ask children to make these and play with them, they will naturally put them together because of their sense of pattern and fit, and they will get something like this.

They’ll notice that this fits in there, like spoons, and this one fits in there, and this one fits in here. So, all those 4 odd numbers fit together to make this little square. That’s interesting, isn’t it? Because the first 4 odd numbers give you a 4 by 4 square. If we take the fourth odd number away, then the first 3 odd numbers, 1, 3 and 5, come together to give you a 3 by 3 square. The first two? The first 2 odd numbers give us a 2 by 2 square. So, there’s a surprise there, that odd numbers hide inside squares.

So, every time I see a square, let’s say this one, I wonder, how many numbers are hiding inside. What do you think?

This idea of odd numbers hiding in squares, meets the expectations in grades 1, 2 and 3, of patterning, of number sense. But really, it’s coming from grade 11 mathematics, where you study sequences and series, where you try to find the sum of the first n odd numbers or the first n even numbers, and you can see how this would help you to do that. So, what we’ve done, is we’ve taken those big ideas and offered a high ceiling and children can engage with them and develop abstractions. Because they can see that it’s not just the first 4: as long as you know how many of them you have, that’s really your answer. Ten odd numbers? It’s 10 by 10.

How to hold infinity in your hand

Another example would be fractions. So, working in a grade 3 classroom, if you want to give children a sense of fractions and their relationships, you might say, let’s try walking out the door.

And, walking out the door, well that’s quite easy, we’ve done it many times before. But what if we used fractions to walk out the door?

So, let’s say I start here, and I walk half way to the door first. And then, I walk half the remaining distance. That would be a quarter. And I walk half of the remaining distance. That would be one eighth. And then half the remaining distance, and half the remaining distance, and you can see that if I keep doing this it would go on forever, because no matter how many steps I take, there’s still a small distance, getting smaller but it’s still there, that I can take half of.

We can then ask children to use squares to represent those fractions using area representations. For example, using a square like this, you can divide it in half and shade this side one colour to represent one-half. Or you can divide it into 4 sections and shade this part to shade one-quarter, and so on. So now the question would be, what if you did that for say the first 5 fractions, 1/2, 1/4, 1/8, 1/16, 1/32.

And then you took those shaded parts and you cut them out with scissors. So you cut out the 1/2, and you cut out the 1/4, and you have those 5 fractions in front of you on your desk. Now imagine putting those fractions together, to make a new shape. Now remember, this is going on forever. So you’re constantly adding more pieces. How big would this shape be if you keep adding fractions to it? Would it fit in this room? Would it fit in your town or city? Would it fit on Earth? How can infinity fit in a finite space?

So what children say if you do this activity with them, is that the fractions fit in a whole. If you ask them what do you mean by a whole, they will say that if you take any one of those squares that you started with, the 1/2 fits in there, the 1/4 fits in there, the 1/8, the 1/16, the 1/32, and so on, all the fractions fit inside this square and they never leave outside, they never spill out. So there’s an understanding that an infinite number of fractions, fit in finite space, and their sum is 1.

This activity would cover the content of area representations of fractions, but it also gives you a sense of what you would study in Calculus, which is infinity and limit.

The mathematical surprise here is that you can hold infinity in your hand.

Video ends with Aboriginal recording artist Tracy Bone & Bob Hallett of Great Big Sea singing “Infinity in my Hand”.