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.

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.