Incorporating Indigenous Ways of Knowing Into Math

I recently spent two thought-provoking days at the Ontario Mathematics Co-ordinators Association's annual conference. The theme was on how we can explore mathematics through Indigenous knowledge systems and our two featured speakers were Isaac Murdoch and Jodie Williams. As often is the case with these conferences, I left with a head overfull of new ideas and thoughts that I needed to order and condense so that I might have a better chance to actually incorporate my new learning. Three things in particular stood out for me:

1) The Importance of Learning from the Land

Whether it was the Egyptians who needed to develop ways of measuring the agriculturally-rich land around the Nile fairly or the Polynesians who learned incredibly sophisticated ways of measuring astral movements and using these to navigate thousands of miles across the Pacific Ocean, so many areas of math originated from a need to understand the world around us. Isaac is a great story-teller and as I listened to his many experiences of learning off the land (navigating his way out of the bush, thinking for a long time about how he was to go about building a birchbark canoe, using the stars of the Plough to help in the design of a lodge), something suddenly dawned on me:

The land is a perfect place to develop and nurture curiosity. And curiosity (which I would think is a transferrable skill) is essential to mathematics.

When we immerse ourselves in the land, we begin to notice and wonder:

• why does moss tend to grow on one side of trees?
• why do the branches of some trees grow longer on one side?
• why do certain plants flourish on one side of my garden and not the other?
• is it true that the sun always sets in the west? 

All these noticings and wonderings are a precursor to a desire to understand different relationships, or to help see patterns. These are important mathematical traits.

2) The Importance of Physical Objects

Isaac also shared the importance of story-telling and how certain physical objects are used as a powerful aide-memoire. For example, the beads that were intricately arranged to form a necklace also revealed a family tree. Similarly the carvings on Isaac's memory stick each had immense significance such that you could sense the memories flowing through him as he held it.

It got me thinking about the tactile nature of such objects and how these are an essential part of learning and understanding math. I have written many times about the importance of the Concrete-Diagrammatic-Symbolic continuum and how often in math, we do our students a disservice if we jump to symbolic without giving them enough concrete or diagrammatic experience. This was most recently made clear when I saw students make good use of algebra tiles to multiply binomials.

I wonder how much students have missed out on these concrete experiences (especially in Math) when they were learning online and how much they would benefit from it now.

3) The Importance of Incorporating Indigenous Knowledge Systems into our Pedagogy

One of the important changes to the new Grade 9 de-streamed math course is the inclusion of how math has historically developed across all cultures. Representation matters, and if our students can see how their culture helped in the development of Math, then they will be more likely to see themselves as mathematicians. Jodie explained how Indigenous knowledge systems are not about learning different math, but rethinking the way that we explore and demonstrate an understanding of the concepts: it's not what we teach but how we teach. In terms of re-thinking our pedagogy, Jodie encouraged us to think about how to make things more experiential. We could:

i) Start with an experience of doing

ii) Encourage students to share their learning as they experience the 'doing'

iii) Once students have become familiar with the experience, then teachers can bring in the math.

In many ways, I see similarities between these ideas and those of Peter Liljedahl's Thinking Classrooms. As such, I think that there are many math teachers here in Ontario who are walking down this path already. I also think that when we create lessons where, for example, we nudge students into developing a measurement formula, then we are also aligning with this pedagogy. My own personal experience is that this always leads to better learning.

I am grateful to OMCA for arranging this conference and to Isaac and Jodie for sharing their wisdom. 

How Coding Revealed a Decimal Misconception

An interesting thing happened in a Grade 9 class recently. I was doing a coding activity looking at so-called Fibonacci numbers where students used a Scratch code to enter two initial values which then generated ten more values. The challenge was to get the twelfth term to be as close to 1000 as possible. I know that there is at least one solution that involves whole numbers but typically students get close to 1000 with two whole numbers and then use decimals to get closer and closer. One student tried 4.5 and 8.4 like this:

It was too low so she then tried 4.5 and 8.5:

This was too big so she asked 'What do I do now?' 

'Maybe pick a number between 8.4 and 8.5,' I suggested.

'But there are no more numbers between 8.4 and 8.5,' she replied.

As soon as she said this, I recognised a classic decimal misconception: sometimes students do not understand the density of numbers and that there are an infinite number of numbers between any two values. To help her rethink this, I drew a number line between 8.4 and 8.5 and asked her if she could now give me a value between these two:

'8.04... no, wait... 8.05,' she replied.

Using some virtual manipulatives, I reminded her that since one-tenth is equivalent to ten-hundredths, and four-tenths is equivalent to forty-hundredths, then 8.4 and 8.5 are equivalent to 8.40 and 8.50 respectively:

As soon as I relabelled these on the number line, the light bulb went on.

'Oh... I could try 8.45.... or 8.46 or 8.41!' This she did:

Now she was suddenly willing and able to use ever more precise decimals. 

'So I could now try 8.455... and then 8.4555 and keep going like that?'

So this one coding activity did more to reveal and then help correct this particular misconception than anything that I can think of that I have used in the past and at the same time gave great insights into the density of numbers (a new expectation in Ontario's new MTH1W curriculum). This particular coding activity occurred towards the end of the semester though so what I am now thinking is that it should be moved more towards the start of the semester.

One other thing about this activity: I noticed that some pairs of values added to give a curious next value. For example, in the first case above, the seventh and eighth terms, 89.7 and 145.2, add to give 234.8999... and not 234.9. I think that this is because the two values that are inputted by the user are converted to hexadecimal values which are then added to give the next value as a hexadecimal. This is then converted back to decimal but there is sometimes a rounding error as can be seen

Algebraic Expressions and Polypad

I tried this activity in Heather Lyon's MTH1W class using Polypad on Mathigon to help illustrate algebraic expressions. First Q: if this is x + 2 what would 3(x+2) look like? Students go to the VNPS to work in small groups:

Some puzzled looks at first but once they began sketching what it might look like, the students saw what was happening:

I could then easily use Polypad to confirm this:

I followed this up with 5(x+4) and we could now talk of more efficient ways of drawing our thinking. All the while I'm nudging them to the array model:

Now, I want to chunk the ones so give this question. Students' array models are now becoming more efficient:

We do this for a few more example. As they are feeling confident, I want to push them so ask them to sketch out what x(2x+3) would look like. Though they have never seen this sort of question before, they connect it to the model that they have just been using:

I can quickly confirm this on Polypad:

Some students choose to use the concrete algebra tiles but I can then show them how this connects to the diagrammatic model. One of these students says 'Now I see it... and that will save time!'

A few more examples follow and while I help a couple of groups, I give an extension to others: the answer is 18x^2-27 x... what is the question? They factor this successfully without any help from me:

They have been working hard so we take a little break and I perform a little mathmagic: think of a number... add 5, double it... add 8... half it... take away the number you first thought of. By using a little chicanery with a pack of cars, I show them the nine of hearts... the number they were all left with!! I then show how I used algebra to 'rig the system'!

With a little bit of time left, we decide to see if they can use this new knowledge to solve equations. I scroll to the top of my polypad to create an equation from the first two expressions we created:

And with no fuss whatsoever, they successfully solve it!

I was thinking how this approach differed to the one I would have used when I began teaching. Then, I would have given ten examples, no visuals, all symbolic and got the students to do more exercises. Now the students (thanks to Mathigon!) are doing the math!

The link to the polypad I used is here.

A Nice Algebra Puzzle

Last week I went into a Grade 9 class that had just begun to learn about simplifying polynomials by collecting like terms. I had an idea for a task that I thought would help them with this so began by showing them this pyramid.

I explained that the numbers in two adjacent squares add to give the number in the square directly above them. With this information, I split them into visibly random groups of 3 and had them work at whiteboards to find four numbers that go in the bottom row that would give 54 in the top square.

After completing this, I then gave them this pyramid:

I wanted to see how comfortable they were with collecting like terms before giving them something more thought-provoking. Some were able to complete this symbolically and others were happy to use algebra tiles to help their thinking:

I then gave them this task:

By now, I could hear how adept the students were at collecting like terms and was impressed at the different ways they went about solving the task:

Next, I gave them this task with the restriction that all the terms on the bottom row had to be different:

Again, I was really pleased how they went about solving this (some symbolically, some using algebra tiles again) and by listening to the students talk, I could tell that they were really understanding this. I even overheard a few groups say how much fun the task was!

As this had taken less time than I expected, I had to then think quick and come up with an extension. I asked them that if the value of the top brick was 57, what would the value of the four bottom bricks be?

This required a bit of clarification for two or three groups, but once they understood it, it allowed them to demonstrate their algebraic skills in solving equations and substitution:

It was great to see all the different solutions as well as see that in some cases, the different terms actually resulted in the same value once the substitution was made.

I finished by asking them to figure out:

a) the side lengths of an isosceles triangle, given that the perimeter is 120 cm and that the two equal sides are double the size of the other other side

b) the dimensions of a rectangle given that the perimeter is 1000 cm and that the length is triple the width.

For the first problem, they did this by trial and improvement, so I walked them through how to set this up algebraically. I was pleased to see them al use this approach for the rectangle problem.

I was really happy with the way they remained engaged throughout these tasks, especially as it was a Friday afternoon and would definitely use these again.

The 12 Days of Christmath: Day 12

 Here are the Day 12 puzzles for the 12 Days of Christmath challenge. Enjoy!!

Primary

Primary (French)

Junior

Intermediate