Developing Formulas (2)

"I deeply worry about a curriculum that pushes students to results and not let mathematics be the organic conversation it deserves to be..." James Tanton

When I began teaching, I (like many others) simply gave students the formulas, and some worked examples, and expected this knowledge to stick. 
It did not work as well as I wanted to. 
When I started showing why the formulas worked, students were far more likely to recall and use the correct formula. When studying area, we would learn about rectangles, then triangles, then parallelograms, then trapezoids (or, if you will, trapeziums (or, if you will, trapezia!)) before getting stuck into circles. Recently, I have been wondering if a more logical order would be rectangles, then parallelograms, then triangles, then trapezoids. Fundamental to all of this is learning why the area of a rectangle is length times width and the best way to get students to develop this idea is to consider arrays (as touched upon in this earlier post ) How I then get the students to develop the formulas for parallelograms, triangles, and trapezoids can be seen below. I must point out, that I do not do all of this in one lesson!

Usually at this point, I am pretty confident that most students will now understand why the formulas work. Part of a balanced Math program must involve putting this knowledge to practice. This practice should involve a good balance of closed questions (the standard text book ones where a diagram is given with different measurements given) as well as open questions e.g. a trapezoid has an area between 60 cm² and 70cm², what could its dimensions be? The question below came from Anne Yeager and I have used the question with many grade 7 and grade 8 classes. It has always generated a lot of different solutions as well as great thinking and discussion amongst the students as they decide which formulas to use and when.

James Tanton (whose quote appears at the top) provides some fantastic resources for Math teachers. In particular, I love his curriculum videos and his Mathematical Essays. Do check out his site here.

Developing Formulas (1)

“It is not the knowledge but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.” Karl Freidrich Gauss

Here in sunny but frigid Ontario, we occasionally hear and read blasts from a variety of sources about how our Math curriculum is useless because it is discovery-based. These sources often call for a return to the 'proven' rote-learning method of yesteryear. These claims are wrong on two accounts: firstly, there is no mention of 'discovery learning' in any of the Ontario Math curricula; secondly, if yesteryear's rote-learning methods were so effective, then we would have a generation of confident and capable mathematicians. I regularly speak with parents at Math Nights and can testify that most of them are mathphobic and attribute this to the rote-learning models they endured.
But there is one key word that is ubiquitous throughout our curriculum:

Develop

 

I see this word as an action that teachers must do with great intentionality to help students truly understand Math concepts. For example, I was in a Grade 5 class last week and the goal was to get students to develop the formula for the volume of a rectangular prism (or cuboid as we say in England). In the past, I might have given the students the formula, then a bunch of examples to copy and then a load of questions to do. I would then get frustrated when, weeks later, students used the wrong formula. Now I realise that because the students hadn't developed the formula, they didn't understand it. If they didn't understand it, they were more likely to forget it or confuse it with another formula. I saw this formula for the area of a triangle many times:

So in the Grade 5 class, we gave each pair of students twelve linking cubes and asked them to make as many solid rectangular prisms as they could, and to write down the length, width, and height for each one. As they went about making these we uncovered some great conversations (e.g. is a 2 by 6 by 1 prism the same as a 6 by 1 by 2 prism?). Most of the students wrote their results in a table and for those who hadn't it was easy enough to convince them why this was a good idea. I then asked them to add another column to their table and label it Volume. Any initial thoughts of  "Oh no, we have to make them all over again!" were quickly dispersed as they realised that the volume was always going to be 12. This gave us this table of values:
 

This allows me to give one of my favourite challenges:

"Right, Math is the study of patterns, and there is a pattern in this table that is waiting to be discovered. Discover it!"

It didn't take to long before Zade thought he had a rule that worked. I asked him to check that it worked for all cases. Before long every group had made the discovery but since Zade was the first to get it, the glory belonged to him:

 

What was powerful about this is that it allowed us to check to see if a 2 by 3 by 4 prism had a volume of 12 (as one student had mistakenly claimed). It was also neat to see that some students were able to connect this to previous knowledge (the area of a rectangle is length times width) and why this appears in this formula (it is the area of the rectangular base). This is important as it allows us to connect to future learning i.e. the volume of any prism is the area of the base times by its height.

It is perhaps important to emphasise that the students did not 'discover' this formula: they developed it with a lot of intentionality on the part of the teacher. It is also important to note that whilst it is great that they now understand what the formula for the volume of a rectangular prism is, they now, as part of a balanced numeracy program, need to practise this new found knowledge through a variety of closed and open questions.

I will follow up this post with how I have got students to develop other formulas in other grades.



Visualise, Verbalise, Verify

This year I've been learning more and more about the importance of Spatial Reasoning in Math. I've learned that Spatial Reasoning is multifaceted and that you can be very good in one area and not so good in another. For example, the one thing that I love to do more than anything else is hiking. Now when I first started hiking and reading maps, I relied solely on man-made features such as paths and cairns and trig points. I knew that the contours on a map represented height but couldn't look at these to see what the hills looked like. Then one day in the hills, something funny happened: I looked at the 2-D map

and in my mind's eye could clearly see the shape of the hills in 3-D:

As a hiker, this is very powerful and now I must say (blowing my own trumpet) that I am pretty awesome at reading maps.
Yet I still have to think very carefully about 'left' and 'right'!
So at a session run last week by Ontario's Ministry of Education, I came across some great advice to help improve Spatial Reasoning:

Visualise, Verbalise, Verify

For example, consider the views of an object as shown below:

Visualising gets a student exercising her mind's eye to try to build a mental image about what the structure might look like. Ideally this should be done individually.

Verbalising gets students describing what they have just visualised; it forces the students to reason, to use spatial and positional language, to communicate with words and gestures. Sometimes the students will agree, sometimes they won't (and this is more fun!)

Verifying is when we allow students to create the structure to check if they are right. In the past, I have jumped straight to this stage but now I realise the importance of the getting students to visualise and verbalise. What is nice about problems such as this is that once a student has proven that their solution is correct, we can ask: "Is there another solution?" For example:

Spatial Reasoning is malleable. I am convinced that if we can get our students to visualise, verbalise and verify, then their Spatial Reasoning will improve dramatically and this will have a knock-on effect on the Math understanding.

Name That Fraction

I came across a fraction misconception last week that I've never seen before (or to be more precise, I've always missed before). We asked students what fraction was shaded:

Most said 'one-sixth' apart from one student who was adamant it was a fourth. Initially, I thought she had miscounted the parts but upon further questioning it was apparent that this student also thought that the following were also fourths: can you see why?

 

 

Basically she is describing the fractions using ordinal or positional language. What was neat about this is that before I had a chance to challenge her thinking, her partner did it for me (referring to the first diagram above):

 

"Well if that's the case, you could count from the right side and it would be called a third and it can't be a fourth and a third!"

 



The original student was pretty adamant that she was correct though which got me wondering what experiences she might need to understand how we name fractions. Perhaps she never had an opportunity (or not enough opportunities) to split a shape into equal parts like this as shown in this previous post . After the lesson I realised that I could also have challenged her fractional thinking by bringing in spatial reasoning and asked her if each of the following are fourths:

Or by asking her to name each of these fractions (will she name them 'firsts', 'seconds', 'thirds' and 'fourths'?)

It got me thinking how important spatial reasoning is in helping students understand fractions.

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There are some other great ideas for thinking about how we can teach fractions in the Ontario Ministry of Education's latest document Paying Attention to Fractions. One of the ideas coming out of this is to avoid solely teaching fraction as a unit but rather to incorporate it as much as possible throughout other strands and subjects all year long.

Or to put it another way, Fraction Immersion.

Face the Facts: Math Minutes Cause Mathphobia

There are some things that I did when I started teaching which I definitely would not do now. At the top of the list is the Mad Math Minute. This might go under various names but essentially the idea is to do as many questions as you can in one minute. These questions were pretty much always calculation questions or recalling number facts. As a kid, I enjoyed these because I did well: I loved the ego-feedback that I got ('Top of the Class again'!). However, I don't think I ever learned anything new from doing these; I might have gotten fractionally quicker at recalling facts but never to the extent that it made me a better mathematician or a better thinker. To that extent, Math Minutes didn't help me. 

Now I am older and (I hope) wiser and I have spoken with so many teachers who have told me that they started to become Math-phobic when they began doing these Math Minutes. They tell me it's not that they didn't know the answers, but that the pressure of answering the questions quickly caused their brain to freeze. This led to low scores which led to low confidence which led to more nerves which led to more low scores and so on. No wonder they ended up hating maths.

Now my anecdotal evidence is one thing but it is backed up by credible research. Jo Boaler's excellent research points out that whilst these timed tests might have been given with the best intentions, the effect is that they lead to the beginnings of Math Anxiety for a lot of students. She refers to research from Sian Bielock that shows how that the stress caused by these tests impedes students' working memory- the area of the brain where we hold our Math facts! This is backed up in the book Learning to Love Math by neurologist Judy Willis. High stress, low interest situations results in a reactive brain (fight, flight, fear) that prevents effective recall of facts.

Curiously, those who lead the Charge of the Rote Brigade will never consider this compelling evidence.

This is not to say that students shouldn't practice Math though. The more they practice the smoother the recall. However, practice doesn't make perfect: practice of the right kind makes perfect.
Good practice, for example, might involve a game situation such as The Product Game which you can see me playing with my daughter below.

Indeed there are many board games and card games which allow students to use and practise their number sense (Monopoly, Yahtzee, cribbage etc.) One which I would certainly recommend is the excellent City of Zombies in which you must use your math skills to prevent a zombie apocalypse. When I see students try games such as these, I see them more engaged, more willing to take risks, and learning more. The opposite of what I see in a Math Minute.

So, if you permit me to use some Yorkshire bluntness:

Stop pretending: Math Minutes help no-one.

Always True/ Sometimes True/ Never True

I was reminded about how much I love this question today when I took part in a crowdsourcing of questions courtesy of Tracy Zager (@TracyZager) via Twitter and Google docs. I have given this type of questions to students from all grades  (K to 12) and what is great is that it gets them reasoning and proving from a very early age. Consider the following:

Addition makes a number bigger

 

Most students (especially in the primary, junior grades) will say 'Always true' and back it up with examples. Yet there will be some who wonder about what happens when you add zero? Does this make the number bigger? And intermediate students will then begin to reason that adding a negative number actually makes the number smaller. So the answer is 'Sometimes true'. 
Sometimes a question helps them broaden their understanding of math terminology:

Two identical triangles can be put together to make a parallelogram

I know some students will say 'Sometimes true' offering the case where two right-angled isosceles triangles join to make a square (in green below) which, they think, is not a parallelogram. Others might make what they think is a more obvious parallelogram (in blue below).

This gives us a great opportunity to learn why all squares are parallelograms (quadrilaterals with two pairs of parallel sides). This leads into an understanding of why the area of a triangle (½×base×height) is simply half the area of a parallelogram (base×height)

Perhaps my two favourite questions I got today were:

A solid that has a square shadow is a cube

A solid that has a circular shadow is a sphere

 

It immediately got me thinking about other solids that might have these shadows. Or what about if I reverse the order of each statement?:

A cube has a square shadow

A sphere has a circular shadow



So here, for your delight, are some other Always true/Sometimes true/ Never true questions:

• A rectangle is a square
• When you cut a piece off a shape, you reduce its area
• When you cut a piece off a shape, you reduce its perimeter
• Bigger objects are heavier than smaller ones
• The diagonals of a parallelogram are unequal in length
• Multiplication makes numbers bigger
• Division makes numbers smaller
• The sum of four consecutive numbers is a multiple of 4
• The sum of three consecutive numbers is a multiple of 3
• The more you roll a dice, the more likely you are to get a 6.
• The sum of two odd numbers is an odd number
• The product of an even number and an odd number is an odd number.

Next, Near, Far, Any.

"A Maths lesson without the opportunity to generalise is not a Maths lesson."

I love this quote from John Mason as it gets to the heart of what it means to think mathematically. Math is the study of patterns and the power of patterns is that they are predictable. Part of predicting a pattern is generalising a rule which we think we might have spotted. Now this is easier said than done, but doable it is providing we give students opportunities for them to generalise.
Last year, the Ontario Ministry of Education published Paying Attention to Algebraic Reasoning. When I did a word cloud of the text, it was neat to see that the word 'Generalizations' was very prominent.

For me, the biggest learning that I got from this document though was a continuum that has helped me guide students as they learn to generalise:

Next----Near----Far----Any

 

Basically, younger kids will start by saying what the next term in a sequence is. As they get better, they will be able to predict what a near term is (e.g. the tenth term). This will most likely be done by finding the in-between terms. Now we can get students to predict a far term (e.g. the fiftieth, the hundredth, the thousandth term). This will necessitate a move away from concrete into abstract thinking as now it will be cumbersome to find all the in-between terms. Now students are ready to generalise for any term.
So I was in a Grade 6 class recently and gave them this pattern to consider:

I wondered if they could tell me how many sticks and/or dots there would be for any term. They had no trouble giving me the next term and a near term:

At this point, they were pretty clear about describing how to find the number of sticks and noticed that you just add two to this number to get the number of dots. What wasn't clear is what you do to the term number to get the number of dots (i.e. add 3). So I then pushed them to think about the thousandth term:

Note how this couldn't be drawn out but they got the 1001 sticks straight away and soon had also told me that 1003 dots were needed. Had they made the link between the term number and the number of dots? I decided to really push the envelope and asked them to generalise for any (i.e. the nth) term. Straight away they said the number of sticks would be n+1 but they needed a bit more help with the number of dots being n+3. Still, not so bad as it was the start of the unit and with further exposure.
Now I was unsure if it was helpful just drawing the pattern on the board: is it better for students to have the pattern modelled with manipulatives at their desk? I was in another class, a split Grade 6/Grade 7 and we gave each student a set of three cubes that were linked together. We asked the students to put them on the desk and tell us how many square faces they could see. When we all agreed that it was 11 we then asked the Grade 6s to find out how many square faces are visible on a set of 10 cubes (near) and the Grade 7s a set of 50 cubes (far). Notice how we skipped the 'next' stage here. Very quickly they were able to use the cubes to help explain their thinking:

The Grade 6s then tried the Grade 7 question whilst the Grade 7s were asked to make an 'any' prediction. We got three possible answers for this:

n+n+n+2

(n×3)+2

3n+2

(Note: In both these cases, I basically gave the student the patterns but would not always do this: it is vital that students create and test their own patterns.)
It was interesting to see, as my colleague Christine pointed out, that the students who said the first answer may be thinking more additively than the multiplicative thinking of the other two answers. With more experience of these types of tasks, students will develop the skills that are needed to generalise.

(Update to original post):
I spent another lesson with the same class and actually showed them the next-near-far-any continuum. I challenged them to see how far along the continuum they could go for this pattern:

It was great to see that so many students are already using their spatial reasoning to help the generalising as best exemplified by this:

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The Ontario MOE's Paying Attention to Algebraic Reasoning can be found here.