Monday, November 30, 2015

How Many Eggs Does it Take to Make a 3-Egg Omelette?: Understanding the Problem

One of the things that we are actively trying to do in our Board is to get our students to fall in love with problem solving. But how can we achieve this? Simply giving students problems to work on isn't enough; if the students don't know how to solve a problem, then what, as their teacher, do I need to do to help?

Early in my career, I would 'help' students by basically doing the problem for them. So when a student asked me, "What do I have to do?", then I'd reply , "Oh, easy. Just multiply the 45 by 213 then subtract the original 1234." And on to the next student. Then the next. And so on, rescuing as many as I could. Now I realise that all I was doing was telling them what calculations they needed to do: I was not developing their problem solving skills.
I am at the point now where, whenever I solve any particular problem, I think very carefully about what I am doing to solve the problem, and how I can model these actions to my students. George Polya's four-step problem-solving model has been valuable in helping me think through this. Polya lists the four steps as:
It is important to realise that Polya's model is not a linear model. Instead, it should be expected that we go back and forth between the stages in order to make sense of the problem, and to try different strategies and to optimise our solutions. 

In a previous post, I mentioned at how students think they look back at the solution when in fact they do not. What has been of greater interest to me recently is this:

Do students really understand the problem?

And if they don't understand the problem, what do we do? This came about initially from giving the following problem to some Grade 4 students:

A farm packs eggs into boxes of 6 and boxes of 8. Every box they pack has to be completely filled. There are a 100 eggs to pack. Show how this can be done.

As a class, we read the question. We re-read the question. We spoke about how we can pack into boxes of 6, or boxes of 8, or both. We agreed that we understood the question. And so the students set out to solve the problem.
Or not, as it turned out.
Some engaged very earnestly in long division (100÷6=16R4, 100÷8=12R4) but didn't know why they were doing this. Others still added 100+6+8 to get 114. One or two did 100−6−8 to get 86.
One student had written 11×8=88, 2×6=12, 88+12=100. 'Great!', we thought and asked him how many boxes were needed. "Ermm, one hundred?" came the reply.

There were some correct answers. One student used a hundreds chart really well to count by eights to 96 (12 boxes) then realised this was not going to give a solution so took one box of eight off to get 88 (11 boxes) then counted up by sixes from there to get to 100 giving a total of 13 boxes. Another realised that 2 boxes of six and one box of 8 would pack 20 eggs. Repeating this five times would mean 10 boxes of six and five boxes of 8 would pack 100 eggs (or 15 boxes in total). These students had made sense of the question.
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Another colleague drew my attention to a question that stumped some of her students:
How many eggs does it take to make a three-egg omelette?
Were these students not understanding because they were reading the question too fast and had missed the all important 'three'? Or were they wondering what an omelette was (let alone a three-egg omelette)? When I tried this in a different class, I was actually pleased that one student asked me what a three-egg omelette was: if you know what this is, then you can answer the question!
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Another way to see if students understand the question is to give a nonsensical one. The following one was first used by Professor Kurt Reusser in his fascinating 1986 research (and later used by Professor Katherine Merseth in her 1993 research) and also referred to by Robert Kaplinsky in this post. I wondered how these students would react to it:
There are 125 sheep and 5 dogs in a flock. How old is the shepherd?
Now these students are very much urban so I wanted to make sure they all knew what sheep were, what dogs were, and what shepherds were. They assured me that they did! So on with the problem.
We saw all manner of answers: 
  • 125+5=130
  • 125-5=120
  • 125÷5=25 (the most common)
  • One or two said a random number like 63 or 41 and I wondered if these students were guessing because they knew that it was impossible to tell and they just wanted to give an answer.
  • A few said that it was impossible to tell.
So what do we do with all this? Well, as a class, we asked them what information do we need (the shepherd's age) and then what information do we have that could help us get the age? At this point, most students began to realise that we had no such information therefore we couldn't answer the question. "It was a trick question!" one student asserted.
Yes. Yes, it was.

So after this class visit, myself and the three teachers I was working with began thinking about different types of questions that we could give our students. We came up with the following:

1) Not-Enough-Information Questions
A classic is 'How much does it cost to run a car for a year?' or 'How much does it cost to travel from Toronto to Halifax, Nova Scotia?' or Dan Meyer's Super Bear question. For this type of question, information is deliberately withheld from the students: they have to think about what the question is asking and therefore what information they need to get. Our thinking is that these tend to be open-ended questions.

2) Too-Much-Information Questions
These are questions such as 'How much does it cost a 40-year old to carpet a 10m by 8m by 3m room with square carpet tiles that are 30cm long and cost $5 each?'. Some students might realise that the 40 is a red herring but still might do something with the 3 in the 3 metres.

3) Just-the-Right-Amount-of-Information Questions
I sometimes call these 'Goldilocks questions', not too much, not too little, just right. One such example is 'If 2.4kg of potatoes cost $7, how much would 6kg of potatoes cost?'. These can be single-step or multi-step problems. Our feeling is that this type of question forms the vast majority of 'problems' that students are asked to do.

4) Self-Answering Questions
These are ones such as the 'How many eggs does it take to make a three-egg omelette?' from above, but also include: 'Which is heavier: a kilogram of feathers or a kilogram of stone?'; 'How long does a thirty-minute bus ride take?'; 'Who wrote Beethoven's Fifth Symphony?'; and one of my Dad's favourites which was 'If one-and-a-half herrings cost three ha'pennies, how much would six herrings cost?'

5) Nonsensical Questions
The 'How old is the shepherd?' is an example of these as is this one: 'A town was founded in 1847 and is at an altitude of 254m above sea level. What is its population?'

Part of the challenge in getting students to 'Understand the Question' is to get them to think critically about all the information that is given to them in the question. 

But what if students encounter nothing but 'Just-the-Right-Amount-of-Information' questions, especially if they are single step problems? 

Is there a real danger that our students will be conditioned to just take the numbers that are in the question and select an operation without doing any deep thinking? And if they don't do any deep thinking, how are we developing their problem solving skills? As part of our inquiry, we are going to make sure that we give our students  a variety of these types of questions. Then, if we ask the students right at the start 'Do you understand the question?', then we might get them thinking more carefully about the information that is contained within the question.

Our list is not meant to be exhaustive by any means, and it is probably the case that there are better ways of classifying problems already out there. We'd love to hear what you think about this and what strategies you have used to get students to understand the question.

Wednesday, November 4, 2015

The Forgotten Art of Decomposition

I have a belief that all of us are curious beings who have an innate desire to see how things work: to some extent or another, we will take something apart to figure out the inner mechanisms then reassemble it. Being able to take an object apart, and to then put it back together again tells us a lot about that object: its features, what it can do, what it can't do. This ability to decompose (and recompose) objects is fundamental to problem solving.

However, I think it is something that is often overlooked in Maths.
For example, look at this number, and think about how many different ways you see it.


How did you see it?
Four tens?
Eight fives or five eights?
Half of eighty? Eighty halves? 
Ten less than fifty? Sixty less than one hundred?
Two twenty pence pieces (in the UK)?
Four dimes? A quarter, a dime, a nickel (Canada and the U.S.)?

Or do you get more dangerous and think:
Four hundred tenths?
120 thirds?
The sum of two squares?
The sum of two prime numbers? A different sum of two prime numbers?
The sum of three prime numbers?

To me, being good at Math necessarily involves the ability to decompose and recompose objects. In number sense, this is much, much more than simply memorising facts. Now don't get me wrong: kids (and adults) do need to learn and understand their additive and multiplicative facts. But I worry that when students memorise these without making connections then we create situations where kids 'know' 7×5 but don't know 5×7. Or they know both of these but cannot do 35÷?=5. Or they cannot look at a number like 40 and figure out its factors.

We can help students learn how to decompose numbers by beginning with some concrete representations. Here, grade 1 students work with deciblocks to see 10 decomposed in a variety of ways.

This in turn can be backed up with diagrammatic representations:

We can also share and model decomposition strategies that we use. I ran a Math Night for parents in one of our schools recently and shared with them the following:

  • knowing that 9 is one less than ten means that adding nine is more simply done as 'add 10-1'
  • multiplying by 9 is more simply done as 'times by 10 subtract the number' so 9×27 is 270-27=243

Think how powerful this is: by decomposition, we have connected multiplying by ten to multiplying by nine. We can then connect these to multiplying by 11, or twelve (ten times plus a double) or multiplying by 20 (times by ten then double). The smiles on the parents' faces after I shared these strategies were great to see and we agreed that there is a big difference between memorising a fact and understanding a fact.

Even when learning facts, it is powerful to see four sixes as the same as two twelves hence 24. In fact whenever you multiply by four knowing this as doubling twice is very powerful. Arrays are fantastic tools to help students understand why facts. Consider this from Jordan Ellenberg in his book How Not To Be Wrong:

"Here's my earliest mathematical memory. I'm lying on the floor in my parents' house...looking at the stereo. Maybe I'm six. This is the seventies and therefore the stereo is encased in a pressed wood panel, which has a rectangular array of airholes punched into one side. Eight holes across, six holes up and down. So I'm lying there, looking at the airholes. The six rows of holes. The eight columns of holes...Six rows with eight holes each. Eight columns with six holes each.
And then I had it-eight groups of six were the same as six groups of eight."

Finally, a little puzzle: Suppose you are being paid mileage at 45 cents per km and you have driven 218 km. How could you work out your expenses without using a calculator or an algorithm? Scroll down to see my method.

50c/km would pay $109, 5c/km would pay $10.90.
Thus 45c/km would pay $109 - $10.90 = $98.10.

Monday, September 28, 2015

Still In Love After All These Years

One of the things that keeps me loving Maths is its ability to surprise me. A couple of weeks ago I posted this puzzle that I had created on Twitter:
I had a solution which I knew was correct but I wasn't satisfied with my approach (which was heavily algebraic). I wondered if anyone in the Twitterverse could provide a more visual approach. I am indebted to Carlos Luna (@el_luna) who shared with me his solution which I've shown at the bottom of this post (SPOILER: it has the answer!). He said he used Carpets Theorem to solve it.
Carpets Theorem? Now I maybe accused of being a touch naive but I have never heard of it. But a quick check online had me up to speed with Carpets Theorem and also faced with this most beautiful result of it:

What can you say about the areas A and B in this trapezoid? 

All these years that I've drawn the diagonals on trapezia and seen these triangles but never once considered how their areas relate. Well, Carpets Theorem says that the areas are identical! And the proof of it is such a thing of beauty I cannot share it with you because that would deny you the enjoyment.

I was gobsmacked, totally gobsmacked, and in the best of ways.

So why do I bring up this personal story? Imagine it from a student perspective: making a Mathematical conjecture and then suddenly seeing that it is true all the time. I was reminded of the importance of this this morning when a colleague tweeted the following:

Can you imagine the look on Noah's face when he clicks that 8+2 is the same as 2+8? It is still the same feeling that I got when I discovered that the two triangles are the same area in the example above. The teacher has deliberately created a task that will guide Noah into making this discovery himself. It is not always an easy thing to do but my belief is that the more we give students opportunities like these, the more they will become better mathematicians.

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Here is @el_luna 's solution: 

Friday, June 19, 2015

Balanced Math = Mastery

I must give a hat tip to the folks at Mastery Maths who always have a way of pushing my thinking. In particular,  this post raised a question that had been simmering away in my mind for a while: what do people mean when they talk about Mastery? 
There is often a lot of noise in various media that students should just focus on 'mastering the basics' and that problem solving is something that should be left to later on. The post above views Mastery as the ability to apply something in an unfamiliar situation. So if a student can do a hundred long multiplications or long divisions but cannot then see which of these to use when solving word problems, it would be illusory to say that the student has mastered multiplication and division. Sometimes I get asked if I think students should learn their number facts. I reply, of course they should but that I think that the wrong question has been asked: instead we should be asking 'How should students learn their facts?'. I like what the folks at Mastery Maths write: 

The challenge is developing skills and understanding concurrently...

In our Board, we have been doing a lot of work trying to get a better idea of what Balanced Numeracy is and what it looks like in a school. A few months ago, Dylan William tweeted this:

When I saw it, I immediately thought how this could be tweaked to give a Math perspective:
I think this still needs some work but I have shared this graphic with many colleagues in my Board and they have found it a useful way of helping to understand why a good Math program will not focus entirely on skills and drills nor entirely on problem solving: they must complement each other. 
In fact, I would argue that true Mastery can only come out of a Balanced Math program.
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Out of curiosity I gave the following computation (mentioned in the post above) to a group of my colleagues: 


Even though I gave this question to them individually with a cheeky but friendly grin, a lot of them admitted to feelings of anxiety: "Would I be able to remember how to do this? Could I use a calculator? What if I get it wrong?". Now these are adults who all told me that they learned Math in an algorithm-driven manner, yet they were not confident with this question. It turns out that they all multiplied the 36 and 175 first to get 6300 which they then divided mentally by the 63 to get the answer of 100. Some even chose not to use the algorithm that they learned at school but instead used a partial products-type method. 

I would argue that if we truly understand multiplication, we might use the commutative law to see that 36×175÷63 can be rewritten as 36÷63×175 or 4÷7×175. This can now be rewritten as 4×175÷7 which is 700÷7 or 100. Or, we might see that since 175÷7 is 25, then 4×175÷7 becomes 4×25 again giving 100. This ability to decompose and recompose computations and numbers is crucial in helping our students develop a mastery of Multiplication but will not happen if all they experience is the standard algorithm.
In fact, the whole notion of decomposing and recomposing objects is something that is crucial in getting students to master mathematics. 
But more on this in a future blog...

Thursday, June 4, 2015

Some of my Favourite Measurement Puzzles

In previous posts, I have shown how we can get students to understand various measurement formulas by developing these formulas with them. Once students understand these, it is important (in the context of a Balanced Numeracy approach) that students be given opportunities to practise using these formulas with straightforward, closed questions (e.g. find the area and circumference of a circle with diameter 10 cm). At the same time, it is also important that students are given opportunities to non-standard, open problems. This is where we can help students develop their problem solving skills and their perseverance. Over the past year or so, I've collected many great puzzles from the Math community on Twitter. Below, you will find some of my favourite measurement puzzles, mainly suitable for intermediate students. Try them yourself and then see if you can get each solution in a different, more beautiful way.  
Answers are in the back of the book.

Friday, May 15, 2015

Having Fun With Math Vines (or the Grapes of Math!)

Spatial Reasoning is such an important component of Math yet it is often overlooked. Too often, Math teaching rushes to abstract symbolic representations of certain concepts at the expense of pictorial representations. Recently, I've been working with my fellow Math consultants to make some Math Vines. These seem to have struck a positive chord with teachers and students alike in allowing them to truly appreciate these concepts. It is worth remembering that many math concepts (e.g. the Pythagorean Rule, completing the square) were first discovered geometrically and only later algebracised. If a picture is worth a thousand words, how much is a Vine worth? 
Over the next few months we are hoping to build a bank of these. They are fun to make (get your students to make some!) though sometimes I do find the six second time limit a bit restricting. Do let me know what you think of these.

This next one we might use as a Minds-On when learning about the area of a parallelogram:

When students are asked to complete the square, do they know what this actually means?

Using snap cubes can help students develop a much deeper understanding of slope and y-intercept.

And this is my first Vine: the difference between two squares!

You can follow our progress on our Vine account, Math Vine a Day.

Friday, April 24, 2015

Clocking On

A few years ago I was in the school office when a Grade 12 student (whom I knew as being bright) came in and asked what time her exam began. I told her 12:30 and she replied "What time is it now?" I pointed to the analogue clock behind me (it was 11:45). She paused, then confessed, "I don't know how to tell the time".
I have heard the argument that kids these days don't need to tell the time using analogue clocks as everything is now digital but I would wholeheartedly disagree with this. In fact, I believe that if students don't learn how to read an analogue clock, then they are at a disadvantage mathematically. Being able to read analogue clocks allows students to use, develop and practice so many math skills:

Estimating ("It's about two o'clock", "It's about ten to two.")

Skip Counting ("It's (5, 10, 15) 20 to three.")

Non-standard skip-counting ("It's (5, 10, 15, 20, 21, 22) 23 minutes past four.")

Fractions ("I can see two quarter hours are the same as a half hour.")

Decomposition ("15 minutes to is the same as 45 minutes past.")

Angles (Since angles measure turn, we can see the hands of the clock continually forming angles. When is the time an acute angle? An obtuse angle?)

All of this is fantastic for developing number sense and my view is that you cannot get anywhere close to this only by reading the digits off a digital clock.

I also feel though that analogue clocks (with their built in spatial nature) are far superior than digital clocks in getting students to understand what time is. They help students see the cyclical nature of time more clearly (something which will be important when they are working with elapsed time.)

The best sort of clocks for students to learn with are geared such that the hour hand moves with the minute hand. They might cost more than cardboard versions but the learning is much better. I also look for clocks that have clear numerals on the face: I often tell parents that they should buy analogue watches for their kids that have all the numbers 1 to 12 and not Roman numerals. I also recommend the Feel Clock app (available at itunes) because it has a very clear display and animates the background to emphasise the difference between a.m. and p.m.  

If students have access to these then they can also attempt problems like this one:
I can see a clock. The two hands are nearly touching but not on top of each other. What could the time be?
I tried this in a Grade 4 class recently. Initially it was interesting seeing how many kids tried to draw their solutions. This does beg the question: why do we get students to draw clocks? It is a tricky thing to do accurately (especially for times other than the o'clocks and half-pasts) and it is not a skill needed in real life. In the time it takes for a child to draw a given time on a piece of paper, I reckon another child could show at least four different times on Feel Clock. Anyway, one student realised that they could use the analogue clocks and before long, all groups were doing so: it allowed them to tackle the problem far more effectively.

Notice how the students have recorded their times digitally: they do need to know how digital times work as this is how time is often shown. In fact, this group saw a pattern in their first three solutions which they used to predict their next six.
Extensions of his question could be:
What times are the hands at right angles?
How many times a day do the hands form a straight line?

Friday, April 10, 2015

Have You Checked Your Work?

"Have you checked your work?"
I reckon this is one of the most common questions that teachers ask their students. And it is usually more of a rhetorical question as we ask it when we know that students have made a glaring error or have come up with an answer that makes no sense.
And usually the students look briefly at the work and say "Yep!" as if to say "Yep, it's still there!"
Checking your work is a vital piece of Polya's four step approach to problem solving. In many ways, I think it should feedback into the first three parts (Understand the Question, Make a Plan, Carry Out the Plan). My sense of things is that this maybe the weakest link for a lot of students.
I have been wondering why this is and now I'm thinking that maybe it is the question itself ("Have you checked your work?") which is causing the problem. Maybe there are better prompts to get the students to reflect on what they have written.
As a case in point, I was in a Grade 9 class recently and they were working in groups on this problem: try it yourself before you read on!!
What was interesting is that many students stopped after they found an answer. They didn't think to consider of there was more than one answer. It was almost as if they stopped out of habit: I have an answer, so now I'll just wait for further instructions.
Biting my tongue, I managed to avoid saying "Have you checked your work?" Instead I asked "Which two sides are equal?" They pointed to the 3x-4 and the 5x-8. I then asked "Are these the only possibilities?" 
Immediately, this had a much better effect than "Have you checked your work?" The students realised that the 3x-4 could be the same as x+6, or the 5x-8 could be the same as the x+6. There were some comments along the lines "They don't look the same!" but they were reminded that they were told that the diagram is not to scale!

But even if they had considered each of the three possible isosceles triangles, and had done the algebra correctly, there was one geometric error that kept on coming up and was not seen by the students: notice the 2cm, 2cm, 8cm triangle:
"How do you know this is an isosceles triangle?"
"Because two sides are the same."
" do you know it is a triangle?"
"Errr...because it has three sides that are joined together, (said in that 'D'uh' tone!).
"OK...if you can draw me that triangle, then I'll give you twenty dollars!"
Sometimes when we ask students to check their work, they might just check the algebra (which in this case was correct) but not realise that what this leads to is an impossible shape. Which is why I really like this question as it will force students to appreciate the triangle inequality: any two sides of a triangle must add up to more than the third side.
But to get students into this habit of thoroughly seeing if their solutions make total sense, we must provide them with rich questions that force them to look for different cases, or to examine the validity of their solutions.
If, however, we only provide them with questions where there is only one answer (and probably just one way to get this), then they won't get into the habit of effectively reflecting on their work. Thus, we cannot expect them to be complete problem solvers.
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The problem above is adapted from the University of Waterloo's CEMC's Problem of the Week.