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!
                                                                   *          *          *
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.