Thursday, January 28, 2016

Measurement to the Fore

I am sometimes concerned that our students' experience of Measurement consists primarily of memorising formulas and conversions at the expense of the the actual skills of estimating and measuring objects. It is all very well if a student can tell me how many millimetres are in three kilometres, but if they cannot measure the length of a floor accurately, then they have not learned about Measurement. 
My experience has been that as students use metre sticks, measuring jugs, and sets of scales, they develop such good measurement sense that they can see that since one metre is 100cm, then they will also understand that 5m is 500cm or that 725cm is 7.25m. In other words, they will use their measurement sense to make conversions. Those who rely solely on the staircase method for conversions will not develop their measurement sense so easily. 
As such, I like giving students activities which actually gets them measuring. It allows me to see if the are using the tools available to accurately measure objects. In two classes this week, I began the following task:
Design a container to hold twelve golf balls.
We gave each pair of students a single golf ball and told them that they could choose to use any further tools they might need from around the classroom. Right from the get go, all students were engaged in the task. They quickly began thinking about how they should arrange the golf balls. Some initially decided to design a bucket (but quickly abandoned this idea!). Others looked at a 12 by 1 arrangement; or a 2 by 6 arrangement; or a 3 by 4 arrangement. A couple of groups even thought about a 2 by 2 by 3 box. As they began to measuring the golf ball we noticed a few misconceptions:

1) Measuring from the edge of the ruler and not the zero mark.
I reckon most teachers have seen some of their students make this mistake or a variant of it (e.g. measuring from the 'one' mark). One way to correct this is explicit teaching: demonstrate to the student that measuring this way leads to inaccuracies and have them move the object (or the ruler) so that they measure from the zero mark.

2) Measuring the circumference and not the diameter
Maybe some students 'see' this attribute (circumference) more readily than the diameter. Having measured this, one group multiplied by 12 to get 158 cm. We asked them how they would use this value to design their container. When they seemed unsure, we showed them a metre stick to help them realise that this measurement was greater than a metre. At this point, they began to realise that they were in a bunker so we prompted them with 'Is there a better measurement to use?'

3) Spheres are difficult to measure!
We could have given the students a similar task with objects that are much easier to measure (e.g. design a box that will hold 12 juice boxes) but that being said, I think it was great that they were challenged; it brought out some creative approaches. For students who really struggled, I put a ruler either side of the golf ball and they measured the distance between these two rulers.

4) Only thinking two-dimensionally
This was fairly common and we dealt with it using this prompt: 'OK, you are phoning me, the manufacturer of these containers. Tell me all the information I need to know to make your container accurately.' This got the students to think about the height of the container.

The next step for both of these classes is to take their plans and actually make the boxes out of card. However, before this happens, the students need to know the old adage, 'Measure twice, cut once'. Most of them measured the diameter as 4cm when really it is about 43mm. They need to factor this exact measurement into their design (and consider if they need to add a little extra) before they actually begin construction.
Once all these boxes are constructed (again, this will involve a lot of hands-on measurement), they can argue which of the boxes is best for the job. I already know one student who is convinced that the 1 by 12 box will be best as it will fit in nicely alongside the clubs in the golf bag: it will be interesting to hear the other students' thoughts on this!
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I once gave students the task of designing a pasta box that would hold 1000 cubic centimetres. The choice boiled down to a 10 by 10 by 10cm cube or a 10 by 5 by 20cm rectangular prism. Many students made the argument that the cube was the better box as it had less surface area and therefore would be cheaper to manufacture. One student though argued against the cube as it was difficult to hold and therefore pouring the past into a pot of boiling water would be an issue. Another student argued that the cube had a smaller 'front' than the rectangular prism (100 compared to 200 square centimetres) so it would not be as visible on the shelf of a grocery store which would impact sales. 
It goes to show that sometimes the mathematically best solution is not the best solution practically. 

Sunday, January 24, 2016

The Subtle Art of Taking a Break

There is, I believe, a common misconception that Maths is a subject which consists entirely of questions that need to be answered immediately. Whilst I acknowledge that that efficient fact retrieval is a huge advantage in solving problems, our students will be at a huge disadvantage if they never experience questions which force them to stop and rethink. Our students need to know what to do when they reach that cul-de-sac. One strategy I think we might do more to encourage is to get our students to actually take a break from the problem.
A couple of months ago, I was in a meeting and doodling away when I accidentally sketched this shape:

I wondered what its area would be, thinking it would be a trivial problem. Five minutes playing around with it didn't yield anything so I put it to one side. I revisited it several times, each time not making any progress. I knew that I could take a purely algebraic approach (using co-ordinate geometry) but where's the fun in that? I was looking for a beautiful proof. Then one day, I took the problem out again and suddenly saw the answer:
 And there have been many times when the solution to a problem suddenly materialises after I have taken a break from the problem. Once I was stuck on this problem from a Number Theory course I was taking as part of my Masters of Mathematics for Teachers at the University of Waterloo:

Show that for any positive integer, n, there exists n consecutive values, none of which are prime.

For example, suppose n=4, I have to show that there are 4 consecutive numbers which aren't prime. I can do that: 24, 25, 26 and 27 are four consecutive numbers which aren't prime. But I had to prove that it works not just for 4 consecutive numbers but for any amount of consecutive numbers. I worked on this for quite some time but without making significant inroads so I took a break, and as it was late went to bed. My son woke me up at half past three in the morning asking for a glass of water. I got this for him, tucked him in and then headed back to my bed. In the six steps it took me to get back to my room, in the middle of the night, I suddenly saw the answer. In fact I saw it so clearly that I knew that I didn't need to write it down anywhere.

Now I am sure there will be some readers who might think, "You got stuck on those problems? But they are easy!" And, now I know the answer, I do wonder why I got stuck. But the fact remains that for whatever reasons, I did get stuck, and consciously taking a break somehow reset my way of thinking.
And I know that I am not alone in this. In the wonderful book about the great mathematician Paul Erdos, The Man who Loved Only Numbers  another great mathematician, Ron Graham, explains how he had a "...flash of insight into a stubborn problem in the middle of a back somersault with a triple twist." 
I would love to know the neurological reasons why this happens. Is it a case of the brain thinking too hard about the problem (as result missing some vital information) and then,  after a break, a rejuvenated, more relaxed brain sees what should have been seen all along?
Whatever the reasons for this, as a teacher, I need to model what I do when I get stuck. I need to get students to understand that sometimes the best way to crack a problem is to leave it alone. I can tell my students that if they get stuck with a question (on a test for example) to leave it, do some other questions and then come back to it: they might then take a fresh, more productive approach to the problem.
But also, in the same way that English teachers will habitually share with their students what book they are currently reading, maybe we as Maths teachers can share with our students what Maths problem we are currently working on. This, for example is what I am working on right now (courtesy of the University of Waterloo CEMC's Problem of the Week.

Wednesday, January 6, 2016

A Concrete-Diagrammatic-Symbolic Development of Division

It is no secret that students find division the most difficult of the four operations to understand. I don't just mean the procedure of the long division algorithm, I mean the concept of what division is as an operation; how it relates to the other operations and how it is used in different contexts.
Here are some thoughts on how we can get our students to truly understand division, based on the Concrete-Diagrammatic-Symbolic continuum. Whilst one of the end-products of this will be that students can use an algorithm to divide two numbers, I truly believe that just teaching an algorithm will not get our students to understand what division is. It must be pointed out too, that a huge part of understanding division necessarily involves seeing how it links to multiplication. Graham Fletcher has created a great video on this. And we should also remind ourselves that knowing your multiplicative facts is more than knowing the answer to 7×8. It is also knowing the answers to such questions as 24÷a=8 and a÷8=7 or even 48÷a=b.

Concrete Division
This can begin in primary grades. Kids might well have real-world experiences of sharing situations (e.g. sharing candy, dealing out cards, cutting a pizza into equal slices). They might not have real-world experiences of grouping situations though so it is a good idea to give questions like 'If an egg box hold 6 eggs, and you have 24 eggs to pack, how many egg boxes are needed?' or 'There are 24 kids in our class and they need to put in groups of three. How many groups will there be?' It is vital that students experience this tactile sense of what division (either sharing or grouping) is. As they become more familiar with this, we can give them problems that will involve remainders so that they can consider what effect this has on their answer (e.g. if an egg box holds 6 eggs, how many are needed to hold 32 eggs?)

Diagrammatic Division
With enough experience of this, they can then represent the division action using diagrams. By this, I don't mean that they need to draw pictures of the actual objects that are in the question but rather use this method which I call Spoke Division:
Suppose you have to do 517÷4.
Firstly, write 517 with four spokes radiating outwards:

Now think of a friendly number that you could put into each spoke. In this case, 100 seems to be a good choice. After taking four hundreds out we are now left with 117:

Now think of a friendly number that you could put into each spoke. Some students might say 10, some might say 20.  Both of these work but will take a little longer. I myself will use 25:

Now we have 17 left, so I can put four more into each spoke leaving a remainder of 1.

Since each spoke has 129, we can say 517÷4=129 R1 or, if you prefer, 129.25.

Initially, I'd be careful about what numbers to use; friendly at first, then building complexity. What I like about this method is that it connects with the students' concrete representation of division and, as such, still feels like division. There will come a point when the spokes method can be developed into something more powerful.

Symbolic Division
Although I learned a version of the standard algorithm growing up, it is not the one I would initially show my students. Instead, I would use the following method, often called partial quotients. 

See how it connects nicely with diagrammatic division. Also notice how it allows the student to use friendly numbers to get the answer. It is not so easy to use friendly numbers in the standard algorithm.

With regards the more abstract standard algorithm, it should be pointed out that different countries have different versions of what this looks like (see this entry in Wikipedia). I myself learned something which I believe is called short division and spent a long time focusing on single digit divisors. When I first saw long division, it seemed (to me at least) to involve an unnecessary amount of writing.

Here in Ontario, not too many teachers seem to have seen 'short' division. It does require the user to mentally compute the remainder at each step (e.g. there are 6 sixes in 40 with a remainder of 4). Will this be tricky for students? I don't believe so, especially if we gradually build up the complexity of such questions.

Double digit division is problematic (unless the double digits are 'friendly': the advantage of working with single digit operators that standard algorithms usually have vanish when trying to do something like 7054÷82. Mentally, I'm trying to do 705÷82. It's doable for sure but potentially time consuming and open to error. Yet using partial products, a student can use friendly numbers:

Some students are fine at using the standard algorithm for double digits. For those who aren't, get them to try partial products; my experience is that this is a game changer for these students.

This concrete-diagrammatic-symbolic development of division takes a long time, years even. It is not to be rushed unnecessarily; I am not convinced that the best way of teaching students about the operation of division is to jump straight to the algorithm of division. 
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Recently, I gave a Math Night for parents at one of the schools in our Board during which I shared our Board's vision of what good Math education looks like. Afterwards, a parent approached me seeking some guidance as how he could help his daughter with long division. He said, "They have to do something like 'Dragons Must Suck Blood' and, to be honest, I don't understand what any of this means. And neither does she." As we chatted more, I gathered that the Dragons Must Suck Blood was an mnemonic to help students remember certain steps of the algorithm (Divide, Multiply, Subtract, Bring Down): clearly it wasn't working. At this point, rather than explain the algorithm in a different way, I showed him how I would develop the concept of division and how this needs to be in place before we try to make sense of the algorithm. It was neat seeing his eyes light up when I showed him the spokes method and the partial products methods and hear him say, "I actually understand those ways!"