The Penny Drops

So I learned something this week that I'm kicking myself for not seeing earlier. Marshall McLuhan famously said 'The medium is the message'. Today I am wondering if:

The medium determines the math.

I was in two Grade 2 classes in two different schools. Both were working on composing and decomposing money amounts and both were given the same question:

How many different ways can you make 31 cents?

The first class had a small amount of plastic coins on their desk and were asked to record their solutions in their math books.
The second class had a larger amount of plastic coins and were also asked to simply leave each solution on their desk. They were also told that they could not use more than eight coins for each solution.
The second group ended up doing a lot more math than the first. They quickly got solutions for 31 cents so we then challenged them with 82 cents. The first group on the other hand were slowed down by the act of recording their solutions. For most, this meant drawing the coins although some did write a number sentence. This wasn't necessarily a bad thing as they were still composing and decomposing money and we later had a good discussion about different ways to represent our thinking.
But it just took them so long!
And it wasn't because there was a difference in the ability levels between the classes; by listening to the students' explaining their solutions I got the sense that there their number sense was very similar. It's just that the second class had more opportunities to use their number sense and this has to be a good thing.
In terms of evidence of student learning, it was easy to take a photo of the students' work:

Some other thoughts:
1) I really liked restricting the number of coins: it forced students away from the time consuming 1+1+1+... solution and made them think in bigger denominations. This is definitely what they will need in the real world.
2) I also love just giving students a handful of coins and asking 'how much?' Again, this is a real world skill and does not require that the students write down their solution.
3) Questions like this one (from Ontario's Grade 3 EQAO test in 2007) annoy me:

This is a perfect example of a question that is meant to be real world but actually is anything but. The ball and skipping rope cost $4.10 in total. Jorge isn't going to dump all his coins on the counter and say 'There you go!' to the shopkeeper and wait patiently for his change. I would hope he more sensibly gives the shopkeeper the toonie ($2 coin), loonie ($1 coin), four quarters and a dime. Or maybe five nickels instead of a quarter, and a nickel and five pennies instead of the dime so that he can get rid of a lot of his loose coins (or shrapnel as some say) as this is what I do. Either way, though, he would not get any change because he gave the right amount because this is what happens IN THE REAL WORLD!

Discrete and Continuous Data

What is discrete data? Is it data that says "Shh... don't tell anyone this"? And what is continuous data?
More importantly, is it possible to get Grade 5 students to understand the differences between the two?
In fact, what is the best way to get any student to learn how to handle data? Filling in blackline masters? Hand-drawing pie graphs after changing the data to percentages and then multiplying each percentage by 3.6 to get the number of degrees and then using a protractor to measure these?
To me, the ability to handle data is so much more than this. It is about:

• asking the right questions to get good data
• thinking about the most effective way of displaying the data, with your audience in mind
• interacting with the data: is it biased? Valid? What are its implications?

For example, I would hope that my students would be able to look at this pie graph...

... and say that it makes no sense to present this data as a pie graph.
I tool part in a lesson study recently with the focus being on handling data. We were going into a Grade 5 class and the teacher was concerned that her students didn't know the difference between discrete and continuous data. What was quite telling was that as adults there was some confusion regarding this too and it took a lot of discussion with a lot of different examples before we all had some level of comfort with these terms.
Our 'Minds On' for the lesson was what does 'handling data' mean? The students were pretty good at saying that data is just information and that we need to try to make sense of this data. We used the following infographic from http://visual.ly/what-infographic-2 to sum this up:

We then got them to create a bar graph about their shoe size by using their actual shoes. We created three such graphs: one with all the students' shoes, one with just the boys' shoes and one with just the girls' shoes (below).

The students then had a great discussion as to why the boys' shoes were more 'spread out' than the girls' shoes.
Next we measured the students' heights and got them to create a living histogram. We used intervals of 10cm e.g. 110-120, 120-130 etc.

Again, we created a histogram just with the boys in and another one with just the girls in and again had great discussions as to why there were differences.
Now came the tricky bit: could we use this activity as a way of getting students to see the difference between discrete data (the shoe sizes) and continuous data (the heights)? We asked the students what was different about the 'shoe size numbers' and the 'height numbers'?  A lot of conversations centred on the idea that the heights were measured but that the shoe sizes weren't. Then one student suggested that for the heights you could always find 'numbers in between' but that you couldn't do this for shoe sizes.
"Yes you can, you can have 3 and a half," shouted out one student.
"But can you have 3 and a quarter, or 3 and an eighth?" I asked. Quickly the class agreed that shoe sizes can only have certain numbers but heights could have any numbers. To emphasise this, we used a number line on Geogebra and simply kept on zooming in on the 140-150 interval. As the decimals changed from tenths to hundredths then thousandths and beyond one student said "Wow, you could go on forever doing that!"
And then another said "Yeah, it's like those numbers are continuous!"
It is very hard to suppress a huge Cheshire-Cat-smile when something like this happens!
We summarised like this:
Discrete data can only have specific values, often (but not always) just whole numbers.
Continuous data  can take any value.
We ended by asking students to give their own examples:
Number of students in a class? Discrete.
Amount of pop in a bottle? Continuous.
Amount of time it takes to get to school? Continuous.
Number of siblings? Discrete.

Nets of Prisms

I think that if you were to ask most people what a net of a cube looks like, they will say a cross shape. Very few though will know that there are in fact 11 different nets. I've challenged both students and adults to find all 11 and it never fails to get everyone engaged. But this is mainly because of the manipulative that we use: Polydrons. Here it is easy to quickly check if a net does or does not work. It is also great to have a good old think about what congruence means and how two 'different' net are in fact the same if one is a reflection or rotation of the other.
I was in a Grade 3/4 split class recently and we were hoping to do this nets of a cube investigation. However after a quick glance at the bucket of Polydrons, I realised that I would have to change things a bit: there were not enough 'square' Polydrons. We would have to find nets of triangular prisms instead. This was going to be a ton of fun for me too as, off the top of my head, I didn't know how many such nets existed.
I started the lesson by showing a net of a cube (the one shaped like a T) and asked them to predict what this 2D shape would fold into. Most predicted a cube (some said a box, some said a square) and everyone was quite happy when I showed them that indeed this was the case.
I then held up a triangular prism and gave them the challenge:

Find as many nets as you can for this shape.

There was one catch: they had to tell how many of what shapes they needed. Rather than giving them two triangles and three squares, I wanted them to think about what was required. They were all able to do this even though I didn't hold up the prism for very long. In fact, I could see a lot of students trying to visualise what was needed (you know that look, when they seem to stare at a spot on a wall and have their hands in front of them holding an invisible prism?)

In a short space of time we got the following:

This was pretty much the first one that most students did.

And each time I responded with "Great! Now make a different one." And they did!

When students came up with congruent nets we had great discussions that led us to agree that even though they looked a bit different, mathematicians say that congruent nets are in fact the same net.

The wonderful thing about Polydrons is that students can easily reason and prove if their nets do work as seen in this clip:







Planning the consolidation with other teachers and principals was fun. I was asked how many nets existed. "Well, I think there are six as that's how many I've found," I said. In a flash, my buddy Mansel had corrected me (with a little help from Google I think!): there are in fact eight. So we set as a challenge to the class: they had found four, now they had to find the other four.

The other thing we wanted to check is do they really know what a prism is? Their own explanations were on the right lines but far from exact. So I used an analogy which has helped many students (and teachers) in the past.

A prism is like a loaf of bread: you can cut it from front to back and each slice is always the same size and shape.

I have found that this draws their attention to two important attributes that in later grades will be very important for working out the volume: the slice of bread is the area of the cross section and the number of slices will represent the 'length' of the prism.

For this class though it was enough to say that the shape of the slice tells us the name of the prism. Holding up a cube, I asked them what this could also be called. After a quick discussion, most groups agreed that it could also be called a square prism.

Or squarular prism as one student said.

I later showed the adults some paper cups that you get in fast food joints for getting ketchup and asked them what is the net of this shape. Most people are surprised when you unwrap the cup. Try it!

I then showed this photo I took recently and asked "Is this a prism? And how might you work out its volume?"



Using the Empty Number Line to Subtract

One of the best bits of advice that I ever got (and now pass on to students, parents and educators) is this:

Good mathematicians look at the numbers first before they decide which strategy to use.

 

I was in a Grade 4 class recently and the teacher was concerned that she had some students who had trouble doing the traditional algorithm. My concern with the traditional algorithm is not that students shouldn't learn it but they should learn when to use it. This is fast becoming a forgotten skill. Ask yourself if you need to use the traditional method to do the following:

 

1) 1000-5

2) 1000-998

3) 400-320

4) $10.00 -$2.25

5) 17-8

6) 800-481

7) 5028-2279

 

I would imagine that most people would do all of these in their heads with the exception possibly of #7 (and maybe #6 too). That being the case, we need to get our students to do so too.

To help students develop this skill, I began by writing the following number string:

I always make the point of writing the question horizontally and not stacking the numbers; it nudges the students into thinking of non-standard approaches. Students wrote their answers on their personal whiteboards and I would say that whilst most just wrote the (correct) answer, there was a core who wrote the algorithm for each question. So I asked the class if it was always necessary to use the algorithm.

Some said yes.

Some said no.

No surprises really. I then asked how they did #5 above. A common method was to subtract the 200 from the 500 to get 300, then subtract the 30 from this to get 270 then subtract the 4 from this to get 266. I represented this on an empty number line:

We then gave students some follow up questions. Some used the empty number line to model their thinking:

Others stuck to the traditional algorithm but it was becoming clear that this was causing issues for them. When attempting 800-481, this student (who at first thought the algorithm was better) had an a-ha moment:

Not successful with the algorithm, but successful using good number sense. He felt pretty good about himself.
There remained a couple of students who insisted that the algorithm was always the best method. One student worked out 100-68 traditionally and when I asked if he could work out 100-60 and 100-70 could only do so with the algorithm:

To me, this is a clear case of when the algorithm is introduced too early it actually undoes number sense.

 

 

Reasoning and Proving Grade 3 Style

I had the most enjoyable hour with a group of Grade 3 students the other day. I had them reasoning and proving using their number sense with a very basic Excel spreadsheet that I made.

Some random numbers were put in the bricks on the bottom row and they were then asked how are all the other numbers generated? I stood back and let them argue for a couple of minutes before they figured that the numbers from two bricks add to give the number on the brick above.
"Can you predict what the top brick will be if the bottom bricks are all 1s?" I asked.
A brief but furious debate ended with "The second row will be all 2s, the third row will be all 4s, the next row will be all...8s so the top row will be 16! It will be 16!"
"Prove it," I said which they duly did to hearty cheers. I followed up with asking for their predictions for all 2s on the bottom row then all 10s and again they were spot on with their predictions.
So I decided to ad lib a bit:

"Right, your target is to get 1000 in the top brick but you must have the same number in all the bottom bricks."

After a flurry of trial and error (or trial, feedback and refining to be exact) they got to the point where 63s on the bottom row gave 1008 in the top brick.
"Too big! It's too big! Try 62!"
And then the following happened:

Now decimals are not on the Grade 3 curriculum so this struck me as being not too shabby.
The students loved it and I consolidated by giving them this question from the Problem of the Week section on the University of Waterloo's CEMC Problem of the Week site..

The Great Divide

Here's a question we gave recently to a Grade 6 class:

Three classes are collecting cans for the Thanksgiving food drive. Class A collected 275 cans. Class B collected 225 cans. Class C collected 253 cans. These cans need to be put in boxes of 24 before they can be delivered to the local food bank. How many boxes are needed?

We chose the numbers deliberately. First we wanted to see who would look at the numbers and add them mentally and who who would automatically put them in columns to add. Then we wanted to see what students would do with the remainder. We were also interested in seeing how students would do the division.
As it turned out, all but one pair solved the problem correctly but they did so in varying ways. Yes, some added the numbers in a traditional way which is fair enough. Others though said "I know 25 and 75 is 100 so I added this to the other 400 to get 500 and then added the 253 to get 753," or a variant of that. Most students thought this was neat and it allowed us to give this bit of advice:

Good mathematicians look at the numbers first before they decide what strategy to use.

So, 506+499... do that in my head. 689+7259+164... maybe do that on paper.

With the division, some used a traditional long division approach. Others used an abbreviated version of this (I think this is called short division).

Notice how this student figured out that there were three 24s in 75.
Other students used a chunking method using friendly numbers (this is what I tend to do if I do a division in my head).

 Some chose to start with a chunk of 10, then another and another. Others chose a chunk of 5, then two chunks of 10 followed by another chunk of 5. One group chose a chunk of 9 to begin with. Which got us as teachers thinking 'We tell them to use friendly numbers but do they know what friendly means?'
So in our Consolidation, we asked the students this very question. They were pretty much in agreement that it meant numbers you could work with easily in your head. We also had students comment that a friendly number for you might not be friendly for someone else. With this in mind, I asked the group who started with a chunk of 9 the following question:
If you could go back in time and give yourself a bit of advice before you start this question, what would it be?
Straight away, they replied "Don't use a group of 9. Use a group of 10 instead as it is way easier."
Other groups, having seen the chunk of 20 approach (above) said that they wouldn't just stick to chunks of ten and would look to use 'larger chunks of 10' i.e. multiples of ten.
So there was some really good thinking and understanding going on. With some well chosen practice questions, this understanding will continue to be solidified.

Grade 6 Constructing Shapes

Some interesting things arose out of a geometry lesson in a Grade 6 class. We wanted to get the kids to construct some shapes using different tools. As teachers we tried to do some of these questions ourselves:

All the questions were engaging but especially the last two. The discussion that led to proving that such shapes are impossible actually uncovered some new geometric ideas for us. We decided to ask students some similar questions:
1) Make a shape with three right angles.
2) Make a quadrilateral with three acute angles.
Most of the students used geoboards but a couple of pairs had access to an iPad. Some answered the first question with a square...

 ...as it has three right angles. More than three was the justification given. A simple tweak to the question "... exactly three right angles..." clarified things quickly enough. Our question was deliberately ambiguous to bring out such conversation.
The second question brought out a lot more thinking. Some groups looked to a create a kite and checked for acuteness by using a corner of a piece of paper:

 

Others came up with a chevron type figure:

(Later, the teachers debated as to whether or not a symmetrical chevron is also considered a kite, something I'd never thought about. Now I think it is, unless someone else can convince me otherwise?)
Overall, the students engaged in some really good conversations and were using some very precise geometric language.
There were a couple of considerations though. One student I was watching found it very difficult to use the geoboard. When asked to make a pentagon, he didn't know where to begin and moved the elastic from peg to peg in the hope that some familiar shape would come up. Eventually he made a pentagon but didn't realise that he had done so. I then asked him how many sides his shape had. He proceeded to count all the pegs that the elastic was in contact with! I'd not seen this before so i handed him a pencil and paper and asked him to draw a pentagon. He did this quickly, and then drew another four all of which were irregular. For this student, the geoboard was not a good tool. Yet. Maybe it will be later on I don't know. But at least he knew that pentagons have five sides and could draw them.
Two other groups of students used the geoboard to make a 'pentagon' like this:

They argued it has 5 sides and traced these with their fingers. We shared this with the rest of the class who offered advice along the lines that 'It can't be a pentagon if the lines cross', or 'It's not a pentagon, it's three triangles.' After some debate the class agreed that polygons can only be made up of straight lines that don not cross and that all polygons must be closed.
Now I could have written this down on the board at the start of class and told the kids to learn this by heart. But this would have prevented any of the rich thinking and debate that actually occurred.