Thursday, December 20, 2012

Super Bear

So it's the last week before the holidays and we're in a Grade 8 class hoping to give them something engaging. Dan Meyer's Superbear task came to mind. We also wanted to take advantage of the fact that this class has a set of iPads at their disposal.
But we were also curious: if we showed the students the above video, what sort of questions would come to mind? Well why not use a backchannel to collate these questions? Having recently seen TodaysMeet in action we decided to give this a whirl.
Now I won't lie to you; I was a touch apprehensive about how the students might use (or misuse) this tool. I have seen adults get a bit silly when using it. Royan Lee (who met with our department last week) gave some good advice: Let them use the tool and let them make the mistakes. And when they do make a mistake then it's an opportunity for learning.
I needn't have worried. Yes, there was a lot of "Wassup?" and "Yo, I'm so hungry I wanna eat those bears" comments but nothing harmful. More importantly, there was a lot of really good questions:
We then generalised these into one question:

How many of the regular bears are equivalent to the superbear?
We were careful to avoid all reference to any attributes such as mass as we wondered how the students would approach this. Some chose to use volume:
 Some chose to use mass:

Others used the internet to help them get the relevant information:
 A couple of groups even considered calories:
What was neat was that by giving them no information in terms of measurements they had to be very active in thinking what measurements they actually needed. How they then got these and how they then used these was very impressive especially in terms of showing their proportional reasoning.
We watched the Act 3 from Dan Meyer's site and there was some disappointment from some students that their answers differed from his but we emphasised that he was using his bears whilst we were using ours!
Finally we asked the students to reflect on how they approached the problem today:
What probably made this lesson work so well was the lack of information that we gave the students: it forced them to think what question needed to be answered and from there how to answer it.
After all, part of being a good mathematician is not so much answering questions, but asking the questions in the first place.
*          *          *          *
Dan Meyer's Three Act Math can be found here:

TodaysMeet can be found here:

Monday, December 17, 2012

The Study of Patterns

“A mathematician, like a painter or poet, is a maker of patterns. If his (or her) patterns are more permanent than theirs, it is because they are made with ideas.”
G.H. Hardy, the famous British mathematician, wrote the above in his book A Mathematician's Apology (though I added the 'or her' in italics). I often ask students and adults what is Math the study of and whilst most say something along the line of 'the study of numbers' very few say what Hardy was getting at:
Math is the study of patterns.
Sometimes it is patterns in numbers (and these patterns can help us compose and decompose numbers as well as operate with them). Sometimes it is patterns in shapes, or how we measure things. Sometimes it is patterns in data. When I tell students this it is very liberating because human beings are born with the capacity to spot patterns (as explained in Professor Brian Butterworth's book The Mathematical Brain).
So we wanted to see how Grades 5 and 6 students would make their own patterns. Our original plan was to give students the first three terms of a pattern and then get them to predict the tenth term. I gave them the first term (shown below) and then went around each pair just to check that they had made it correctly on their desk.
What we noticed happening though was that students started doing the second term themselves, even though I had not told them to.
I know. The cheek of it!
But here are some of their examples:

 This was the pattern we had expected. What we got was more interesting. Students could now be asked to look at other students' work and to describe the pattern as well as predict what the tenth term would look like. The students' conversations were full of reasoning and proving based on the variety of patterns that they had created. It was so much better than forcing them all to work with the same pattern.
So we were a bit surprised when we tried this exact same approach in a different class the next week. Just about all the students created the L-shaped pattern and the conversations weren't as rich:
"What's your pattern rule then?....Oh it's the same as mine. Wow."
So we quickly opened it up and asked them to create their own patterns without giving them the first term. This is what we got:

Now, the conversations were a lot richer. I am convinced that this was a result of the greater variety in the patterns.
So the lesson learned was this:
Students must be allowed to create their own patterns so that they will develop the ability to ask questions about other students' patterns.

Tuesday, December 11, 2012

A Must-Fix Misconception

I love the quote on the homepage (highlighted below)

I don't think it is any secret that division is the least understood of the four operations. There is often a lot of noise in the media (from people who have never had to teach math to elementary students) that students must learn long division. Right now (as I'm typing this) I'm trying to remember the last time that I had to do long division: it was way, way back. Most situations I can deal with using good number sense to get a good estimate. If an accurate answer is required, I might pull out a calculator (why bark yourself when you have a dog? as we say in Yorkshire). And if I have to do a bunch of such calculations, I'd use a spreadsheet.
But by focusing on long division, a more pressing concern is overlooked: students don't understand division. And learning the standard long division algorithm does not teach students (or adults) what division is. Students understanding of division should begin concretely by actually doing some division in the two most common situations: sharing and grouping.
For example, if these cubes represented 23 cookies:
and if these are shared between three people, then each person gets 7 cookies with two left over. This is a sharing situation:
If the cookies are to be put into packs of three then we will get 7 packs with two left over. This is a grouping situation:
 Both situations, though different, are represented by the same number sentence (23 ÷ 3 = 7 with 2 remaining). Students must experience both sharing and grouping situations that involve remainders so that they develop a solid schema of division situations. Students can then develop from concrete representations of division to diagrammatic representation to symbolic representation. Not just jump straight in to long division.
There are also some other misconceptions that are ignored by focusing on long division. I asked some Grade 6s to complete the following two statements:
Multiplication makes numbers...
Division makes numbers ...
 The whole class agreed that multiplication makes numbers bigger and that division makes numbers smaller. Rather than tell them that this was not necessarily true, I wrote down the following two questions:
8×10=…      8×2=…     8×1=...     8×0=…
After they gave me the answer to 8 times 1, I asked again "Does multiplication always make bigger?" Puzzled looks quickly gave way to smiles of recognition. "No, sometimes it makes the same!" was the agreement. After we agreed that 8 times zero is zero students were calling out "...and sometimes multiplication makes smaller." One student summarised it neatly thus:
"Multiplication can make bigger, stay the same or make smaller... it just depends on the number you multiply by."
I repeated this process for division using the following string:
10÷5=...      10÷2=...    10÷1=...    10÷½ =...
The class quickly agreed that division could also make bigger, stay the same, or make smaller depending on the number you divide by. Their huge misconception had been drawn out into the open and addressed. To emphasise why this is important, I should tell you that I have given this question to adults on many occasions:
½ x = 24
and sometimes as many as two-thirds of them have told me that the answer is 12. They go on to reason that to find x they have to do 24 divided by a half which gives 12 which makes sense since  division makes smaller!
We consolidated this learning by playing a game called Target. The Target game is very effective in getting students to understand that multiplying doesn’t always make a number bigger, and likewise, division doesn’t always make a number smaller. Students play this game in pairs with the use of one calculator. A starting number (e.g. 37) is chosen as is a target number (e.g. 100). The first player uses the calculator to multiply (only multiplication is allowed) the start number by any number to attempt to get the target number. If he or she gets 100 exactly or ‘100.something’ then that student wins. If not, the other student gets the calculator and tries to multiply the new number by another number to get the target number. Play continues this way until a winner is found. It is most engaging!

Tuesday, December 4, 2012

88.2% of statistics are made up.

The British comedian, Vic Reeves, once quipped that "88.2% of statistics are made up." I thought of this when I saw the picture below:
Getting our students to be statistically literate must be one of our moral imperatives. This would partly involve developing our students' ability to ask the right questions in order to collect the right data so that they can display it in the clearest way. But we also need to help our students develop the ability to question any data that is presented to them, to see through the noise of misleading data and to reflect on such things as the sources of the data.
We gave the following question to a Grade 7/8 split class:
You are the CEO of a company that makes plastic water bottles. You need to create two graphs. One is to convince shareholders that you are making a profit. The other is to convince the Ministry of Environment that your company is environmentally responsible.
We purposely avoided giving them actual data to work with as we were more interested in how they would take into account that they have to present information to two very different audiences. Some used a single line graph and plotted time against the number of bottles sold:
Others created a double line graph, plotting their company's profit against another's over time:
For the second graph, some plotted the number of their bottles that were being recycled whilst others plotted the amount of 'pollution' over time, again compared to another company:
We then allowed students to go on a gallery walk and give feedback on other students' work.
They were reminded beforehand (as recommended by Dylan Wiliam) that the best sort of feedback is not ego-centred ('Great job') but task-centred and generally they were true to this. In the example below, students were asking how the author could possibly predict future sales. I like that!

In our consolidation we posed some further questions to the students:
1) Does it make more sense to have time measured in months or years?
2) Does it make more sense to have 'profit' or 'Number of bottles sold' on the y-axis?
3) How do you measure 'pollution'? Or how do you know how many of your bottles are recycled?
The ensuing conversation brought out a wealth of reasoning, proving and reflecting.
Afterwards, the teachers and principals who were observing remarked how this lesson could have easily have been a Media Literacy lesson.
Some recommended resources:
1) has a huge amount of infographics that will be sure to kickstart some great data discussions.
2) I have never, since I started working in 1990, ever had to draw by hand a graph of any sort. All those hours spent converting data to percentages, then multiplying these by 3.6 to get degrees and then using these to draw (with a protractor) a pie chart were hours spent in vein. The Create-a-Graph site is very user-friendly and allows students to enter their data and choose the most appropriate way to display their data. After all, people in the real world will use technology to create graphs and charts, not be forced to draw them by hand.
3) Darrell Huff's How to Lie With Statistics although written in the 1950s is still wonderful reading. If you read it (I believe it can be downloaded for free now) you will become a better teacher, 100% guaranteed!