Monthly Archives: March 2013

Discount Decisions

Following this post, I’m blogging some lesson ideas which address some of the problems with how mathematical modelling has been taught in primary schools.

When I showed this picture to my class, I explained the concept of money-off vouchers and asked them – which one should I use? I probed a bit further and made sure that they understood the very basics of economics – that as a shopper, buying something for the cheapest price possible is desirable. I told them nothing else at this point. I had to be sure that they understood the crux of the modelling. The vast majority were adamant that the voucher to use was the £10 off one. I pushed for explanations and not many were forthcoming. The best I got was: “If it cost £10, you’d get it for free.” I clung on to that magical word ‘if’. A couple of children at this point were physically struggling with the cognitive dissonance until one bravely piped up with: “It depends!”

After a bit of probing and development of their articulations, we settled on the idea that sometimes, the 10% voucher is better and sometimes the £10 off voucher is better. I still pressed them for an answer though, in order to sharpen their thinking further. I asked: “What information do you need to know?” After a bit of discussion, they agreed on needing to know the item I intended to purchase and its price. I told them that I wanted to buy a mobile phone, but as I had not chosen the one I wanted, I was not sure how much I would spend yet.

At this point I needed to make sure that they were able to do the maths. I knew that all of my class could subtract £10 from any amount of money, and I knew that they could all find 10% of an amount of money. What was new to all of them was to work out a 10% discount. I showed them how to do this. We drew up success criteria etc. They practised. A lot. Then I gave them the prices of the phones (screen shots from Tesco’s website – I had simplified the prices for some children and I had a great question for those that grasped the concept quickly).

I asked – If I were to buy this one, which voucher should I use? What about this one? This one? etc. Some needed a little more time than others on this so here was the tricky question. At which price should I switch vouchers from £10 off to £10 off? Thanks to a post and comments on Dan Meyer’s Blog, I also had this question ready (although unused in the lesson): If you can use both vouchers to buy the phone, does it matter which one you use first?

An alternative approach to what I did could have been giving them a worksheet of questions like this:

A mobile phone costs £79. You have a £10 off voucher and a 10% off voucher. Which voucher should you use?

I think, had I given my class this ‘word problem’ in this manner, not as many of them would have been successful. They would not have done as much thinking and reasoning; they would not have understood the concept of the modelling as much; they would not have done as much necessary practise of calculations.

Once again, thanks to Dan Meyer (@ddmeyer) for his input on 3 Act Maths.

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B, I, N, G, ohh… (Or, why bingo doesn’t work)

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Bingo. A favourite pastime of many and for some teachers, a staple in their repertoire of activities.

I can see why it has some appeal. It’s easy to do; it’s a generic game that can be tweaked for many different topics; the children like it. But let’s be clear: these should not be the reasons for curriculum design!

The problem with bingo

Lets take the scenario that you want children to practise quick recall of, say, the 7 times table. A typical use of bingo would be to ask children to split a whiteboard up into 6 parts and write a multiple of seven in each box. The teacher then asks a series of 7 times table questions while the children tick them off. The winner being the first child to tick off all their numbers.

First, consider the experience of the child who does not fully know the 7 times table. They wrote down 7, then 14. Then they get told to hurry up so that game can start. So they write something like 30 (not doing good maths). As the teacher is keen to start the game, maybe they miss this…

Next, consider the experience of the child who knows the 7 times table fairly well. They sit patiently (not doing maths) while the teacher hurries some other children along.

The game starts. The teacher reads some questions – both agonisingly slow and unhelpfully fast at the same time. This is when a raft of triumphant sibilances ripple around the room as some children get closer to winning. Those that are struggling to keep up know where they can look for ‘help’ and look to see what others have crossed off (not doing maths).

The teacher notices that some children have missed an opportunity to cross off a number. “You’ve missed one!” The child has no idea which one and, feeling watched, crosses one off by guessing (not doing maths).

All the while, the children who need the most improvement in learning the 7 times table have done one or two calculations, possibly incorrectly, with no feedback.

Eventually, a child calls bingo – always a higher attaining child. That child then reads out the numbers while the teacher confirms it. The rest rue their missed opportunity and perhaps beg the teacher to play again. See? They love bingo!

One child got feedback on their calculations. One. The most calculations that any child did could be as low as 6. Six. And this child will have undoubtedly known all this anyway. “I can’t believe that so and so stilldoesn’t know their times tables!”

We must question the validity of the things we do. If children are to learn times tables they must do far more practise than bingo offers them. They must get feedback on their thinking and then appropriate intervention.

Don’t get me started on loop cards…

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Writers’ Toolkit – Discussion

These are photos of our writers’ toolkit for discussion writing. Note that the intention goes beyond “provide a balanced argument”. There are different intentions for different parts of the text. See this post for more on creating writers’ toolkits.

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Giraffe

Following this post, I’m blogging some lesson ideas which address some of the problems with how mathematical modelling has been taught in primary schools.

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I asked my class “What questions pop into your head?”. After the expected “What is that man doing to that poor giraffe?”, we got some interesting ones.

What is the difference in weight between the man and the giraffe?

How many baby giraffes would weigh the same as an adult giraffe?

Now, I had an objective for this lesson, but I didn’t say anything about it straight away. I wanted my class to be able to read scales, but saying this at the beginning can kill lessons stone dead. Instead, I had a question of my own.

How much does the baby giraffe weigh?

Here we had to clarify why they can’t just put the baby giraffe on the scales. I asked them for an answer. Someone should say that they need to know more, but just in case, I asked them “What information do you need to know to answer the question?” I questioned further, asking what they might do with that information or why they think it’s important. We settled on the necessary requirements-weight of both man and giraffe, and weight of man alone.

I told them I didn’t know these weights, but I did have pictures of the scales when this was happening. But before I gave this to them (different scales for different children) I made sure that they knew how to read the scales. I modelled how to work out the size of the intervals; we recorded the success criteria, they practised on unrelated scales.

Then, they returned to the giraffe problem. I gave them the information they asked for earlier. The children worked through the problem, some quickly, some slower. At this point I had some related, but differently worded problems as ‘sequels’. Yes, these were more like the traditional word problems, but the children knew very well the context by now. Here are some of the prepared questions.

An adult giraffe weighs 3 times as much as the baby. Weight of adult and baby together?

In a year’s time the giraffe is weighed again. The man weighs the same but the giraffe’s weight has increased by 10%. Weight of man and giraffe? Weight of giraffe?

Also, the good quality original questions that children asked at the beginning could be answered.

Here, I could also formalise the lesson – talk lesson objectives and work on the accuracy of mathematical vocabulary. Name the lesson.

This way, there is little literacy demand at the beginning of the lesson. The purpose is made clear from the outset before any maths is introduced. The children were not simply given information; they had to work for it. I directly taught them how to read scales and they practised. They had a variety of question types.

With thanks to @ddmeyer for the concept of 3 Act Maths. This is far from polished and there are further opportunities to develop. I haven’t developed yet how to show children the answer to the original question. Seeing is believing and it validates the maths that they have been using.

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The Problem with Word Problems

Mathematical modelling in the primary classroom has long been an area that I have wanted to develop yet didn’t really know how. I’ve seen (and, admittedly, taught) a probably familiar looking lesson many times – ‘word problems’ tagged onto the end of teaching children some concept or other. Underline the important information; decide which operation is needed; calculate; answer the question. Sure, some children get it, but many, as we know, slip through the net.

I first came across Dan Meyer (@ddmeyer) when I watched his TED talk about a year ago. Recently, I had the chance to attend one of his workshops and even though his work is very much aimed at teaching the secondary age range, I felt that there was plenty that could be applied to improve mathematical modelling in primary schools.

The problem with word problems

Here’s a typical word problem that requires some mathematical modelling that you might find in a primary classroom:

A rectangle has a length of 15cm and a width of 8cm. What is the area of the rectangle?

There will definitely be some children that have trouble decoding and comprehending this. The literacy demand may play some role in children being unable to work trough this type of problem. All the necessary information is given from the outset which is not how the world tends to work. The purpose of the problem comes last. The child will read some words without knowing the purpose for it until the end. Children may be given a whole raft of almost identically worded problems with slightly changed numbers.

One way of addressing these problems

How to address these problems? Dan Meyer’s blog post explains in good detail, but here’s a simplified version to get started with. First, remove the literacy demand and make the context concrete. Image or video works great here. Ask “What questions pop into your head?” I’d have a question ready that I’d like children to work on, but children may think of questions that have some mileage. Make sure children know the question that they’ll be working on – the purpose comes before any of the maths or specific information. Ask “What information is needed to answer this question?” With skilful further questioning, make children work for the necessary information, revealing it when they have shown an understanding of what it may be used for or why it is important. Once they have the information they need, it’s time for the maths. Make sure that they know how to do what they need to do. Model; generate success criteria and get them to practise as necessary before returning to the problem. Children will soon have an answer – have ready a few related but different questions as opposed to repetitively worded problems.

My next few posts will be some examples of these principles that I have tried out.

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