Tag Archives: 3 act maths

Spacing, Interleaving and Retrieval Practice in Primary Maths

In the last few weeks there has been a flurry of posts written on spacing, interleaving and retrieval practice. It seems that this flurry has in part been triggered by @miss_mcinerney’s Touchpaper problems. Two that stand out are Joe Kirby’s and Mark Miller’s. Both digest the research before summarising with great clarity what seems to be optimal conditions for learning. I first came across the ideas reading David Didau’s blog, and have been working on Year 6 maths planning to benefit from the effects of spacing, interleaving and retrieval practice. It’s split into 2 parts: longer term curriculum design and shorter term lesson planning.

Curriculum Design

OVerview

This screenshot is a section of the Year 6 Spring Term overview. The overview is split into units of work which consist of two topics. Sometimes, these topics compliment each other in order to show children links between areas of maths: working walls depict these links and they are referred to often. Other times, there is no link between them. This is a first draft of a curriculum overview and although there are probably more meaningful combinations of topics, it will take some time to reflect and switch things around. In this instance, I’m not sure how significant the benefit would be to deliberate too much over this.

The superiority of spaced rather than blocked practice is well known, and this overview plans for spacing in two ways. Some topics are repeated regularly as additional teaching blocks. The Pareto Principle, or the ‘law of the vital few’ describes the imbalance of effects of different causes. The theory applied to this situation would suggest that twenty percent of the content of the curriculum provides eighty percent of the value: there are certain topics that have much greater value than others. Knowing number facts such as times tables as well as being able to calculate quickly and reliably would certainly be within that twenty percent. As such, these vital few topics are repeated often.

Day to Day Planning

The other way that spacing is set up is through the switching between the two topics in each unit of work. Deciding when to switch is contextual – a natural break in one topic is the switching point.  For example, a few days on converting betweeen fractions and decimals before switching to working on calculating unknown angles would provide a few potentially fruitful opportunities.  It gives the teacher a bit of time to assign any extra practice (perhaps for homework) to help some children to be ready for ordering fractions and decimals.  It also gives the teacher a chance to delay feedback for a couple of days, which could be well worth experimenting with, as David Didau suggests here.

But what of the topics that are not in the vital few? These need to be spaced too if they are to be encoded into long term memory. A relic from the National Strategies is the oral / mental starter which could be tweaked to provide spacing and retrieval practice. Each lesson, an old topic is selected to work on where children use a model or image to practise recalling a concept, before working through a series of questions to practise recalling procedural knowledge. This not only spaces out learning but gives the teacher the opportunity to see what children can still do or what they have forgotten; to give feedback on known and likely misconceptions; and plan for revision sessions.  In the example below, children had, within the last few weeks, been working on calculating the area of compound shapes.  The success criteria that we developed at the time was shown on the screen and children used the images to recall the steps needed.  After that, they had the opportunity to practise.  The questions got progressively more difficult from left to right and children either chose to start from ‘column 1 or 2’ or were directed to the appropriate questions:

 MM PerimeterMM PErimeter 2

Factual recall is crucial in order to think with clarity about a concept. For example, if children are to be able to compare fractions, decimals and percentages, they have to be able to quickly recall conversion facts. For situations like these, the mental maths session would include individual use of flash cards, like these.

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Children look at the prompt then say the decimal and percentage conversion. They turn the card over to check and make two piles. One pile of facts that they can reliably recall accurately, and one pile of facts that they have not yet internalised. When putting the cards away, the ‘wrong’ pile gets put on top to practise first next time. Often, having practised an area of maths, a short problem solving task is presented for children to work through, like in the screen shot below.

FDP Q

What next?

My organisation of the spacing is still fairly arbitrary. Whether there are optimal spacing times is not yet clear and certainly, trying to engineer optimum times would be difficult and perhaps not worth the opportunity cost, especially if it turns out to be non linear.

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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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