Counters & bar models used to unpick a classic PS question

This question is taken from the Y3 Autumn term White Rose Progress Check assessment:

I’ve really enjoyed exploring this question type (although, I have to admit, never yet with children as young as Y3). I want children to see and feel the structure of this type of problem, building up to being able to answer a question like the example above in small steps. Then, by working through a series of related questions, children will learn how to use efficient problem-solving strategies. They will also come to see that questions with different ‘surface features’ can have a very similar mathematical ‘deep structure’.

To start with, using double-sided red/blue counters, children attempt the question below:

Often, children start with 8 counters – 4 red, 4 blue. Then, they turn over two blue counters. They realise (with a nudge) that the difference between the number of red/blue counters is incorrect. With a bit more cajoling, we see only one counter needed turning over. At this point I line the counters up above/below each other. I suggest, rather than starting with the correct number of counters, we could start with the correct difference. Have 2 more red counters than blue counters; keep adding a red & blue until you have 8 counters.

That technique, or other methods, are then be practised using the question below. We note that this question is worded slightly differently, but see that the red/blue counters can still be useful:
This time, many children start by laying out four blue counters. We note that ten more counters are needed (5 blue, 5 red). Other children get 14 counters and experiment with how many to turn over. We look at these different approaches. Then, I draw a bar model around the counters (like for the original example), drawing a dotted line to highlight the difference of 4 counters.

Now it’s time for a worked example and another ‘different surface, same deep structure’ question. In this case, I model how to answer the question using the ‘start from the difference’ technique:
Having shown that the difference between the prices is 10p, the cost of the rubber can be calculated by halving 30p (a common incorrect answer to this question is pencil=30p, rubber=10p).

Children then attempt questions that have a very similar structure, still regularly using the counters. Some children are given slightly extended challenges:
Here’s another lesson example of how to break down the problem-solving process. I See Problem-Solving – UKS2 is designed to give teachers the tools to teach problem-solving systematically too. Work will start on the LKS2 version in January 2019. I can’t wait!

Advertisements

Learning to Problem-Solve (and the role of I See Problem-Solving)

It’s easy to give children maths problems to solve; it’s much harder (and very time consuming) to systematically teach problem-solving so children become more competent problem-solvers. Here’s some thoughts on how my teaching of problem-solving has evolved and how I See Problem-Solving fits within this vision.

I have always loved engaging children in rich maths problems. The EEF research (recommendation 3) highlights the importance of engaging children in non-routine maths tasks. However, it also points out that such tasks can create a heavy a cognitive load for many children. Now I try I think more carefully about the sub-skills involved in solving a problem, and build up to the introduction of a problem-solving task using more carefully chosen examples. Consider this question:

Here’s how I built up to using this task. First of all, we clarify how to calculate the sum of four numbers and the difference between the smallest/largest numbers using this question:

Then we model the process of finding all possible answers with this prompt (answers: 5&5, 4&6, 7&3). We also highlight a likely mistake: 8&2 (the difference between the smallest/largest number no longer 4).

Then the final ‘pre-task’ question is introduced, focusing on the most difficult part of the calculation:

Now we get into the main task (sum of 4 numbers is 23, difference between smallest/largest = 4, all numbers different, how many ways?). Where appropriate, children use whiteboards and counters to access the task. When we find the answer 3, 4, 7, 9 we consider whether there is another solution that can be found keeping the 3&9 as the smallest/largest numbers.

As the lesson progresses, we explore systems for finding all possible answers and some children move on to the explain/extend tasks (which are variations on the main task). If you want to use Task 13, check out the free sample resources on this page. The solution to the problem is shown step-by-step by the pre-made worked example.

One little caveat: some children may benefit more from going straight into the question; others may need this extra scaffolding before getting into the main task. As ever, it’s about knowing your children.

There have been lots of great maths problems written. I hope that I See Problem-Solving takes things to the next level by presenting related problems and reasoning tasks in a coherent order, and by clearly representing the mathematics within the tasks.

I also hope that this blog gives food for thought about how to introduce problem-solving tasks so more children experience success. Enjoy using the resources!

A career of improving teaching skills

Over the summer I read ‘Peak’ by Anders Ericsson, a fascinating book that examines the training that leads to expert performance in various fields. Ericsson studied world class performers (chess players, musicians, sportspeople, doctors etc) and describes the ‘deliberate practice’ that they have engaged in to develop their skill.

Ericsson argued that once we have achieved competence in something, simply ‘doing it more’ rarely leads to improved performance. Instead, a tennis player practises by hitting hundreds of backhands from kicking serves; chess masters train by studying key moves from previous games; a radiologist looks at difficult-to-interpret scans from previous cases to improve their diagnoses.

With my teacher hat on, I took away two main reflections from this book:

1. Focus on improving one small aspect of my teaching at a time
Teaching is wonderfully complex. So many things can affect the success of a lesson. I typically think about whether lessons have gone well and then adapt my teaching. Now, though, I will also really zone in on one specific thing (e.g. managing the transition into independent learning), and spend an extended period of time developing that skill.

I remember once focusing for a half-term on having the best possible routine during the morning register. I analysed everything, from the logistics of my classroom layout to the little games and activities that were provided for the children. I would secretly time how long it would take children to be settled, pick through how children engaged with our little morning tasks and constantly make small tweaks to that part of the day. In 6 weeks I had a routine that I used successfully (without much further thought) for many years.

2. When making changes to my teaching, seek specific feedback
I love reading research and getting new ideas. When I first try out a new technique, it’s common that my first attempt(s) don’t go that well. For example, after reading ‘How I Wish I’d Taught Maths’ (Craig Barton) I tried out using ‘hinge-point’ questions as short mid-lesson assessments. At first I wasn’t skilled at exactly when/how to use these questions. I’d always arrange for another teacher to be in my class at those moments (even for just 5 minutes) so after we could unpick what worked and what could be improved.

Equally, I remember my first term in year 1. I would plan lessons with my partner Y1 teacher, but knew that her class were getting better outcomes from those lessons than mine. I learnt so much from popping my head in her classroom and watching what she was doing differently to me at specific moments. Or let’s say my focus is on the engagement of a target group of children during the plenary. I might use a TA to make specific observations about the actions of those children so I have better feedback on the success of a particular approach.

By constantly making small improvements to specific parts of my teaching, I hope that in 20 years’ time I will still be getting a bit better at doing my job every day!

I See Problem Solving – UKS2: providing challenge

I See Problem Solving – UKS2 is designed to transform the teaching of problem-solving in mathematics. Its design addresses Recommendation 3: teach strategies for solving problems from the recent EEF report. It will give all children the opportunity to understand and answer non-standard questions, whilst also providing appropriate challenge. This blog focuses on how extra challenge is provided to deepen and extend children’s learning in each task.

Each task begins with the main prompt question. For some children, answering this question may be their ‘Everest’; others will need more challenge.

First of all, a number of the tasks include a ‘how many ways’ prompt, with the challenge coming from finding all possible answers. Here’s a typical example of a ‘how many ways’ task:

And here are two other initial prompts:

Each task comes with an Explain prompt, where children have to unpick likely misconceptions, solve related problems, rank questions by difficulty or agree/disagree with different opinions. The prompts here are similar to those from I See Reasoning – UKS2 and will extend the thinking from the initial prompt. Here are the Explain prompts for the above examples:

For many children, the real challenge will come from the Extend prompt for each task. These tasks are related to the initial prompt, but the challenge is taken to the next level. These are the corresponding Extend examples (and two of the friendlier ones!):

This blog explains how extra support is also provided in each task.

I See Problem Solving – UKS2 includes a huge range non-standard problem-solving tasks spanning right across the curriculum. It costs £24.98.

I See Problem Solving – UKS2: providing support

I See Problem Solving – UKS2 is designed to transform the teaching of problem-solving in mathematics. Its design addresses Recommendation 3: teach strategies for solving problems from the recent EEF report. It will give all children the opportunity to understand and answer non-standard questions, whilst also providing appropriate challenge. This blog focuses on how extra support is provided to help children to ‘see’ the structure of the problems and to experience success.

The tasks are designed to be used at the end of a sequence of lessons, so children have developed their basic skills in that curriculum area. To start with, children are given the initial prompt – a question where there is not an obvious ‘standard’ approach to work out the answer. Here are two examples of the initial prompts:

There is then a ‘support’ prompt for each task which the children may choose to use. This will help children to understand the mathematical structure of the task. It may show a part-completed bar model, give some suggestions or offer a ‘way into’ the task. This will help all children to access the task and be more likely to taste success.

Then there is a ‘worked example’ to accompany each task. This is a series of images that shows the solutions modelled step-by-step, helping children to see the ‘deep structure’ of each problem. You will be able to download this for free as a PDF and/or as a PowerPoint file (available at iseemaths.com when the product is released). Here is a page from each of the worked examples from our two example tasks:


This blog explains how deeper levels of reasoning and extra challenge are then built into each task.

I See Problem Solving – UKS2 includes a huge range non-standard problem-solving tasks spanning right across the curriculum. It costs £24.98.

One of my favourite investigations

This is one of my very favourite mathematical investigations from I See Reasoning – UKS2: there’s a great pattern to explore. When I was in Y6 it was one of my ‘go to’ tasks for this time of year. Here’s our first discovery:

Despite having the same sum, the numbers give different products. And the further apart the numbers get, the smaller the product. But look at this:

There’s a pattern to how the products decrease: 1 less, 4 less, 9 less, 16 less. This is a pattern of square numbers. How odd! I wonder… is this the case for this example only? Or would it work for any example where the sequence starts from a square number? So the exploration continues, and we see that the pattern is repeated (e.g. 10×10=100, 11×9=99, 12×8=96, 13×7=91).

Eventually, I would challenge the children to use this knowledge to perform calculations. For example, consider 23×17. We know 20×20=400, so it follows that 23×17=391 (9 less than 400).

A beautiful pattern to explore!

 

The Vision: I See Problem-Solving

My philosophy has always been simple: be firmly rooted in educational research; find ways to apply evidence-based principles in the classroom; share the ideas that work best with other teachers. This approach led to me write the I See Reasoning eBooks, and it has driven the creation of my next resource, I See Problem-Solving UKS2.

In reference to problem-solving in maths, the latest EEF research states:
‘Explicit training appears essential; these sub-skills do not appear to derive from practice without direction and oversight.’

It also says: ‘Teachers should deliberately select visual representations that are appropriate to the problem’ and continues ‘provide prompts to encourage learners to monitor and reflect during problem solving.’

My belief is that I See Problem-Solving UKS2 will help us to explicitly teach problem-solving skills, helping all children understand the mathematical structure of each question. The resource will unpick a wide range of multi-step problems from right across the UKS2 maths curriculum.

Each task centres on a main question, like the example below:

Tasks are made more accessible by the ‘scaffold’ prompt which children have the option of accessing. This might be a part-completed worked example, a related example or some other prompt to break down the question:

The ‘explain’ prompt will provide a context for deeper thought or discussion, for example using ‘explain the mistake’ examples:

There is also an ‘extension’ prompt to provide further challenge based on the same task:

The mathematical structure of the problem is shown step-by-step and very visually by the ‘worked example’. This will be made available as a PDF or as a PowerPoint file. The worked examples can be displayed like ‘flip books’, showing each stage of the problem. Click here to see the worked example for this task – click through the pages rather than scrolling up and down for maximum effect!

I See Problem-Solving UKS2 is currently in production. It is being trialled by a large team of teachers who are sending me feedback on how the tasks can be improved. To join the team, register here.

I’m writing the resource over the 2018 summer holidays, will trial a few tasks in early September then I will aim for a September release – watch this space!