Learning to Problem-Solve: number sequences and negative numbers

This is the first in a series of blog posts about how to systematically teach problem-solving skills using I See Problem-Solving, outworking the EEF research (recommendation 3) about using rich problems to learn mathematics.

Here’s the task that the class were given part-way through the lesson:

Before we get to this point, I want to break down the sub-steps involved in answering this question. First, a little pre-teach group are given this open task to bring back some prior learning:

Then we show the first part of the question that we are building up towards answering and these three example sequences. The children calculate (or spot) the first and then the second negative number in each sequence:

Now the children have a go at this short task. They identify that -4 is the second negative number in the first two sequences. I explain that, when writing the first two sequences, I actually started from -4 and added in equal steps, rather than starting from the positive number and subtracting (which would be more akin to trial and error):

Now the children are equipped to deal with the task. We work to find all the possible answers, noting that the sequence must decrease by more than 3 but less than 7. There ‘support’ prompt for children who need help, and some children also attempt the ‘explain’ or ‘extend’ tasks:

The free I See Problem-Solving Worked Example is used to show the three possible solutions. The following day, we pick up on a few misconceptions and look for ways to become more efficient, including looking at the example above and considering how we could add a multiple of 4 and 5 rather than the repeated adding.

I’m trying to make problem-solving accessible for all children, whilst ensuring that every child is challenged. I hope you find I See Problem-Solving super-helpful. The LKS2 and KS1 versions are in production!

Also in this series:
Equals Means Same As
Sum of the Digits Place Value Challenge

Learning to Problem-Solve: sum of the digits

This is the second in a series of blog posts about how to systematically teach problem-solving skills using I See Problem-Solving, outworking the EEF research (recommendation 3) about using rich problems to learn mathematics.

When children have a really secure understanding of place value, I love using sum of the digit challenges. Here’s the task we’re coming to later:

The build-up focuses on the meaning of the sum of the digits. We start by ordering 74, 312, 214 & 47, and identifying how many digits in each number. Then we work out the sum of the digits for each, noting that the largest number, 312, had the smallest digit sum. To consolidate this skill, we have a go at this:

We also find numbers where the sum of the digits is 8. Example numbers that the children come up with include 53, 44000, 123500 and we even get an infinity sign for repeated zeros! Next, a quick recap on finding multiples:

After this, we are into the main task (number with sum of digits of 13, multiple of 4). Discussions were held about where to start: listing the multiples of 4, or finding all the 2-digit numbers with a sum of the digits of 13? The key question, it was decided was ‘which narrows down the possible answers more?’ Once the answer was found (76) it’s onto the explain and extend tasks:

 

We also made the point that, for the example above, we don’t need to cross out those beautiful workings out!

I’m trying to make problem-solving accessible for all children, whilst ensuring that every child is challenged. I hope you find I See Problem-Solving super-helpful. The LKS2 and KS1 versions are in production!

Also in this series:
Equals Means Same As Task
Number Sequences and Negative Numbers

Learning to Problem-Solve: equals means same as

This is the third in a series of blog posts about how to systematically teach problem-solving skills using I See Problem-Solving, outworking the EEF research (recommendation 3) about using rich problems to learn mathematics.

I’m in Y5, building up to a task which requires children to understand the meaning of the =, < and > signs. To help model this idea, I show the children an image that they may have seen in KS1 from the Early Number Sense I See Maths page: at first the circles are white; then we see them coloured red and blue:

A range of other visual representations are used to show equivalence, including the image below to represent 4×3=7+5:

And this one to show 4×3>7+3:

Then the children write part-number sentences using different operations that are equal to 8, 10 and 12. They are positioned on the correct board. After that, children move the statements to make balancing number sentences, and sentences using the < & > signs:

Now we are ready for the main task. The support feature gives a clue: start by thinking about where to position the 8. Some children progress to the explain task, spotting different mistakes:

There is a super-challenging extend task that some children will get to tomorrow. We continue to model = as balance using scales and Numicon.

I’m trying to make problem-solving accessible for all children, whilst ensuring that every child is challenged. I hope you find I See Problem-Solving super-helpful. The LKS2 and KS1 versions are in production!

Also in this series:
Number Sequences and Negative Numbers
Sum of the Digits Place Value Challenge

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!

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 – there are so many variables! At any one time, though, I try to have one very specific thing that I focus on improving, and spend an extended period of time developing that one 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.