For the last year year we have been presented with a mathematical problem as the opening item on every news bulletin. We have been presented with information about the rate of spread of Covid-19, or the number of cases in our area, or the probability of the spread of the virus being impeded by taking a range of measures. The problem we have to solve is how to adapt our behaviour to minimise the risk to ourselves and to our families.
An activity that I have adapted to work with students on this problem uses a simple mathematical model. If I am working with a group of 30 students I will give them each a small, folded, piece of paper. One of the group has a cross on their paper, the rest are blank. I invite the class to mingle and then pair up. They open the paper. If they have ‘met’ the student with a cross on their paper they have ‘caught’ the virus and put a cross on their paper. We repeat this for several rounds and notice how quickly the virus has spread. We can then adapt the model to take health measures into account. We might suggest that social distancing reduces the chance of catching the virus by 50%. So, on this occasion if we meet up with an ‘infected’ colleague we roll a dice and only ‘catch’ the virus if we roll an odd number. We could suggest that wearing a mask reduces the chance of catching the virus to one in six. Now we only catch the virus if we roll a ‘one’. A simple mathematical model but one which can help our students in their decisions of how to act back in their real world. For older students they can devise their own simulations based on the actual data.
This is why I think teaching mathematics through problem solving (as opposed to teaching problem solving as a discrete part of mathematics) is vital. It helps our students interpret the world and helps them make good decisions about how to operate in the world.
I would argue that young children have an aptitude to solve problems. They notice mathematical problems around them and enjoy exploring the problem as much as feeling as though they have solved the problem. I became convinced of this when I was looking after my grandchildren, aged 8 and 3, during the first lockdown. Let me draw on two examples.
Which car will jump the furthest
My 8-year-old grandson loves his Hotwheels track. One morning he asked, “which sort of cars will jump the furthest?” You will notice that he is already generalising. Not asking which particular car will jump the furthest but which ‘type’. This meant that there was classifying to be done before we could start. The picture below shows the experiment he designed, including a sand pit so that he could measure accurately.
How can I fill a 100 grid?
I bought my 3-year-old grandson a set of numicon and a one hundred grid. Without any prompting he set himself the problem of how many different ways he could fill the container. I showed him images of other children working on the problem that had been posted on twitter and he loved being part of a global community working on the same problem
Here we have two different problems. One set in a real world be it playful and the other set in the mathematical world. I would also argue that through this problem solving approach I was meeting the requirements of the International Baccalaureate whose list of the aptitudes for learning suggest that students should be:
- Exploring, wondering and questioning
- Taking and defending a position
- Using critical thinking to understand a concept
- Making and testing theories
- Experimenting and playing with possibilities
- Solving problems in a variety of ways.
Can I ask you to pause to reflect for a moment and ask yourself to what extent do you meet these requirements in your classroom? I edit the journal Mathematics Teaching. One of the articles from MT which I keep returning to was written by Laurinda Brown. In this article she explored what we might mean by ‘being mathematical’ and suggested that by adopting a problem solving approach in our teaching we could support our students in becoming mathematical thinkers. She suggested a list of ‘mathematical’ behaviours:
- Being systematic and organised
- Being able to analyse situations and make generalisations
- Predicting and testing predictions
- Being precise in our definitions
- Developing precise mathematical vocabulary
- Being aware of the big picture whilst working in a particular direction
- Being able to pose our own questions
- Sharing ideas and working independently
- Challenging other people’s ideas by asking ‘What if …’ questions
I find this list useful when I interrogate my own teaching. I often look around classes that I am working I and make a mental note of the behaviours I am noticing. If I notice, for example, that students are not being precise in their definitions, I will think how I can support students in developing this skill.
I will finish this blog with two examples from the classroom.
How many bricks?
I wonder how many bricks are in this wall. One way to approach this is to ask students the following questions:
- Tell me a number you are certain is greater than the number of bricks in the wall?
- Tell me a number you are certain is smaller than the number of bricks in the wall?
We can then take an average of the midpoint of all these estimates before getting a closer estimate by counting one row and multiplying by the number of rows of bricks.
Number grid.
| 1 | 3 | 5 | 7 | ||
| 2 | 6 | 10 | 14 | ||
| 4 | 12 | 20 | 28 | ||
| 8 | 24 | 40 | 56 | ||
I have used this number grid with students in primary schools, secondary schools and in universities. I would encourage you to explore it for yourself and use it with your students whatever age you teach. Firstly, what do you notice. I leave this deliberately open – I am not expecting you to notice anything in particular. This is the first lesson in problem solving for your students. It is their problem – I do not have a recipe I want them to follow or an answer in my head for them to guess. Some answers I have heard:
- The odds go across the top
- I can see doubles
- The first column is powers of two
- The bottom right number in a 2 x 2 grid is the sum of the other three cells. (Is this always true I wonder? Why might this be?)
As you can see this question is easily a lesson’s worth of activity. Another question – does every number appear if I extend the grid vertically and horizontally to infinity? Early thoughts:
- Well, all the odd numbers are in the first row. So do all the even numbers appear in the other rows?
- Why does it look as though there are more even numbers than odd numbers?
A final question. If every number does appear how many times does it appear. Once or more than once?
Two more examples. Again, one set in the real world and one in the mathematical world.
I hope you will agree that the examples I have shared meet some of the requirements for mathematical thinking set by the IB and by Laurinda Brown. And if we are supporting our students in becoming mathematical thinkers through taking such a problem solving approach they will be better able to see the world through mathematical eyes and make important decisions about the actions they might take. Rather than teaching them
how to solve word problems we are supporting them in solving world problems.
Dr Tony Cotton 17/5/21
Tony is the editor of Mathematics Teaching, the journal of the Association of Teachers of Mathematics. For more information go to www.atm.org.uk . He is also the author of Understanding and Teaching Primary Mathematics, published by Routledge. This blog is written in a personal capacity.
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