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Mathematical concepts – context is key Mathematical concepts – context is key

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Authors: Dave Tout
Mathematical concepts – context is key

In the typical mathematics classroom, especially in the middle years of schooling, we tend to use one model to connect maths with the real world; we start by teaching the maths content and skills, we then get students to practice and do some maths, and then we next might apply some of those skills into a real world context by using learning activities such as word problems.

However, when you look at other models, such as those used in international assessment frameworks like PISA, you will find that a different process is modelled.

Below is the diagrammatic conception of mathematical literacy developed for PISA 2012 (OECD 2013) which is based around the need to assess students’ capacity to transfer and apply their maths knowledge and skills to problems that originate outside school-based learning contexts. 

The diagrammatic conception of mathematical literacy developed for PISA 2012.

Source: OECD (2013). PISA 2012 Assessment and Analytical Framework: Mathematics, Reading, Science, Problem Solving and Financial Literacy. OECD Publishing.

The processes outlined in the model are key components of solving a real world problem, where the starting point is the problem in its context, not the maths.

  • Formulating situations mathematically involves identifying how to apply and use mathematics to the problem being posed in the real world—it includes being able to take a situation and transform it into a form amenable to mathematical treatment;
  • Employing mathematical concepts, facts, procedures, and reasoning involves using mathematical concepts, procedures, facts and tools to derive a mathematical solution;
  • Interpreting and evaluating the mathematical outcomes involves reflecting upon mathematical solutions or results and interpreting or evaluating them back in the context of the initial problem.

This PISA process contrasts with the typical model highlighted above of: teach some maths; practice some maths; apply some maths.

The process in the real world requires a set of different skills undertaken in the reverse order – starting with the initial ability to identify the maths in the context and formulate it as a mathematical problem. Then, the second step is to do the maths, and employ skills and knowledge. Students then need to interpret and evaluate the outcomes of the maths and reflect on how the maths result(s) applies and fits in with the real world.

The first task in this PISA cycle is not something we normally address in our maths classes, yet it is possibly the most complex and difficult of all the processes.

Most of the emphasis in a typical maths classroom is on the first two processes: teach some maths; practice some maths. If we have time we may then apply that maths often through a word problem—but the maths skills have already been identified and formulated for the student (and as I said in Connecting Maths to the Real World, word problems are often not realistic or connected to real-life issues).

So, what can we do?

  • Use a problem solving, investigative, open-ended approach—use real texts and real situations—make connections between maths and the real world;
  • Start from the real world—teach students how to identify and extract the maths from the messy, real-life situations that they are likely to face (what I often call excavating the maths);
  • Part of this is to make the maths explicit and then, when the need arises, or gaps in knowledge appear, teach the maths that is required.

For the engaged and the disengaged student, and the competent and less competent student, the context can provide the challenge, the motivation and the purpose for understanding and using mathematics. However, in order for this to work in the classroom, we need to explicitly integrate the above PISA cycle of skills into teaching and learning.

In the next article I will discuss further how such an approach might work.

References

OECD (2013a). Mathematics Framework. In OECD (2013). PISA 2012 Assessment and Analytical Framework: Mathematics, Reading, Science, Problem Solving and Financial Literacy. OECD Publishing. Retrieved from http://www.oecd.org/pisa/pisaproducts/PISA%202012%20framework%20e-book_final.pdf

In the typical mathematics classroom, especially in the middle years of schooling, we tend to use one model to connect maths with the real world; we start by teaching the maths content and skills, we then get students to practice and do some maths, and then we next might apply some of those skills into a real world context by using learning activities such as word problems.

However, when you look at other models, such as those used in international assessment frameworks like PISA, you will find that a different process is modelled.

Below is the diagrammatic conception of mathematical literacy developed for PISA 2012 (OECD 2013) which is based around the need to assess students’ capacity to transfer and apply their maths knowledge and skills to problems that originate outside school-based learning contexts. 

The diagrammatic conception of mathematical literacy developed for PISA 2012.

Source: OECD (2013). PISA 2012 Assessment and Analytical Framework: Mathematics, Reading, Science, Problem Solving and Financial Literacy. OECD Publishing.

The processes outlined in the model are key components of solving a real world problem, where the starting point is the problem in its context, not the maths.

  • Formulating situations mathematically involves identifying how to apply and use mathematics to the problem being posed in the real world—it includes being able to take a situation and transform it into a form amenable to mathematical treatment;
  • Employing mathematical concepts, facts, procedures, and reasoning involves using mathematical concepts, procedures, facts and tools to derive a mathematical solution;
  • Interpreting and evaluating the mathematical outcomes involves reflecting upon mathematical solutions or results and interpreting or evaluating them back in the context of the initial problem.

This PISA process contrasts with the typical model highlighted above of: teach some maths; practice some maths; apply some maths.

The process in the real world requires a set of different skills undertaken in the reverse order – starting with the initial ability to identify the maths in the context and formulate it as a mathematical problem. Then, the second step is to do the maths, and employ skills and knowledge. Students then need to interpret and evaluate the outcomes of the maths and reflect on how the maths result(s) applies and fits in with the real world.

The first task in this PISA cycle is not something we normally address in our maths classes, yet it is possibly the most complex and difficult of all the processes.

Most of the emphasis in a typical maths classroom is on the first two processes: teach some maths; practice some maths. If we have time we may then apply that maths often through a word problem—but the maths skills have already been identified and formulated for the student (and as I said in Connecting Maths to the Real World, word problems are often not realistic or connected to real-life issues).

So, what can we do?

  • Use a problem solving, investigative, open-ended approach—use real texts and real situations—make connections between maths and the real world;
  • Start from the real world—teach students how to identify and extract the maths from the messy, real-life situations that they are likely to face (what I often call excavating the maths);
  • Part of this is to make the maths explicit and then, when the need arises, or gaps in knowledge appear, teach the maths that is required.

For the engaged and the disengaged student, and the competent and less competent student, the context can provide the challenge, the motivation and the purpose for understanding and using mathematics. However, in order for this to work in the classroom, we need to explicitly integrate the above PISA cycle of skills into teaching and learning.

In the next article I will discuss further how such an approach might work.

References

OECD (2013a). Mathematics Framework. In OECD (2013). PISA 2012 Assessment and Analytical Framework: Mathematics, Reading, Science, Problem Solving and Financial Literacy. OECD Publishing. Retrieved from http://www.oecd.org/pisa/pisaproducts/PISA%202012%20framework%20e-book_final.pdf

As a maths teacher, how are you connecting maths and the real world in your classroom?

Do you use activities that reflect the real-life situations that your students are likely to face beyond the school gates?

What processes do you use for the formulation aspect of solving a problem?

As a maths teacher, how are you connecting maths and the real world in your classroom?

Do you use activities that reflect the real-life situations that your students are likely to face beyond the school gates?

What processes do you use for the formulation aspect of solving a problem?

Michelle Crowfoot 06 November 2014

As I have recently finished analyising NAPLAN and am looking at introducing the new curriculum in 2015. ..this article has given me some serious food for thought..It may support our ideas around moving children beyond the middle bands but also applying a more practical application to maths and therefore increasing engagement through relevance

Jim 17 November 2014

At the time of President Obama’s inauguration in 2009 there was much discussion about the size of the crowd present in Washington that day. His supporters claimed it was the ‘biggest ever’ at an inauguration, while detractors quoted much lower estimates. ‘The Age’ newspaper had a large overhead photograph of the crowd in National Mall in Washington. My Year 10 class was given the task of creating its own crowd estimate. They researched the dimensions of the Mall, and estimated the crowd density (close packed in front of the big screens and sparser elsewhere). Their consensus figure was ~1.1 million, which fell neatly in the middle of the politically-motivated extreme values. They found that crowd size estimates are a part of the broader political debate: how many people actually tuned out for that rally on the weekend ... At the end of the exercise one student asked the expected ‘Is this part of our assessment?”. When told that is wasn’t she was a bit surprised: ‘So we just did it because it was interesting to know the answer’.
Lately a colleague in the humanities area made a succinct statement of what should be the basis of assessments:
1. an authentic context
2. a question worth asking
3. an answer worth having.

Jim Spithill, ACER

Dave Tout 17 November 2014

Thanks Michelle for the comment and hope the food for thought grows into something rich for your learners. And thanks Jim for the response. I like the succinct statement about the three key elements of a good assessment. I think this connects very neatly in with what I have been trying to get at, esp in my previous article (Connecting Maths to the Real World), and I think it was how we tried to view the development of the assessment questions for PISA 2012 too - but it isn’t easy to do this in our teaching or assessment! It’s a serious challenge, but a good one. I’ll see what I can say in the next article to help.

Nicole 18 November 2014

It starts in early childhood. Mathematical language needs to be part of our daily lives with children so they learn to become part of a numerate world. Our role is to guide children to participate confidently and actively. Intentional teaching as part of the EYLF promotes many strategies to lay good foundations in numeracy but what we can’t overlook is everyday problem solving abilities that assist children to become creative independent thinkers.

Louky 29 April 2015

I work in a support unit and as part of teaching our Life Skills maths program we apply maths to everyday situations. For example, our students take orders for our soup kitchen, calculate change, cook, measure quantities, tally orders, expenses & profit made. All done within a tight timeframe. This is how maths should be taught.

Dave Tout 30 April 2015

Hi Louky - that sounds like great practice in how you can (easily) connect maths with the real world and give learning and knowing maths a purpose and meaning. What age are the learners?

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