Real Life Maths
"Sir, when will we ever use this in real life?", is a question I never tire of hearing... And for years, I have not had a good reply. My response has often been to try to shoehorn a supposed "real-life" context into my lesson, and believe me "shoehorn" is the correct way of describing it. Did David Beckham really consider the properties of quadratic graphs when lining up a free kick? Advocates of real life problems argue that they have a motivating factor, but as we will see in some of the papers that follow, this must be weighed up against the potential for misinterpretation. More recently, I have given up. I simply tell the students we are doing mathematics for the love, challenge and beauty of doing mathematics, and everyone has gotten on a whole lot better ever since. So, it was with some interest that I sought to see what educational research had to say about the use of real life contexts in mathematics.Research Paper Title: Relevance as MacGuffin in Mathematics Education
Author(s): Dylan Wiliam
My Takeaway:
In this brilliant paper, Dylan Wiliam looks at the use of "real life" maths problems in the classroom. Wiliam shows examples of where the context adds nothing at all to the question: " Alan drank 5/8 , of his pint of beer. What fraction was left?", as well as examples where the context can actually lead students to get the "wrong" answer: "a disc jockey plays two records. The first lasts 2 5/8 minutes and the second lasts 3 1/4 minutes. Find the total time taken to play both records" - the point being that DJs are likely to talk or play jingles between songs, and hence students may (sensibly) choose to account for this in their answer. There is a fascinating insight into how, when presented with a real-life problem, many girls seek to relate the problem to their existing knowledge, supplying any information that they feel is missing from their own experience, while boys are often content to tackle the problem in isolation from their previous experience. Finally, Wiliam suggests three criteria that should be applied to ensure that the contexts used in problems are useful:
1) Commonality - metaphors and contexts can be really useful to aid understanding, but they must be commonly shared by all students, otherwise they could lead to confusion
2) Match - does the task (and possible interpretations of the task) match the core mathematical activities you want to convey? Will students need to suppress/ignore attributes to engage in the mathematical activity you intended?
3) Range - how far does the model take you along your journey to understanding a topic? Wiliam cites the example of negative numbers: "Envisaging negative numbers as temperatures above or below zero may be very useful for understanding ordering of integers, and may even provide good pictures for adding directed number. However, contriving plausible scenarios where temperatures below zero are subtracted from each other, or are multiplied is rather more difficult!"
My main takeaway from this was to think really carefully about what the context is adding to the question. If the effect is neutral (as in the pint of beer question), then why bother with it - indeed we have learned from the Redundancy Effect in Cognitive Load Theory that pointless information imposes an unnecessary load onto fragile working memory? If it is negative (such as with the DJ, or if it is likely to disadvantage some students), then remove it. But if a metaphor or a framing can be used to impose extra clarity on the problem, without altering its underlying structure, then this would be valid. I will certainly be checking my examples more carefully in the future.
My favourite quote:
In the vast majority of classrooms, relevance is a MacGuffin—a device to motivate learners; to convince them that the activities they are given are somehow of the real world, even though they do not appear to be connected to it. Students know this, and as a result their thinking in mathematics lessons becomes divorced even from their thinking elsewhere in school, let alone the world outside school. Mathematics lessons thus become literally mindless—an activity in which students come to believe that thinking is not helpful.
Research Paper Title: The effect of using real world contexts in post-16 mathematics questions
Author(s): Chris Little and Keith Jones
My Takeaway:
Before we get into the study itself, the authors have a really nice way of summing up the dilemma of using real world contexts: "On the one hand, by making a connection between the abstract world of mathematics and everyday, or scientific, contexts, we are reinforcing the utility of mathematics as a language for explaining the patterns and symmetries of the ‘real’ world. On the other hand, if we manipulate and ‘sanitise’ real-world experiences to enable them to be modeled by a pre-ordained set of mathematical techniques, then the result can appear to be artificial and contrived". I like that! Anyway, in this interesting study, alternative versions of the same questions (all on sequences) were presented in explicit, algebraic, word and pattern contexts, and set to a sample of 594 Year 13 students (aged 17-18) in a one-hour test. The results showed that that setting sequence questions in real-world contexts does indeed add to the overall demand, though a context can on occasions provide ‘mental scaffolding’ to help the solver to use context-specific heuristic strategies. My key takeaway is that we need to think carefully what our aim is. If it is to ensure our students are comfortable with the basics, then there is little point setting questions in a real-world context. Interpreting the question and deciding what strategy to use all impose additional cognitive load to the student's working memory, and as we have seen in the sections on Cognitive Load Theory and Cognitive Science, that may mean that no learning actually takes place. However, the researchers did find that some real-world contexts provided additional support for students in guiding them towards the correct method, so it may not be as clear cut as we would like. Of course, if your aim is assess if students can apply their knowledge and skills to different contexts, then using real-world questions may be a way to do that. I guess we just need to be aware how we interpret students' results in that case - do they struggle because the fundamentals are not in place, because they do not understand what the question is asking, or because they have been misled by the context itself?
My favourite quote:
The potency of algebraic formulae lies in their universality and blindness to individual contexts, and, in resorting to context-specific thinking to solve these questions, students are avoiding the necessity to transfer and abstract from context to mathematical model, which is, arguably, the heuristic strategy intended by the questions.
Research Paper Title: Do realistic contexts and graphical representations always have a beneficial impact on students’ performance?
Author(s): D. De Bock, L. Verschaffel, D. Janssens, W. Van Dooren, K. Claes
My Takeaway:
This is an amazing study which looks at the effect of using video to aid mathematical explanation. One hundred and fifty-two eighth graders (13–14-year olds) and 161 tenth graders (15– 16-year olds) participated. They were give a paper-and-pencil test about the relationships among the lengths, areas and volumes of different types of rectilinear and non-rectilinear figures. The problems were administered in or out of an authentic context and setting, and either with or without an integrated drawing instruction, leading to four combinations of groupings. The "authentic context" involved students watching clips of a screen version of ‘Gulliver’s Travels’, a world in which all lengths are 12 times as small as in our (and Gulliver’s) world. In the drawing instruction groups, students were provided with a drawing of the (geometrical) object introduced in the problem, and they were asked to complete the drawing by making a reduced copy using the given scale factor. How did the groups perform on the test? Students who watched Gulliver’s Travels and who received the video-related items performed significantly worse than the students from the other groups. Surprisingly, students who had to make a drawing performed significantly worse than students from the non-drawing groups, although the authors admit this is probably due to the nature of the linear scale factor task itself. To explain the video results, one possibility suggested really resonated with me: students perceive video as a less difficult medium than written materials, and therefore are inclined to invest less mental effort in working with information transmitted by this easy medium as compared to media that are perceived as difficult. As we have seen in the Cognitive Science and Memory sections, students remember what they think about, and sometimes thinking needs to be difficult. When watching a video (or indeed any other form of novel media we might use in a lesson, such as a website, app, or music) are students thinking about the task itself or the form of media? Any increase in engagement and motivation must be weighed up against this.
My favourite quote:
Some of the qualitative findings indicate that the reason why these two new forms of help did not yield the expected positive effect, was that they conflicted with students’ implicit norms, expectations and beliefs about doing mathematics, especially about their appreciation of formal and informal strategies and of drawings as a valuable modelling tool. From that point of view, classroom interventions that are only partial and instantaneous, and that are unable to influence or alter these more fundamental attitudes and beliefs, have little chance of success. Most likely, only a long-term classroom intervention, not only acting upon students’ deep conceptual understanding of proportional reasoning in a modelling context, but also taking into account the social, cultural and emotional context for learning, can produce a positive effect in defeating the illusion of linearity.
Research Paper Title: The role of context in linear equation questions: utility or futility?
Author(s): Chris Little
My Takeaway:
Four linear equations questions were presented, each set in a context. Two of these questions were : 1) In 18 years time, Halley will be five times as old as he was 2 years ago. How old is he now? 4) The largest angle of a triangle is six times as big as the smallest. The third angle is 75°. Find the size of the three angles. The author argues that these questions may be regarded as ‘applications’ of linear equations, in the sense that they involve formulating and then solving them. However, they have no practical utility value - none of the results provide significant information about the context. So, what is the point in the contexts? They are certainly not motivating. But they are unlikely to introduce any of the misinterpretations highlighted in previous papers in this section. The author makes a really interesting point: whilst the underlying algebraic structure of all four problems is the same, the contexts may actually encourage students to try different methods to solve them - trial and improvement, for example. I have seen this a lot with my students - often the context prevents students from spotting the topic that the question intends to test, and as a result the students end up trying something different. My takeaway here is similar to the paper on post-16 students - think about what we want our students to achieve. Do we want them to practise solving equations? If so, then maybe it is best to cut out the context and just have the equations. If we want them to practice recognising when it might be appropriate to formulate a question in terms of a linear equation, then we have seen in the sections on Cognitive Science and Cognitive Load Theory that students will only be in a position to do that - and crucially, to learn from it - when the basic knowledge of how to solve linear equations is secure in long term memory. Otherwise, the context may inhibit their learning.
My favourite quote:
Some researchers into word problems (Greer, 1997) have advocated introducing more elements of realism into classroom tasks, for example by adding irrelevant information, which then has to be discounted by the solver. However, expecting students to engage in genuine mathematical modelling activity before they serve an apprenticeship in formulating algebraic equations, and learning abstract, analytical methods for solving them, is perhaps itself unrealistic. Many students find the process of translating real-life numerical concepts into algebraic variables demanding enough, without being deflected by realistic ‘noise’.
Research Paper Title: Intellectual Need and Problem-Free Activity in the Mathematics Classroom
Author(s): Evan Fuller, Jeffrey M. Rabin, Guershon Harel
My Takeaway:
I discovered this interesting paper following my interview with Dan Meyer on my podcast. The authors argue that intellectual need is necessary for significant learning to occur. Among other concepts, it introduces the need for computation. This is defined as the need to find more efficient computational methods, such as one might need to extend computations to larger numbers in a reasonable “running time.” The key here is that in order for the need for computation to be met, students need to experience the “longcut” before they learn the shortcut, for only then will they appreciate the power of that shortcut. Otherwise the techniques we teach seem like just another trick in the endless series of tricks students call maths class. This forms the basis of Dan Meyer's "Headach and Aspirin" series, where he asks the question: If Math Is The Aspirin, Then How Do You Create The Headache?. Dan argues that real world applications of many maths skills we teach in school are a lie. So if our theory is “maths is interesting only it’s real world,” then we will struggle to find interest in many of the things we teach. Instead, we should ask ourselves, “Why did mathematicians think this skill was worth even a little bit of our time? If the ability to factorise quadratic expressions is aspirin for a mathematician, then how do we create the headache?”. This approach has been incredibly influential in my teaching, and has enabled me to offer approaches into topics that I would never have thought of. I would strongly recommend checking out Dan's factorising quadratic expressions blog post to see how the headache and aspirin approach works. I will never introduce factorising quadratics in the same way again.
My favourite quote:
The need for computation is not a student’s psychological motivation to solve drill exercises on algorithms, but her intellectual recognition that realistic and compelling problems require the development of efficient computational methods for their solution