The Great Mugg \u0026 Bean Mystery?

$Page 1: The Great Mugg \u0026 Bean Mystery?$
Learning and Teaching Mathematics, No. 6 Page 29

Learning and Teaching Mathematics, 6, 29-35

The Great Mugg & Bean Mystery?

Marc North St. Annes Diocesan College, Natal About the author: He is a teacher at St. Anne’s Diocesan College, a private school in Hilton (near Pietermaritzburg in KZN). This year he is teaching Mathematical Literacy in Grades 12 and working as a consultant doing Mathematical Literacy teacher training. He has authored/co-authored three Mathematical Literacy textbooks (Grades 10, 11 and 12) and is the examiner for Mathematical Literacy for the Independent Examinations Board (IEB). He has a wife (Kate) and two small children – Joshua (4 years old) and Elizabeth (2 years old), and enjoys eating and running. He is also passionate about Mathematical Literacy and believes whole-heartedly that it is the next best thing to coffee and pancakes. This article has been produced with the permission of the Mugg & Bean restaurant group.

The Mystery? In July last year (2007) I attended the AMESA conference in White River at Uplands College. Having been given strict instructions by my son to bring him back a present, on the last day of the conference I headed off to Nelspruit to do some shopping before my airplane trip home that afternoon. During the course of my shopping I spotted a Mugg & Bean restaurant and given that they offer a “bottomless coffee”, I decided to stop for a cup or two. When I asked the waiter for a cup of coffee, he proceeded to bring me a mug rather than a cup and, in true Mugg & Bean style, the mug was enormous.

Mugg & Bean Cup

Mugg & Bean Mug (Note: I have deliberately made the pictures above different sizes to try to show as accurately as possible the difference in size between the Mugg & Bean mug and the Mugg & Bean cup. In other words, the different size of the pictures provides a realistic impression of the actual size of the mug in relation to the cup.)

Page 30 Learning and Teaching Mathematics, No. 6


This got me thinking – why does the Mugg & Bean offer customers a mug of coffee when the mug clearly holds more coffee than an ordinary cup? Surely, by offering their customers a mug of coffee, they would be losing money because their customers would drink more? In an attempt to answer these questions I decided at this point to do an experiment. I asked the waiter for an empty cup and for a large glass of water. Despite several puzzled glances from other people sitting in the restaurant, I proceeded to pour some of the water from the glass into the cup, filling the cup almost to the brim. I then poured the water from the cup into the mug. Step 1

Pour water from the glass into the cup

Step 2 Pour water from the

cup into the mug The result: the water from the cup filled approximately two-thirds of the mug. In more mathematical terms, the volume of the cup is approximately two-thirds the volume of the mug. What could possibly have possessed the Mugg & Bean to offer their customers’ mugs that hold approximately one-third more coffee than a cup? To answer this question, let’s think a little bit about stomach (or bladder) capacity when drinking coffee. Given how much I hate cold coffee, when I eventually started drinking the mug of coffee, I continually drank (and drank and drank …) and, despite the ludicrous amount of coffee in the mug, I managed to finish the coffee in a relatively short amount of time. It definitely took me longer to finish the mug of coffee than it normally takes me to finish a cup of coffee, but not by much. Having drunk my mug of coffee in a relatively short amount of time (and having received my caffeine fix for the week), by the end of the mug there was certainly no more space left in the various orifices of my body to accommodate another enormous mug of coffee. And so, I stopped drinking coffee, asked for the bill and, feeling very full, headed for the nearest toilet. Now let me consider what usually happens when I arrive at the Mugg & Bean and am presented with a cup of coffee. By the time I leave I have drunk a minimum of two cups and sometimes even three cups of coffee. The reason for this? A Mugg & Bean cup of coffee contains less coffee than a mug. This means that I am able to drink a cup of coffee at a comfortably pace without having to worry about the coffee getting cold or without having to gulp down the coffee. It also means that I am able to drink a cup of coffee without feeling too full at the end of the cup. And so, a few minutes break after my first cup, I am fairly soon in the mood for another cup. And, if my stomach and bladder can cope, I sometimes even make it through for a third cup.



What does this mean mathematically?

No. of mugs I can drink (comfortably) = 1 No. of cups I can drink (comfortably) = 2

1 cup ≈ 32 of a mug

∴ 2 cups = 32 of a mug +

32 of a mug

= ( 2 + 32 ) of a mug

3

= 4 of a mug 3

= 1 31 of a mug

In other words, if I drink two Mugg & Bean cups of coffee instead one Mugg & Bean mug of coffee, then I

will drink approximately 31 more coffee. Or put another way, if I drink a Mugg & Bean mug of coffee, I will

drink 31 less coffee than if I drink 2 cups of coffee.

What could this mean? Well, perhaps the Mugg & Bean are trying to get people to drink less coffee in order to help them to reduce the amount of caffeine that they take in at a sitting? Perhaps they are doing it because many of their customers genuinely enjoy drinking coffee out of a mug? Or perhaps, just perhaps, they are doing it to maximise profits by ensuring that people drink less of their “bottomless” coffee? The only question that remains to be asked now is whether the Mugg & Bean are in fact in partnership with the toilet companies! The Mug & Bean Mystery(?) and Mathematical Literacy In discussions with Mathematical Literacy teachers, there is a great deal of difference of opinion in how we should be teaching Mathematical Literacy and what we should be focussing on in our teaching. Some teachers argue that exploring real-life contexts are the most important aspect of the subject. Others argue that the mathematical content of the subject is primary. Still others argue that a balanced focus on context and content is important (Graven and Venekat). In my own view, as important as both context and mathematical content are, there is another goal that is primary in the teaching of Mathematical Literacy − namely, the development of problem-solving skills. I will explain this by referring to the issues of context, content and problem-solving skills in relation to my experience at the Mugg & Bean. Context: To begin with, I need to make it explicitly clear that this article is in no way an attempt to discredit the Mugg & Bean. The Mugg & Bean is one of my favourite restaurants and I enjoy not only their bottomless coffee but also their generous and scrumptious helpings of food (and no, I am not being paid by Mugg & Bean to say all of these wonderful things). In fact, this article is not about the Mugg & Bean at all. The Mugg & Bean is simply a particular real-life context in which I discovered a problem relating to volume. The article could just as easily have been written about any number of coffee shops or restaurants which serve coffee in different sized cups or in both cups and mugs. In other words, the problem is not confined to the context of the Mugg & Bean and prior knowledge of or experience with the Mugg & Bean is not a prerequisite for being able to understand and solve the problem.



The same principle applies in the teaching of the subject Mathematical Literacy. It would be naïve to think that in the Mathematical Literacy classroom we will be able to expose our students to all of the real-life contexts that they will encounter in their lives and the mathematics that exists within those contexts. Rather, we will expose students to real-life contexts that are not only relevant to them, but that also allow us to highlight certain mathematical principles. As such, context in the Mathematical Literacy classroom is simply a tool to help students to make sense of specific mathematical content and skills that they are learning about. Mathematical Content: In mathematical terms, the Mugg & Bean cup versus mug problem was about comparing the volumes of the cup and the mug. The first thing to notice is how simple the mathematics was that I needed to have an understanding of in order to solve this problem. At no point did I need knowledge of complex mathematical calculations to find a solution to the problem. In fact, I actually did not have to do any calculations at all. Rather, what enabled me to solve the problem was an understanding of what volume is and of techniques for estimating the volumes of different objects. The same principle applies in Mathematical Literacy. As the National Curriculum Statement for Mathematical Literacy (NCS) states in referring to the “Purpose” of the Mathematical Literacy:

“In the teaching and learning of Mathematical Literacy, learners will be provided with opportunities to engage with real-life problems in different contexts, and so to consolidate and extend basic mathematical skills.” (My emphasis) (DoE. 2003. p.9)

The second point to consider is that in the same way in which the context provides a tool for helping to make sense of the mathematical content that is being taught or learned, the mathematical content is also simply a tool that enables us to make sense of the contexts in which we encounter the mathematics (Brombacher,). In the context of the Mugg & Bean, knowledge of volume enabled me to solve the problem of whether or not I should drink coffee from a cup or a mug when visiting the Mugg & Bean. What this implies for the Mathematical Literacy classroom is that as important as it is for our students to understand basic mathematical concepts, the understanding of these concepts is not the be-all-and-end-all of their learning. Rather, knowledge of mathematical content is simply a tool that our students can use to help them to make sense of real-life contexts. Problem Solving: As important as both context and mathematical content are in the Mathematical Literacy classroom, I believe that to be a mathematically literate person implies much more that simply working with mathematical content in real-life contexts. Rather, as the OECD / PISA study points out,

“The extent of a person’s mathematical literacy is seen in the way he or she uses mathematical knowledge and skills in solving problems.” (my emphasis) (2003. p.30)

The NCS echoes a similar sentiment when defining the subject Mathematical Literacy and providing a description of the purpose of the subject:

“It enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems.” … “Furthermore, Mathematical Literacy will develop the use of basic mathematical skills in critically analysing situations and creatively solving everyday problems.” (my emphasis) (DoE. 2003. p.9)

In other words, being mathematically literate implies having the ability to realise that a problem actually exists in the first place; it involves being able to identify the appropriate mathematics needed to solve the problem; it involves being able to do the necessary mathematical calculations; and it involves being able to



interpret the results of the calculations in order to come to some sort of decision or deduction about the original problem. In short, being mathematically literate implies having the ability to problem solve. The OECD / PISA study refers to this process of problem solving as “mathematisation” and suggest that in solving problems in real-life situations people will often employ the following steps or cycle: Real-Life Problem

(Adapted from: OECD. 2003. p.38)

Identifying mathematics in the problem

Turning the problem into a mathematical problem

Solving the mathematical problem

Reinterpreting the mathematical solution to make sense of the real-life problem

As an example of how a person might employ this technique for solving a problem, consider a painter who needs to paint a room and must decide how many litres of paint to buy.

The real-life problem is about painting a room and buying paint.

The mathematics inherent in the problem involves surface area and converting units of measurement from m2 to litres.

To help him to make sense of the problem, the painter might draw separate rough drawings of each of the walls in the room.

To solve the mathematical problem, the painter will calculate the surface area of all of the walls in the room and subtract the areas of any doors and windows. The painter will also use a conversion ratio to determine the number of litres of paint needed.

If the mathematical solution provides a decimal answer, then the painter will need to realise that it is impossible to buy a decimal number of litres of paint and that he will have to round up to the nearest litre as rounding down will leave him with too little paint.

It is my belief that alongside exposing our students to various real-life contexts and teaching a variety of mathematical concepts, our primary aim in the teaching of Mathematical Literacy should be the development in our students of a set of skills that will enable them to “mathematise”. Implications for the Teaching of Mathematical Literacy: If the ultimate goal of the subject Mathematical Literacy is to assist our students in the development of a set of skills (competencies) that can be used to solve problems that they may encounter in their daily life (OECD. 2003, cited in Brombacher), then this has implications for the way in which we should be teaching the subject and the way in which the subject will be assessed. If the ability to problem solve or “mathematise” is seen as key in the teaching of Mathematical Literacy, then any national exam for the subject will need to assess to what extent students possess this ability. This would mean that the exam would contain several problems based in real-life contexts in which students would be required to identify the mathematics inherent in the problem, use appropriate mathematical



content and techniques in order to solve the problem, and then make sense of the solution in terms of the real-life context in which the problem is situated. And so to be true to the purpose of the subject and to prepare our students properly for the matriculation examination, in our teaching we need to teach our students how to problem solve and “mathematise”. It is my belief that this can be achieved by exposing students in a gradual way to various real-life contexts in which they are required to utilise a variety of mathematical content and skills in order to answer questions relating to the real-life context. It is with this belief in mind that I propose the following approach for the teaching of Mathematical Literacy at each grade. Grade 10

In Grade 10 primary focus is on teaching specific mathematical content and competencies, and the application of that content and competencies to a variety of different contexts. This will help students to develop an understanding of how a single routine or concept can be used to solve problems in a variety of situations.

Grade 11 In Grade 11 there is a dual focus: the primary focus is still on learning and consolidation of essential mathematical content and competencies, and the application of content and use of skills in a variety of contexts. There is a secondary focus on exposing students in a limited way to contexts in which they are required to utilise several strands of content or skills to solve problems. In other words, there is move towards teaching students how to combine knowledge and skills to problem solve. Grade 12 In Grade 12 there is still a dual focus, but primary focus is now on exposing students to a variety of problem solving scenarios in which use of several content strands and competencies are needed to solve problems; secondary focus is on the learning of new and consolidation of old content and competencies.

Context

Context

Context

Content/ skills

Context

Context

Context

+Content/ skills

Context

Content/ skills

Content/ skills

Content/ skills

Context Content/ skills

Content/ skills

Context

Context

+Content/ skills



Parting Words To sum up, for me, being mathematically literate is about more than just working with mathematics in real-life contexts. It is about having the ability to interpret contexts and use mathematical content in order to make decisions that will impact positively on your life. Being mathematically literate is about decision-making in daily life and the use of mathematics to achieve this goal. One final comment: in case you are wondering, I spent so much time at the Mugg & Bean thinking about the “Cup versus Mug” mystery that I arrived late at the airport and missed my plane. The extra night’s accommodation in White River, plus another airplane ticket, plus an extra extra-large present for my wife (given that I missed my flight on our wedding anniversary!), cost me well over R1 000,00. Not so clever for someone who claims to be mathematically literate, hey? References: Brombacher, A. (2007). Mathematical Literacy − A Reader. Cape Town: Bateleur Books. Department of Education (2003). National Curriculum Statement Grades 10 – 12 (General) Mathematical Literacy. Pretoria: Government Printer. Organisation for Economic Co-Operation and Development (2003). The PISA 2003 Assessment Framework – Mathematics, Reading, Science and Problem Solving, Knowledge and Skills. Graven, M and Venekat, H. Teaching Mathematical Literacy: A Spectrum of Agendas, in Setati, M., Chitera, N. & Essien, A. (Eds) (2007). Proceedings of the 13th Annual National Congress of the Association for Mathematics Education of South Africa. p.338 – 344.

Documents

The Great Mugg \u0026 Bean Mystery?