The Mathematical Experience
This page is a discussion about 'What makes a good task?' The aim is to present the features of good activity that create powerful, effective and long lasting mathematical experiences for students from which they discover, learn and draw confidence through engagement. It is not intended as a return to college but more as a means to bring some issues into focus and start debates. It is by no means comprehensive and only serves to raise the issues with brief rather than detailed discussion!
A presentation to illustrate what we mean
I think that most of us are able to remember some of our most profound learning experiences, when circumstances conspired to help us gain a crystal clear understanding of an idea. This happens probably more often than we perceive and continues to happen way beyond our school days. It often happens when we have new experiences that lend new perspectives and present links that we have never noticed before. Along with this it's equally likely that we can all remember less positive experiences when we have tried and tried and still not been able to understand an idea. A principal aim as a mathematics teacher is to try to create and facilitate as many of the former as possible. To do so it is necessary to consider the elements that contribute to creating those experiences based on our own experiences as learners and teachers, those of students in our classes and those of colleagues. The implication is not that all of these elements need to be present but that we should consider them when we design tasks and activities.
Engagement & focus
These terms are often thrown about, but it is worth spending a moment to think about what they mean and what the implications are. For example as a parent, I have often wondered why my young children could always beat me at a game of memory and concluded that it is not just their young fresh minds, but the fact that I am guilty of not focusing enough on the game, I might try and carry on another conversation at the same time, or be thinking about something I am going to do later. In short their minds are ‘engaged’ by the game and as a result, they are totally focussed on the task at hand in a way I am not There is effectively no contest. This, in turn, makes me wonder about how often I am completely engaged and focused on a task and when that happens. The following are some answers,
- Something about the activity pulls you away from distractions around you
- You have an instinctive desire to do the activity
- The activity grabs your attention
- The goal or challenge appeals to you
- The activity is enjoyable, it gives you pleasure
On writing it the list seems obvious, but I wonder how often these things are considered in the design of tasks with the same importance as a learning objective.
When a task or activity truly engages you it has the effect of consuming your mind for a period of time. If a task is well designed it creates that level of engagement and focuses it on a particular goal so that the mind is as effective as it can be for that period of time. I often feel that this is our only concern as teachers and that everything else will come if we create engaging tasks that encourage focus. It is truly remarkable what we can all achieve if we have total focus. In many ways, what follows is mostly about achieving this level of engagement and focus.
No - engagement does not imply learning, but does learning ever happen without it?
Modelling Quadratics - Find curves in monuments, buildings and water jets and fit a parabola to them using dynamic geometry software
The use of context in mathematics teaching is a fascinating topic for debate. During a postgraduate research project it became clear to me that it is a mistake to view context simply as ‘real world applications of mathematics’. Mathematics can be taught in the abstract, a semi reality or through real world applications all with equal levels of engagement and success. Below are some examples
- Abstract concepts can be explored through games, puzzles and well phrased questions where the game, puzzle or questions provide the context.
- A semi-reality can be constructed to give an indication of a possible application and/or to provide a context for exploring the mathematics.
- Real world applications can be used to demonstrate the need for and purpose of the mathematics.
Within each of the above there are pros and cons
A game, puzzle or question has to have a suitably engaging level of challenge associated with it so that it does not draw the question, ‘Why do I have to do this’. For example, if I were to ask, ‘What is the largest prime number you can think of?’ this question is likely to prompt some serious thinking amongst anyone who knows what a prime number is, for a short while at least, but should soon give rise to the question, how can you prove it is prime, which opens up a whole world of fascinating mathematics. The challenge for teachers here is get as much engagement out of this before anyone asks, ‘who cares?’. Of course, it is a good question and it can be answered, but at the point when we start answering that question, we change the nature of the activity.
Semi reality, limits the reality of a context so as to keep the level of Mathematics appropriate or simply to create engagement. For example, "a long lost relative rings you up and announces that they would like to share their fortune with you and that you must choose if you would like to receive $1000 a year or 1$ in the first, $2 in the second, $4 in the third and so on doubling each year". The context is ‘semi real’ but designed to engage and to be a simple exploration of the exponential function. The point here is that discussion usually centres around which is the best option, not about how ‘unreal’ the context is. The danger with ‘semi reality’ is when it is used as an attempt to put mathematics in to relevant contexts but the limits on the reality reduce the relevance to a degree where the exercise has little point. The classic example is the question that gives you the maximum capacity of a lift/elevator and asks how many times it will need to go up and down to carry a given number of people.
Mathematics is an extremely powerful tool for understanding our world and our universe and as such setting contexts to demonstrate this can be equally powerful. Using real data for statistical analysis might be more useful than using fictional data for example - but something abstract like Anscombes quartet is equally worth exploring. The danger with real applications is two fold, firstly, real world problems are rarely clean and straightforward and so bring mathematical complications that are always interesting but may provide barriers. Secondly, real does not necessarily imply relevant or interesting for any given group of students!
In summary there is no ‘best way’ to use context, only some considerations that should be made about why we use context and what it brings with it.
Predict the future - See real world data add power to the teaching of moving averages
Interest and relevance
Following on from the previous paragraph about real world contexts it is important to consider ‘interest’ and ‘relevance’ for students in task design. It can, of course, be a great challenge to understand what students perceive as interesting and relevant and one obvious solution to this is to ask them regularly about it and certainly to pay attention to have the receive the ideas presented to them. As a result, there is no list that can be provided on this but only a need to think it through first.
For example a lesson comparing mobile phone tarifs through linear functions is more likely to be relevant to a class full of mobile phone owners than a similar lesson on taxi tarifs. This of course depends on who pays the bills! Looking for a correlation between height and armspan, would be more interesting to classes using data about themselves than data from another class or fictional data.
Sprint or endurance - students think about their own strengths and attributes
In the opening paragraph of this page, I talk about remembering key moments in our education! Creating memorable experiences is important, being able to remember the occasion and the circumstances in which you learned something helps to remember that which you learned. Of course, the difficulty here is that if everything we did was truly memorable then less of it would stand out as so, but that should not prevent teachers from aiming to create memorable experiences. Memorable experiences usually have a feature that is in someway unique. This may be related to the type of task, resources .used or the location. There may be some audio or video association in the form of a song or youtube clip. Often there is a product from the experience that reminds us when seen, a photo, video, display or model or something like that.
When designing a task it is always worth thinking in advance about what the unique feature of that task may be that will help it to become a memorable one.
Dancing Vectors -Teach vectors through a dance routine where dance moves are vectors!
Human Loci - Let the students be the points that obey the given rule.
Human Coordinates - Students act as coordinates in a huge grid of chairs
Building mathematical links
Nothing is quite so profoundly understood or as memorable as it is when you discovered it for yourself! That is not to say that you weren’t guided to that discovery, but that it was you that eventually put two and two together, if you’ll forgive me. As teachers, most of us have experienced those moments when students start conjecturing in the class about what they think is going to happen, what they think might be a generalisation. Students will then argue with each other by citing particular cases, then modify the conjecture and so on and so on… These are truly great moments for teachers but mostly for students. When a student does this for themselves, they realise that they are empowered to investigate and discover at will and that it is they who are most in control of what they do and don’t learn, do and don’t understand. It may not happen all the time, but it is a good aim to create these circumstances as part of the whole picture.
A quick example might be to ask students to draw several 30, 60, 90 triangles, (preferably using dynamic geometry) and measure the side lengths. If students divide the side opposite the 30 degree angle by the hypotenuse they will always get 0.5. This instantly gives meaning to the idea that sin 30 = 0.5. It usually gives rise to the question, what if I divide this side by that side etc, will that give me the same answer too? What if the angles are different? These and other questions lead to the basic principles of trigonometry, without students ever being told, sin30 = 0.5. All the teacher adds is the conventional names for the ratios. Moreover, beyond just learning the mathematical content, they are learning a way of working. The Scientific method of experimentation, recording data, hypothesising, testing hypothesis which then, in mathematics, can eventually lead to proof. They are learning a working methdology, which can be applied in almost all fields of endeavour, to great effect.
Discovering Circle Theorems - Exploration and discovery with dynamic geometry
Discovering SOHCAHTOA - An investigation to discover the 3 trigonometric ratios using dynamic geometry software.
Having talked already about discovery, it's worth talking about what exactly students are trying to discover. This brings us to questions about learning objectives and syllabi etc. Most of what happens in a maths classroom though is about degrees of generalisation and mostly the conjectures that students make in the process of discovery are generalisations. Generalising is at the core of our subject and so activities should, where possible, provide opportunities for students to make and refine conjectures, to articulate and test generalisations and consider the use, limits and conditions of those generalisations. The ability to recognise pattern is arguably at the heart of the human capacity to evolve faster than its animal competitors.
Operating Fractions - Find values that always make generalisations work!
Much emphasis has already been placed on what students do for themselves during the course of an activity and the more they do for themselves, the more decisions they are responsible for making, the more ownership they have over the activity. Often ownership gives rise to engagement and, in turn, focus. The difficulty is giving just the right amount of freedom to individuals without either opening them up to pathways too difficult or restricting them to the point of disinterest.
Freedom for ownership can be achieved in small ways though. For example, the above activity about drawing triangles was not so prescriptive that students would all have the same triangles. A student or group could decide to draw an enormous one or a tiny one and this could be encouraged as part of understanding there are an infinite number of possibilities.
The Rice Show - Huge piles of rice represent different numbers of people!
Visualising Indices - What does 3 to the power of 4 look like?
Richness of mathematics is probably a relative concept and as such difficult to define, but if students are prompted to investigate or explore an idea then it is hopeful that there are numerous interesting discoveries for them to make along the way. This implies that a degree of relative richness is inherent in any given activity.
One such example might be in exploring the sets and subsets of quadrilaterals. Starting with the statement ‘All squares are rectangles’ what other similar statements can be made and how can they be structured in a logical order? Can a diagram be drawn that illustrates all these statements? The task is approachable to anyone who knows and understands the properties of quadrilaterals, but the final structure of the subsets is much harder to get at and turns up some things.
Skewey Squares - Squares, areas and Pythagoras' theorem
3-2-1 Blast Off - Calculate the height reached by a water powered rocket using right-angled trigonometry.
Without wanting to open the 'Learning styles' can of worms the topic of variety is definitely something to be considered. Whatever your view, as much as this is possible, it can only be a good thing to be aware of the impact of different media and forms of communicating and representing ideas. What the 'learning styles' episode has done for me as a teacher and my classes is remind me how important a variety of experiences are in terms of engagement , creating memorable experiences as broadness. Teachers should consider the huge and increasing (with technology) variety of possibilities there are for different ways of learning.
Being able to explain yourself can be seen as having reached the ultimate level of understanding. It's one thing to follow an idea and put it into practice, but another altogether to explain it to a third party, to find the words and examples you need to make it clear. As teachers, we know this more than most. There are times when we can't find the words and examples and need to think in greater depth to find something satisfactory! For these reasons, it is important to ask students to try and do this as often as possible, to create an atmosphere where trying is all that is expected, given how difficult a skill it is. Students should write or present their conclusions, explain their working and the processes they used. It's good to build opportunities for students to do this into either the task itself or the follow up.
Finally here for teachers is the need to consider the balance between instruction and facilitation. There is no implication that one is preferable to the other but only that the balance should be considered. How does the teacher best contribute to the running of the activity so that the above features are best achieved. This is likely to be different from teacher to teacher and class to class, and the perfect answer almost certainly does not exist, but optimization should certainly be sought!
Adding & Subtracting Fractions - Adding and subtracting negative numbers made easy with this series of games and activities.