Being a mathematician: the importance of proficiencies. ACARA (Australian Curriculum and Assessment Reporting Authority) sets out, as one of three overarching aims for the mathematics curriculum that it should aim to ensure that students: are confident, creative users and communicators of mathematics, able to investigate, represent and interpret situations in their personal and work lives and as active citizens While there are still questions about the final form of the curriculum, the aims are very much in line with what is being advocated globally for mathematics education, and a sound basis for thinking about developing the curriculum. The scoping document goes on to note that: The curriculum is written with the expectation that schools will ensure that all students benefit from access to the power of mathematical reasoning and be able to apply their mathematical understanding creatively and efficiently. The mathematics curriculum provides students with carefully paced, in-‐depth study of critical skills and concepts. It encourages teachers to facilitate students to become self-‐motivated, confident learners through inquiry and active participation in challenging and engaging experience. Key emphases here, particularly as they relate to the proficiencies are ‘creatively’, and ‘active participation in challenging and engaging experience. The research into flow -‐ the state of engagement that represents optimal engagement -‐ shows that such a state is reached when there is an optimal balance of challenge with the skills necessary to meet the challenge. This is a point I will be returning to later in considering the interplay between fluency and other proficiencies. To bring about this vision, the curriculum has two dimensions -‐ the content strand and the proficiencies. The content strand contains few surprises -‐ list the content strands. One challenge to schools and teachers with respect to the content is ‘filling in the blanks’ -‐ then content strands specify a number of end of year targets, but the fact that something is not mentioned in a particular year should not suggest that content strand is not addressed in other years. For example -‐ flesh out the stuff on multiplication here. Perhaps more challenging to some mathematics teaching is the inclusion and importance of four proficiencies that cut across the content strands. Fluency Understanding Problem solving Reasoning The proficiency strands describe the actions in which students can engage when learning and using the content. Two important things to note here. The proficiencies are seen as being just as important when learning content as when using it. In other words, this challenges the popularly held view (myth) that you learn about addition, equivalent fractions, algebraic manipulations or whatever first and then apply it to solve problems, or reason about it. This is such a strongly held view, at it’s most voluble in the ‘back to basics’ cries that it is worth looking at some of
the evidence against it, as I do below. Without challenging this view the proficiencies are likely to be ‘put off’ to some later (and again often mythical) time when learners are thought to be ‘ready’ to engage with them. The other thing to note is the call to think about proficiencies as ‘actions’. This also poses challenges. The everyday use of ‘proficient’ carries some connotations of a level of expertise. We would not describe someone stumbling through a rendition of ‘chopsticks’ as a proficient piano player. But like learning to play the piano, becoming proficient means engaging in certain actions even when one is not yet proficient in them. Becoming fluent is your scales means stumbling through; playing a sonata may initially mean picking out the notes without a sense of rhythm. At the risk of stretching the analogy, being fluent in your scales helps the playing of the sonata, but there is more to the playing than simply being able to ‘do’ scales (and working on the sonata feeds back into developing the scales) -‐ there is no waiting to learn the ‘basics’ of the scales before being allowed (and encouraged) to play a tune. Becoming a proficient piano player means working with all of the proficiencies -‐ scales, reading music, playing sonatas -‐ from the beginning. Becoming a proficient mathematician means working with all of the proficiencies -‐ fluency, problem solving, reasoning and understanding -‐ from the beginning. And by mathematician here I mean anyone using mathematics in his or her life. Everyone is a mathematician. I do have some difficulty with understanding as a ‘action’ -‐ I can develop understanding, I can draw on understanding, I can demonstrate understanding, but I’m not clear how I ‘do’ understanding. I prefer to think of understanding as being largely the result of doing the other proficiencies -‐ engaging in problem solving, reasoning about the ‘why’ of mathematics and being fluent in the ‘how’ of mathematics are the building blocks of understanding. But I do agree with the overall sentiment of proficiencies being actions, not states. Taking such a stance means moving from seeing school mathematics as a body of knowledge for learners to acquire to seeing it as an activity to engage in. Or, in the words of Brent Davis, moving from seeing mathematics as preformed to mathematics as performed.
The actions of proficiency Fluency Fluency includes, but is not just about, recall of ‘facts’, which would include fluency in number bonds (addition and subtraction within numbers to 20 and multiplication up to 10 x 10 and associated division facts). Being fluent in number bonds includes some memorizing of ‘facts’, but it is not just about memory and drill and practice: understanding and reasoning play a big part in becoming fluent. And it is not simply a case of being fluent for the sake of it. There is evidence that success in being fluent in basic number calculations ((addition and subtraction within numbers to 20) is strongly correlated with later mathematical success. However, that should not be taken as an indication that ‘drilling’ children in basic number calculations is the key to promoting later success. Myth: Children need to be fluent in the ‘basics’ before they can reason mathematically. Evidence: There is a relationship between becoming fluent and reasoning, each supporting and being supported by the other. The exact nature of the relationship between becoming fluent and the other proficiencies is not entirely clear, for example, it may be that being interested in solving mathematical problems encourages learners to become fluent in basic calculation, but it is clear that fluency, reasoning, problem solving and understanding are intertwined and not related sequentially.
Take, for example, learning 'basic' number facts (addition and subtraction with answers up to 20). The research evidence shows that children use three approaches to answering calculations like 5 + 6 or 13 − 7. Retrieval -‐ simply knowing the answer. Children can recall that 5 + 6 = 11 (3 seconds being the benchmark time for retrieval) Counting -‐ either counting all (putting out five objects, another six and counting the total) or counting on from one of the numbers (and coming to appreciate the counting on from the larger number is more efficient) or counting back in the case of subtraction. Decomposition -‐ splitting one or both of the numbers to make use of number facts that they can retrieve. So thinking of 5 + 6 as 5 + 5 + 1 or double six minus 1. Or using 14 − 7 = 7 (from rapid retrieval of doubles) to deduce that 13 − 7 must be 6. These are not entirely distinct strategies: decomposition can only be used with the use of retrieval. The evidence is that there is no clear hierarchy to these strategies: children will make use of all three, even until quite late in primary school. Drilling children to try and improve retrieval doesn’t make them any more likely to use that method and often they will be more accurate using a counting method. Contrary to what might seem like ‘progress’ trying to move children on to use only retrieval does not appear to be that successful. Children who use a mix of the methods seem to be most successful. This illustrates one key aspect of the proficiencies that is different from the ‘division’ of the content: the proficiencies feed into and off each other, and cannot be ‘taught’ separately. Other aspects of fluency would include recall of definitions, but fluency goes beyond simply ‘knowing' number facts or definitions. Being mathematically fluent also involves choosing methods and procedures and working flexibly. For example, a student might mentally calculate 3004 − 2997 by counting on from 2997 to 3004, or decomposing 3004 into 3000 and 4, and using the retrieval of 7 = 3 = 10 to figure out that 2997 + 3 = 3000, so the total difference must be 3 + 4 = 7. But applying the same approach to 2005 − 8 (counting on from 8 to 2005 or adding 1992 to 8 and then another 5) would not be as fluent. Working flexibly would mean counting back in this cases, or knowing that a sensible approach is to decompose 8 into 5 + 3, take 5 from 2005 and then subtract 3. Note again, that this is not distinct from reasoning about the relationship between the numbers and understanding subtraction both as ‘taking away’ and ‘finding the difference’.
Reasoning What are some of the actions associated with reasoning? These might include: Explaining thinking Deducing and justifying strategies Adapting the known to the unknown Transferring learning Reasoning or calculating Which is larger? ½ or 3/7 4/9 or 3/7 This is a simple example of how choice of examples might call into play either a ‘reasoning’ or ‘calculational’ approach to number situations. In the first comparison you might choose to compare the fractions by putting them each over a common denominator of 14. But you can also reason through to an answer, along the lines of ‘well, half of 7 is 31/2 so 3/7 must be smaller than ½ (if there are seven bars of chocolate and I’m given three of them, then I’ve got
less than ½). Note that this reasoning can be followed through without any knowledge of equivalent fractions. On the other hand 4/9 compared to 3/7 is less easily reasoned through in this way and a calculation of equivalent fractions might be more efficient. An implication for teaching here is that numbers need to be carefully chosen depending on whether or not we want children to adopt a calculational or reasoning approach. For example, ‘near doubles’ is an effective calculational strategy for adding pairs of near numbers: 35 + 36 is 71 because it must be one more than double 35 (another variation on decomposition and the associative rule: 35 + 36 = 35 + (35 + 1) = (35 + 35) + 1 Trying to teach ‘near doubles’ with small numbers is probably not the most effective approach. Going from 3 + 3 to 3 + 4 on the argument that the second can be derived by adding 1 to 3 + 3 is not likely to make much impact on children who are already confident with the answer to 3 + 4. On the other hand, starting with 50 + 50 = 100 and going on to 50 + 51 or 49 + 50 May lead more students to reason rather than calculate. We could take this further and work with an initial calculation that isn’t easily found, but use it to deduce other answers. 78 + 78 = 156 78 + 79 = 77 + 78 =
Numbers as relations Ivor collects pencils. Ivor had 7 pencils. His dad gave him 4 more. Ivor gave 6 of his pencils to his sister. How many pencils did Ivor end up with?
Ivor collects pencils. Ivor’s mum gave him 7 pencils. His dad gave him 4 more. Ivor gave 6 of his pencils to his sister. Did Ivor end up with more or fewer stickers than he started with? How many?
Both of these ‘problems’ (yes, problems is in scare quotes because they are the type you only meet in school maths, but bear with me) can be answered by calculating 7 + 4 − 6 (and yes, there are other ways). Yet children find the second problem much harder than the first. In fact, since we are not told how many pencils Ivor started off with, some children will say that the second problem cannot be answered. Why do they find it more difficult? The difficulty is in the nature of the answers. In the first problem, five represents a quantity -‐ the number of pencils Ivor ends up with. But the answer five to the second problem represents a relationship -‐ how many more pencils Ivor had than when he started. We often switch between using numbers as quantities and numbers as relations without paying attention to the distinction. But awareness of the distinction is important in supporting children’s developing number sense. Take a simple calculation like 7 − 3 = 4 If we read the ‘−’ as ‘take away’ then the answer four is a quantity -‐ it is the quantity remaining when three things (stickers, marbles, zingbats) are removed from seven. When we ‘take away’ we can hold up the particular quantity that is left behind. Reading ‘−’ as ‘difference between’, the four then represents a relationship -‐ how many more there are in a collection of seven objects then there are in a collection of three. Unlike ‘taking away’, when comparing two collections there is no particular quantity of four from the
larger collection that specifically makes up the difference. Four expresses an abstract relationship between seven and three. I often invite teachers to show what picture they might draw to help a child understand 7 − 3. Almost invariably people draw seven objects and strike out or somehow bracket off three of them. It is very rare for someone to draw a set of seven and another set of three and point up the difference. Even adults seem to prefer the concrete, the quantity, over the abstract, the relationship. To fully develop children’s number sense we need to make sure that they get a rich diet of operating with numbers as relationships as well as quantities. Myth: Children’s thinking has to have reached a certain ‘stage’ before they can reason mathematically. Evidence: Children have the quality of thinking available to them that adults have for reasoning. It is lack of experience that accounts for children’s thinking being less accomplished, not an inability to reason. Although Paiget’s theory of ‘stages of thinking’ is no longer talked about that much, it does still seem to be held that children cannot reason in the same way that adults can. Long term Piagetian researchers now think that he got this part of his theory wrong and that the structures for thinking that we use throughout our lives are very much in place from an early age. Experience is what is needed, not waiting for children to ‘develop’ into a particular form of thinking. The evidence that the structure of students’ thinking is essentially the same as that of adults has implications for thinking about progression in mathematics. We need to be asking where the informal origins of later mathematics might lie in the primary curriculum and make these more explicit. The links between algebra and number are particularly strong here. For example, one fundamental ‘rule’ of algebra is the associative law: a + ( b + c ) = (a + b ) + c While not suggesting that this is introduced to seven-‐year-‐olds, this is precisely the thinking that they are informally drawing upon when using decomposition to add five and six by using their retrieval of five plus five: 5 + 6 = 5 + (5 + 1) = (5 + 5) + 1 Missing box calculations can provide a way of making this reasoning explicit to young children without needing to go into algebra. 8 + 7 = 8 + 5 + [ ] First presented with a number sentence like this some children will calculate to find the answer. They will add eight and seven to get 15, add eight and five to 13 and then work out that they need to add two onto the 13 to make it up to 15. But once they get talking about their methods, some children will begin to understand that thinking of the seven as five plus two means that the extra ‘bit’ that needs to be added to the right-‐hand side must be two. Working with missing number calculations with numbers that are too big to calculate can extend such reasoning. 378 + 245 = 378 + 241 + [ ] 436 + 157 = 432 + 155 + [ ] 436 + 198 = 436 + 200 -‐ [ ] As this last example shows, such reasoning is not only powerful for its own sake, but it can help children develop effective mental calculation strategies.
Problem solving
Actions here include Making choices Modelling situations mathematically Communicating solutions Myth: Children are only interested in mathematics that is directly applicable to their everyday lives. Evidence: Many children do find mathematics a fascinating object of study, finding satisfaction in number patterns for their own sake, gaining pleasure through tussling with a problem and then having a moment of insight, seeking solutions that are pleasing in their simplicity. There is some confusion in the use of ‘everyday’ concepts and how they can contribute to the curriculum. For example a problem like: Four hungry girls share three pizzas equally. Eight hungry boys share six pizzas equally. Do the girls get to more than the boys, less or the same? Traditionally a problem like this might be given to students to answer after they have done some work on fractions and division with the expectation that they quickly recognise that each girl gets 3/4 of a pizza, each boy gets 6/8 of a pizza, which is equivalent to 3/4 and so they each get the same. Next question. The main thing here is that the ‘story’ of pizzas and hungry children doesn’t serve much purpose: students quickly learn that the ‘point’ of such problems is strip out the mathematics and work some procedure. It could just as easily be about builders sharing bricks and most learners would not stop to think about the near impossibility of four builders sharing six bricks. A problem like this can also be used differently: To introduce students to thinking about fractions and equivalences. The context of hungry children and pizzas then is starting point and a carefully chosen one. Not because children are intrinsically motivated by food and the context might make ‘unpalatable’ fractions digestible. No, the context is chosen because children do know about fair shares and slicing up pizzas -‐ they can solve the problem without any formal knowledge of fractions (and if you don’t believe that, then, as Terezhina Nunes once said, show me four children who, given three bars of chocolate to share out fairly, hand the bars back saying ‘it can’t be done’). Children have ‘action schemas’ for solving such problems, they can find ways to record this with pictures, diagrams and, perhaps, symbols, and teaching can then draw on these informal solutions to draw out the formal mathematics of fractions. From being one of 20 ‘problems’ on a worksheet to complete in a lesson, such a problem can become a ‘rich task’ taking up a whole lesson if children work on it in pairs and then carefully selected solutions are shared with the class. The ‘richness’ of tasks is less in the tasks themselves then when and how they are worked on. We don’t need loads more problems for students to work there -‐ there are more than enough out there. But we do need to think about when they might be best used and how to work with them and children so that they are treated as rich problems rather than exercises to be quickly worked through and marked right or wrong. Children, from birth, are proficient problem solvers. By two or three they have solved what are probably life’s two biggest problems -‐ how to walk and talk. The fact that they do not solve problems using the mathematics that we might want them to eventually use must not blind us to the fact that that they can solve problems. Young children can share out 12 pies fairly between 3 bears or figure out how many dogs live in a street of 4 hours if there are 3 dogs in each house, and they can do this long before they’ve heard about division or multiplication. New mathematics can arise through problem solving just as much as problems
solving can draw on existing mathematics. We just need to be clear which we are focusing on. And the problem itself cannot do that.
Understanding Actions here include Asking ‘Why’ as well as ‘how’ Working with and moving between different representations Connecting ideas
Into practice
So what are the practical considerations when teaching for these proficiencies? I suggest attending to a model of three aspects that I call the ‘teaching tripod’: • Tasks • Tools • Talk. When it comes to legs, tripods are particularly stable – two legged tables won’t stand up on their own and four legged tables can easily develop a ‘wobble’ on an uneven surface. Tripods stand steady. By attending to each of the elements in the teaching tripod lessons can be structured to be sufficiently open to allow children to bring their mathematical proficiencies into play, but also sufficiently structured to allow some degree of control over the direction of the mathematics. 6
Task and activity I want to distinguish between tasks – what teachers set for children to do – and the subsequent mathematical activity that children, collectively, engage in to carry out the task. If students are going to develop mathematical understanding, and reasoning there needs to be a certain ‘gap’ between the task that a teacher sets and the students’ subsequent mathematical activity. Teaching which strives to narrow the gap between task and activity, for example through careful instructions that learners have to follow, may lead to short-‐term success but not long-‐term learning. Problem solving opens up this gap (if we set aside expectations that there is a ‘right’ way to solve a problem), which is why I am arguing for it to provide a starting point from which to build mathematical understanding rather than the endpoint of mathematical application. Working with problems as vehicles for learning requires a style of teaching that begins with engaging the students with the problem. We need to be providers of mathematical tasks that are likely to lead to rich mathematical activity. Some writers prefer to talk about ‘rich mathematical tasks’. I am not convinced by calling a task rich There is a danger that talk of rich tasks is interpreted as tasks having to be elaborate or complicated: richness arises by encouraging learners to take a creative stance to solving the problems. If children are to learn from their problem-‐solving experiences then the problems have to be chosen in the expectation that children will need to sustain activity on them for some time. We need to question the popularly held view that ‘speed’ is a key marker of mathematical ability. While fluency may require a certain amount of rapidity (rapid recall rather than instant) learners have to be able to tolerate a certain amount of ambiguity when thinking about and solving problems, which is different from the sort of precision that is needed in finding exact answers to calculation. The students’ solutions to problems are not the end of the story, but only the beginning of the development of the formal mathematics. An important, if not the important, part of the lesson is the drawing together of strands of mathematics from the various solutions the learners offer. A crucial role of the teacher is to decide which of the students’ solutions will provide the opportunity for rich dialogue with the class about the mathematics. It is not just a
question of asking the volunteers at the end of the lesson. Nor is it simply a matter of choosing the student with the most sophisticated solution – it may well be that such a solution may be too far removed from what the other students’ solutions have done to be up to making sense of it, and is just their thinking about their own solutions in the light of the dialogue. Opening up the gap between task and activity I have found Marion Walters and Stephen Brown’s work to be invaluable in developing rich mathematical activities (Brown and Walter 1990). The basic tenet of their work is to list the elements of a problem (including rather obvious ones) and from this list of ‘what is’ ask ‘what if not’: how changing some of the givens might lead to further mathematical activity. For example, a simple question is 25 + 25 = 50 What is
What if not
25 and 25
Other doubles, for example, 36 and 36 Other multiples of 5, for example 25 and 45
Two numbers added
What sets of three numbers add to 50? Sets of four numbers
Adding to 50
What other pairs of numbers add to 50? What pairs of numbers have a difference of 50? What pairs of numbers have a product of 50? What pairs of numbers divide to give 50?
Both less than 50
Can you have a pair of numbers that add to 50 where one of the numbers is bigger than 50?
Whole numbers
What if we include fractions?
Not all of the resulting questions are equally curious, but some may be worth pursuing. Playing around with doubling multiples of three can lead to insights into the relationship between multiples of three and multiples of six for example.
Models, tools and artifacts
While there is general agreement about the importance of ‘models and images’ in helping children learn mathematics, there is less clarity about why they are important and how to work with them. It is not the models or images themselves that are important, but the way that these support children’s mathematical activity. Models and images have to be worked with, not simply presented to the children. My preference is to talk about tools, as tools are only useful when someone is using them. Images are often interpreted as being self-‐evident. Models are a step on the road to tool use. We are better off introducing children to a small number of models and working intensively with these over time. My advice about models is that simpler is better. Were I teaching again I would equip my class with base-‐ten blocks, interlocking cubes, a wide variety of 2D and 3D shapes and lots of plain paper and colored pencils or pens. And that’s about it. From models to tools Tools are but ‘artifacts’ to the novice; they start their psychological life as simply a physical presence but with no personal meaning. To a young child an analogue clock may make a good wheel, plate, or drum, but it is not a ‘clock’ to the child in the sense of something that we mark time with. The child is aware of the physical presence of artifacts, but not aware of their
meaning in activity. Artifacts become tools when they come to serve the purpose and meaning with which experienced users imbue them. As teachers we introduce artifacts – and I am including marks on paper or the board here – that have meaning for us as teachers but have to come to be filled with meaning by the child. Such meaning cannot simply be explained; it comes about through joint activity. The work of the Dutch Realist Mathematics Education at the Freudenthal Institute provides a framework for thinking about how we can help, and monitor, children moving from artifacts to tools. They distinguish between: • models of • models for • tools for Artifacts start life in the classroom as ‘models of’, in the sense that the teacher uses these with meaning and through joint activity, the children begin to also ‘read’ meaning into the artifacts. To take a simple example, children will have informal strategies for solving addition problems in context. Adding four and three they may hold out four fingers and then another three and count the total. The teacher could provide a model of this by drawing a number line, counting along to four on the line and then counting on three more ‘steps’ to land on seven. This ‘model of’ the addition is not a transparent mirroring of what the children did: they were counting objects, the teacher is counting jumps. Through the joint activity of the children solving the calculations in their own way and the teacher sharing her model of this, then, over time, the children begin to appropriate the teacher’s model and begin to use it for themselves: the number line has been transformed from a model of what the children to a model for them to use themselves. The Dutch research shows that over more time the children come to be able to work with the number line as a model for addition, but without needing to make actual marks on paper: it becomes a tool for thinking with. There is a subtle distinction here between watching a lesson where a teacher models an addition on a number line: it can look as though the teacher is merely recording what the children are doing – she isn’t – or using the number to ‘explain’ addition – it can’t. The movement from a ‘model of’ to ‘model for’ to ‘tool for thinking’ takes time and children will not all move through these stages at the same rate.
Talk Tasks and tools will not bring about mathematical understanding or reasoning without lots of talk. Talk that supports mathematical activity is distinguished by • emphasizing listening as well as speaking • recognizing the difference between discussion and dialogue • focusing on mathematical reasoning as much as answers. Discussion or dialogue At the heart of learning to listen is the distinction between being involved in a discussion or in a dialogue. Robin Alexander espouses the importance of ‘dialogic teaching’ in which the classroom participants (including the teacher) are engaged in productive talk about what is being learned. Dialogue comes from the Greek dialogos. ‘Logos’ –’the word’, and ‘dia’ – ‘through’; hence meaning is created through the word – not in the words. Knowing comes about through the building and bartering, the back and forth exchanges of a dialogue, as opposed to knowledge being packaged up in words and passed across. And a dialogue can involve many more than two people as often supposed. David Bohm, the physicist turned philosopher, contrasts the origins of dialogue with those of discussion. Discussion shares a common etymological root with percussion and concussion (that must be why so many discussions leave me with a headache). Discussion, Bohm suggests, is about breaking things up, with ‘point scoring’ often involved. In discussion the participants usually have a pre-‐ determined position that they hold to. Discussion is about trying to establish this position (or shift another’s). Discussion is largely a win–lose game play. Dialogue has a different play to it.
It’s not about trying to win – it’s an exchange of views, an attempt to understand the other better, rather than to try and impose one’s view upon the other. Dialogue, rather than discussion, in mathematics lessons can be a challenge – after all there are still right answers in mathematics. Answers depend on questions and it’s generally taken that preparation of ‘good’ questions is an essential part of effective mathematics teaching. Contrary to what we might expect, teacher questions close down options while statements can lead to more dialogue, deeper involvement, broader participation and richer arguments. Getting children to talk about whether they consider a statement such as All squares are rectangles is always, sometimes or never true, can produce a richer dialogue than posing the question. Is a square a rectangle?
