Deep content knowledge in mathematics – Part 2
Robert Stevens, and his colleagues, explore instruction, hands on learning and pedagogy to support the development of deep content knowledge in mathematics.
Deep content knowledge in mathematics is philosophical knowledge, that is, the problematic and the controversial.
Deep content knowledge in mathematics can be cultivated by a stronger emphasis on dialogic instruction and hands on learning (the use of manipulatives to help render abstract mathematical concepts more tangible). This is because deep content knowledge in mathematics is philosophical and abstract. Mathematical reasoning, in common with all reasoning, is rooted in dialogue.
Deep content knowledge can be cultivated by encouraging a slower, more collaborative and reflective approach to learning mathematics.
Why give emphasis to dialogue?
Mathematical reasoning and dialogue
Why is a dialogic approach combined with hands on learning most appropriate for students learning deep content knowledge in mathematics? Deep content knowledge is philosophical knowledge – the essentially contestable concepts that lie at the heart of a discipline. In mathematics these include:
- concepts of infinity – different orders of infinity – infinity comes in different sizes – Aleph null, Aleph one and so on (Matson, 2007)
- concepts of the infinitesimal - the assumption that any line (for example, a piece of string) or stretch of time is infinitely divisible generates paradoxes such as Zeno’s paradoxes that are deep philosophical problems without agreed answers (Huggett, 2018)
- concepts of zero. Zero is a deep philosophical concept as reflected in divergent answers to the question – what is zero to the power of zero? Some mathematicians argue it is zero, others argue it is one, while others claim it is indeterminate. The indeterminacy of a number is an interesting philosophical idea in itself
- concepts of mathematical sequences in the living world - The Fibonacci Sequence - the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it. This is a recurring pattern in the geometry of living systems (for example, the arrangement of petals on flowers, the patterns of spines on pineapples, patterns on a pine cone, breeding patterns of rabbits) (du Sautoy, 2008)
- concepts of space - the question of whether space is a relationship between objects or has properties of its own is another philosophical issue at the heart of geometry. Is space Euclidian (flat) or Riemannian (curved)?
- concepts of topology – for example, one sided objects such as the Moebius Strip (Lamb, 2016).
There are numerous mathematical problems and conjectures that mathematicians explore and disagree about. These philosophical concepts lie at the heart of mathematics. The best way to understand these concepts and the philosophical issues they embody, is through dialogue, using a dialogic pedagogy.
The idea that deep content knowledge is higher order knowledge in terms of the SOLO taxonomy is complementary to its being philosophical knowledge. The stages of the SOLO taxonomy (Biggs & Collis, 1991, p. 65) well describe systematic philosophical thinking, bringing to bear and integrating an increasing range of considerations relevant to an issue.
‘Introducing SOLO Taxonomy’ (2mins 35 secs) by Pam Hook (2017) from HookED provides a brief introduction to SOLO Taxonomy.
An argument in favour of a dialogic approach to teach deep content knowledge in mathematics is that the paradigm of mathematical reasoning itself – the proof – is not an internal mental operation but has its roots in dialogue. The technique of mathematical proofs emerged in the context of political debates in Athenian democracy. For example, in Plato’s ‘Meno’, Socrates shows a slave boy how to double the area of a square – a mathematical proof. The dialogue might have gone ‘If you agree with this then this other thing follows. Do you agree? – Yes, I agree Socrates’.
With the move from orality to writing it became a much more regimented technique. Diagrams become important. On the other hand, many dialogical features of proof remain in place, even in the written medium. The main purpose of a mathematical proof is to produce explanatory persuasion. Proof is the end product of a dialogue where there is an audience receiving the proof but when there is nothing to object to the audience remains silent. There is just tacit agreement with all the steps of the proof.
Two participants in fictive dialogue are the prover and skeptic. The job of the prover is of formulating the proof. The role of the skeptic is to ensure the proof is correct, that it is explanatory and that the steps are clear.
In contemporary mathematical practice, the role of skeptic is filled by referees of articles to a journal. This is a dialogical structure. Proof is always going to be a triadic notion – the prover, the proof itself, the receiver (to whom the proof is intended) (Novaca, 2018).
Mathematical proof was born of dialogue and still retains implicit dialogical features.
In their recent book ‘The Enigma of Reason’,Mercier and Sperber argue that human reason is first and foremost a social competence. Reason can bring huge intellectual benefits, but it does this through interaction with others (Mercier, 2017). Reasoning, including mathematical reasoning occurs in interaction with others – in a community of inquiry. It is the explicitly social context of these pedagogies that contributes to their effectiveness in cultivating critical and creative thinking. The practice of critical and creative thinking is deeply embedded in these pedagogies.
Considering how to best enable students to develop skills in reasoning is closely related to the notion of how knowledge is classified. In mathematics education, some scholars propose that knowledge accords with two types. With a research interest in how children develop concepts in mathematics, Bethany Rittle-Johnson proposes that two forms of knowledge relate to the acquisition of mathematical knowledge, these being conceptual knowledge and procedural knowledge.
Conceptual knowledge is referred to as ‘knowledge of concepts, which are abstract and general principles’ (Rittle-Johnson, 2017, p. 184) and procedural knowledge is described as ‘knowledge of procedures – what steps or actions to take to accomplish a goal’ (Rittle-Johnson, 2017, p. 184).
In defining these types of knowledge for learning about mathematics, Rittle-Johnson is concerned with how relations might be developed between knowledge types. Developmental relations between these knowledge types are explored through the employment of the concept of procedural flexibility. Procedural flexibility is defined as ‘knowing more than one type of procedure for solving a particular type of problem and applying them adaptively to a range of situations’ (Rittle-Johnson, 2017, p. 184).
Types of reasoning used by students in mathematics are positioned by Rittle-Johnson in terms of flexibility in the application of skills. One might expect, however, that there are other modes of reasoning that come into play in mathematics besides agility in conceptual reasoning, that contribute evidence towards gauging student understanding.
Pedagogies and student identities
Dialogic pedagogy is associated with identities as students of mathematics that contribute to deep content knowledge.
Jo Boaler suggests that knowledge is inextricably linked to the manner in which it is learned and the practices in which it is embedded. Students’ knowledge is constituted by the pedagogical practices in which they are engaged. Practices such as working through textbook exercises or discussing and using mathematical ideas shape the forms of knowledge produced (Boaler, 2002).
Besides learning knowledge in mathematics classrooms, students learn a set of practices. These come to define their knowledge and even who they are – their identities - as learners.
The following table contrasts student identities associated with different pedagogies in mathematics.
Differing student identities associated with different pedagogies in mathematics | |
working through textbook exercises | discussing and using mathematical ideas |
Competitive | Collaborative |
Fast | Slow |
Receptive | Reflective, deliberative |
Algorithmic | Systematic |
Received knowers | Knowledge producers |
Receiving questions and ideas | Generating questions and ideas |
Risk averse | Risk taking (Sharma, 2015) |
Fixed mindset | Growth mindset |
Philippa Foot suggests that ‘in doing philosophy one should not try to banish or tidy up a ludicrously crude but troubling thought, but rather give it its day, its week, its month, in court… It chimes, of course, with Wittgenstein’s idea that in philosophy it is very difficult to work as slowly as one should’ (Foot, 2001, p. 2.). The same applies to deep learning in any discipline.
Boaler (2013) notes that students with these differing identities achieved at similar levels on tests, but ‘they were developing very different relationships with the knowledge they encountered’. If we assume that timed tests tend to measure surface knowledge and that developing surface knowledge is a condition for developing deep knowledge, students engaged in the dialogic classrooms are likely to develop a more creative and critical relationship with mathematics than students receiving direct instruction. Their consequent differing identities as learners of mathematics contribute to students in the dialogic classroom developing deep (philosophical) content knowledge in mathematics as well as surface knowledge.
Researching on mistakes and mathematics, Boaler suggests that teacher’s practice concerns the ways in which they treat mistakes in mathematics classrooms. Mistakes are important opportunities for learning and growth, but students routinely regard mistakes as indicators of their own low ability (Bolar, 2013).
If as a system, we aim to cultivate deep content knowledge (in mathematics for example), we will need to develop structures to facilitate students who pride themselves as being strong collaborative, slow, reflective, systematic, creative and critical thinkers. The current system, the pinnacle of which is a three-hour exam at the end of 12 years of schooling, appears to privilege learners who are who can remember the ‘right answers’, or algorithms to reach them and reproduce them quickly in a competitive test. Collaboration in an exam is cheating. They are timed and most questions have one right answer. There is little scope in a timed exam for slow, systematic reflection. Critical, creative and systematic thinking typically take time.
It is significant to note that machines are capable of applying algorithms in making mathematical calculations at a speed far faster than humans. Machines are less proficient in answering more open-ended questions. Should we be encouraging students to focus the bulk of their efforts in mathematics on making calculations and applying algorithms that machines can do with greater speed and accuracy?
It is a common practice in schools for mathematics students to be streamed in accordance with ability. Boaler argues that streaming is informed by and encourages a fixed mindset where students see their performance as related to fixed ability rather than as something they have some control over (Boaler, 2013).
Streaming fosters competitive identities in mathematics. Streaming is inimical to dialogic pedagogies involving students collaborating in a community of inquiry, since a community of inquiry thrives on diversity of participants rather than grouping participants according to a common perceived ability. As we have seen, dialogic instruction facilitates the cultivation of deep content knowledge in mathematics.
What spaces best support these pedagogies?
Futurist David Thornburg identifies three archetypal learning spaces – the campfire, cave and watering hole – that schools can use as physical spaces and virtual spaces for student and adult learning (Davis & Kappler-Hewitt, 2013, p. 25).
Stephen Collis, Director of Innovation at the Sydney Centre of Innovation in Learning, explains David Thornburg’s learning spaces in ‘Learning spaces – Different spaces and their purposes’ (3mins 38 secs.) (CORE Ministry Video, 2016).
The ‘campfire’is a space where people gather to learn from an expert and suits more teacher centred and explicit instruction. The experts are not only teachers and guest speakers, but also students who are empowered to share their learning with peers and other teachers.
The ‘watering hole’ is an informal space where people can share information and discoveries, acting as both learner and teacher simultaneously. This shared space can serve as an incubator for ideas and can promote a sense of shared culture. This sort of space supports dialogic approaches.
The ‘cave’ is a private space where an individual can think, reflect and transform learning from external knowledge to internal belief. It also acknowledges the need for privacy and to be by ourselves sometimes. Most learners need some time to themselves and some need more time alone than others.
With flexible furnishings, for example, chairs and tables that can be readily moved around the room or up and down, a campfire can be transformed into a watering hole, or into a series of caves.
We would add other metaphors to Thornburg’s by including the:
- ‘sand pit’ for creative interactive pedagogies with access to manipulatives
- ‘yarning circle’, a seating arrangement for a community of inquiry - usually a circle or an inner and outer circle and
- ‘amphitheatre’, a seating arrangement for a large presentation.
Direct Instruction is best supported by the amphitheatre for larger groups and by the campfire for smaller groups.
Dialogical Instruction is best supported by the watering hole and the yarning circle.
Hands on learning is best supported by a sand pit.
The cave is a vital space to support each of these pedagogies. It is a space where students can reflect or practice a skill.
What technologies best support these pedagogies?
Technologies facilitating Direct Instruction include microphones. Lectures can be videoed to reach a wider audience (for example, TED talks). Worked examples can be developed on computer software that can provide immediate feedback on responses.
No special technology is required for the implementation of Dialogic Instruction. Whilst technology is not required or essential in this model of pedagogy, the addition of online collaborative communities could enable the process to continue post the formal lesson and also provide opportunities for students who lack the confidence to contribute to the verbal component of the lesson to express their views.
No special technology is required for hands on learning. However, an object to think with may well be an artefact and a piece of technology. Papert would suggest that a computer, or a computer-generated object, might be a useful object to think with – a tangible representation of an abstract idea.
What curriculum best supports these pedagogies?
A curriculum that best supports these pedagogies would emphasise the philosophical aspects of mathematics - essentially contestable concepts that lie at the heart of the subject. What kinds of considerations should be brought to bear in justifying a mathematical truth? Emphasis should be given to students giving reasons for their mathematical ideas. The curriculum should facilitate student dialogue about topics such as the nature of numbers (are they real or invented?), why does mathematics apply to the world if it is known about a priori? What is the nature of a mathematical proof? Is it a mental operation or the product of dialogue? Has Euclidian Geometry been replaced by Riemannian Geometry? Is any line or stretch of time infinitely divisible? If so, how do we avoid the paradox of all lines therefore being the same length? Or the indeterminacy of a finite line having no last divisor? (Zeno’s paradoxes).
Mickael Launay describes the Mandelbrot set as one of the most dazzling mathematical gems of the 20^{th} century (Launay, 2018). The Mandelbrot set is a set of sequences generated by starting with 0 and in which every term is equal to the square of the previous term. Some of these sequences are unbounded – they fly off towards infinity - and others are bounded and include no number greater than 2. If for example we choose 2, the sequence would be: 0, 2, 6, 38, 1446… [2=0X0+2, 6=2x2+2, 38=6x6+2, 1446=38x38+2…] On the other hand, if we choose -1 then the sequence would be 0, -1, 0, -1… 0 would be 0,0,0,0… The Mandelbrot set is the set of real numbers that generates a bounded sequence. If we map this by placing real numbers on the horizontal axis and the imaginary numbers on the vertical axis we get an intricate pattern of fractals. Fractals are prevalent patterns in nature, for example coastlines and fern leaves.
The Mandelbrot set, while not included in NSW Mathematics Syllabuses, could be used to teach complex numbers, imaginary numbers, fractal geometry, even infinity, since many of the bounded sequences are infinite without repetition. Good teaching may not involve going through the syllabus lock step, but rather developing rich tasks that involve using a range of different outcomes.
‘The Mandelbrot Set – The only video you need to see!’ (21 mins 18 secs) by TheBITK (2016) describes the incredible mathematical formula explaining fractals and geometry.
If the syllabus is seen as something that the teacher interprets rather than sequentially delivers, then the current mathematics syllabus is rich enough to support the range of pedagogies necessary to cultivate deep content knowledge, such as inquiry based pedagogies.
Mathematics can be taught as a component of an integrated course in Science, Technology, Engineering and Mathematics (STEM) (Education Council, 2015). Teaching mathematics in the context of a STEM integrated course may encourage a greater breadth of knowledge in mathematics, and facilitate connections between mathematics, science and technology. On the other hand, it may not encourage greater depth in mathematical knowledge. Indeed, the mathematical component of a STEM course may be diluted, depending on whether it is taught by a mathematics teacher. An integrated STEM course may not be the best context for philosophical reflection on the big ideas that are at the heart of the discipline.
How can deep content knowledge be best assessed?
To support the cultivation of deep content knowledge in mathematics, it would be helpful to place less emphasis on timed tests and allow students more time to think through mathematical/philosophical problems. More emphasis could be placed on mathematical reasoning in assessments. The SOLO model could be used here as a framework for assessing the quality of the reasons students provide.
Deep content knowledge in mathematics, is philosophical knowledge - the problematic and the controversial.
Deep content knowledge in mathematics can be cultivated by a stronger emphasis on dialogic instruction and hands on learning (the use of manipulatives to help render abstract mathematical concepts more tangible). This is because deep content knowledge in mathematics is philosophical and abstract. Mathematical reasoning, in common with all reasoning, is rooted in dialogue.
References
Biggs, J.B. & Collis, K.F. (1982). Evaluating the quality of learning – the SOLO Taxonomy. New York: Academic Press.
Boaler, J. (2002). The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms. Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics. Norwich, England.
Boaler, J. (2013). Ability and mathematics: The mindset revolution that is shaping education. Forum, 55(1), 143-152.
CORE Ministry Video. (2016). Learning spaces – Different spaces and their purposes.
Davis, A. & Kappler-Hewitt, K. (2013) Australia’s campfires, caves, and watering holes. Learning & Leading Technology, 3.
Du Sautoy, M. (2006). The story of maths. United Kingdom: 3 hours.
Education Council. (2015). National STEM school education strategy: A comprehensive plan for science, technology, engineering and mathematics education in Australia.
Foot, P. (2001). Natural goodness. Oxford: Clarendon Press.
Hook, P. (2017). Introducing SOLO Taxonomy- HookED.
Huggett, N. (2018). Zeno’s paradoxes. The Stanford encyclopedia of philosophy.
Lamb, E. (2016). A few of my favourite spaces: The Mobius Strip. Scientific American.
Launay, M. (2018). It all adds up: The story of people and mathematics. Harper Collins.
Matson, J. (2007). Strange but true: Infinity comes in different sizes. Scientific American.
Mercier, H. & Sperber, D. (2017). The enigma of reason: A new theory of human understanding. Cambridge, Massachusetts: Harvard University Press.
Novaca, C.D. (2018). Proof and beauty. The Philosopher’s Zone. D. Rutledge, Australia. ABC Radio National: 25 minutes.
Rittle-Johnson, B. (2017). Developing mathematics knowledge. Child Development Perspectives, 11(3), 184-190.
Sharma, S. (2015). Promoting risk taking in mathematics classrooms: The importance of creating a safe learning environment. The Mathematics Enthusiast, 12(1-3), 290-306.
TheBITK. (2016). The Mandelbrot Set – The only video you need to see!
How to cite this article – Stevens, R., Liyanage, S., Liondos, N., Woo, E., Ali Kan, A., Blue, J., De Marcellis, L., Birungi, A., Brady, K., Tregoning, M. & Coupland, M. (2019). Deep content knowledge in mathematics – Part 2. Scan, 38(7).