Deep content knowledge in mathematics - Part 1

Robert Stevens, and his colleagues, explore deep content knowledge in mathematics as philosophical knowledge, supported by instruction and hands on learning, which may involve a major transformation in the way mathematics is taught.

Introduction

Schools are institutions charged with the responsibility of educating students by providing children and young people with opportunities to learn in particular courses of study. Nevertheless, the thesis that content knowledge in subject domains is a foundational principle of curriculum is a premise that is sometimes challenged in contemporary education systems. However, the content of disciplines—as variously endorsed, modified and practised within specialised fields—is pivotal to how knowledge is conceived of, taught and learned in schools and other education contexts, such as tertiary institutions, and to assessing student understanding in subject domains. As public documents organised in subject domains, NSW syllabuses attest to the importance of content knowledge in the school curriculum and a thorough knowledge of how to teach subject content is seen as preliminary to proficient teaching (NSW Government Education Standards Authority, 2018). The importance of content knowledge in a subject domain is supported by the NSW Government, where Goal 5 of the NSW Department of Education’s Strategic Plan 2018-2022 states that ‘all young people have a strong foundation in literacy and numeracy; deep content knowledge; and confidence in their ability to learn, adapt and be responsible citizens’ (NSW Department of Education, 2018).

What is deep content knowledge?

Content knowledge can be defined as knowledge about the actual subject matter that is to be learned or taught. Content knowledge refers to the facts, concepts and principles that are taught and learned, as distinct from related skills such as reading, writing, or critical and creative thinking.  Content knowledge is knowledge that… as distinct from knowledge how… (procedural knowledge).

In addressing the question of deep content knowledge in mathematics, one account is that deep content knowledge is philosophical knowledge.

Matthew Lipman (1991) says that philosophy deals with essentially contestable concepts - concepts that lie at the heart of any discipline when it is presented as a living thing rather than simply as a body of established knowledge. Lipman writes:

‘Philosophy is attracted by the problematic and the controversial, by the conceptual difficulties that lurk in the cracks and interstices of our conceptual schemes ...The significance of this quest for the problematic is that it generates thinking. And so when we encounter those prefixes, “philosophy of science,” “philosophy of history,” and so on, we are grappling with the problematic aspects of those disciplines... It is when a discipline conceives of its integrity to lie in ridding itself of its epistemological, metaphysical, aesthetic, ethical and logical considerations [the philosophical, in short] that it succeeds in becoming merely a body of alienated knowledge and procedures’ (Lipman, 1991, pp.33-34).

In applying Lipman’s account of philosophy to the domain of mathematics education, a curriculum inclusive of the philosophy of mathematics entails seeking conceptual difficulties in ways that generate thinking. According to this account, an emphasis on deep content knowledge is therefore foregrounded through the development of reasoning skills.

Another account is that deep content knowledge is higher order knowledge.

The Structure of the Observed Learning Outcome (SOLO) taxonomy, (Biggs & Collis, 1982; Biggs & Collis, 1991; Biggs, 1995) provides a systematic way of describing how a learner's performance grows in complexity when mastering varied tasks. The SOLO taxonomy postulates five levels of increasing complexity in growth or development of concepts or skills:

SOLO Stage

Description

Prestructural

The task is engaged, but the learner is distracted or misled by an irrelevant aspect belonging to a previous stage or mode

Unistructural

The learner focuses on the relevant domain and picks up one aspect to work with

Multistructural

The learner picks up more and more relevant and correct features, but does not integrate them

Relational

The learner now integrates the parts with each other, so that the whole has a coherent structure and meaning

Extended abstract

The learner now generalises the structures to take in new and more abstract features, representing a new and higher mode of operation

(Biggs & Collis, 1991, p. 65)

Implicit in the SOLO model is a set of criteria for evaluating the quality of a response to (or outcome of) a task utilising general capabilities. The quality (or richness or complexity) of a response to a complex task varies with the relevance of the considerations brought to bear on the task, the range or plurality of those considerations and the extent to which these considerations are integrated into a whole, and generalised to or related to, broader contexts.

Underlying the SOLO taxonomy is the idea that performance in a rich task consists of not overlooking any important consideration - so bringing to bear on the issue a range of considerations relevant to the issue and considering these in relation to each other.

Hattie (2012) interprets the SOLO levels as meaning:

  • [Pre-structural: no idea]
  • Unistructural: an idea
  • Multistructural: many ideas
  • Relational: relating ideas
  • Extended abstract: extending ideas.

What does the SOLO taxonomy have to do with deep content knowledge?

John Hattie suggests that the Uni-Structural and Multi-Structural are about surface knowledge, and relational and extended abstract are about deeper knowledge.

He comments, ‘Together, surface and deep understanding lead to the student developing conceptual understanding’ (Hattie, 2012, pp. 60-61).

The SOLO model

Uni-structural

An idea

Surface

Multi-structural

Many Ideas

Relational

Relating Ideas

Deep

Extended Abstract

Extending ideas

John Pegg (2010) suggests that higher-order knowledge commences at the relational level. This arises through the ability to integrate information (Pegg, 2010, pp. 36-37).

Combining Hattie and Pegg's account we could say that deep content knowledge is higher order knowledge.

In The Colors of Infinity – Arthur C. Clarke [53 mins 10 secs] a number of notable mathematicians illustrate how simple formulas can lead to complicated results aligned with deep content knowledge.

What pedagogical practices are conducive to the cultivation of deep content knowledge?

There is a long history in education concerning the relations between pedagogical practice and content knowledge that are enacted through teaching and learning. Many interpretations have influenced the formation of curriculum content and how frameworks of content are then interpreted in classrooms. Cultural critic and theorist of critical pedagogy, Henry Giroux (2004), proposes that at the very least, an understanding of critical pedagogy entails recognition of the social, moral and political dimensions of knowledge and how agency in teaching and learning produces transformation (Giroux, 2004, p. 34).

This section of the paper examines research literature as it relates to the cultivation of deep content knowledge in mathematics learning. To begin the review, a practical example is presented as a way of demonstrating the nuances that occur in teaching and learning, and how the functional agency of a teacher is critical in setting up opportunities for depth of learning through the pedagogical decisions that are put into practice. A range of pedagogical approaches is then examined.

The significance of pedagogy through an example of learning to count

Counting forms a critical foundation in early years learning for very good reasons. When you’re able to count, you can describe how much, compare quantities, combine collections, separate collections, and begin to solve problems using early additive strategies, and so on.

Counting seems like a reasonably simple act. You could ask a young child to count to 10 and she may recite: ‘1, 2, 3, 4, 5, 6, 7, 8, 9, 10.’

An adult may observe this behaviour and declare: ‘She can count to 10’. This is not an inaccurate summary, however, it is not the entirety of the story. Being able to meaningfully count includes more than mere recitation. Reciting the number sequence suggests that a student is aware of the stable-order principle: knowing that counting involves an unchanging sequence of number words. This is just one component piece of a complex jigsaw puzzle of concepts and skills. Amongst other things, students also need to be supported into knowing that:

  • counting involves matching the sequences with quantities, developing one-to-one number correspondence whereby each object or unit being counted must be given one count, and only one count only;
  • the last number word produced describes the ‘many-ness’ of the entire quantity (or the cardinal value of the collection) and doesn’t just name the last item that was counted. Teachers need to intentionally make connections to conservation and the order-irrelevance principle when supporting students to develop the concept of cardinality;
  • it doesn’t matter what is counted, the process is the same. The abstraction principle means I can count physical things, visible things, big things, small things, invisible things…the process remains the same and the emphasis remains on ‘how many?’. This forms the basis of the commutative property and later extends into the multiplicative world, empowering students to be able to work algebraically in ways that make sense to them.

Learning to count may seem like a simple idea but a rich, complete understanding is both complex and nuanced. Linking collections to number names and numerals forms part of the big idea called ‘trusting the count’ (Siemon et al., 2012) and is inextricably intertwined with other early numerical skills and understanding such as subitising, knowledge of spatial patterns, reasoning, communicating, making meaning from numerals, and early additive thinking. ‘Children must construct these ideas through a variety of experiences and activities’ (Van de Walle et al., 2014, p. 101) and as such, teachers need to design appropriate tasks that take place inside learning environments focused on working mathematically. The pedagogical skills and decisions of teachers are equally as important as the content knowledge they have.

In addition to knowing how to support students develop both the procedural and conceptual understandings needed to be able to count, teachers also need to be aware of common misunderstandings students develop, and, develop horizon knowledge which includes understanding where ideas begin, where they extend to and how they relate to a larger landscape of mathematical competencies (Ball & Bass, 2009). For example,

‘in countries like Australia and the United States, there has been an over-emphasis on counting at the expense of building strong visual images of the numbers to 10, in terms of their parts – that is, as relative quantities. This has resulted in a significant proportion of students in the middle years of schooling developing an over-reliance on some form of counting and/or additive thinking to solve problems that involve multiplication or division (Siemon et al., 2016, p. 297).

When we consider the importance of deep content knowledge through the seemingly simple task of learning to count, it is perhaps unsurprising that research shows that ‘teachers' mathematical knowledge matters and significantly predicts gains in students' achievement’ (National Council of Teachers of Mathematics, 2010). Without deep content and specialised knowledge, educators are unable to make discerning, targeted decisions about where to invest their time, energy and resources in order to support the students they work with. Thames and Ball (2010) are succinct in recognising the nuances required in teaching and the relations between pedagogical decisions and mathematical content knowledge, in teaching and learning:

‘Teaching is not merely about doing math oneself, but about helping students learn to do it. This is challenging and requires specialised, skilled ways of knowing the domain… mathematical knowledge for teaching is a kind of complex mathematical understanding, skill, and fluency used in the work of helping others learn mathematics’ (Thames & Ball, 2010, p. 228).

Pedagogies that relate to the cultivation of deep content knowledge in mathematics

Glenda Anthony and Margaret Walshaw (2009) discuss an effective pedagogy of mathematics teaching. They identify ten effective pedagogical practices:

  • an ethic of care
  • arranging for the learning
  • building on students’ thinking
  • worthwhile mathematical tasks
  • making connections
  • assessment for learning
  • mathematical communication
  • mathematical language
  • tools and representation
  • teacher knowledge.

They note that effective teachers:

  • provide students with opportunities to work both independently and collaboratively to make sense of ideas. Students’ learning opportunities are supported through independent, whole-class, partnered and small group situations
  • support students in creating connections between different ways of solving problems, between mathematical representations and topics, and between mathematics and everyday experiences. These practices are supported by encouraging students to make and test conjectures about mathematical ideas and concepts where students demonstrate multiple solutions, representations and flexible thinking
  • facilitate classroom dialogue that is focused on mathematical argumentation. This is explored through examining conjectures, disagreements and counterarguments as well as teaching how to communicate mathematical ideas and thinking
  • shape mathematical language by modelling appropriate terms and communicating their meaning in ways that students understand. Concepts and technical terms need to be explained and modelled in ways to support students’ true understanding of the terms and their meanings
  • carefully select tools and representations to provide support for students’ thinking. Tools provide vehicles for representation, communication, reflection and argumentation. These tools support students at all stages of their learning, and are also important
  • have clear ideas about how to build procedural proficiency and how to extend and challenge student ideas and adjust a lesson according to the needs of the students (Anthony & Walshaw, 2009).

Direct and Dialogic Instruction

Hattie distinguishes between Direct and Dialogic Instruction.

Through direct instruction students learn from:

  1. watching clear, complete demonstrations of how to solve problems with accompanying explanations and accurate definitions
  2. practising similar problems sequenced according to difficulty and
  3. receiving immediate corrective feedback.

Through dialogic instruction, students learn from:

  1. actively engaging in problem solving, persevering to solve novel problems
  2. participating in a discourse of conjecture, explanation and argumentation
  3. engaging in generalisation and abstraction, develop efficient problem-solving strategies and relating their ideas to conventional procedures; and to achieve fluency with these skills and
  4. engaging in some amount of practice (Hattie et al., 2017).

An example of dialogic instruction in a community of inquiry is Philosophy for Children. Lessons are stimulated by a question asked by a student, a current news story, a picture book for younger students or a short film clip. Children are invited to say whether anything interested them or puzzled them about the stimulus. From this, a whole class discussion ensues relating to life’s big questions. Students learn how to respectfully disagree because the focus is explicitly on taking issue with a claim rather than taking issue with a person (‘I disagree with your argument’ rather than ‘I disagree with you’) (Jensen & Kennedy-White, 2014). Students take it in turn to speak and to facilitate an orderly discussion a Speaker’s Ball was used. The person with ball is the speaker and everyone else is a listener. The ball is rolled from person to person, as indicated by a show of hands.

Differences between the direct and dialogic methods are the types of tasks students are invited to complete and the role of classroom discourse, collaborative learning and feedback (Hattie, et al., 2017).

Hattie has found that Direct Instruction has an effect size 0.59. Dialogic Instruction has an effect size of 0.82 - double the effect size of 0.4, which is generally regarded as one year’s teaching for one year’s growth.

Hattie observes that the higher effect size of dialogic instruction does not mean that teachers should always choose this approach over another. It should never be an either/or situation. Rather it should be a both/and situation. The art of teaching involves teachers choosing the right approach at the right time to ensure learning and understanding how both dialogic and direct approaches have a role to play throughout the learning process, but in different ways.

Nor should teachers confine their practice to direct and dialogic pedagogies. The pedagogy chosen should be fit for purpose.

Hattie suggests that direct instruction best contributes to surface knowledge and dialogic instruction best contributes to deep knowledge.

Hands on learning in mathematics

The concepts at the heart of Mathematics are highly abstract. Naturally, teachers and students seek ways to make these concepts more tangible. This can be achieved through hands on learning such as by drawing. It could be said that learning Mathematics involves learning to draw better.  There are other ways besides drawing, the use of manipulatives more generally, that can facilitate rendering abstract mathematical concepts more tangible.

According to an enactive perspective on cognition, the brain is not composed of computational machinery locked away inside the head, representing the external world to provide knowledge upon which we can act. Rather, in action – whether reaching and grasping, pointing, or gesturing – the brain partners with the hand and forms a functional unit that properly engages with the agent’s environment.

We, like other animals, are action oriented. Our ability to understand the world comes from a pragmatic engagement with it, along with other people. The mind consists of a structural coupling of brain-body-world constituents. The brain is not the sole cognitive resource we have available to us to solve problems.  The body and the world also play their part.

This all has implications for how we teach and learn. Teaching should not involve transmission of content from teacher to students like a jug filling a mug. Learning involves active engagement with the world: working with other people, manipulating physical objects or representations of them – not necessarily inside our heads.

Mathematics should also be taught by students manipulating physical objects, such as counters, an abacus, blocks, pieces of string, flowers, snails, bubbles, puppets, computer simulations.

A meta-analysis of studies compared the use of manipulatives, or hands-on practical apparatus in teaching mathematics, with teaching that relied only on abstract mathematical symbols (Carbonneau et al., 2013). The researchers found statistically significant evidence that manipulatives had a positive effect on learning with small to moderate effect sizes. This is compelling evidence in favour of using manipulatives - though the ways in which they are used is hugely important.

Conclusion

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).

Cultivation of deep content knowledge in mathematics is likely to involve a major transformation in the way that mathematics has been taught for centuries from a didactic to a dialogical approach.

References

Anderson, L. & Krathwohl, D. (2001). A taxonomy for learning, teaching and assessing: A revision of Bloom’s Taxonomy of educational objectives. Boston, MA: Allyn & Bacon.

Anthony, G. & Walshaw, M. (2009). Effective pedagogy in mathematics. Educational Practices Series, 19. International Academy of Education.

Ball, D. & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners’ mathematical futures. 43rd Jahrestagung fur Didaktik der Mathematik. March 1-4, Oldenburg, Germany.

Biggs, J.B. (1995). Assessing for learning: Some dimensions underlying new approaches to educational assessment. Alberta Journal of Educational Research, 41(1), 1-17.

Biggs, J.B. & Collis, K.F. (1982). Evaluating the quality of learning – the SOLO Taxonomy. New York: Academic Press.

Biggs, J.B. & Collis, K.F. (1991). Multimodal learning and the quality of intelligent behaviour. In H. Rowe (Ed.), Intelligence: Reconceptualization and measurement. Melbourne: ACER.

Carbonneau, K.J., et al. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380-400.

Giroux, H. (2004). Critical pedagogy and the postmodern/modern divide: Towards a pedagogy of democratization. Teacher Education Quarterly, 31(1), 31-47.

Hattie, J. (2012). Visible learning for teachers: Maximizing impact on learning. London: Routledge.

Hattie, J., et al. (2007). Visible learning for mathematics grades K-12: What works best to optimise student learning. California, USA: Corwin, A Sage company.

Jensen, B. & Kennedy-White, K. (2014). The case for philosophical inquiry in K-12 classrooms. Scan, 33(2).

Lipman, M. (1991). Thinking in education. New York: Cambridge University Press.

National Council for Teachers of Mathematics. (2010). Mathematics professional development. Professional Development Research Brief.

NSW Department of Education. (2018). Department of Education Strategic Plan 2018-2022. NSW Government.

NSW Government Education Standards Authority. (2018). Australian professional standards for teachers. NSW Government.

Pegg, J. (2010). Promoting the acquision of higher-order skills and understandings in primary and secondary mathematics. Australian Council for Educational Research Conference. Crown Conference Centre, Melbourne: ACER Press.

Siemon, D., et al. (2016). Teaching mathematics: Foundations to middle years. Oxford, UK: Oxford University Press.

Siemon, D. et al. (2012). Working with the big ideas in number and the Australian Curriculum: mathematics. In B. Atweh, M. Goss, R. Jorgenson & D. Siemon, Engaging the Australian National Curriculm: Mathematics – Perspectives from the field (pp.19-45). Mathematics Education Research Groups of Australasia.

Thams, M. & Ball, D. (2010). What mathematical knowledge does teaching require? Knowing mathematics in and for teaching.Teaching Children Mathematics, 17(4), 220-229.

Van de Walle, J., et al. (2014). Teaching student-centered mathematics: Developmentally appropriate instruction for grades K-2. Pearson.

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 1, Scan, 38(5).

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