# M2 - Models of Mathematics Learning

## Introduction

This second module will investigate how people learn mathematics and continue to investigate the nature and attributes of students. You will engage with literature and video elements to examine the major theories and research relevant to tertiary mathematics education. It is important to note that this is an open area of study and that there are a range of other theories and approaches that have not been discussed.

### Learning Outcomes

This module explores how students learn, both generally and in the discipline of mathematics. At the end of this module you will be able to:

- explain different learning theories as they relate to mathematics
- describe different approaches to mathematics learning and how they can inform your mathematics teaching
- outline different learning styles and describe ways in which your teaching may be adjusted to cater to them

### Module Structure

The module proceeds as follows:

- Perspectives of learning
- Models of mathematics learning
- General education frameworks relevant to mathematics

## Perspectives of learning

There are a range of views of learning that you may consciously or inadvertently adopt in your mathematics teaching. For instance:

- Behaviourism - learning and teaching is essentially about changing behaviours through stimulus materials and consequent rewards/punishments
- Constructivism - learning and teaching is about designing activities that enable students to build upon prior knowledge to construct internally consistent mental models
- Socio-constructivism - learning is a collaborative, negotiated experience that is most effectively achieved through peer and group interaction
- Authentic learning - learning is best facilitated through realistic, relevant, complex, and problematised tasks.

Other perspectives of learning including cognitivism, constructionism and connectionism and connectivism. Watch the following two videos that explore some of these perspectives on learning.

(Ianr3128, 2007)

(CustomTeach, 2009)

It is important that teachers adopt perspectives and approaches that align with what we know about learning. This is expounded in the video Why We (Should) Teach the Way We (Should) Teach: How Learning Theory Should Impact the Design of Classroom Experiences. In this presentation Dr Nancy Casey, Associate Professor of education emphasises that "In any teaching-learning situation, teachers must be guided about what we know about learning and how learners learn ... they must design their teaching to be responsive to the needs of the learners" (St. Bonaventure University, n.d.). An abridged version of Dr Casey's presentation is provided below (IAmBona, 2009):

Casey stresses that teachers at all levels need to acknowledge that the important thing is not the teaching, but the learning. For instance, it would be erroneous to assume that because you have transmitted the content to students in a lecture, they have learnt the material.

### Task 2.1 Perspectives of learning for your classes

- Select three perspectives of learning that most interest you to investigate in more depth. Outline the key features of the three perspectives of learning that you have selected and compare and contrast the relative advantages and disadvantages of each. Which approach do you believe to be most important in mathematics learning? Why?
- Which approach/approaches do you see yourself adopting in your classes? Explain.

## Models of mathematics learning

Several experienced mathematics education researchers have presented models and heuristics to guide mathematics teaching. It is an interesting and open question - what are the different ways in which students learn mathematics? Three models of mathematics learning that will be examined here are Schoenfeld's model of mathematical problem solving, Skemp's Instrumental versus Relational Understanding, and Threshold concepts.

### Schoenfeld's approach to mathematical problem solving

Alan Schoenfeld identifies four types of knowledge and skills that are required to be successful in mathematics. They are:

- Resources - factual and procedural knowledge of mathematics.
- Heuristics - problem solving strategies and techniques (deductive reasoning, working backwards, drawing diagrams etc).
- Control - the ability to make decisions about which strategies are used when and how.
- Beliefs - a mathematical epistemology or "world view" that determines attitudes and approaches towards a problem.

For example, when solving a projectile motion problem, students need to understand the basic concepts of vectors, differentiation etc (resources); they need to have strategies that guide how they go about finding the unknown quantities (heuristics); they need to be able to reflect upon whether their approach is leading them closer to a solution and know when to adopt an alternative approach (control); and they need to see mathematics as a useful tool that they can confidently use to solve projectile motion problems (beliefs). Success in mathematical problem solving depends on combining all of these four areas, and deficiencies in any one of these areas will inhibit a student's progress

A previous participant in this module made this comment:

*Control is the biggest thing in my third year group theory unit. When confronted with having to do a proof, the students need to be able to recognise what knowledge they need to apply and when to apply it. I can help with this by showing them the thought processes behind the proofs that I do in class, pointing out how to start one, think about what you know about the topic, and how to put the knowledge together to get a proof.*

**Further reading**

- Good summary of Schoenfeld's Mathematical Problem Solving (Kearsley, n.d.)
- Schoenfeld, A. (1985).
*Mathematical Problem Solving.*New York: Academic Press. - Schoenfeld, A. (1987).
*Cognitive Science and Mathematics Education.*Hillsdale, NJ: Erlbaum Assoc.

### Skemp's Instrumental versus Relational understanding of mathematics

Richard Skemp draws the distinction between learning and teaching that takes a relational approach and that which takes an instrumental approach. Relational understanding refers to knowing both what to do and why - an understanding of all of the parts, how they relate, and why they are applied in the manner they are. On the other hand instrumental understanding refers merely to being able to apply a series of steps without knowing why they are being applied in that way or what they mean - 'rules without reasons'. Quite often when students learn mathematics they experience (perhaps encouraged by the approaches of their teachers) temporary success by forming an instrumental understanding of the topic. However this is at the detriment of their long-term success compared to their potential if they had formed a relational understanding.

For instance, when learning differentiation students might easily learn 'the rules' of calculating derivatives but have little concept of when to use each rule and what the derivative means in practice. The consequence of forming this instrumental rather than relational understanding is that they cannot sense the utility of differential calculus and they are unable to solve problems with it

Previous participants in this module identified that to promote relational understanding, teachers need to consciously set tasks requiring relational understanding, and to reward it (or not reward instrumental understanding as highly). The rewards of relational understanding suggested were not only marks(!) but also the satisfaction of the “Aha!” kind of experience, and the better recall that comes from relational understanding. Here is one of their suggestions:

*Try to change the assessment tasks to include tasks that require a relational understanding. For example, “Write a paragraph that a peer could understand, including diagrams where appropriate, explaining ...”*

**Further reading**

- Skemp, R. (1976). Relational Understanding and Instrumental Understanding,
*Mathematics Teaching*,*77*, pp. 20-26

### Threshold concepts in mathematics

It is valuable when planning your unit and learning activities, to spend some time considering the "threshold concepts" in your discipline. This is a relatively new insight into learning that has been developed by Meyer and Land (2003) who describe a threshold concept as "akin to a portal, opening up a new and previously inaccessible way of thinking about something. It represents a transformed way of understanding, or interpreting, or viewing something without which the learner cannot progress" (p. 1). Understanding threshold concepts can be a kind of initiation into the predominant way of thinking, perceiving and practising in a particular discipline (Meyer & Land, 2005).

Threshold concepts are:

- transformative (they trigger a shift in perception)
- irreversible (they cannot be easily unlearned or discarded)
- integrative (they expose previously hidden or unrecognised connections and interrelations)
- bounded (they open up a new conceptual space, and to move beyond it may take engagement with a further threshold concept)
- troublesome (appearing complex, alien, counterintuitive or incoherent).

(Meyer & Land, 2005).

For instance, in mathematics, a complex number consisting of a ‘real' and ‘imaginary' component is conceptually difficult to grasp and almost seems absurd, yet an understanding of complex numbers underpins solutions to many mathematical problems and has applications in both the pure and applied sciences (Meyer & Land, 2005). In pure mathematics the concept of a limit is another threshold concept. For instance, the fact that the limit as x tends to zero of the function f(x)=(sine x)/x is in fact one (1) is counter intuitive, but being able to calculate such limits "is the gateway to mathematical analysis and constitutes a fundamental basis for understanding some of the foundations and application of other branches of mathematics such as differential and integral calculus". (Meyer and Land, 2003, p. 2). A previous participant identified understanding a scalar function of two variables as a surface to be a threshold concept in vector calculus.

Well before the idea of threshold concepts gained acceptance across many disciplines, the mathematician David Tall had described the “difficult transition” to *Advanced Mathematical Thinking* that must take place during undergraduate mathematical education (Tall, 1992).

Given the centrality of such concepts within sequences of learning and curricular structures, their troublesome nature for students assumes significant pedagogical importance (Meyer and Land, 2003, p. 5). Thus it is critical that teachers dedicate adequate time and focus to developing students' understanding of threshold concepts. Possible strategies include extra time dedicated to instruction (including provision of applied examples), time for students to experiment with problems, and peer explanation tasks.

**Further reading**

- Meyer, J., & Land, R. (2005). Threshold concepts and troublesome knowledge (2): Epistemological Considerations and a Conceptual Framework for Teaching and Learning.
*Higher Education*,*49*, pp. 373-388. - Tall, D. (1992). The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity and Proof. In Grouws D.A. (ed.)
*Handbook of Research on Mathematics Teaching and Learning*, New York: Macmillan, 495–511. A PDF of this article is available online.

### Task 2.2: Ways your students learn mathematics

Post responses to the following questions on the **discussion board:**

- Which of the four areas outlined by Schoenfeld do you feel will cause or does cause students in your classes the most difficulty? What strategies can you apply as a teacher to address this?
- You notice that some of the students in your class are only aiming for an instrumental rather than relational understanding of the mathematics being covered in your unit. What can you as the teacher do?
- What are the key threshold concepts in unit/s you are teaching? What can you do to support students to acquire these threshold concepts?

## General education frameworks relevant to mathematics

As well as models of learning that are specific to mathematics, there are also general models of learning that can be applied to learning and teaching mathematics. Two that will be examined here are Anderson and Krathwohls' (2001) Taxonomy of Learning, Teaching and Assessing, and the Structure of Observed Learning Outcomes (SOLO) Taxonomy.

### Anderson and Krathwohl's taxonomy of learning, teaching and assessing

Anderson and Krathwohl (2001) revised Bloom's taxonomy of learning objectives to derive their comprehensive framework for Learning, Teaching and Assessing. The framework not only considers the type of cognitive process in which students engage (as did Bloom's taxonomy) but also the type of knowledge being addressed. The Anderson and Krathwohl framework is represented in Table 1 below.

The Cognitive Process dimension relates to the type of mental activity that the student undertakes. The six levels can be essentially summarised as follows:

- Remember - retrieving relevant knowledge from long-term memory
- Understand - constructing meaning from instructional messages (oral, written, visual)
- Apply - carrying out or using a procedure in a given situation
- Analyze - breaking material into its constituent parts and determining how the parts relate to one another and to an overall structure or purpose
- Evaluate - making judgments based on criteria or standards
- Create - putting elements together to form a coherent or functional whole; reorganising elements into an original pattern or structure.

The Knowledge dimension relates to the nature of the subject matter being considered (learnt). The four levels of knowledge are:

- Factual (declarative) knowledge - discrete pieces of elementary information, required if people are to be acquainted with a discipline and solve problems within it.
- Procedural knowledge - the skills to perform processes, to execute algorithms and to know the criteria for their appropriate application.
- Conceptual knowledge - interrelated representations of more complex knowledge forms, including schemas, categorisation hierarchies, and explanations.
- Metacognitive knowledge - knowledge and awareness of one's own cognition as well as that of other people.

This interactive tool from the Iowa State University provides some cross-disciplinary examples of what Anderson and Krathwohl's taxonomy means in practice (Iowa State University, n.d.).

In terms of mathematics, remembering factual knowledge might be being able to define the meaning of an Eigenvector, applying procedural knowledge might be the ability to find the roots of a quadratic equation, analysing conceptual knowledge might be solving a prime number proof, and creating metacognitive knowledge might be finding new ways for students to understand how they best learn mathematics.

As stated in the name, Anderson and Krathwohl's (2001) framework is not just a reflective tool, but can be used to guide learning, teaching and assessing in a subject area (for instance mathematics). Firstly, the framework can be used to plan learning objectives, to help define and ensure an appropriate range of cognitive processes and knowledge types for a mathematics unit or course. For instance, upon mapping intended learning outcomes in a mathematics subject using the Anderson and Krathwohl framework it may become apparent that the majority of objectives address lower order cognitive processes (remember, understand, apply) rather than higher order cognitive processes (analyse, evaluate, and create). Secondly, Anderson and Krathwohl's (2001) framework can be used to plan curriculum and learning activities, to make sure that there are range of mathematical tasks and knowledge areas being addressed. For instance, considering the activities planned for a mathematics subject in light of Anderson and Krathwohl's Taxonomy may reveal that, although metacognitive thinking was an objective, it was not addressed in any of the learning tasks. Thirdly, the framework can be used during evaluation of student performance, to gauge the different sorts of processes and knowledge being assessed. When setting a mathematics exam Anderson and Krathwohl's taxonomy may provide a point of reference to ensure that there is an adequate range of questions (and that they match the mathematics learning outcomes identified for the unit).

By providing a framework for conceptualising objectives, activities and evaluation, Anderson and Krathwohl's Taxonomy of Learning Teaching and Assessing enables educators to determine whether learning objectives, learning activities and assessment are aligned. These three aspects of mathematics subjects are discussed at length in later modules.

### Task 2.3 Learning, teaching and assessing in mathematics

- Attempt to provide some examples of mathematics learning tasks that exemplify different combinations of learning processes and knowledge types. In your opinion, are there any types of mathematics thinking or learning that are not represented in the framework?
- Which areas of the Anderson and Krathwohl's (2001) taxonomy do you think are least represented in most mathematics curricula? Does this deserve special attention or rectification?

### The SOLO Taxonomy

Whereas Anderson and Krathwohl's (2001) Taxonomy of Learning, Teaching and Assessing considers the type of learning being addressed by a task, the SOLO Taxonomy (Biggs & Collis, 1982) provides a framework for assessing the quality of students' responses to tasks. Here SOLO stands for Structure of the Observed Learning Outcome. The SOLO Taxonomy is used to evaluate the extent to which students have acquired all of the component information items required to understand a concept and the extent to which these knowledge items are correctly interrelated, by examining the structure of their response to a task. According to the SOLO Taxonomy (Biggs & Collis, 1992) there are five levels of understanding that a person may demonstrate for any given concept:

- Prestructural - students cannot provide even one correct response item in relation to a problem.
- Unistructural - a single correct response item in relation to a problem.
- Multistructural -several correct response items but not an entire set and not entirely interrelated with one another.
- Relational - an entire set of fully interrelated response items in relation to a problem.
- Extended abstract - students can provide a complete set of fully interrelated response items and interrelate that information beyond the bounds of the concept or process being considered.

These levels of understanding can be represented diagrammatically, as shown in Figure 1.

For instance, for the topic of differentiation of polynomials, a prestructural understanding is when students do not understand even the first concepts about differentiation of polynomials; a unistructural understanding might be the ability to calculate the derivative of a polynomial term but with no understanding of what this represents; a multistructural understanding may be where students can attempt to differentiate a polynomial term by first principles but have a systematic error in how they calculate limits; a relational understanding might be where they can correctly differentiate polynomials using first principles and explain this in relation to a graph of the polynomial; and an extended abstract understanding may be where they can relate the differentiation of polynomials to differentiation of other function types.

While the SOLO Taxonomy (Biggs & Collis, 2002) is mainly used as a framework for evaluating the quality of student assessment responses, it can also be used when setting objectives (to identify the level of understanding required) and when designing learning activities (to gauge whether the component knowledge items have been articulated and whether the learning tasks support interrelation of component knowledge items).

### Task 2.4 Assessing the quality of student learning in mathematics

- Pick an undergraduate concept in mathematics and design an examination question to assess it or think of a recent exam or assignment you have marked. Provide an example of:
- a prestructural response
- a unistructural response
- a multistructural response
- a relational response
- an extended abstract response.

- Now think of the marking scheme you would normally use for such a question. How effectively does your marking scheme distinguish between the different types of responses?
- Did the question you considered invite responses at the higher SOLO levels? Can you change it so that it relates to the same content, but so that students are required to show relational understanding (either as Skemp or SOLO describes it)?
- Are there aspects of mathematics learning that the SOLO Taxonomy does not adequately capture? Explain your answer.

## Review and conclusion

There is a range of perspectives from which mathematics educators can view education and learning (for instance behaviourist, constructivist, socio-constructivist etc) and this influences the approaches they adopt in their classes. While the way in which students learn mathematics is a vast and somewhat open area of research, there are some models that provide guidance and insight for teachers. Schoenfeld's elements of mathematical problem solving ability (resources, heuristics, control, beliefs), Skemp's emphasis on relational rather than instrumental mathematical understanding, and the importance of addressing threshold concepts provide reference points for reflecting on mathematics teaching practice. Anderson and Krathwohl's (2001) Taxonomy of Learning Teaching and Assessing provides a framework for reflecting on the mathematics learning objectives, activities and assessment tasks that teachers integrate into their units. It is a tool which enables teachers not only to evaluate whether they have an appropriate range of mathematical knowledge levels and cognitive processes represented, but also to check that there is alignment between objectives, tasks and assessment. The SOLO Taxonomy (Biggs & Collis, 1982) provides a general framework for assessing the quality of students' performance that can be applied across a wide variety of assessment tasks.

Look back at your answer to the question from Module 1 - *What is learning: how do students learn mathematics?* Would you like to develop it further in any way?

While well known, these models of mathematics learning are still open to interpretation, and teachers need to decide the extent to which they will apply them in their teaching. As you progress through the rest of the modules, reflect upon how they might be used to design curriculum and assessment. For instance, these models may directly inform your thinking in the next module.

## Assessment task one

The ideas and readings presented in modules one and two, and your reflections on your own teaching in light of them, have prepared you to write a teaching philosophy statement for the first assessment task for the unit. Details of this task and the assessment rubric are in the unit outline. ## References

- Anderson, L. W. (Ed.), Krathwohl, D. R. (Ed.), Airasian, P. W., Cruikshank, K. A., Mayer, R. E., Pintrich, P. R., Raths, J., & Wittrock, M. C. (2001).
*A taxonomy for learning, teaching, and assessing: A revision of Bloom's Taxonomy of Educational Objectives*(Complete edition). New York: Longman. - Biggs, J., & Collis, K. (1982).
*Evaluating the quality of learning: the SOLO taxonomy,*New York: Academic Press. - Center for Excellence in Learning and Teaching. (n.d.)
*A Model of Learning Objectives*. Retrieved from Iowa State University: http://www.celt.iastate.edu/teaching/RevisedBlooms1.html - CustomTeach. (2009, March 29).
*Theories of Learning*Video file. Video posted to http://www.youtube.com/watch?v=Vq9XIrNGgoQ. - IamBona. (2009, May 8).
*Learning Theory's Impact on Teaching*Video file. Video posted to http://www.youtube.com/watch?v=CnnjX9RrGq8. - Ianr3128. (2007, April 3).
*Introduction to Learning Theories*Video file. Video posted to http://www.youtube.com/watch?v=hsX5Tq3WTBw. - Kearsley, G. (n.d.).
*Mathematical Problem Solving*. Retrieved February 21, 2011, from http: http://tip.psychology.org/schoen.html - Meyer, J., & Land, R. (2003).
*Threshold concepts and troublesome knowledge: Linkages to ways of thinking and practicing within the disciplines (Occasional Report 4)*. Edinburgh: Enhancing Teaching-Learning Environments in Undergraduate Course Project. - Meyer, J., & Land, R. (2005). Threshold concepts and troublesome knowledge (2): Epistemological Considerations and a Conceptual Framework for Teaching and Learning.
*Higher Education*,*49*, pp. 373-388. - Schoenfeld, A. (1985).
*Mathematical Problem Solving*. New York: Academic Press. - Schoenfeld, A. (1987).
*Cognitive Science and Mathematics Education*. Hillsdale, NJ: Erlbaum Assoc. - Skemp, R. (1976). Relational Understanding and Instrumental Understanding.
*Mathematics Teaching*, 77, pp. 20-26. - St. Bonaventure University. (n.d.)
*Learning Theory's Impact on Teaching*Video file. Video posted to http://blip.tv/file/2088678. - Tall, D. (1992). The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity and Proof. In Grouws D.A. (ed.)
*Handbook of Research on Mathematics Teaching and Learning*, (pp. 495–511). New York: MacMillan. Retrieved from http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1992e-trans-to-amt.pdf

## Further reading

- D'Souza, S. & Wood, L. (2003). Rationale for collaborative learning in first year engineering mathematics. New Zealand Journal of Mathematics, 32 (Supplementary Issue), pp. 47-55. Retrieved from: http://epress.lib.uts.edu.au/research/bitstream/handle/10453/5661/2003001066.pdf?sequence=1
- Engelbrecht, J. (2010). Adding structure to the transition process to advanced mathematical activity,
*International Journal of Mathematical Education in Science and Technology*, 41(2), pp. 143 — 154. Retrieved from: http://tsg.icme11.org/document/get/530 - Mayer, R.E. (1992). Cognition and instruction: Their historic meeting within educational psychology.
*Journal of Educational Psychology*, 84(4), pp. 405-412. - Schoenfeld, A.H. (1992). Learning to think mathematically: Problem solving. metacognition, and sense-making in mathematics. In D. Grouws (Ed.),
*Handbook for Research on Mathematics Teaching and Learning*, (pp. 334-370). New York: Macmillan. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.87.7976&rep=rep1&type=pdf - Tall, D. (1994, July). The Psychology of Advanced Mathematical Thinking: Biological Brain and Mathematical Mind. Paper presented for the Working Group on Advanced Mathematical Thinking, at the
*Conference of the International Group for the Psychology of Mathematics Education*, Lisbon, July 1994. Retrieved from: http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1994e-biolbrain-mathmind.pdf - Tall, D. (2005).
*Advanced Mathematical Thinking, from:*http://www.warwick.ac.uk/staff/David.Tall/themes/amt.html - Thompson, A. (1992). Teacher's beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 127-146). New York: Macmillan.