In this first module we outline the background to teaching mathematics at university, with a particular emphasis on the role of the teacher, the nature and attributes of the students, and the importance of instilling a passion for mathematics in your students.

Our aim is for your teaching to become more: efficient, effective and enjoyable

Learning Outcomes

At the end of this module, you will be able to:

• explain perspectives of teaching mathematics at university
• describe the variety that may exist within the student body you teach
• explain the importance of instilling a passion for mathematics in students.

Module Structure

The module proceeds as follows:

• The context of teaching mathematics in Australia
• What does good and poor teaching look like?
• What do you believe about learning and teaching?
• Seven principles of learning
• What expert teachers know

The context of learning and teaching mathematics in Australia

University education in Australia is changing. Many students are coming to university to improve their prospects of getting professional employment. The emphasis of learning has moved from content where a student gains knowledge to being where a student becomes a professional. All univesities now state graduate/generic skills/attributes/capabilities as part of what a student will achieve in their university studies.

This change in emphasis has made the job of teaching more complex and interesting. You teach the discipline content and the graduate skills the student will require after university. We have always done this however it has been implicit rather than explicit. Now governments, industry, graduates and students are requiring that we demonstrate that students achieve both discipline-specific knowledge and graduate skills.

For those of us teaching in quantitative disciplines, this is a boon. Most graduate attribute statements include the quantitative and analytic skills that we are teaching!

This unit of 12 modules will help equip you for this new teaching environment. The unit will enhance your teaching as well as show you how to prepare a portfolio for your teaching career.

Our aim is for your teaching to become more:

efficient, effective and enjoyable

What does good and poor teaching look like?

We can understand the elements of good teaching by examining examples of both good and poor teaching, deconstructing the interaction and the relationship between the teacher and the learner. The video below provides an example of poor teaching. Focus primarily on the teaching, rather than the content.

Task 1.1 Attributes of poor teaching

In the following five minute video extract from The Big Bang Theory, Sheldon (a physics researcher) is teaching Penny (a waitress and a physics novice) about the physics research that her boyfriend is undertaking. Penny has actively sought out help to learn.

Watch the video, then read the questions below and re-watch the video, pausing as necessary to answer the questions, being as specific as possible.

Extract from "The Big Bang Theory", Sheldon teaches Penny Physics, (SuperTbbt, 2009)

Questions:

1. What does the teacher believe about teaching and learning? How does that belief influence his teaching? How does he change his approach in response to feedback?
2. What was the student experience of the teacher's teaching? Specifically what was she feeling? How did the teacher's actions support or impact her motivation?
3. What was the teacher's expectation of the student performance?
4. How did the teacher check the developing understanding of the student, and how the student was constructing knowledge?
5. What things does the teacher do which you would not want to do if you were in his position?

Task 1.2 Attributes of good teaching in a mathematics / science context

Compare and contrast the teaching and learning interaction above with the thoughts and examples of Harvard University Professor Eric Mazur, in From Questions to Concepts.

(BokCentre, 2008)

Questions:

1. What does Professor Mazur believe about teaching, and how does that influence his teaching.
2. What is the student experience of learning led by Professor Mazur?

What do you believe about learning and teaching?

Lecturers and tutors of university mathematics come from a wide range of backgrounds and experiences, both professionally within the discipline of mathematics and from their past experience as learners exposed to a wide diversity of teaching styles and approaches throughout their secondary and post-secondary education.

During that time, you will have developed some key mental models that shape your current approach as a lecturer or tutor - particularly in relation to fundamental concepts including your beliefs in relation to what mathematics is, what teaching is, and how learning occurs.

Task 1.3 Reflections on learning, teaching and mathematics

Answer each of the three questions below, writing between one and three sentences. It is important to realise there is no right or wrong answer. This activity is not directed towards what you think an examiner thinks, or what an educational psychologist might think; it is all about what you think and believe in relation to mathematics, teaching and learning.

Take 10-15 minutes to consider and write your answers. If it helps, you can also consider why you have formed these views, and what influenced your views.

1. What is mathematics?
2. What is teaching?
3. What is the goal in teaching mathematics?
4. What is learning: how do students learn mathematics?

Post your answer to the last question on the discussion board.

Common myths about teaching undergraduate mathematics

In an article based on a conference plenary address, Why the Professor Must be a Stimulating Teacher (2010), Alsina identifies a series of myths and practices in relation to the teaching of mathematics at an undergraduate level. From personal experience in Spanish universities, Alsina outlines myths in the teaching of mathematics at an undergraduate level, and makes suggestions as to changes going forward in teaching, assessment and technology. The myths are:

• The researchers-always-make-good-teachers myth
• The self-made-teacher tradition (i.e. maths teachers do not need training to be teachers)
• Context-free universal content (a core curriculum is necessary irrespective of the major being pursued)
• Deductive organization (learners learn through representations of deductive reasoning in general form with proofs)
• The top-down approach (Maths taught in the most general form is the most useful, and students will be able to apply it to specific contexts. Real world examples are not necessary.)
• The perfect-theory presentation (Maths is almost complete, and mathematics is a process of getting the right answer which is always possible. Exploration and difficulties are not presented.)
• The ‘master class'/formal lecture paradigm (teaching is about transmitting knowledge to the students)
• The mature students myth (students can be assumed to be motivated, prepared and aware of the need for maths in their training. Diversity of backgrounds is ignored)
• The routine individual-written assessment (the written end-of-unit exam is the ideal way of assessing whether students have mastered the content delivered in lectures)
• The non-emotional audience (students enrolled in a maths course have a singular goal of learning maths, and come without any emotional, personal, or individual problems).

You may recognise these myths, and even agree with some of them, to some extent. Bear them in mind, and your own responses to Task 1.3, as you proceed.

The influence of teacher beliefs on student performance

Although the research context was junior mathematics, Carter and Norwood (1997) found teacher beliefs in relation to mathematics had a significant influence on student beliefs about success factors in mathematics, specifically that working hard to solve problems and striving for understanding would lead to success.

Carlson (1999) investigated the mathematical behaviour, experiences and beliefs of successful graduate mathematics students. She found that consistent beliefs of the graduate students included:

"mathematics involves a process that may include many incorrect attempts; problems that involve mathematical reasoning are enjoyable; individual effort is needed when confronting a difficulty; students should be expected to "sort out" information on their own; and persistence will eventually result in a solution to a problem." (p. 224)

Task 1.4 Reflections on the power of beliefs

1. How do you feel your beliefs impact the beliefs of your students in relation to mathematics?
2. What is your role in helping students establish these beliefs and how can this be achieved?

Task 1.5: Exploring paradigms of undergraduate education

The following article by Barr and Tagg (1995) examines the paradigms that pervade undergraduate education. Although written some time ago, this often cited piece provides a useful context to examine the role of teachers in an undergraduate setting, and the differences between the dominant Instruction Paradigm and a Learning Paradigm of undergraduate education. After reading the article, answer the questions below.

Questions:

1. Which paradigm of undergraduate education do you consider appropriate for your teaching? Why?
2. In the terms outlined in the article, how would you describe your current teaching practices?
3. How would your students describe your teaching paradigm?
4. How do you measure the success of your teaching? What are the goals you set or the measures which you value?

Task 1.6: The variety of student knowledge, understanding and skills

In teaching undergraduate mathematics it is easy to overestimate the level of skills in the student body, and to erroneously assume that if students have certain mathematical understandings then they are readily transferrable by those students to other contexts. For example, Trigueros and Ursini (2003) found that first year mathematics undergraduates evidenced a wide range of misconceptions and approaches characteristic of algebra beginners in secondary school, and that students responses can often be characterised as reactions to symbols present in an expression (exponentials, the equals sign etc) rather than a considered mathematical response.

The Trigueros and Ursini article (2003) confronts teachers with the reality of the range of applied mathematical skills present in their class, and the challenge of applying known mathematical contexts in non-routine cases. Please read the Trigueros and Ursini article and answer the following questions.

1. Describe some of the problems that the undergraduate students evidenced in dealing with variables.
2. Provide examples where students possessed the mathematical "knowledge" to solve a problem, but were unable to apply that knowledge in the context of the question posed.
3. What implications does this diversity in understanding have for the teaching of undergraduate mathematics? Did any of your responses in Task 1.3 allow for these?
   
4. Do you have any strategies (yet) as an undergraduate mathematics teacher to identify misconceptions and help students develop skills to apply mathematical knowledge in non routine contexts?

Seven principles of learning

The body of research in the cognitive, learning and neurological sciences has yielded a range of insights into how people learn, how they organise and internally construct knowledge, how they differ in their preferences and learning styles, and how people develop expertise. In Evaluating and Improving Undergraduate Teaching in Science, Technology, Engineering and Mathematics (Fox & Hackerman, 2003), the US National Research Council identified seven principles of learning (p. 20):

1. Learning with understanding is facilitated when new and existing knowledge is structured around the major concepts and principles of the discipline
   In the context of mathematics, this requires identification and teaching of the big ideas and core concepts of the discipline rather than students learning a series of disconnected facts. Mathematical knowledge can be more readily transferred to new contexts when the big ideas and core concepts are understood rather than isolated, issue-specific problem-solving processes being mastered. Students must build knowledge structured around the major organising principles and core concepts of the discipline and understand how new knowledge is related to those major concepts.

2. Learners use what they already know to construct new understandings
   Students existing knowledge (both correct, and incorrect), beliefs and skills shape their approach to learning and their understanding of new knowledge. In the context of mathematics, this means teachers must determine what students already know about a subject, probe to identify and confront misconceptions, and plan ways to build the new knowledge on top of, and networked into, existing knowledge of the students. This student-specific construction of knowledge filtered through their existing knowledge and conceptions is the opposite of the transmissive model of teaching, which relies on a belief that teaching is telling and that students internalise the message as understood by the teacher who is transmitting that knowledge.
3. Learning is facilitated through the use of metacognitive strategies that identify, monitor, and regulate cognitive processes
   Metacognitive strategies - thinking about thinking, learning, and problem solving - are key skills in human learning, and especially significant in mathematical problem solving. These strategies include assessing current mastery levels, monitoring performance, consciously connecting new knowledge to existing knowledge, deliberate selection of thinking or problem-solving strategies, and thinking strategies of planning, monitoring and evaluating. In the context of mathematics, this requires explicit instruction focused on the development in students of a range of metacognitive skills for managing learning and problem solving, the modelling of thinking skills, the sharing of internal thought processes by "thinking out loud", and probing students' existing metacognitive strategies.
4. Learners have different strategies, approaches, patterns of abilities, and learning styles that are a function of the interaction between their heredity and their prior experiences
   As students have different skills, and learn at different rates in response to different learning activities and approaches (learning styles), there is no universal teaching strategy for the effective development of higher mathematical skills. It follows that a single mode of assessment task will advantage some learners and disadvantage others. Importantly, the different skills and learning preferences of the teacher will affect the range of teaching approaches with which the teacher is most comfortable, and will influence the teacher's beliefs in what constitutes effective teaching - i.e, constrained by the view of what would be effective teaching if the teacher were the student. In the context of mathematics, this means that teachers needs to be aware of teaching assumptions influenced by their preferences and backgrounds, to be alert to the range of abilities and learning styles in the classroom, and to design and deliver a range of learning activities across different learning styles in both unit content and assessment processes.
5. Learners' motivation to learn and sense of self affect what is learned, how much is learned, and how much effort will be put into the learning process
   There is strong research evidence that learners' beliefs about their own abilities in a subject area are strongly connected with success in that subject. In the context of mathematics, building learners' self belief in relation to their ability to understand and engage with the core concepts and principles of the definition is a key component of increasing student performance. In any classroom, some students will believe that their mathematics ability is pre-determined, whilst others will believe their ability to learn is a function of effort expended. A belief in pre-determined skill level, which cannot be altered by effort, is a constraining self-fulfilling prophecy which will significantly impair learning, problem solving and performance.
6. The practices and activities in which people engage while learning shape what is learned
   The way in which knowledge is learned, and the range of uses to which that knowledge is put in the learning process, are important parts of the knowledge that is learned. Where knowledge is learned in a narrow, or artificial, context, students find difficulty abstracting that knowledge and transferring and applying it to other contexts. Whilst the knowledge can be applied again in the context in which it was taught, students are unable to see connections and applicability to other contexts, novel problems, or different disciplines. In the context of teaching mathematics, diverse real world applications of knowledge and problem solving can be used to support students in building these broader connections, especially with problem-based and case based learning and assessment.
7. Learning is enhanced through socially supported interactions
   Conceptual learning is enhanced where students have opportunities to interact and collaborate with others, both as part of the learning processes and tasks, and as part of assessment tasks. The process of students interacting with other students in sharing, challenging, and building understanding is particularly effective with conceptual learning, and can be used by teachers as a less teacher-centred teaching strategy. In the context of mathematics, activities include joint problem solving, peer explanation of concepts, solo problem solving followed by sharing and comparing strategies, and observed problem solving with students articulating their thought processes aloud. The sources of student learning are not limited to the teacher, and one of the roles and challenges for the teacher is how to leverage this additional learning source in the classroom.

Task 1.7: Considering your current teaching and experiences of teaching

1. How, and to what extent, does your current teaching address each of these seven principles of learning?
2. Consider the teaching of one of your best higher education mathematics teachers. How, and to what extent, did their teaching practice address each of these seven principles of learning?

What expert teachers know

Whilst the research on learning tells us about how students learn as individuals constructing knowledge, and how they learn from each other, the other major influence in the classroom is the teacher: the knowledge and activities of the effective teacher.

Shulman (1987) identified the following specific knowledge that expert teachers know:

1. the academic subjects they teach
2. general teaching strategies independent of discipline (classroom management, effective teaching, evaluation)
3. applicable curriculum materials and programs for their subject
4. subject specific knowledge for teaching: teaching different types of students, and teaching particular concepts
5. the characteristics and cultural backgrounds of learners
6. the settings in which students learn: pairs, small groups, classes etc
7. the goals and purposes of teaching.

In the context of science, technology, engineering and mathematics (Fox & Hackerman, 2003) the following five criteria for teaching effectiveness were identified (p. 27):

1. knowledge of subject matter
2. skill, experience and creativity with a range of appropriate pedagogies and technologies
3. understanding of, and skill in using, appropriate assessment practices
4. professional interactions with students within and beyond the classroom
5. involvement with and contributions to one's profession in enhancing teaching and learning.

Task 1.8: Assessing your knowledge

Using a scale of 1-10 where 10 is very strong and 1 is very weak - rate your current level of knowledge and skills across each of the dimensions of knowledge and teaching effectiveness outlined above.

Is there one "correct" expert?

Although in our discipline there is general agreement about correctness, when it comes to teaching approaches there are no such absolutes! For example, Dubinksy and Krantz have quite differing perspectives on the teaching of undergraduate mathematics. Many people find both views enriching and helpful for their teaching. Steven Krantz, Professor of Mathematics at Washington University, St. Louis invited Ed Dubinsky (then a Professor at Georgia State University) to write a reflection in his book How to Teach Mathematics (Krantz, 2000). Krantz's book was a bestseller, and included an appendix in which he sought contributions from those who agreed, and those who disagreed, with his views on the teaching of mathematics expressed in his book. Dubinksy's reflection is included within that appendix. In his reflection, Dubinsky (2000) provides a useful analysis and discussion of constructivism in the context of undergraduate mathematics education, and a commentary on "traditional" teaching of mathematics. Note that Krantz's is personal reflection and Dubinksy is using evidence-based research. Read this as a counterpoint and reflect on the implications of this debate on your teaching practice.

Review and conclusion

Your ideas about mathematics and learning will influence how you teach.

For yourself, write down three ideas you have learnt in this module, and one way in which you will change your teaching or that has challenged your beliefs.

In the next module we will look at models of learning mathematics.

References

• Alsina, C. (2010). Why the professor must be a stimulating teacher. In D. Holton (Ed.), The Teaching and Learning of Mathematics at University Level: An ICMI Study (pp. 3-12). The Netherlands: Kluwer Academic Publishers.
• Artigue, M. (2010). What can we learn from educational research at the university level. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI Study (pp. 207-220). The Netherlands: Kluwer Academic Publishers.
• Barr, R., & Tagg, J. (1995). From teaching to learning: A new paradigm for undergraduate education. Change, 27(6), pp. 12-25. Retrieved from http://www.athens.edu/visitors/QEP/Barr_and_Tagg_article.pdf
• BokCentre. (2008, January 20). From Questions to Concepts: Interactive Teaching in Physics Video file. Video posted to http://www.youtube.com/watch?v=lBYrKPoVFwg
• Carlson, M. P. (1999). The mathematical behavior of six successful mathematics graduate students: Influences leading to mathematical success, Educational Studies in Mathematics, 40(3), pp. 237-258.
• Dubinksy, E. (2000). Reflections on Krantz's How To Teach Mathematics: A different view. In S. Krantz, How to Teach Mathematics (2nd ed.) (pp. 197-214). Providence, Rhode Island: American Mathematical Society.
• Fox, M. A., & Hackerman, N. (Eds). Evaluating and improving undergraduate teaching in science, technology, engineering, and mathematics. Washington, DC: The National Academies Press.
• Holton, D. (Ed.). (2010). The teaching and learning of mathematics at university level: An ICMI study, The Netherlands: Kluwer Academic Publishers.
• Krantz, S. (2000). How to Teach Mathematics (2nd ed.) Providence, Rhode Island: American Mathematical Society.
• Norwood , K. S., & Carter, G. (1997). The relationship between teacher and student beliefs about mathematics, School Science and Mathematics, 97(2), pp. 62-67. Retrieved from http://www.merga.net.au/documents/MERJ_15_1_Handal.pdf
• Schoenfeld, A. H. (2010). Purposes and methods of research in mathematics education. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 221-236). The Netherlands: Kluwer Academic Publishers.
• Selden, A., & Selden J. (2010). Tertiary mathematics education research and its future. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 237-254). The Netherlands: Kluwer Academic Publishers.
• Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), pp. 1-22.
• SuperTbbt. (2009, December 28). The Big Bang Theory - Sheldon Teaches Penny Physics Video file. Video posted to http://www.youtube.com/watch?v=AEIn3T6nDAo.
• Trigueros, M., & Ursini, S. (2003). First year undergraduates' difficulties in working with different uses of variable. In A. Selden, E. Dubinsky, G. Harel, & F. Hitt (Eds.), Conference Board of the Mathematical Sciences Issues in Mathematics Education (Vol. 12), Research in Collegiate Mathematics Education (Vol. V, pp. 1-29). Providence, Rhode Island: American Mathematical Society.

Further Reading

• Clark, M. and Lovric, M. (2009). Understanding secondary-tertiary transition in mathematics, International Journal of Mathematical Education in Science and Technology, 40(6), pp. 755-776
• Radloff, A. (2006). Applying principles for good teaching practice at CQU - Can we? Should we? Retrieved 28 February, 2011, from: http://iris.cqu.edu.au/FCWViewer/view.do?page=8827.
• Schoenfeld, A.H. (1989). Explorations of students' mathematical beliefs and behavior. Journal for Research in Mathematics Education. 20(4), pp. 338-355.
• Selden, A. (2005). New developments and trends in tertiary mathematics education: or, more of the same?, International Journal of Mathematical Education in Science and Technology, 36(2), pp. 131-147.
• Speer, N.M., Smith III, J.P., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. The Journal of Mathematical Behavior. 29(2), pp. 99-114.
• Wood, L. N. (2010). Graduate capabilities: putting mathematics into context. International Journal of Mathematical Education in Science and Technology, 41(2), pp. 189-198.