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Mathematics Questionnaire Learning

Learning Module

Instructional Module for Mathematics Education

Introduction

This research is concerned with the attitudes and development of skills of a group of mathematics students studying a third level module on non-linear systems. It arose as a means of evaluating a teaching style developed by the authors over the past two years. There were 17 students on the course. We chose a third level module so that the students had adapted from a school mode of learning to a university mode. This would enable us to compare their concepts of learning within higher education to the experience on our module.

Research Methodology

We decided to investigate the students' concepts of learning using a revised version of the questionnaire used in Berry & Sahlberg (1996). The questionnaire consisted of two parts:

  • An open-ended task;
  • A set of 12 statements to provide student ratings.

The open-ended task consisted of asking students to write down, in five minutes, their own answers to the question "What is Learning?".

The answers to this open-ended task are used as our prime source of evidence for identifying the students' views of how they learn.

The second task was a Likert-type rating scale and is included in the questionnaire to provide extra evidence to support the students views on how they learn. We were particularly interested in the students' view of the role of the lecturer (statements 1, 3, 8 and 11) and their view of the importance of co-operating in learning (statements 4 and 5).

The teaching style is designed to downplay the lecturers role, and encourage discussion and co-operation between the students. A post-module evaluation was carried out using two open questions:

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  • What do you like most about this module?
  • What do you like least about this module?

The aim of this questionnaire was to investigate the views of the students' on the learning and assessment style. We decided not to use focused type questions to avoid the tendency for students to tell us 'what we wanted to hear'. For example, a student who has only experienced a transmission model of learning with little or no co-operation with their peers would identify with this model in the pre-module questionnaire.

If such a student believed that this is the best model then we expected negative comments such as 'not enough lecturing' in their answers to the second question. The primary aim of this research was not to change students' concepts of learning--that would be an ambitious task when they have five other modules with a more traditional presentation style.

Hence, using the same concepts of learning questionnaire as a post-test was not appropriate. The aim of the research and the purpose of this paper is to present an evaluation of an innovative, interactive, whole class teaching style in mathematics in higher education.

Findings

For the academic year of the evaluation there were 17 students enrolled on the module. Each student completed the pre-module questionnaire investigating their concepts of learning and a post-module questionnaire on their attitude to the module.

Pre-module questionnaire

The open question 'what is learning' provided a range of responses showing views of learning from a passive teacher-centred transmission to a more active student-centred concept. Here are examples of student responses.

  • Learning is lectures;
  • Learning is the intake of information and the ability to memorise this information and apply it when necessary.

These responses are typical of students with a passive view of learning in which lectures and memorising are important features of the process.

Learning is gaining new knowledge on an unfamiliar subject or it enables you to use what you already know in another way. I learn from others and mistakes I make. In this response the student recognises the importance of constructing new knowledge from what is known and that she is a partner in the process with other students.

Learning is discovering 'things' I do not already know--where 'things' may be facts or may even be physical and emotional. But learning is like building blocks and one fact or item of knowledge does not stand on its own--unless for a crossword puzzle! If I didn't keep learning I would die of boredom but I don't think this is true for everyone--some people seem happier not knowing.

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For this student, 'discovering' is an important part of the process and linking new things to old through the comparison to building blocks suggest to us an active personalised view of learning. Each individual answer in task 2 was scored -2, -1, 1 or 2 according to the Likert-type rating scale. Items were scored so that they all indicated the same direction as the steps to good learning criteria, i.e. +2 indicates an active transformation view, whereas -2 indicates a passive transmission model.

We want to take a close look at just two of the features of learning that are important for an evaluation of the pedagogy for the module. These are the role of the teacher as identified by statements 1, 3, 8 and 11 and the importance of co-operation as identified by statements 4 and 5.

A summary of the responses to these six statements is shown in Table I. The table shows that this cohort of students believe that the teacher plays a dominant role in their learning. We would expect that Statement 3 (Learning is most effective when the teacher tells me what I need to know) and Statement 11 (Learning, in most cases, is transferring knowledge from the teacher myself) are typical beliefs of most mathematics students in higher education for whom the lecture/problem class learning environment is common. The teaching style evaluated in this research project is opposed to this dominant view of the role of the teacher and it was necessary in the early weeks of the module to constantly reassure the students.

The responses to Statement 8 (I learn best by doing lots of exercises after watching a teacher doing examples) show that all the students believe in the importance of mimicking in the learning of mathematics skills. The responses to the co-operation in learning Statements 4 (I learn more by working with other students) and 5 (I learn better when the teacher is teaching me than working with a group of other students) was more encouraging. The need to co-operate and discuss mathematics with their peers was a key component in the teaching style. That most of the students recognised this gave us some encouragement.

Post-module Questionnaire

For most students it was the style of coursework that featured strongly in their responses. In most of their studies the coursework element of each module is the summative assessment element and not a formative activity designed to develop understanding, knowledge and skills. For the module discussed in this paper the students were encouraged to view all of their tasks as formative coursework in the sense of their work done on the course.

The first two comments are typical of all the students' views. They show the positive side of the course style providing self-pacing to keep on schedule and the negative side of fear at giving presentations to their peers.

  • I thought that the style of coursework was good, it ensured that I kept up to date with all the exercises and for people to attend lectures.
  • I like the idea of doing it on the board but it is very nerve racking. At first, I dreaded these sessions, they didn't do anything for my confidence or my feelings.

In their open writing statements we were looking for words or phrases that fitted one of the levels on the steps to good learning. For example, in the next comments the students comment on the positive aspect of practice and the opportunity for reflection on their own work (and correction of their work) from seeing the work of others.

  • The writing up of the question provided us with tremendous practice which could only be advantageous to us. By writing up one on the board meant we could learn from our mistakes and be shown different approaches, which sometimes proved useful.
  • Means you have to do more exercises therefore better.
  • This was very useful and enabled me to see where I was going wrong and what I was doing right in a constructive way.

The next comment suggests a recognition of the gradual continuous understanding of the content of the course. However, note the negative second sentence. The style of coursework was good--made me do lots of work and helped me to understand the work throughout the course rather than trying to understand everything all at once!

The time I spent affected my other subjects. The next three students still see coursework as summative assessment. The second comment suggests that for this student the main purpose of the learning style used in the module was the writing up of questions for marks!

  • I like this style of coursework as opposed to having one large piece as if I made a mistake on one piece then marks can be made up on the remainder whereas if you make a mistake on one large piece then that is the end of it.
  • I thought that the coursework was fine, but to spend 7 or 8 hours on the exercises in order to write up one question meant I couldn't spend time on other subjects and seemed pointless.
  • I actually think it would have been useful to receive a couple of larger pieces of coursework in addition to the weekly questions. A larger piece would've enabled more complex questions combining several areas of the course.

The Case Study of Janet

We complete this section with the feelings of one particular student, a mature female who we have called Janet (not her real name). Her pre-course questionnaire consisted of a personal letter at the beginning of the module showing that, at the start of the module, Janet was very negative and hostile to the whole teaching philosophy of the module. Please stop insulting our intelligence.

Your assertion that lectures are 'not the best method of learning' is not supported by fact, is it? Last year I achieved respectable results in 11 modules. My lowest mark (extremely low by my standards) was for your 'trendy teaching' mechanics module ... and the so-called discussion classes degenerated into playtime for the most immature.

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Your classes are designed for the exclusive benefit of frivolous teenagers--to hell with the rest of us. It is not appropriate for you to dictate that we must work in groups like infants in primary school. I had a responsible career for 25 years before I came here; I will think for myself and do my own work, otherwise I can derive no satisfaction from my studies. The post-module feedback shows a complete transformation:

I did not appreciate the interactive teaching style in the first year--I think some of the younger students misinterpreted its purpose and behaved inappropriately. However, as the difficulty of the work increased it was obvious that this method was more effective than the standard lecture class. I enjoyed the weekly homework--we covered everything we needed for the exam. In other modules I felt that the feedback from one written assignment was insufficient--the gaps in my knowledge/comprehension were never corrected. Even if no marks had been awarded for the coursework in this module I think the style was beneficial.

Janet was a mature student with fixed views on teaching and learning. She was very conscientious and on other modules worked very much on her own. Even in this module she did not co-operate fully with other students to form a group. However, her presentations of solutions were of the highest standard and she joined in the whole class discussions. For Janet the module teaching style almost discouraged her from studying the module. The final statement on the post-module questionnaire is what most module leaders wish to hear.

Discussion and Conclusions

This short research study reports on an attempt to break new ground in teaching mathematics in higher education by developing a pedagogy based on a transformation model as opposed to a transmission model that is common in university mathematics faculties.

The evaluation of a mathematics module in higher education usually focuses on the teacher rather than on the learning experience. At the University of Plymouth most modules use a post-module questionnaire with statements such as:

  • Lectures are well prepared and well structured
  • Tutorials are useful and reinforce the material taught
  • Lecture notes are clearly structured
  • Lecturer is easily heard and communicates clearly
  • Lecturer uses teaching aids (OHPs, etc.) effectively

Clearly encouraging a view of good teaching/learning as passive. However, we would argue that good learning is active, co-operative and transformative. The pedagogical approach described in this research report has these features and increases student autonomy.

The case study of Janet is typical of the views of students passing through our modules. They begin with a view of learning that depends on the quality and quantity of our presentations, and their 'success' requires them to mimic our worked examples. Changing their views, encouraging them to be responsible for their own learning by actively acquiring new knowledge and skills in co-operation with their peers is a challenge for them and us.

During the first few weeks of the module the students are reluctant to give a presentation and discuss mathematics openly. They look to us for a 'have I got it correct?' answer. We have to be encouraging and cajoling! We have to avoid the standard fall-back position of 'let me show you how to do it'.

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The findings from the pre- and post-module questionnaires are perhaps best summarized by the writings in the case study of Janet. Not all the students are willing to express their views as strongly as Janet, but the authors have sensed such feelings during the first few weeks of using the teaching style of each module we teach.

It is disappointing to us that mathematics students do not appear to value peer discussion as an important part of the learning process. This seems so at odds with the work of professional mathematicians for whom the discussion of their research ideas and problems is common in Faculty tea rooms, seminars and conferences. (Mathematicians are probably the worst 'shop-talkers' at any social event!).

The results of the evaluation of this mathematics module on non-linear systems are typical of our experiences over the past 3 years. Students give positive feedback and we have noticed an improvement in their written solutions in the formal examination. One feature that comes out strongly is the awareness of learning mathematics by doing instead of watching. In a traditional module students may complete 12-20 problems as part of coursework.

In this module the students have attempted and discussed 120-150 problems. Another feature is the continuous learning and filling in gaps in understanding. For each student there is the view that 'I am the learner' and 'I am doing it within a community of learners'.

We would argue that students are gaining a deeper understanding of mathematics through a continuous development of concepts and skills. The contrast between a deep approach to learning and a surface approach is described by Ramsden (1988). In the deep approach the main intention is on understanding through vigorous interaction with the teaching material. One advantage for the mathematician of such an approach is to provide a solid foundation on which to build.

The surface approach is typical for the learner whose sole intention is the completion of a task such as an assessment component. Too often, we experience students who perform very well in modules at one level, but do not have sufficient understanding on which to build. This is a familiar example of a surface approach to learning.

An important change that has occurred during our experiments is the role of the lecturer. We believe that good learning is enhanced through increased student autonomy. Our students have been given more responsibility for their own learning through the optional homework exercises and in-class discussions. The improvement in their written and oral mathematical skills from previous cohorts is a measure that good learning is taking place.

Enhancing student autonomy has changed our role from one where the student is dependent on the teacher towards an environment of a community of learners where the students look to each other, as well as the teacher. In a sense we are hoping to become expert witnesses drawn into a discussion. However, a total transition has not yet occurred; the students still expect us to act like teachers with control of the situation but not totally responsible.

Our experience is that the fully interactive teaching style described in this paper can be used with classes of about 30 students. We have found that for a first year calculus module of 70 students the 'short introduction--group activity--student presentation' session is too easily disrupted with the tendency for some students to remain as passengers. For this module, we revert to a more traditional lecture mixed with small group activities. However, we retain the 'homework--student presentations' style for the two other sessions by splitting the class into two groups. The essential features of co-operation, reflection and discussion are maintained.

References

BERRY, J. (1996) An Introduction to Non-Linear Systems (London, Hodder and Stoughton).

BERRY. J. & SAHLBERG. P. (1996) Investigating pupils' ideas of learning, Learning and Instruction, 6, pp. 19-36.

BERGER, P. (1996) Mathematics versus computer science: teachers' views on teacher roles and the relation of both subjects, in: PEHKONEN, E. (Ed.) Proceedings of the MAVI-3 Workshop, Research Report 170, University Helsinki.

DE CORTE, E. (1993) Learning theory and Instructional Science, paper presented at the Final Planning Workshop of the ESF programme "Learning in Humans and Machines", St Gallen, Switzerland.

JOHNSON, D.W. & JOHNSON, R.T. (1987) Learning Together and Alone; Cooperative, Competitive and Individualistic Learning (London, Prentice Hall).

RAMSDEN, P. (1988). Studying-learning: improving teaching, in: RAMSDEN, P. (Ed.) Improving Learning: New Perspectives (London, Kogan Page).

REYNOLDS, B.E., HAGELGANS, N.L., SCHWINGENDORF, K.G. VIDAKOVIC, D., DUBINSKY, E., SHAHIN, M. & WINBUSH, G.J. (1995) A Practical Guide to Cooperative Learning in Collegiate Mathematics (The Mathematical Association of America Notes Number 37 (Washington DC, MAA).

RICHARDS, J. (1991) Mathematical discussions, in: VON GLASERFELD, E. (Ed.) Radical Constructivism in Mathematics Education (Dordrecht, Kluwer).

ZUCKER, S. (1996). Teaching at the University Level, Notices of the American Mathematical Society.


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