I am being inspired by reading Katherine Safford-Ramus’s book, “Unlatching the gate. Helping adult students learn mathematics (2008).

She asks ” Do these ideas sound familiar?

- Practice makes perfect
- Students should be rewarded for correct responses
- Specific goals should be stated for every grade level
- Testing will inform us as to the success of our educational programme.

If so then you are already familiar with some of the ideas of the theorists called behaviorists” (p.37). This is also known as stimulus response psychology (S-R psychology), whereby a certain stimulus causes a conditioned response in the subject (or a change in behaviour). In the area of mathematics education there is a common condition in our students called “maths anxiety”. As Safford-Ramus states , “Math anxiety, as it is experienced by many of our adult students, is a clear example of classical conditioning. Past failure in maths (probably at school) causes mind numbing anxiety attacks when faced with having to do maths in their tertiary setting. This response to the stimulus of a mathematical problem takes a lot of work to eradicate on the part of the teacher and student alike. Ideas for eradication could include:

- acknowledgement of the fear
- safe, welcoming, non-judgemental environments
- early success
- small incremental progressions in difficulty
- positive reinforcement (praise, food etc)

Safford-Ramus goes on to explain that operant conditioning techniques are useful in the treatment of anxiety. For example “assessment tasks that assure student success can be made incrementally more difficult so that students cease associating math assessment with failure and begin to look forward to tests as a way to demonstrate what they have mastered rather than a qualified indicator of deficiency” (p.41).

When we use programmed instruction (PI) or computer-aided instruction (CAI) in our courses we are applying operant conditioning. Skinner (1968) saw many advantages of this over traditional classroom instruction. Computers can provide immediate responses to questions put forward so individuals can progress at their own pace and provides opportunities for exercise or drill.

Here are some examples where I have made use of PI in my teaching;

Skills practice for drug calculations for nursing and midwifery students:

These sorts of websites cannot be used in the absence of other teaching methods for the majority of students. They are a useful addition to the classroom but they are not the whole story.

I have to admit that my own experience of mathematical education was very much grounded in the behaviourist methods and I progressed through algebra and calculus at school into basic mathematical and statistical methods at university by applying rules and methods to find the correct answer without ever really having a clue what the application of this maths was about. Passing the course became a means to an end rather than an expansion of my mind and leading to a deeper understanding of mathematics. I had an instructional understanding of mathematics, rather than a relational understanding (Skemp 1976). It has not been until more recently that I have had to develop a relational understanding of maths so I can help my students with their foundational understanding of the concepts. It is a more concrete understanding and way of teaching and involves a cognitive approach rather than a behavioural one.

Behaviourism may work very well for students mastering a particular skill (learning by rote e.g. the times tables) but what happens when they are faced with a completely new type of problem? The student needs to develop *insight, “*which is a Gestalt description of the form learning takes when there is a sudden reorganisation of the field of experience to create a new idea” (Gagne, 1985 as cited in Safford-Ramus 2008). The psychologists who began to move away from the behaviourists are known as the cognitive theorists. They believe that learning involves the reorganisation of experiences in order to make sense of stimuli from the environment.

The cognitive theories include Social Learning Theory. This is applicable to mathematics teaching particularly when applied to the concept of modelling behaviour. According to Safford-Ramus a form of modelling that is especially pertinant for mathematics instruction is that of *abstract modelling*. This generates behaviour that goes beyond what they have actually seen or heard (Banbura, 1977 as cited in Safford-Ramus 2008). Examples would be problem solving behaviour and mathematical communication skills in students. A teacher employing this theory would hope that by modelling good problem solving strategies, students may begin to imitate these over time.

Another Cognitive Theory is Information-Processing Theory. This theory supports the practice of consciously connecting new material to old. “Asking students to see similarities to, as well as differences from, a series of problems invites students to call on previous knowledge and expand on it” ( Safford-Ramus 2008). Where this method of learning can be obstructive is when a student has an incorrect understanding of something previously learned and continues to call on this faulty knowledge confounding the understanding of the new knowledge. I see this time and time again and it is really important to identify these gaps in understanding as soon as possible. This is where a really good diagnostic assessment tool comes in handy. If you drill down into the student’s answers in this assessment tool it will highlight incorrect thinking which can then be fixed.

Finally I am going to discuss the Theory of Constructivism. This is another cognitive theory and has great influence on contemporary mathematics education. Safford-Ramus (2008, p48) briefly describes constructivism as ‘the notion that all knowledge is constructed by individuals acting in response to external stimuli and assimilating new experiences by building a knowledge base or altering existing schemas”. One constructivist theorist, Jean Piaget described a *staged development* theory of learning, where each stage depends upon the completion of the preceding one. One such stage named the *concrete-operational* stage is when children begin to use deductive reasoning but they often require tangible objects to manipulate to draw conclusions. We as maths teachers in tertiary education settings are encouraged to use manipulatives with adults who are struggling with mathematics, suggesting they are still functioning at this stage. My experience concurs and concrete materials have been very useful to explain abstract concepts. Opponents to Piagetian theory hold that students may construct erroneous rules that it will be difficult to deconstruct.

Vygotsky is another constructivist who’s theories I personally model much of my teaching on. His idea of the ‘Zone of Proximal Development” (Wilson et al., 1993 as cited in Safford-Ramus 2008) makes a lot of sense to me and the method of “scaffolding” is one I use all the time. There are four basic steps to the process:

- The student observes the teacher modelling an activity
- The student tries the activity under the guidance of the teacher.
- The teachers prompt with cues only when needed.
- The student is free to practice the skill independently (Wilson et al., 1993 as cited in Safford-Ramus 2008, p.51)