Mathematics Education :

I have been tutoring mathematics for as long as I remember, and this has been for at least 45 years, since I was in my first year of university studies in Mathematics and Computer Science in 1970.

Whereas I had always believed that I would become a professional mathematician, my life took a turn towards IT when I enrolled at university and discovered that Computer Science was being offered for the first time in South Africa, and during my interview my professor said that “computers may be the thing of the future”. On reflection it seems he may have been right!

Education and Artificial Intelligence

Having spent the past 45 years professionally engaged in the ICT industry I never lost this contact with mathematics tutoring, and this led me to register for a PhD in Mathematics Education, in which I hoped to be able to combine the worlds of ICT and Mathematics Education for the benefit of society.

I am of the belief that teaching mathematics is a hard problem, and that parts of the responsibility can be shifted to computerised systems, in effect artificial intelligences which are skilled at teaching.

My research focus has been on diagnostic assessment for mathematics, concerning the discovery of what a learners knows and does not know in terms of the cognitive structure. When a learner is presented with a mathematics problem, they may get this right or wrong for a variety of cognitive reasons:

  • they know what they are doing but make a mistake – commonly called a “slip” or a “silly mistake”
  • they do not know what they are doing but they get the right answer – which can be called a “lucky guess”
  • they know what they are doing and get the right answer – expected
  • they do not know what they are doing and get the wrong answer – expected

However, the notion of whether a learner actually knows or not is not a simple dichotomy but is best seen as a continuum, and I have referred to this as a Development Stage model which considers increase proficiency in a small area of mathematical knowledge and how this proficiency develops.

My theory is that a learner learned by advancing through these Development Stages, in response to the inputs provided by the teacher, in the form of problems, feedback, and comments. My research is concerned with how to accelerate learning by providing the right problems and the right feedback at the right time, and in particular how to identify the fine-grained status of the learner’s knowledge which we need to have access to so that we can find the right problems. This is a hard problem since there are many possible types of knowledge structures which a learner will develop during the learning process, and is a problem is not directed to the specific needs of a learner, then presenting this problem may be a waste of time for both the learner and the teacher.

For an AI to be able to become a mathematics teacher, or to provide support (as a tool) to a teacher, it needs to be able to know and to do the following:

  • understand the scope of the knowledge within a domain of mathematics, including the terminology
  • understand what constitutes total proficiency in the domain, in terms of the types of problems which should be able to be solved
  • understand the common problems experienced by learners, the misconceptions, and other intermediate constructs
  • be able to identify what constructs a learner has developed at their current status of learning, by asking diagnostic questions
  • be able to generate new problems, or select a specific problem from a bank of problems, which is perfectly suited to the needs of a learner
  • use this in a cyclic manner for a learner until it is established that the learner most likely is proficient

My work is now focused on formulating an AI-based framework for mathematics teaching, and in building prototypes for this framework for selected classes of problems.

This is my web site for providing some useful mathematics to learners, to teachers, and to parents, and is focused on mathematics topics in the curriculum. Whereas this is initially focused on secondary school mathematics for Grades 7-12, there are plans to extend this to include primary and pre-primary mathematics, and also into university and graduate-level mathematics.

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