Mathematics is my profession, my hobby and my passion. My wife has a very strong mathematics background so our home life is pretty immersed in this subject. In an academic sense, I have worked with mathematics for nearly 60 years; ever since beginning university studies in 1961. Before that time, mathematics was something I did without realising it.
I grew up in a small community, on a farm in rural Iowa, United States. I did well in mathematics, but in this small school, we didn’t go deeply into some of the Year 11 and 12 topics taught today.
On reflection, the smooth flow of any mathematical prowess I had at a school, can be attributed to mentally playing with numbers on the farm. My ‘times tables’ weren’t any problem because I was always counting in groups. I knew fours, eights, and even 16s because that is the way we built 64-bale hay stacks. The mowers cut either 7- or 9-feet wide swathes of hay (lucerne) so I would work out the total widths of what I cut after every pass around the paddock. Fence posts were set at equal intervals, so my calculations for fencing materials involved multiples of five and six paces. Fencing involved a lot of geometry and algebra; but I didn’t know it at the time.
From 1961 to 1975, I studied full-time to complete three degrees. The first was a teaching degree, which initially lacked a particular focus. This soon changed as I became interested in mathematics. By the end, I completed every undergraduate mathematics subject available at the university, so I obtained a Master’s degree in pure mathematics. This was the space race era and teachers of mathematics were in high demand. So, with my background, I taught for several years before pursuing my doctorate.
My students seemed to lack prior mathematical starting points that I had. I knew they must have some experiences so I got them to talk. Our first discussions weren’t very mathematical, but gradually we wove our way to include more mathematical topics into our conversations. I had a discussion with my practice teacher supervisor in 1964 (which I still recall vividly today). He told me that I needed to ask better questions.
“Don’t focus so much on the answer,” he said. So, I learned to use how, why, when, where and what — do you think?
During this time as a teacher, the methods could be described as cut and paste. The teacher gave the steps and the students replicated some — over and over again. The focus was on the procedure and the main question was ‘What answer did you get?’ After reflecting, I backed up and started again.
I started a very focused career in mathematics education at a university in Brisbane, Australia in 1975. Early in this position, I was fortunate to obtain a grant allowing me time to tutor school students who needed assistance. At the same time, I developed teaching methods that helped prevent students’ misconceptions. I encouraged students to describe their interests, which allowed me to probe where they were successful.
We talked together about ‘things’; objects or pictures to show mathematical ideas. I developed activities and special tools to help students see the ideas. This research gave me many ideas for mathematics subjects I taught at the university and the materials I wrote for schools.
At age 70 and with an inquisitive mind working at full speed, I walked out the door of the Queensland University of Technology and co-founded Origo Education with James Burnett, who was one of my past students. Origo publishes mathematics resources for primary school students and their teachers and parents. I find this is a great outlet for sharing what I have learned about teaching maths to young students over my long career.
My students today are nearly all from the secondary school. The approach is very much the same; start where they are, whether it is mathematical knowledge or other areas of interest. Get them to talk and show me.
Older students can verbalise more and I have them explain the thinking they use to solve a problem. When they come to me, I search through the content that seems to be an issue to focus on topics that can be represented with pictures. I ask them to draw a picture to illustrate the situation and describe the thinking they use. We could call this conversational mathematics.
When written work is included in our conversations, this form of mathematics can be highly rigorous and very demanding. But at the same time, it is friendly. I believe that most people can do more mathematics than they might think, so I have been sure to include this approach in the resources that Origo produces. By including more mathematics discussions in a conversational friendly way, more young people can be successful at mathematics. This is particularly true for individuals who need more support to learn important mathematical skills.
Now 76, I saw an opportunity for Origo to evolve to meet the needs of parents who had been forced into home schooling as a result of coronavirus as well as those changes to employment as a result of the digital age. It’s a new way of thinking and I enjoy thinking outside the box and off to the side.
When it comes to mathematical games grandparents, parents and their children can enjoy together, I suggest dominoes. Not only is it an enjoyable game, but it gets everyone focused on practising lots of different maths concepts.