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Helen is an independent researcher in mathematics education from the USA. She is currently a visiting researcher at the University of Cambridge investigating secondary mathematics teachers’ interactions with a large database of instructional resources. Additionally, she is conducting research on the mathematical symbol known as the minus sign and its role in the pre-university curriculum.
1. What’s your earliest memory of doing mathematics?
I don’t remember anything about school mathematics prior to introductory algebra in lower secondary school, which I thoroughly enjoyed. Interestingly, my first algebra teacher was a soft-spoken, first-year female teacher. This would have been around 1960 and I think it was unusual at that time to have a female maths teacher at that level.
Outside of school, probably around the age of 9 or 10, I remember having puzzle books. Primarily logic puzzles – these were a favourite pastime.
2. How has mathematics education changed in the time you have been involved in it?
Changes that spring to mind are (1) new understandings about teachers’ mathematical knowledge and how this knowledge can support learning, and (2) the impact of new technologies on what is possible in mathematics teaching and learning.
When I started teaching mathematics in the 1990s, I discovered that despite my teaching credentials, my teaching experience, and having a mathematics background, I was not good at teaching mathematics. Fortunately, this was in the wake of the NCTM Standards Movement in the USA. At this time, three prominent figures – Magdalene Lampert, Deborah Ball and Hyman Bass – were investigating what was involved in teaching school mathematics. I enrolled in courses with them to find out what I was lacking and discovered the specialized knowledge of mathematics then being mapped as mathematical knowledge for teaching. This proved to be an important missing ingredient in my teaching. The work in which Ball and her colleagues were engaged was my own first encounter with what were then new ideas about the mathematical knowledge recommended for teachers. Usually attributed to Shulman’s 19861 paper, this new thinking about teachers’ content knowledge had widespread influence within the mathematics education community. The work mentioned was further developed by educators both in the USA and internationally. Currently, ideas about the need for teachers to acquire specialized content knowledge and learn how to apply this knowledge in classrooms are firmly embedded in mathematics education.
With regard to the second change in mathematics education, decades of fast-paced technological change contributed to the growth of what is now an enormous, somewhat unwieldy body of resources* that teachers can draw on to support their teaching. For example, dynamic geometry software appeared in the 1980s. Wikipedia currently compares 14 basic versions and lists other related types of dynamic geometry. The second phase of the Internet contributed to the expansion of the resource pool by making many resources interactive and by adding numerous new types of resources to the mix. Consequently, teachers had opportunities to capitalize on new ways of conducting their work. Significantly, they might choose to network with others in the global mathematics education community. Technological change, and resulting new opportunities for teachers, continue apace.
*NOTE: The enormous pool of resources is not unique to England, but teachers in England may be relying more on the newer types of resources and less on traditional curriculum materials such as textbooks in comparison to teachers in countries where more traditional materials dominate. For example, many mathematics teachers in England do not use textbooks or may use them infrequently, which makes England an interesting site for resource investigation. My research investigates resources available to teachers and teachers’ opportunities to acquire specialized content knowledge in mathematics.
3. Tell me about a time in your career when something totally flabbergasted you.
There were a number of occasions when a student’s perspective changed my views about teaching mathematics. This happened more often after I deliberately began using a teaching technique called access student thinking, which amounts to asking students questions about their work. Sometimes, however, students’ thinking became apparent without me even intending to access student thinking.
One early example I recall, before I began purposefully interviewing students, was when a teacher asked me to work with a group of 5 and 6-year-old children who were advanced in their numeracy (arithmetic) skills. She wanted me to enrich their mathematics experience. I began with a problem from Marilyn Burns2 about a group of cows and chickens, asking the children how many legs and tails were in the group of animals. The children were completely stumped, had no idea how to figure out the answer, and I was stunned. Obtaining a correct answer via symbol manipulation when provided with a mathematical exercise in symbol form did not mean the children understood the underlying mathematics. Facility with symbol manipulation is not the same thing as facility with problem solving. I learned, among other things, the importance of a multiple representations approach to teaching and learning mathematics.
4. Do you practise mathematics differently in company?
Yes. Working with others requires the use of important communication skills. I have to articulate my thinking and reasoning, and I sketch a lot more, incorporating diagrams or other representations. Can’t shortcut anything. Working with others also necessitates time and space for two-way communication. Good listening skills are not required when working alone.
5. Do you think a brilliant maths teacher is born or made?
I don’t believe there is a born/made dichotomy. Undoubtedly, some people have more innate teaching ability than others, just as some people have more athletic ability than others. From my view, the knowledge and skills leading to successful mathematics teaching (whether or not it would be considered brilliant teaching) can be acquired, beginning with content knowledge that enables the teacher to unpack mathematics in ways that make it accessible to learners. In their training programs and throughout their career, teachers need opportunities to extend their own understanding of mathematics, to deepen their knowledge of learners and how they learn, and to gradually add to their toolbox of strategies and routines. As they enhance their knowledge and skills, teachers may need support on how to best apply these in their teaching.
Presently, we need to know more about how new resources and technologies can be accessed, developed and used advantageously by teachers. Additionally, we need to recognise that teachers need opportunities to become familiar with new thinking from the larger mathematics education community.
Supporting teachers, finding ways to reach them and to help them obtain what they need and can use well – all of this contributes to successful teaching. In my opinion, within the mathematics education community, nationally or internationally, we all share responsibility for making teachers successful.
6. What’s the most fun a mathematician can have?
I assume the question means having fun with mathematics and that the definition of fun relates to entertainment and enjoyment beyond the satisfaction of professional work. I personally find it enjoyable and intellectually engaging to tackle mathematics contests or tests published for school students, middle grades and above, and to see how I fare. I also like problem-solving in general, but if I cannot solve something in a reasonable amount of time, it ceases to be entertaining. Games such as SUDOKU, where I can try and beat my own record, appeal. Whether or not these are the most fun a mathematician can have? Hmm…
7. Do you have a favourite maths joke?
Since I did not have a favourite maths joke, I asked a 9-year-old girl and her 7-year-old brother if they could tell me any maths jokes. Both offered jokes without hesitation, which surprised me.
9-year-old: Why is 6 afraid of 7?
Answer: Because 7 comes after 6 and 7, 8, 9.
7-year-old: A man was walking to school and passed 3 people going in the opposite direction. Each of those 3 people was walking 3 dogs. How many people and dogs were walking to school?
1Shulman, L.S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2), 4-14.
2Burns, M. (n.d.). Marilyn Burns Math Blog.
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