In this monthly column, we speak with a notable member of the Western Canadian mathematics education community about their work and their perspectives on the teaching and learning of mathematics. This month, we had the pleasure of speaking with Dr. Gale Russell of the University of Regina.
Gale Russell is a Saskatchewanian through and through. She was born and grew up in Saskatoon, and after completing a B.Sc. (Honours) in Mathematics and a B.Ed. (Great Distinction) at the University of Saskatchewan, she began teaching in the community of Raymore. There, she taught all of the secondary level mathematics courses as well as some Arts Education classes. Gale was also a representative to the local teachers’ association, ran a successful drama club, held regular meetings of a calculus club, and was an on call “jewelry coach,” while also continuing to pursue her other passion – playing the bagpipe. During her time in Raymore, Gale also became involved in being a pilot teacher of the then-renewed high school mathematics curricula (the former Math 10, 20, A30, B30 C30), and was later regularly seconded by the Ministry of Education to be an implementation leader around the province for those curricula. After six years in Raymore, Gale moved to Rosetown, where she taught secondary mathematics while continuing her other activities at the school level and for the Ministry, and playing in a pipe band.
After two years and one month in Rosetown, Gale was made the first full-time permanent Educational Consultant for K-12 Mathematics at the Ministry of Education in Regina. In this role, Gale was actively involved in facilitating professional development throughout the province, in reviewing resources, in curriculum framework renewal with the Western and Northern Canadian Protocol (WNCP), and in writing the most recent mathematics curricula. Also during this time, Gale obtained her M.Ed. from the University of Regina, focusing her research on teachers’ and students’ conceptions about zero. After 11 years and 11 months at the Ministry, Gale left to pursue her PhD in Education at the University of Saskatchewan, focusing her research on the kinds of knowledge and ways of knowing valued within mathematics and the teaching and learning of mathematics. Only one day ago, Gale successfully defended her dissertation on this topic, thereby completing all of the requirements for her PhD. For the past two years (and continuing onward), Gale has been working in the Faculty of Education at the University of Regina as an Assistant Professor of Secondary Mathematics Education. She continues to play her bagpipes and has two small dogs, Euclid and Chevy.
First of all, thank you for taking the time to have this conversation during this busy time of the year! Could you talk a little bit about the courses are you currently (or have just finished) teaching at the University of Regina?
Over the past two years, I have taught courses at the elementary level (EMTH 310), middle level (EMTH 217), and secondary level (EMTH 300, 351, and 450). As EMTH 310 is the only mathematics methods class that the elementary pre-service teachers (currently – note the optimism) take, it is a fast and furious class in which I try to teach the students the elementary curriculum content using the pedagogical strategies that the new curriculum (and research) supports. I also strive to have the students understand why the teaching of mathematics needs to change and to become more aware of other issues that are present in or impact mathematics classrooms (standardized testing, math anxiety, gender and cultural gaps, and so on). Actually, the description I just gave can be applied to the other classes I teach, only they are a little less intense because the students take at least two EMTH courses in the middle and secondary programs (not that we couldn’t use more time in all the classes!).
Probably the biggest challenges that I give to my students in any of my classes are to contemplate what we have to directly teach to students and why, and how to engage in meaningful mathematics teaching and learning through the use of open tasks (inquiry, problem solving… pick the lingo of your liking) and class discussions. I also try to give my students experiences in the classroom or out in a school that they likely haven’t had before, such as having an Elder come to our class for a day, visiting a classroom that is taught entirely through inquiry, and visiting Campus Regina Public to see an example of an alternative approach to high school and integration. I also like to engage the students in current issues in education, whether it be responding to a community paper column making claims against the teaching of mathematics or taking on the role of a BC mathematics teacher who, after 14 lost school days, has to start teaching a particular course on the following Monday. I try to make my classes as real and problematic as teaching can be, and I’m honest and open about my planning, my errors, and my quick changes in plans with my students so that they can better come to understand the process of teaching and the impacts upon it.
I understand that your first Bachelor degree was in mathematics. What drew you to education, and then to research in the field of mathematics education?
I actually knew I wanted to be in education when I was in high school; however, it seemed like every second person (from my graduating class of over 600) was going into education and I started to doubt whether I really wanted to be in this field. So, I started off as an English major pursuing an Arts and Science degree. For some reason, still unknown to me, I decided to change to a math major for my second year. Throughout my four years in the Arts and Science program, I paid my bills by tutoring mathematics, both to individuals and to groups. This experience reinforced what I had originally knew – I love helping people learn. From there it was a natural step to go into an education program.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]I’ve always been an “I want to understand why” kind of person, and I was finding that I often couldn’t find satisfactory answers in the literature and research. For me, that meant that if I was going to continue teaching, I needed to start finding out the answers for myself.[/perfectpullquote] My desire to do research in the field of mathematics education probably started in Grade 1, when I was drawn to helping a classmate with her math and was puzzled by why she was having so much difficulty with it. As I grew older and progressed through my K-12 schooling, Bachelor of Science, and Bachelor of Education, and started teaching in rural Saskatchewan, these kinds of experiences continued to occur. I have always been an “I want to understand why” kind of person, and I was finding that I often couldn’t find satisfactory answers in the literature and research. For me, that meant that if I was going to continue teaching, I needed to start finding out the answers for myself. Of course, no answer ever stands without the support of questions, and I found myself hooked. To be perfectly honest, my choice of my first research topic (elementary teachers’ and students’ understandings of zero) was a knee-jerk reaction to being told that the concept of zero was too difficult for young children. Based upon my experiences with young children, I couldn’t see how this could be true, and the notion that this concept was developed late in mathematics history was unfathomable to me. I ventured into that research to find out what was going on (and, I will admit it, to prove that this notion was false). Once I started doing this kind of research, there was no looking back, and no shortage of ideas of where to go next.
I know that part of your research involves investigating the relationships between the teaching and learning of mathematics and culture, especially Aboriginal cultures. Although interest in this intersection is growing, I think that many still feel that mathematics—especially the Western conception of mathematics—is a “universal” language that is untainted by culture (“acultural,” if you will). Could you speak a bit about the ways in which mathematics is influenced by culture, and perhaps also the ways in which it transcends it (if at all)?
A few years ago, I would have said that my research is about the relationships between the teaching and learning of mathematics and culture, but now I tend to look at it more as the relationship between different kinds of knowledge and ways of knowing and the teaching and learning of mathematics. I’m still concerned with culture, and I do tend to turn to Indigenous cultures to inform and contrast with my thinking; however, I have come to view true mathematics (versus just Western or academic mathematics) as acultural. Mathematics, and how you know, understand, and work with mathematics is a consequence of the kinds of knowledge and ways of knowing that you value. Mathematics is, in fact, tainted not so much by culture, but rather by worldview. Within a specific culture, you may see tendencies or trends in the ways that people think about, do, and use mathematics, but even within that culture it is likely not universal. We have to be careful here, because when talking about worldviews, particularly when sharing a cultural name, it is easy to equate culture to worldview, and members of a culture to people holding a particular worldview, but this is far from the case. Just because I am Western does not mean that I have a Western worldview.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]I would argue that what is universal about mathematics is that all humans consider and work within situations and solve problems that deal with mathematical notions, such as quantity, patterns, shape and space, data, and likelihood. What is not universal is how we think about, represent, and report on these mathematical notions.[/perfectpullquote] I would argue that what is universal about mathematics is that all humans consider and work within situations and solve problems that deal with mathematical notions, such as quantity, patterns, shape and space, data, and likelihood (and likely many more things I have not listed here). What is not universal is how we think about, represent, and report on these mathematical notions. A deep understanding of quantity is not restricted to those who have a number system with open and closed operations on different kinds of numbers. A deep understanding of quantity is also present in communities where they only have number words for 1, 2, and 3, and have no written representation of them. For example, for addition to be accomplished, in other words to aggregate two quantities or augment one quantity by another, does not require an operational symbol or an equal sign (as Western mathematics would have us believe). What it does require is an understanding of quantity as you represent it and an understanding of the situation that is leading you to determine the result of aggregation or augmentation (you don’t even need to know those two words). It has been clearly shown that infants under nine months of age understand that 2 – 1 = 1, they just don’t know the representation “2 – 1 = 1.”Mathematical representations, algorithms, procedures, processes, and labelling, which is what many people think of when speaking of (Western) mathematics, is not universal. In fact, I would argue that it is the assumption of singularity in representing and working with (Western) mathematics that makes mathematics the much misunderstood, dreaded, feared, misused subject that it is. More importantly, I would also argue that the assumption that Western mathematics is universal has limited the ability of everyone to understand mathematics in more meaningful ways – and I mean everyone: both those who struggle with math and those who believe that they are math experts.
So, in its broadest interpretation, mathematics is acultural and universal; however, that mathematics does not currently exist en masse anywhere (to my knowledge) because of the limitations that have been placed upon what is accepted as mathematical based upon a particular worldview’s assumptions about what mathematics looks like and how it functions in the world. Basically, mathematics, as you and I were taught it, and as we have taught it, is hegemonic and oppressive. Although the origins of this particular view of mathematics is often associated with Greece and the ancient Greek mathematicians and philosophers (which would suggest a single culture), historically it is known that much of what constitutes this mathematics (and even the mathematicians themselves) came from other parts of the world and was appropriated by a group of “mathematicians” at the time. This practice changed over time, in that the origins of new mathematics is formally recognized; however, there is much mathematics that was left behind in the accumulation of Western mathematics stemming from all cultures. It is this part of the broader field of mathematics which was never integrated in to Western mathematics that ultimately denies the universality of Western mathematics.
Today, there is a very positive movement in Saskatchewan towards incorporating Aboriginal perspectives and ways of knowing in schools in order to make content accessible and meaningful to all students. However, I think that many mathematics teachers still feel at a loss when it comes to integrating Aboriginal content and perspectives in their classrooms. What advice would you offer such teachers, and what resources would you point them to?
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]We need to shed our desire to place all mathematics and ways of representing mathematics in abstracted hierarchies.[/perfectpullquote]As I will talk about again in the next two sections, I have come to think of the kinds of knowledge and the ways of knowing that are valued within mathematics classrooms as the key. We need to shed our desire to place all mathematics and ways of representing mathematics in abstracted hierarchies. Although it is true that we want our students to learn the rigours of Western mathematics, they need to be doing so in a context in which other ways of thinking and doing mathematics are valued, seen as contributing to mathematical knowledge, and as relating different ways of knowing mathematically. Based upon the response of a number of Elders that I had the privilege of meeting with during the curriculum renewal process, I strongly suggest that Saskatchewan teachers focus on the four K-12 Goals of Mathematics in the curriculum documents. In particular, because of editing over the course of time and curriculum release (the most recent mathematics curricula were released over 6 years: 2007 saw the release of K, 1, 4, 7; 2008 saw the release of 2, 5, 8; 2009 saw the release of 3, 6, 9; and then the Grade 10, 11, and 12 curricula were released in 2010, 2011, and 2012, respectively), go to this area in the curriculum documents for one of these grades: Grades 3, 6, 9, 10, 11, or 12. Once there, although all of the goals are important, focus in on the goal of Understanding Mathematics as a Human Endeavour. The bullets there will give you a good start to incorporating Aboriginal perspectives and ways of knowing. Often, as teachers we think of the notion of incorporating First Nations and Métis content, perspectives, and ways of knowing as being related to specific content, such as teepees, but that’s not what this is about. This is not only about the kinds of thinking and responses we accept within our classrooms, but also about how we promote, relate and (non-hierarchically) value them. [perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]Conversely, be very, very, very cautious of incorporating Aboriginal content. When this is done in mathematics classrooms, the content tends to become a mathematical artefact rather than a cultural one.[/perfectpullquote]Conversely, be very, very, very cautious of incorporating Aboriginal content. When this is done in mathematics classrooms, the content tends to become a mathematical artefact rather than a cultural one. For example, many people in the past have used the teepee to illustrate a cone (and thereby feeling they have incorporated Aboriginal content), while a teepee is neither mathematically nor (most importantly) culturally a cone. The best way I know of to bring in Aboriginal content into the classroom is to use the Understanding Mathematics as a Human Endeavour goal as a way to create an open space in which students will choose to bring in Aboriginal (and other cultural) content once they feel it is a safe space to do so. You can also invite Elders in to speak with your students, but remember they will likely be speaking mathematically from their worldview, which can seem disjointed from what often happens in Western mathematics classrooms. That’s not a bad thing, however, if you know to embrace the difference and celebrate it.Gradually, some websites have also been emerging that are based on some very dedicated work related to mathematics with different Indigenous communities. Feel free to look at these for ideas, but always remember that this Aboriginal content may very well be foreign to your students. An example of a website to look at is Lisa Lunney Borden’s (St. Francis Xavier University, NS) Show Me Your Math (http://showmeyourmath.ca). On this website, you will see lots of documentation about the Show Me Your Math program that she started in one Mi’kmaw school and which has now been adopted by many other schools across Nova Scotia. You will also find some sample inquiry projects there, but again, remember that these are projects related to Mi’kmaw culture and communities, so you need to make sure that you are acknowledging and situating them appropriately, or choosing to find something similar but more directly related to your students, school, and community.
Let’s talk about your own work in a bit more detail. I understand that you are in the process of, and very close to, completing your PhD in curriculum studies (how exciting!). What questions or issues are you exploring as part of this work, and what do you have planned in terms of your research program going forward?
My main research question in my dissertation is “What kinds of knowledge and ways of knowing are valued within mathematics and the teaching and learning of mathematics?” Starting with “Jagged Worldviews Colliding” by Leroy Little Bear (found in Marie Battiste’s Reclaiming Indigenous Voice and Vision), I complied a theoretical framework based upon two worldviews: an Indigenous Worldview and the Traditional Western Worldview. These two worldviews each define a distinct view regarding the kinds of knowledge and ways of knowing that are of value. In a very brief summary, and very Traditional Western Worldview way of just the facts, these two worldviews are characterized by the valuing of the following:
- Traditional Western Worldview: Linear, singular (right way, correct answer), hierarchical, abstract, compartmentalized, de-contextualized, rational, written knowledge.
- Indigenous Worldview: Knowledge for, of, and through relationships, contextualization, physical, emotional, spiritual, intellectual, intuitive, experiential, and cultural knowledge, diversity of knowledge and ways of knowing, connection to place, and multiple forms of representation (including oral).
In a nutshell, my work looked first at my own mathematical experiences through the two worldview lenses to take note of the kinds of knowledge and ways of knowing that I was valuing. I then considered a number of areas within mathematics and the teaching and learning of mathematics, including the philosophies of mathematics (and there are a lot of them!), the math wars, Indigenous students and mathematics, ethnomathematics, and risk education (as a consideration of possibilities for curricula development), and analyzed their alignment or misalignment with each of the worldviews.
What this research and analysis has led me to are the following conclusions:
- Although the two are distinct worldviews, an Indigenous Worldview includes all of the ways of knowing and kinds of knowledge valued within the Traditional Western Worldview, while the reverse is not true.
- Teaching and learning, as well as just thinking about mathematics from a grounding in an Indigenous Worldview provides a likely solution to the math wars in which both sides can survive and thrive. Likewise, this same grounding could help Indigenous students (as well as all others) in their struggles to learn and succeed in mathematics, would give value to previously unvalued mathematics, and would provide students, teachers, and society with broader and richer ways of thinking and knowing mathematically. I called such a grounding the Transreform Approach to the Teaching and Learning of Mathematics, as it acknowledges and values the traditional and reform approaches, the so-called middle ground between the two, and approaches to mathematics that so far may not have even been considered.
- To engage in the Transreform Approach, we also need to re-view and re-new our philosophies of mathematics.
So, where to go now? I have done some preliminary work with a group of teachers that ultimately provided rich data about how easily values from the Traditional Western Worldview can slip in and take over (I call it “intrusions”) how one is teaching, without the teacher ever realizing it. This same study also identified how group discussion and reflections can help to thwart such overtaking.
From here, I want to take a step back and consider the question of “What ways of knowing and kinds of knowledge are being valued in mathematics classrooms in Saskatchewan?” Then, I hope to build on this information through an exploration of how to effectively help teachers transition in their worldview groundings and to analyze the impacts (good and bad) upon student achievement and affective responses.
As is typical of the person I am, I also have a number of side research interests. Ultimately, I believe that there will be ties into my worldview research, but I am also really interested in the questions of “Why do pre-service teachers pick a particular level (elementary, middle level, secondary) to specialize in, and why are they not picking a different one,” “What is ‘common sense’ when it comes to the teaching and learning of mathematics,” “What instances of polysemy (multiple meanings for a single word) are occurring in mathematics classrooms in communities where Aboriginal English is spoken,” and “How to engage communities and parents in understanding changes in the teaching and learning of mathematics (and to help support them in supporting their children with mathematics).” Of course, there are many others that are floating around my mind, but it gives you a bit of a sense of the diversity of where my interests lie.
Your work has been published in a variety of journals, books, and conference proceedings, including the Canadian Journal of Science, Mathematics and Technology Education and our very own vinculum (the journal of the Saskatchewan Mathematics Teachers’ Society). Which of your publications would you recommend to mathematics teachers in the province who are looking to grow in their practice and their understanding of the teaching and learning of mathematics?
So this is a tough question to answer, because it depends upon what people are interested in. I would first, however, recommend a publication which is not mine, but that I mentioned previously – that of Leroy Little Bear:
Little Bear, L. (2000). Jagged worldviews colliding. In M. Battiste (Ed.), Reclaiming Indigenous voice and vision, pp. 77-85. UBC Press: Vancouver, BC.
For me, this was my big eye-opener in coming to understand what people were talking about when they said things like “First Nations and Métis have different ways of knowing.” Although it’s not explicitly talking about mathematics, I found myself writing in the margins comments such as “this is how the teaching and learning of mathematics has been done” and “this is how research says it should be done.”
If people are interested in a discussion of the math wars and the struggles of Indigenous students in light of the two worldviews, then I recommend “The Marginalisation of Indigenous Students Within School Mathematics and the Math Wars: Seeking Resolutions Within Ethical Spaces” by Egan Chernoff and myself. This article can be found in Volume 25, Issue 1, pp. 109- 127 of the Mathematics Education Research Journal. If, instead, people are interested in how changing the kinds of knowledge and ways of knowing that are valued in mathematics classrooms could impact curriculum and lives, then I suggest reading “Risk Education: A Worldview Analysis of What is Present and Could Be” which can be found in The Mathematics Enthusiast, Volume 12, Issues 1-3.
Thank you, Dr. Russell, for taking the time to share your research and perspectives with our readers. We’ll be following your future work with interest!
Ilona Vashchyshyn