“If a teacher’s own knowledge of the mathematics taught in elementary school is limited to procedures, how could we expect his or her classroom to have a tradition of inquiry mathematics?”
– Liping Ma (1999)
After reading Knowing and Teaching Elementary Mathematics, a study conducted by Liping Ma, I began to consider three things. First, how does the notion of stereotype play into our understanding of Chinese mathematics and American mathematics? Second, how do Canadian elementary teachers fair in regards to their understanding of elementary mathematics? Third, if Canada is down there with the States, how do we change? And fourth, is it possible to be both “limited by procedures” and limited by inquiry mathematics? This blog post will address the first three questions, and a future blog post will address the latter. Before getting into any of these questions, however, a brief summary of the study is necessary.
In her study, Ma details the apparent deficiencies of American teachers in elementary mathematics, compared to their Chinese counterparts. She does so through a series of well thought-out interviews with a number of participants, both who teach in the United States and China. Overall, she determined that many more Chinese teachers have a “profound understanding of fundamental mathematics” [PUFM] (p.120). PUFM is the combination of four principles: connectedness (the ability to make connections among and between mathematical concepts), multiple perspectives (the ability to be flexible towards mathematical approaches), basic ideas (the ability to be aware of principles of mathematics), and longitudinal coherence (the ability to recognize the interconnectedness of the entire mathematics curriculum and its applications). Although both groups had similar procedural knowledge in understanding many elementary topics, the Chinese group had a deeper PUFM, enabling them to encourage thought-provoking discussion amongst students to facilitate mathematical growth. Interestingly enough, when comparing the PUFM of the Chinese teachers and pre-service teachers, it became evident that the pre-service teachers had little PUFM. Rather, it was the current Chinese teachers who seemingly developed it while teaching, through colleagues, students, and self. Chinese teachers were “stimulated by a concern for what to teach and how to teach it, inspired and supported by their colleagues and teaching materials” (p. 143).
When I first began reading this study, I was initially struck by the obviousness of the entire study. Of course Chinese teachers are better at teaching mathematics than American teachers. It was the moment that I thought this that I began to look at both the study and my prejudice more critically. Why do I believe this to be true, before reading the entire paper? We (western society) have a general tendency to think of Asians as being good at math and their schools to be extremely rigid and rote in practice. Did this study support those notions? Partly. The Chinese mathematics teachers had a more in-depth understanding of mathematics and could, therefore, pass that understanding off to their students through facilitation. It seems, however, that the stereotype that the classrooms were more rigid that American classrooms is false. In fact, in the chapter about subtraction with regrouping, it seemed that the Chinese teachers were utilizing inquiry with manipulatives to teach it effectively, which included a large group discussion following the activity- a pivotal part to the discovery learning model. This instructional method was evident in all areas studied by Ma. So, although Chinese students are continually outperforming American students on international examinations, the reasonings do not lie in the rigidity of the mathematics curriculum in China. Rather, it seems they are a result of the PUFM of the teachers, which enables teachers to effectively utilize an inquiry model.
Another though that came up during this study, almost as immediate as my initial prejudice, is how well Canadian teachers would do in a study like this. Do Canadian teachers have PUFM? If not, why? Are we able to place American and Canadian teachers in the same deficit category? In the United States, in order to be a secondary mathematics teacher, you must obtain a mathematics degree and do a teacher-preparation course, which varies in length. To be a elementary teacher, you need a four-year degree in elementary education. The process to become a teacher is similar in Canada. However, during school, students are subjected to many more standardized tests in the US than in Canada, including SATs and NCLB. Unfortunately, rote memorization is often synonymous with studying for standardized tests, while critical thinking gets left on the curb. Since Canada is using fewer standardized tests (and none at a national level in high schools), where do Canadian preservice and in-service teachers fall? Unfortunately, I do not have answers to these questions, as the literature is quite limited on the topic. More limited is the information on how pre-service teachers, who have gone through the latest remodelling of the provincial mathematics curriculum in Saskatchewan understand mathematics and whether or not this is different than those who experienced the former version of the curriculum. These are important questions that need to be discussed because of the cyclic tendencies of teaching and learning. As Ma puts it,
It seems that low-quality school mathematics education and low-quality teacher knowledge of school mathematics reinforce each other. Teachers who do not acquire mathematical competence during schooling are unlikely to have another opportunity to acquire it (p. 145).
In order to effectively dismantle the cycle, Ma makes several suggestions, one of which is teacher preparation. Ma writes: “[Teacher preparation] may serve to break the circle”(p. 149). Currently, there is an understanding of elementary mathematics as low-level and basic. It is when we start to unpack what these words actually mean, and recognize that they are incredibly limiting with our preservice teachers and the students they will teach, can we begin to chip away at the succession of misunderstanding on mathematics. Like Ma, I believe that there are a number of different steps that need to be taken in order to ensure the success of a new tradition and cycle in mathematics education. However, possibly unlike Ma, I believe that this solution begins with teacher education. A teacher education program that focuses not only on mathematics content, but also attitudes of and abilities in mathematics is important. Because elementary education in Canada is not subject-specific (like the secondary programs), one cannot assume that pre-service students will appreciate or enjoy mathematics. Some may even have math anxiety, as many people do. This needs to be combatted during the four years of university, or it can be passed on to the students of future teachers.
Many questions arose as I read through Ma’s study. The questions raised are important to furthering the success of our elementary education programs in Canada. Currently, public opinion is quite dichotomous in regards to discovery-based mathematics. There are many academics, parents, and teachers who support inquiry-based learning. On the other end though, there are academics, parents, and teachers who are looking to “go back to the basics” in an effort to reintroduce fundamental mathematics through algorithms and formulae. I argue for a less binary examination of the problem and and more spectral look. It is not a matter of teaching one way or the other; it is a matter of teaching in ways the effectively reach the maximum number of students in a classroom. As Ma puts it:
…the change of a classroom mathematics tradition may not be a ‘revolution’ that simply throws out the old and adopts the new… the two traditions may not be absolutely agnostic to each other. Rather, the new tradition embraces the old” (p. 153).