What are the expectations we have on mathematics teachers across Canada? Is there a difference in expectation for secondary mathematics teachers and elementary mathematics teachers? If so, why?
What are the expectations we have on our children throughout their academic careers, particularly related to mathematics? What types of skills do we expect our students to develop as they go through mathematics in Canada? Are these skills categorized based on age, grade-level, and/or comprehension, or should they be valued on a continuum, where students develop naturally and without a five-hundred page document telling them what they should know?
Many of these questions may seem obvious. We expect greatness from teachers. In particular, we expect our mathematics teachers to instil mathematical prowess in students, often in the form of mathematical reasoning and problem solving. Our secondary mathematics teachers should understand the mathematics that they are teaching, similar to our elementary mathematics teachers. Yet, their knowledge banks should look different. A secondary mathematics teacher should have a deep understanding of algebra, fractions, and graphs. Our elementary mathematics teachers should know addition, multiplication, and basic shapes. The reasoning is simple, almost redundant. Teaching demands understanding- history teachers know history, biology teachers know biology, etc.
What this frame of thinking leaves out is the inherent properties of mathematics. What is mathematics but a gigantic structure, one that contains a foundation- pivotal ideas that must be understood before the full value of mathematics can be realized. If mathematics demands a foundation, then perhaps the teaching of mathematics demands more than the understanding of the curriculum. If a math teacher is teaching fractional division of rational numbers, is it not important to comprehend fractions, division, and the ever-elusive idea of ‘the whole’? How does this relate to the division of rational expressions? Does it relate at all?
In terms of the inherent properties of mathematics, who decides them? Regardless of whether or not mathematics is invented or discovered (although this is a fascinating topic to discuss), determining how mathematicians determine what a mathematical foundation looks like is important and charged with cultural bias. For the West, the structure that represents mathematical knowledge is a pyramid, with basic arithmetic skills and algebra at the bottom, and pure math resting on top. For other cultures, however, the power of mathematics lies in its applicability. Who is right? Is someone right? Are we both wrong?
If we can all agree that there exists a foundation for mathematics that is itself mathematics, when is it appropriate that children should understand a particular concept. Is it appropriate at all? At some point, a line must be drawn. Why? Because if mathematics has a foundation, then it is important to teach it to children before they can fully understand the complexity and beauty of mathematics. What if, however, through learning concepts beyond the foundation, that understanding the foundation is made more obvious? If this is true, then what is a mathematics curriculum but a prescribed way of learning disguised by the theory of multiple intelligences.
For what multiple intelligences does not account for is the sequence at which a learner learns, discovers, and understands. If it is true that different students better learn when taught through (a) particular method(s), such as visually or kinesthetically, then the sequence of learning may also matter. Furthermore, the foundation of mathematics could be more complex than initially thought.
What are the consequences of this on teaching? Is it enough that elementary mathematics teachers understand ‘elementary mathematics’ and high school teachers understand a ‘little bit more than elementary mathematics’? Currently, professional teaching programs across Canada reflect the idea that elementary mathematics teachers need to know less math than their secondary counterparts. In addition, as reflected
by many programs, it is imperative that secondary mathematics teachers understand calculus. Why? Because through knowing calculus, teachers will better understand high school mathematics. In fact, elementary mathematics teachers are even encouraged to understand a little beyond the mathematics they are teaching. Why?
The questions asked in this post are not rhetorical; rather, they are important questions that have been and need to be researched in order to encourage change in mathematics teaching in school. Whether they have solutions remains to be seen, but as my math philosophy goes: “The solution is the least important step to understanding mathematics. Rather, it is the process that holds significance. Above all, however, is the idea that there are always multiple processes to the same solution.”