by Erin Baldinger, University of Minnesota; Shawn Broderick, Keene State College; Eileen Murray, Montclair State University; Nick Wasserman, Columbia University; and Diana White, Contributing Editor, University of Colorado Denver.
Mathematicians often consider knowledge of how algebraic structure informs the nature of solving equations, simplifying expressions, and multiplying polynomials as crucial knowledge for a teacher to possess, and thus expect that all high school teachers have taken an introductory course in abstract algebra as part of a bachelor’s degree. This is far from reality, however, as many high school teachers do not have a degree in mathematics (or even mathematics education) and have pursued alternative pathways to meet content requirements of certification. Moreover, the mathematics education community knows that more mathematics preparation does not necessarily improve instruction (Darling-Hammond, 2000; Monk, 1994). In fact, some research has shown that more mathematics preparation may hinder a person’s ability to predict student difficulties with mathematics (Nathan & Petrosino, 2003; Nathan & Koedinger, 2000). Nevertheless, the requirements for traditional certification to teach secondary mathematics across the country continue to include an undergraduate major in the subject, and many mathematicians and mathematics educators still regard such advanced mathematics knowledge as potentially important for teachers.
Given this, it is important that, as a field, we investigate the nature of the present mathematics content courses offered (and required) of prospective secondary mathematics teachers to gain a better understanding of which concepts and topics positively impact teachers’ instructional practice. That is, we need to explore links not just between abstract algebra and the content of secondary mathematics, but also to the teaching of that content (e.g., see Wasserman, 2015). In November 2015, a group of mathematicians and mathematics educators met as a working group around this topic at the annual meeting of the North American Chapter of the Psychology of Mathematics Education. We began to probe the impact understanding connections such as those described above might have on teachers’ instructional choices. For example, how does understanding the group axioms shift teacher instruction around solving equations? How does understanding integral domains shift teacher instruction around factoring? Through answering questions such as these, mathematicians and mathematics educators can better support teachers to connect advanced mathematical understanding to school mathematics in meaningful ways that enhance the quality of instruction.
In the remainder of this blog post, we explain and discuss three frequently cited examples of connections between abstract algebra and high school mathematics.
Example 1: Solving equations
Solving equations and simplifying expressions is a technique used in multiple settings within mathematics. It uses the precise axioms of a group, but this is often not made transparent to students
What would you do to solve this “one-step” equation? Many students are taught to subtract 5 from both sides to isolate the variable x, and they might write something like this (crossing out the 5s on the left hand side):
x + 5 = 12
x = 7
However, on closer inspection, a variety of algebraic properties come to bear that the above work suppresses. (See Wasserman  for a more complete elaboration and discussion.) An expanded version might look like this, with justifications for each step.
(x + 5) + -5 = (12) + -5 (Additive Equivalence)
x + (5 + -5) = 12 + -5 (Associativity of addition)
x + 0 = 12 + -5 (Additive Inverse)
x = 12 + -5 (Identity Element for addition)
x = 7 (Closure under addition)
Similarly, if attention is given to algebraic properties used to solve equations, the solution to an equation of the form 5x=12 might appear as follows:
⅕*(5x)= ⅕*12 (Equivalence)
(⅕*5)x = ⅕*12 (Associativity of multiplication)
1*x = ⅕*12 (Multiplicative Inverse)
x = ⅕*12 (Identity Element for multiplication)
x = 12/5 (Closure)
These solution techniques can be related to students’ learning of matrix algebra in a course on linear algebra. Specifically, students learn, under appropriate conditions, to solve matrix equations of the form AX = B using these same steps.
In each case above, the last four steps being used – the ones “hidden” from view in the one-step cancellation process – are the precise axioms for a group. In the first case, we’re working on the additive group of integers, in the second on the nonzero multiplicative group of rational numbers, and in the last under the group of n by n square matrices with nonzero determinant (i.e., invertible) under matrix multiplication. Thus, these are three a priori separate problems, all united by the same algebraic structure of a group – and that structure becomes evident in the algebraic solution process. Wasserman and Stockton (2013) discuss one vignette for how such knowledge might be incorporated into secondary instruction.
Example 2: Simplifying expressions
As a related example, consider the following two samples of student work:
In each case, clearly a form of “cancellation” is being attempted. But what, technically, results in “cancellation”? And what remains after the cancellation is complete? Do sin and sin-1 make “1”? Is the “x” still an exponent? While we recognize this “cancellation” as attending to both the inverse elements and the meaning of the identity element in the group of invertible functions, these are subtle issues that are often not clear to students, and they are often taught in isolation, without the underlying structure being made apparent.
In using the above two examples to illustrate, we do not intend to imply that teachers should require students to make explicit each and every use of a mathematical property when they solve equations. Rather, we aim to draw attention to the importance of recognizing the consistency going on across all of these examples of solving equations. Moreover, it is the collective power of individual properties – as they form the group (or ring/field) axioms – that allow for algebraic solution approaches and also help reconcile the meaning of “cancellation” in these different contexts as an interaction of both inverse and identity elements.
Example 3: Polynomials and Factoring
As another example of the connection between abstract algebra and secondary mathematics, we consider the problem of multiplying two polynomials. (See Baldinger [2013, 2014] for additional examples of this type.) In high school, students learn that the degree of the product of two nonzero polynomials is the sum of the degrees of the factors. Yet this does not hold in all types of algebraic settings. Consider, for example, the product of the following two polynomials when working modulo 7 versus modulo 8.
As mathematicians, we of course recognize that the the degree of the product of two polynomials is the sum of the degrees of the factors — when the coefficients are elements of an integral domain, but that this relationship need not hold in other settings. Students, however, may be mystified when they first encounter an example like this in modular arithmetic, as their prior conceptions and understandings are being challenged, and they are thus being asked to deepen their understanding of the underlying structures that permit a result to hold in one setting, but break down in another.
This example also ties directly into student misconceptions. For example, we teach students in high school that if the product of two polynomials is zero, then to solve we set each one separately equal to zero. Yet this does not hold with nonzero numbers. For example, working in polynomials with real coefficients, we know that f(x) * g(x)=0 implies either f(x) = 0 or g(x) = 0. Yet it is not the case that if f(x) * g(x) = 4, then either f(x) = 2 or g(x) = 2.
The three above examples represent just a few of the many connections between abstract algebra and secondary mathematics. There has been a longstanding debate in the mathematics and mathematics education communities concerning the knowledge secondary mathematics teachers need to provide effective instruction. Central to this debate is what content knowledge secondary teachers should have in order to communicate mathematics to their students, assess student thinking, and make curricular and instructional decisions. This debate has already led to many fruitful projects (e.g., Connecting Middle School and College Mathematics [(CM)2] (Papick, n.d.); Mathematics Education for Teachers I (2001) and II (2012); Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations (Heid, Wilson, & Blume, in press). A common thread in these projects is the belief that mathematics teachers should have a strong mathematical foundation along with the knowledge of how advanced mathematics is connected to secondary mathematics (Papick, 2011). But questions remain regarding what secondary content stems from connections to advanced mathematics, which connections are important, and how might knowledge of such connections influence practice. Our working group hopes to continue to explore these connections and contribute to our collective understanding of teacher education.
Baldinger, E. (2013). Connecting abstract algebra to high school algebra. In Martinez, M. & Castro Superfine, A. (Eds.). Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 733–736). Chicago, IL: University of Illinois at Chicago.
Baldinger, E. (2014). Studying abstract algebra to teach high school algebra: Investigating future teachers’ development of mathematical knowledge for teaching (Unpublished doctoral dissertation). Stanford University, Stanford, CA.
Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers (Issues in Mathematics Education, Vol. 11). Providence, RI: American Mathematical Society.
Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II (Issues in Mathematics Education, Vol. 17). Providence, RI: American Mathematical Society.
Darling-Hammond, L. (2000). Teacher quality and student achievement: A review of state policy evidence. Educational Policy Analysis Archives, 8(1). Retrieved from http://epaa.asu.edu
Heid, M. K., Wilson, P., & Blume, G. W. (in press). Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations. Charlotte, NC: Information Age Publishing.
Monk, D. H. (1994). Subject matter preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125–145.
Nathan, M. J. & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of students’ algebra development. Cognition and Instruction, 18, 209–237.
Nathan, M. J. & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Education Research Journal, 40, 905–928.
Papick, I. (n.d.) Connecting Middle School and College Mathematics Project. Retrieved March 7, 2015 from http://www.teachmathmissouri.org/
Papick, I. J. (2011). Strengthening the mathematical content knowledge of middle and secondary mathematics teachers. Notices of the AMS, 58(3), 389-392.
Wasserman, N. (2014). Introducing algebraic structures through solving equations: Vertical content knowledge for K-12 mathematics teachers. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 24, 191–214.
Wasserman, N. (2015). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science, Mathematics and Technology Education (online first). DOI: 10.1080/14926156.2015.1093200
Wasserman, N. & Stockton, J. (2013). Horizon content knowledge in the work of teaching: A focus on planning. For the Learning of Mathematics, 33(3), pp. 20–22.