# How Math Knowledge Leads to Better Math Teaching

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A study delves into the relationship between an educator's math knowledge and the quality of their teaching in actual elementary school classrooms

What kinds of teacher knowledge most effectively promote student achievement in mathematics? What kinds of math skills do teachers need in the classroom? Harvard Graduate School of Education Associate Professor Heather Hill and her colleagues have developed a teacher assessment tool aimed at measuring exactly such teacher knowledge and skills.

Hill and colleagues’ new assessment not only requires teachers to solve standard mathematics problems, but also taps teachers’ understanding of the mathematical knowledge that is specific to teaching children. For instance, in addition to knowing how to compute, a teacher may need to know why and how specific mathematical procedures work, how best to define a term for children at a particular grade level, or how to interpret, remediate, or prevent students’ errors. Teachers’ expertise in these pedagogically relevant sub-topics, as assessed by this test, has been shown to affect students’ learning outcomes.^{1}

Now, in a recent study, Hill and her colleagues have dug more deeply into the relationship between teachers’ mathematical knowledge for teaching and the quality of their teaching in actual elementary school classrooms. The research team administered their written assessment of mathematical knowledge to ten teachers, and video-recorded nine class sessions from these teachers over a two school-year period. Analyzing teaching practices and comparing them with written assessment scores revealed not only the association between knowledge and instruction, but also the ways in which mathematical knowledge matters to classroom practice.^{2}

To illustrate the types of skills the study examined, let’s take a math quiz! What is the sum of these three numbers?

387 + (950 + 5)

To solve this problem, you might sum up the two numbers in parentheses first, then add the third. In elementary school, students learn that the total (1,342) is the same regardless of how the numbers are grouped together. This “associative property” of addition is behind the fact that a modified version of the problem is equivalent to the first:

(387 + 950) + 5

In this particular case, the regrouping hardly makes the calculation any easier. To a student just learning the associative property, the example doesn’t do much to demonstrate the usefulness of this property.

Now consider this slightly different problem:

(387 + 950) + 50

This calculation is decidedly easier – applying the grouping rule, we can see that 950 + 50 = 1000, and the rest of the solution falls into place. From a mathematical perspective, the associative property can be used in all three of these addition problems. But from a pedagogical perspective, the last example not only demonstrates the property – it also highlights how applying the rule can be useful. In teaching, then, not just any example will do; instead, teachers must design examples carefully to illustrate key features of the mathematics that students study.

This particular example of adding three numbers may at first seem rather insignificant. However, we can imagine that over the course of a school year the cumulative effect of such instructional opportunities – whether they are consistently seized or missed – can be great.

Hill and her colleagues found that teachers’ deep and flexible understanding of math concepts, reflected in their higher scores on the assessment, helped them provide richer learning opportunities for students. Teachers with more mathematical knowledge for teaching were more likely to supply mathematical explanations, to use better concrete models of mathematical processes, and to “translate” more accurately between students’ everyday language and mathematical language.

Furthermore, teachers with a deeper understanding of why numbers do what they do are more likely to understand children’s sometimes ambiguous comments in class. A student who is just learning a new math concept, such as how to express a 50-50 probability as a percentage, may express only a partially complete thought. A skilled teacher will leverage his or her knowledge to interpret what the child may be saying and ask clarifying questions. Rather than passing over such an ambiguous statement, the teacher can use it as a springboard for discussing the idea in more depth.

In addition to providing enhanced learning experiences, teachers scoring higher on mathematical knowledge for teaching also make fewer mistakes in class. While everyone makes occasional errors, such as miscopying numbers from a text, more highly skilled teachers were less likely to make computational mistakes and to use math terms inappropriately.

Each of the teachers included in the study reflected a unique mix of mathematical knowledge, years of teaching experience, beliefs about teaching (e.g., whether it should be skill-based, “fun,” activity-oriented, etc.), and access to instructional resources. Coupled with the small size of the sample, Hill cautions against generalizing about any one-to-one correspondences between mathematical knowledge for teaching and teacher effectiveness. However, this study offers new insights behind the generally accepted notion that subject knowledge matters for math teaching, even in the earliest grades.

Hill and her colleagues recommend that schools rely on specialists for math instruction. Just as we would expect classes like music and physical education to be taught by instructors with some specialization in the domain, students may also benefit by sharing math teachers who are most comfortable with the subject. In schools where a generalist approach to teaching is preferred, teachers should have frequent opportunities to discuss the goals and strategies of their math teaching with one another or with a math coach. Focusing on teachers’ deep understanding of mathematics, and of students’ math reasoning, may help bring about real improvements in elementary math teaching.

^{1} Hill, H.C., Rowan, B., & Ball, D.L. (2005) Effects of Teachers' Mathematical Knowledge for Teaching on Student Achievement. *American Educational Research Journal*, 42, 371-406.

^{2} For a more detailed description of the methods, see: Hill, H.C., Blunk, M. Charalambous, C., Lewis, J., Phelps, G. C. Sleep, L. & Ball, D.L. (2008). Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction: An Exploratory Study. *Cognition and Instruction*.