Mathematical Knowledge in Education Differentiating

Mathematical Knowledge in Education

Differentiating Types of Mathematical Knowledge and Relevance to Education

Ball, D.L., Lubienski, S., and Mewborn, D. (2001). "Research on teaching mathematics:

The unsolved problem of teachers' mathematical knowledge." (In Handbook of Research on Teaching. New York: Macmillan).

Generally, mathematics proficiency among teachers corresponds to higher achievement in their students. While that conclusion has been supported by a substantial volume of empirical research, much less empirical research has been devoted to trying to understand how and why teacher achievement in mathematics benefits student outcomes, or what it is about mathematics, specifically, that generates this apparent relationship. Most importantly, there is a need to understand whether and to what extent teacher mathematics achievement in different aspects of mathematics matters with regard to the positive effect on learners.

According to the authors of this article, there is a fundamental difference between teaching mathematics and teaching through mathematics. In many ways, that distinction helps explain why, in general, mathematics proficiency among teachers tends to correspond to better learning outcomes. More particularly, understanding that distinction may help explain why the positive benefit of mathematics knowledge among teachers is much more evident in connection with their academic study of mathematical method than in connection with their academic study of advanced mathematics. Furthermore, it could explain why advanced mathematical achievement among teachers also corresponds to higher incidence of negative affects on some learners whereas that is not true in the case of teachers whose high achievement in mathematics relates more to their non-pedagogical content knowledge than to their pedagogical content knowledge.

In principle, the value of teaching mathematics is much broader than the value of the substantive material, particularly in contemporary society that provides instant and accessible electronic calculation to solve the types of mathematical problems that could typically arise in everyday adult life. Study after study suggests that teachers who are more knowledgeable about mathematics tend to promote learning better than teachers who are less proficient in mathematics.

However, there is evidence that suggest that this relationship is much more complex than simply a direct transfer of pedagogical mathematical knowledge. For example, one unexpected finding is that the benefit of greater mathematics proficiency exists in the first grade. Presumably, all teachers are equally proficient at first-grade addition and subtraction; moreover, the academic study of mathematics in greater depth (i.e. post-calculus) should not have any impact on the level of teacher understanding of first-grade mathematics concepts. Similarly, there is no intuitive reason that either the mathematical proficiency of teachers or their highest level of mathematical study should translate to better teaching of elementary mathematical concepts. In that background, the correspondence between teachers having studied mathematical method and the highest identifiable benefits to learning seem to explain the basis of the phenomenon.

Specifically, mathematics (especially at the elementary level), can be taught rigidly and by rote rule or by conceptual understanding. Apparently, teachers with more extensive experience in studying mathematical method are better equipped to deliver mathematics lessons in a manner conducive to inspiring student initiative, stimulating the process of mathematical (and logical) reasoning, and more generally, to promote an intellectual inquisitiveness and resourcefulness that translate to higher student achievement. By contrast, teachers with greater pedagogical content knowledge may not necessarily translate that knowledge into teaching methods that accomplish more than simply teaching mathematics.

The Relevance and Significance of Mathematical Knowledge in Teachers

Hill, H.C., Rowan, B., and Ball, D.L. "Effects of Teachers' Mathematical Knowledge

for Teaching on Student Achievement." American Educational Research

Journal; (Summer 2005), Vol. 42, No. 2: 371-406.

In a similar study, the importance of non-pedagogical mathematical content knowledge among teachers was evident. This study highlights the practical implications and difficulties of teacher improvement programs whose purpose is to implement the conclusions about the importance of certain aspects of mathematics competency in teaching. It also relied partially on the evidence that certain aspects of mathematics competency in teachers corresponds to better learning and achievement even in the elementary grades where mathematical concepts are insufficiently complex to allow any knowledge difference in teachers to manifest itself in learner achievement.

According to this study, the recognition of correspondence between teacher knowledge of or academic achievement in mathematics and student benefit is largely useless without a more in-depth understanding of how and why teacher knowledge or achievement in mathematics translates into positive learning outcomes for students. The authors suggest that is particularly true with respect to using that information to design professional improvement programs for teachers. For example, it is important to know how to measure mathematical knowledge in teachers in connection with improving their performance.

Among the various criteria, the authors discuss the relatively low importance of the performance of teachers on mathematical proficiency tests and of their highest level of the academic study of mathematics in their own educational histories. As in the previous study, the authors identified teacher knowledge about mathematical method and demonstrations of deeper understanding of mathematical processes as the most crucial determinants of the translation of their mathematical proficiency into positive learning outcomes for their students. Similarly, it is important to understand whether it is something specific about mathematics knowledge that corresponds to better teaching abilities or if it is simply the case that some of the same intellectual characteristics that promote better or more advanced mathematical knowledge also happen to be conducive to better natural abilities as teachers.