Forms of mathematical knowledge

Anyone interested to understand how mathematics is learned should at first understand what mathematical knowledge consist of. The book Forms of Mathematical Knowledge: Learning and Teaching with Understanding describes various types of knowledge that are significant for learning and teaching mathematics. It defines, discusses and contrasts various types of knowledge involved inthe learning of mathematics. It also describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas. The book is a collection of articles/papers from well known mathematics educators and researchers.

Top in the list of forms of knowledge presented in the book is a discussion about intuition and schemata.
While there is no commonly accepted definition, the implicitly accepted property of intuition is that of self-evidence as opposed to logical-analytical endeavor. Now, what is the role of intuition in the learning of mathematics?

While in the early grades teachers are awed by intuitive solutions by our students, those handling higher-level mathematics would find intuitive knowledge to constrain understanding of mathematics. In the book, the author of offered examples of these. He also defined the concept of intuitions and described the contribution, sometimes positive and sometimes negative, of intuitions in the history of science and mathematics and in the teaching process. The author argues that knowledge about intuitive interpretations is crucial to teachers, authors of textbooks and mathematics education researchers alike. The author further argued that intuitions are generally based on structural schemas.

My favorite article in the book is about the description of mathematical knowledge as knowing that, knowing how, knowing why and knowing-to

Knowing why, meant having “various stories in one’s head” about why a mathematical result is so. For example, when partitioning an interval into n subintervals, one might recall that n+1 fenceposts are required to hold up a straight fence of n sections. Knowing why and proof are different — in many cases, the proof doesn’t reveal why. As an example, the author suggested that when primary teachers ask why (-1)(-1)=1, they want images of temperature or depth, not a proof, or even a consistency argument that negative numbers work like positive numbers.

Knowing to means having access to one’s knowledge in the moment — knowing to do something when it’s needed. For example, in evaluating a limit, a student might just know to multiply by a certain quantity divided by itself. This kind of enacted behavior is not the same as writing an essay explaining what one is doing — it often occurs spontaneously in the form of schemas unsupported by reasons, whereas explanations require supported knowledge.

Forms of Mathematical Knowledge: Learning and Teaching with Understanding is a must-read for teachers, educators, and those doing research in mathematics teaching and learning.