In her groundbreaking book ‘Knowing and Teaching Elementary Mathematics’, Liping Ma documents the gap in the level of ‘profound understanding of fundamental mathematics’ (PUFM) between Chinese elementary school teachers and their American counterparts. She describes PUFM as “an understanding of the terrain of fundamental mathematics that is deep, broad, and thorough” (p. 120), and she goes on to say “PUFM goes beyond being able to compute correctly and to give a rationale for computational algorithms. A teacher with PUFM is not only aware of the conceptual structure and basic attitudes of mathematics inherent in elementary mathematics, but is able to teach them to students … The teacher who explains to students that because , one should move the second row one column to the left when using the standard multiplication algorithm is illustrating basic principles (regrouping, distributive law, place value) and a general attitude (it is not enough to know how, one must also know why).” (p. xxiv)
This general attitude, that it is not enough to know how, one must also know why, motivates all of my teaching, and is the central philosophy of this website. Arguably the most wonderful aspect of mathematics is that everything that is true in math, from the addition algorithm learned in second grade to Fermat’s Last Theorem, is true for a reason. Elementary school mathematics teachers should be able to provide grade-appropriate reasons for each topic they teach, and hopefully the content of this site will help teachers attain that goal.
In my estimation one of the biggest roadblocks that students face in their pursuit of PUFM is that at the elementary level, there is a lack of importance placed on clear language and precise definitions. During my first semester as a university professor I taught a Number and Operations course for pre-service elementary school teachers. When introducing the topic of division by zero I asked the class to discuss the problem . To my surprise the majority of the class thought the answer was zero. I figured they had simply forgotten the correct answer as perhaps it had been some time since they had seen such a problem, but when I asked for their reasons several students said that a teacher specifically told them that the answer was zero.
This has since become my favorite problem, and the reason I like it so much is that I believe it justifies one of my core teaching philosophies; that at all levels of mathematics, but especially the elementary level, clear, precise language is crucial. The answer to the above problem can be explained in about 30 seconds if one simply uses the correct definition of division. That is all; no fancy, upper-level mathematics is needed (see Clip 1 xiii for the solution). For some reason, many elementary teachers do not emphasize precise language, and as a result their students often harbor misunderstandings they will carry with them throughout their lives.
As another example of a situation in which simply using correct terminology can help avoid misunderstanding, I was recently teaching a class for in-service teachers and we were discussing geometry. A teacher mentioned that the way she described a rhombus to her class was “a rhombus is a square that got hit by a bus”. Now I’m sure her class found that description amusing, and I’m sure this ‘definition’ allowed students to correctly identify this shape:
as a rhombus. But what about this shape?
This is quite clearly not a ‘square that got hit by a bus’, so students that were given that as a definition would incorrectly respond that this shape is not a rhombus. However this is a rhombus; as the correct definition (in early elementary school) is that a rhombus is a four-sided figure in which all sides have the same length. (Note that in later grades the definition of rhombus is refined to ‘a rhombus is a quadrilateral in which all sides are congruent’. However this definition is equivalent to the correct definition given in early grades, it simply uses slightly more formal language.)
There are scores of other examples at the elementary level where simply using correct, precise terminology can lead to deeper understanding of a topic, and you will notice this as a central theme throughout the clips. The other central theme is a pursuit of PUFM; at every turn we are most concerned with why something works, as opposed to simply providing a procedural ‘trick’ that adds nothing to profound mathematical understanding. Note: In certain cases, such as when discussing the infinitude of primes or the formal proof of the Fundamental Theorem of Arithmetic, the precise mathematical reasons for why these results are true are not grade appropriate. In these cases I give suggestions for how to handle these discussions with an elementary school audience, and for those interested I include formal proofs in the supporting documentation.