Teaching Mathematics Conceptually: An Example

By Roger Howe

I have been worrying a lot about mathematics education for over a quarter century now. While many university mathematicians who get involved in mathematics education focus on the need for new teaching methods, I have been drawn to examples of failure of the US curriculum to deal properly with basic ideas.

One of the first such ideas I identified was place value, or to be more precise, the base ten place value notational system for whole numbers (and later, decimal fractions).  This is the bedrock of school mathematics, and it is used in almost everything  that is done day-to-day with mathematics. We ought to try to get this as right as possible, and to have students learn it as well as possible. Yet mathematics education research indicates that we fail rather badly to do so.

Susanna Epp and I wrote a lengthy discussion [Epp, Howe 2006] of the details of the principles and techniques that constitute base ten arithmetic. This was organized around what we called the five stages of place value. These are summarized by example as follows.
$$352 \ \ = \hskip 2.4 in$$
$$\ \ \ \ \ \ = \ 300 \hskip .6 in + \ \ \ 50 \ \ \ \ \ \ \ \ \ + \ \ \ 2 \ \ \ \ \ $$
$$\hskip .25 in = \ 3 \times 100 \hskip .38 in + \ \ 5 \times 10 \ \ \ \ + \ \ \ 2 \times 1$$
$$\ \ \ \ \ = \ 3 \times(10\times10) \ + \ \ 5 \times 10 \ \ \ \ + \ \ \ 2 \times 1$$
$$\ \ \ \ \ \ \ \ = \ 3 \times 10^2 \hskip .4 in + \ \ 5 \times 10^1 \ \ + \ \ \ 2 \times 10^0.$$
The first stage, $352$, is just the standard base ten notation for the number. The second stage, $300 + 50 + 2$, indicates that each digit in the number stands for a number of a special kind,  and the number itself is a sum of these special numbers. These numbers have a mathematical description – they are digits times powers of 10 (as is made explicit in the fifth stage), but there has been no simple reference term for them in the mathematics education literature.

Recently, the textbook [Beckmann 2017] has used the name  place value parts for these numbers, and we will adopt this term here. The place value parts are in some sense the atoms of the base ten system, and general whole numbers are like molecules, obtained by combining atoms. The lack of a simple term has impeded focusing on the place value parts as the basic building blocks of the system.

The place value parts have multiplicative structure, and the third and fourth stages make this structure explicit. The third stage displays each place value part as a digit times a base ten unit – a place value part with digit 1. The fourth stage exhibits the base ten units as also being products, of a certain number of factors of 10, which is the base of the system. The fifth stage then uses the standard notation of exponents to indicate each base ten unit by simply recording the number of factors of 10 used to make it. It exhibits the number as what might be called a polynomial in 10. To indicate the relation between the place value parts and the size of numbers, we also refer to the exponent as the order of magnitude.

Together, the five stages reveal the basis for the extraordinary power of base ten notation: it is using all the power of algebra – addition, multiplication, and exponentiation – simply to write numbers, and it employs the clever convention of place value, which allows each place value part of a number each to be indicated simply by its associated digit. This of course is mediated by 0, the symbol for zero, one of the all time great mathematical inventions.

Of course, mathematicians understand this very well, and translate almost unconsciously between the first stage and the last. However, there is evidence in the mathematics education literature that most students do not master this structure. In the 1980s, C. Kamii published several papers [Kamii 1986], [Kamii, Joseph 1988], showing that 3rd and 4th grade students did not understand the significance of the digits in 2- or 3-digit numbers. More recently, E. Thanheiser [Thanheiser 2009, 2010] has shown that many, probably the large majority, of students were arriving in college not understanding the third stage. This is relevant to U.S. mathematics education not only as a measure of current quality, but also for projecting future quality, since as Thanheiser points out, teachers must know the third stage if they are to teach addition and subtraction conceptually. The recent book [Newton 2018] gives examples showing how US K-12 mathematics instruction may fail to develop base ten structure adequately.

It took me several more years to appreciate that, in fact, although mathematicians may take the five stages as common sense, looked at from the educational point of view, each successive stage represents a significant conceptual development, that can take a year or more to get children to understand. The first stage starts in first grade, or even Kindergarten, when students are introduced to 2-digit numbers. In fact, the second stage (aka expanded form) is often stated at about this time,  and may or may not be used in a conceptual fashion. The book [Newton 2018 ] shows that, even the meaning of the digits in two-digit numbers is often taught inadequately. The third stage has to wait until students can deal with multiplication, so at least until 3rd grade if you follow the Common Core State Standards for Mathematics (CCSSM), although it can be dealt with implicitly with manipulatives somewhat earlier. The fourth stage requires considerable comfort with multiplication, and in particular, should involve appreciation of the Associative Rule/Property for Multiplication, which is arguably the subtlest of the Rules of Arithmetic, so could probably not be readily absorbed before 4th grade, perhaps 5th. The fifth stage requires a grasp of powers and exponents. For powers of 10, CCSSM calls for using whole number exponents in 5th grade, but the general idea of whole number powers comes in 6th grade. Thus, preparing a student to grasp the five stages in a conceptual way requires most of elementary school. And the evidence is, that we largely fail to do that.

The CCSSM pay considerable attention to the issues of place value, and in so doing provide a considerable advance over previous versions of mathematics standards. In particular, one of the summary headlines for Number and Operations in Base 10 in 5th Grade is “Understand the place value system”. However, it does not explicitly formulate the five stages.

One advantage of working with the five stage scheme is that it helps one to focus on the place value parts. A crucial feature of the base ten system is that the operations of addition and multiplication are simply combinations of operations with two place value parts (supplemented with regrouping – replacing 10 of some unit with 1 of the next larger unit). This follows from the Rules of Arithmetic (aka, Properties of the Operations). For addition, the two parts should even have the same order of magnitude.) Moreover, the results of such operations are given by the basic number facts (addition facts or multiplication facts), times an appropriate base ten unit. But it hard to teach or understand these points if there is no name for the place value parts.

The place value parts are also the key to comparison and estimation. Here the salient fact is that any place value part of a given order of magnitude is larger than any part of a smaller magnitude. It is at least 10 times as large as any part of two or more orders of magnitude less. (In fact, although place value parts of only one magnitude difference may be approximately the same size, for example 1,000 and 900, about half the time, the larger part will be 10 or more times as large.) It follows that any number is well approximated by the sum of a few of its largest few place value parts, and for practical purposes, can (and should) often be replaced by this. This is of course the topic of rounding, but provides the perspective that the place rounded to is not so significant at the number of places kept (called significant digits in the context of scientific notation).

Perhaps the value of thinking in terms of place value parts is revealed most clearly in long division, the most troublesome topic in whole number arithmetic. Long division in the whole numbers is about the operation of division with remainder. Given a number $n$ and a divisor $d$, we are looking for a quotient $q$ that is the largest whole number such that $qd \leq n$. That is, $n = qd + r,$ with the remainder $r$ being less than $d$. How do we find $q$? If $p = a 10^{\ell} $ is the largest place value part of $q$, then $(a+1)10^{\ell} > q$, so that $(a+1) 10^{\ell} d > n$. That is, $p$ is the largest place value part such that $pd < n$. The converse also holds. So to find the largest place value part of $q$, we should look for the largest place value part $p$ such that $pd \leq n$. Inspection of the long-division algorithm will reveal that this is exactly what it does. The smaller place value parts of $q$ are then found by repeating the process with $n’ = n – pd$, and continuing in this manner. Being able to use the language of place value parts might make the process more transparent.

A common current mantra is that teachers should understand and teach mathematics conceptually. What does this mean for whole number arithmetic? I would contend that it mostly amounts to

• i) understanding the five stages of place value;

along with

• ii) knowing the sums (and differences, when appropriate) of two place value parts of the same order of magnitude; and how this is a consequence of the Rules of Arithmetic together with the structure of the place value pieces;
• iii) knowing the products of two place value pieces, and how this is a consequence of the Rules of Arithmetic together with the structure of the place value pieces;
• iv) understanding how the Rules of Arithmetic combine with ii) and iii) to allow multidigit computation; and
• v) understanding the relative size of place value parts, and how this enables efficient approximation of multidigit numbers with numbers having few non-zero place value parts.

This starts with knowing the five stages, so I would like to see this as a standard part of math preparation for elementary teachers. In seminars I have run with practicing teachers through the Yale Teachers Institute, I have been surprised with the alacrity they show in adapting the five stages to their classroom. One teacher has taken the trouble to write and testify how the five stages seem to have helped her (high school) students.

It seems possible that the five stages can provide an avenue to teaching whole number arithmetic (and beyond this, decimal fractions) in a student-friendly and conceptual manner. Let’s make the five stages of place value a standard part of teacher preparation.

References

1. [Beckmann 2017]
   Beckmann,S. (2017). Mathematics for Elementary Teachers with Activities (5th Edition), Pearson Education, London and New York, 2017.
2. [Epp, Howe 2006]
3. [Kamii 1986]
   Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education,. 1, 75-86.
4. [Kamii, Joseph 1988]
   Kamii, C., & Joseph, L. (1988). Teaching place value and double-column addition. Arithmetic Teacher, 35(6), 48-52.
5. [Newton 2018]
   Newton, X, Improving Teacher Knowledge in K-12 Schooling: Perspectives on STEM Learning, by Xiaoxia A. Newton, Palgrave Macmillan 2018
6. [Thanheiser 2009]
   Thanheiser, E. (2009). Preservice elementary school teachers’ conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40, 251–281.
7. [Thanheiser 2010]
   Thanheiser, E. (2010). Investigating further preservice teachers’ conceptions of multidigit whole numbers: Refining a framework. Educational Studies in Mathematics, 75(3), 241-251.