In my work with teachers and their students, I embrace opportunities to develop my own understanding of the profound fundamentals of elementary mathematics. Here's what I noticed about my practice a few months ago: I caught myself teaching decimal fraction operations conceptually and then reverting to procedures-only instruction when it came to decimal operations. The questions I had to ask myself helped me identify ways to scaffold instruction of decimal procedures with strategies that make the operations easier to understand. Below you will find the questions I asked and the answers that evolved.Why Are Decimal Operations So Difficult for Students?Why do so many students struggle with decimal operations? Why is it that they get to middle school and continue to struggle with what we consider to be simple of arithmetic? What is it about dealing with decimal points that gets in the way of real problem solving? And what can we do about this?To understand the conceptual underpinnings of decimals, students must connect their thinking to fractions. After all, decimals are fractions based on iteratively dividing one unit into ten parts (Lamon, 2005). Empson and Levi write that "...the development of children's understanding of decimals simultaneously draws on their understanding of fractions and our base-ten place value system" (Empson & Levi, 2011). A study conducted with college undergraduate students revealed that even at that level, students preferred to use fraction notation rather than decimal notation in most contexts, stating that it helped them better conceive the relationships inherent in fraction understanding (DeWolf, Bassok, & Holyoak, 2014).Because students often lack full understanding of decimals as base-ten notation for fractions, they tend to learn decimal operations in a procedures-only format. Added to the problem is the tendency for teachers' misconceptions to parallel that of their students, often causing the teacher's misconceptions to become a source of students' faulty reasoning (Kastberg & Morton, 2014).Place Value - How are decimals and fractions related?In fourth grade, the common core standards provide guidelines that distinctly connect decimal numbers to fraction notation (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). These standards are appropriately placed in the Number and Operations - Fractions domain, further promoting the strong connection. For example, fourth graders should demonstrate decimal place value using fraction notation for the digits to the right of the decimal point:435.62 = 400 + 30 + 5 + 6/10 + 2/100This connection promotes understanding and coherence between base-ten numbers and fractions.Connections to fractions provide great opportunities for fourth-graders to develop conceptual understanding of the digits to the right of a decimal point. However, the fifth- and sixth-grade standards shift decimal work back into the Numbers and Operations in Base Ten domain. In fifth grade students develop place value and operation concepts for multi-digit decimal numbers, and by the end of sixth grade, they should achieve fluency with all four operations.Although the fifth- and sixth-grade standards do not explicitly express the connection between base-ten and fraction notation, full understanding of the operations may be more fully attained if the spirit of the fourth-grade standards is carried through all operations. Let's now turn our attention to the operations and how one might assist students in connecting the fourth-grade concepts to operational work in the next two grades.Addition and Subtraction - Why does "lining up the decimal point" matter? Far too often, we teach students to "line up the decimal points," and in their compliance, they do so, albeit with little understanding. One way to help students see the need for "lining up the decimal point" is to use fraction notation to help them identify the value of each digit. After all, lining up the decimal points ensures that students add like terms (tenths to tenths, hundredths to hundredths, etc.). Using fraction notation to represent the numbers in expanded form, as indicated in the example below, may assist students in grasping this idea of adding like terms.2.75 + 23.2(2 + 7/10 + 5/100) + (20 + 3 + 2/10)In this example, students can see that "the ones go with the ones and the tenths go with the tenths," leaving stand-alone values in the tens and hundredths places. Connecting this notation back to the original expression assists students in seeing that by lining up the decimal points, the like terms are lined up and are ready to be added.Multiplication - Why does the "digit shift" work?Once conceptual understanding of fraction multiplication is achieved (e.g., attending to principles such as scaling and applying the identity property), students can use their understanding of fraction multiplication to scaffold their understanding of multiplying multi-digit decimals. Once again, to build on previous understandings, using fraction notation to represent the decimals can help students understand why it is that the "digit shift" works in the standard algorithm.0.42 x 0.642/100 x 6/10 = 252/1000 = 0.252As seen in the example above, multiplying the denominators, 100 x 10, results in a product that has a denominator of 1000. This provides the foundation for understanding why one must "shift the digits" three places to the right in the product of 0.42 x 0.6 - multiplying the "hidden" denominators in the decimal fractions, which are always powers of ten, results in a new denominator that is also a power of ten.Division - Why does the "digit shift" work?I recently encountered a problem that asked why the following process works:2.63 ÷ 0.72 = 263 ÷ 72The correct response was that when you multiply both the dividend and the divisor by the same amount, the multiplicative relationship between the two (and, therefore, the quotient) remains the same. Because I had spent so much time reflecting on the relationship between decimal and fraction notation in the past few months, this was a mind-blowing moment for me. Never before had I thought of using fraction notation and the identity property to help students understand the digit shift in decimal division. In my mind's eye I was envisioning the following:2.63 ÷ 0.72 = 2.63/0.72 x 100/100 = 263/72 = 263 ÷ 72This example combines the fifth-grade standard of representing division as a fraction and ongoing work with equivalent fractions. By using this notation, students can use their understandings of the relationship between fractions and division and of the identity property to scaffold their understanding of multi-digit decimal division.ConclusionIn this post, I made a case that relating fraction notation to decimal number concepts and operations plays a foundational role in helping students transition to understanding the most frequently-used algorithmic procedures. Students use their prior knowledge of fractions and base-ten numbers when performing decimal operations. By using transitional notations to help students relate fraction and decimal operations, they may very well shift from procedural compliance to true understanding - from blindly following procedures to truly understanding the mathematical reasoning behind the procedures.
