*Parent: What did you study in math today?*

*Child: Fractions.*

*Parent: What did you learn?*

*Child: About one-tenth of what I was supposed to!*

My mathematics teaching career began during the 1970s in an individualized middle-school setting. Working with students one-on-one taught me the importance of using diagnostic testing and error analysis to develop pinpoint instruction and intervention. In that setting, I formed my belief that simply marking an incorrect answer to a math problem as being WRONG provides limited information to both teacher and student. The teacher needs to know the nature of the error in order to properly inform instructional activities. The student needs to know what he/she understands as well as know of any specific misconceptions he/she may have.

Years later when I taught developmental mathematics in a learning laboratory at the community college level, I discovered that many of the systematic errors made in basic math (whole number computation, fractions, early algebra) at this level were similar to those I had found during my early teaching experience. A community college student who computes ¼ + 3/4 and obtains 4/8 probably has had the same misconceptions about the meaning of a fraction throughout grade school and high school. This motivated me to examine the academic research on the subject to find ways to address this problem.

According to Siegler (2003), “… children who lack … understanding frequently generate flawed procedures that result in systematic patterns of errors. The errors are an opportunity in that their systematic quality points to the source of the problem and this indicates the specific misunderstanding that needs to be overcome” (p. 291). My strong desire to teach for conceptual understanding based both on what the student knows and on what the student does not know, led me to write a collection of diagnostic tests with targeted interventions.

Consider the subtraction problem below from a diagnostic test, with the four given possible answers. The correct answer is (C). On a well-written diagnostic test, the foils are all based on common error patterns. The choice (D), Not Here, should be included so that students are encouraged NOT to guess when they are unsure about an answer. On a diagnostic test, guessing into a correct or incorrect answer provides no useful information.

**83 **

** – 57**

(A) 34 (B) 36 (C) 26 (D) Not Here

For foil (A), a teacher should ask him or herself, “How was the answer 34 obtained?” Inspection of the problem (or a discussion with the student) likely will reveal that the student evidently subtracted the smaller digit (3) from the large digit (7) in the ones position. For foil (B), it appears that the student only partially renamed the minuend. Instead of renaming 83 as 7 ten and 13 ones, the student “renamed” 83 as 8 tens and 13 ones.

Beattie and Algozzine (1982) note that when teachers use diagnostic tests to look for error patterns, **“testing for teaching begins to evolve”** (p. 47). There are several ways to provided targeted intervention for the above problem. The use of base-10 blocks or play money on a place-value mat is my preference. Students should display the minuend using the 8 tens and 3 ones.

An intervention for foil (A) would involve posing these key questions: “Do we have enough ones to take 7 ones from 3 ones?” (No.) “So what should we do?” (Break apart one of the tens, and combine the resulting 10 ones with the 3 ones. Now we have 13 ones.) “Now do we have enough ones to take 7 ones from 13 ones? (Yes.) It is important to note that we should never tell students, “You cannot subtract a larger number from a smaller number.” Although such language may be expedient, it generally leads to misconceptions in later grades when students use integers to subtract larger numbers from smaller numbers.

For foil (B), key questioning should focus on the renaming process. Students need to understand that when the renaming is done correctly, *the renamed minuend is equal to the original minuend*. Ask: “Does 8 tens and 13 tens = 83?” (No.) “So something is wrong here. What did you do to obtain the 10 ones that were combined with the 3 ones?” (I traded one of the tens for 10 ones) “Good. So since you traded away one of the tens, we now have only 7 tens left, not 8 tens. So what should be the renamed minuend?” (7 tens and 13 ones.)

For both foils, a question related to reasonableness could also be asked: “Is it reasonable that subtracting a number close 60 from 83 would produce a number that is greater than 30?” (No. The answer should be less than 30.)

Let’s revisit the error pattern ¼ + ¾ = 4/8. Some students make this error because they apply whole-number concepts to fractions. They may read the fraction ¾ as “three fours” and may be struggling with how the numerator and denominator are related. It is interesting to note that in Korean, Chinese, and Japanese (and other languages), a fraction, say, *three fourths,* is read **“of four parts, three” **— with the denominator being stated *first.* Based on research with first and second graders, Miura and Yamagishi (2002) concluded that “the Korean vocabulary of fractions appeared to influence conceptual understanding and resulted in the children having acquired a rudimentary understanding of fraction concepts prior to formal instruction” (p. 207).

Language can also be used to illustrate that the denominator tells the unit being counted. For example, a fraction such as ¾ can be thought of as a measurement, say, 3 miles. Explain that in 3 miles, the unit of measure is *miles, *and we are talking about 3 of those units. In ¾, the unit of measure is *fourths,* and we are talking about 3 of them. So, just as 1 **mile** + 3 **miles **= 4 **miles**, we can use the word name “fourths” to show that

1 **fourth** + 3 **fourths** = 4 **fourths.**

Yet another way to address the error is to ask questions related to reasonableness: “Which is greater, ¾ or ½?” (3/4.) “So, is it reasonable that adding ¼ to ¾ would produce a fraction equivalent to ½? (No. The answer will be greater than ½.)

Of course, some errors that occur in students’ work are “random”—generally due to carelessness or incorrect recall o facts. However, I have found that many more errors are due to misconceptions and the use of incorrect strategies—often strategies that were engrained through rote memorization rather than through conceptual understanding. In such cases, I have found that providing (additional) meaningless drill is not the answer. (Would simply assigning ten more problems akin to 83 – 57 or ¼ + 3/4 really help?) Rather, a targeted approach based on teaching for conceptual understanding that addresses the nature of the error seems to provide a more efficient and long-lasting solution.

In her case study, Bray (2011) concluded that teachers “would benefit from a greater awareness of common student errors and how these errors are related to key mathematics concepts” (p. 35). Bray believes that teachers need support in developing teaching practices that use student errors in the classroom as springboards for class discussion. I fully agree, and I encourage teachers to seek resources that provide powerful diagnostic and intervention tools based on identified error patterns. The use of such resources will clearly save teachers time—thus enabling them to diagnose the nature of student errors and provide meaningful instruction based on student conceptual understanding.

**References**

Bray, W. S. (2011). A collective case study of the influence of teachers’ beliefs and knowledge on error-handling practices during class discussion of mathematics.*Journal for Research in Mathematics Education, 42*(1), 2–38.

Beattie, J. & Algozzine, B. (1982). Testing for teaching. *Arithmetic Teacher, 30*(1), 47–51.

Miura, I. T., & Yamagishi, J. M. (2002). The development of rational number sense. In B. Litwiller & G. Bright (Eds.), *Making sense of fractions, ratios, and proportions. *Yearbook of the National Council of Teachers of Mathematics (pp. 206–212). Reston, VA: National Council of Teachers of Mathematics.

Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), *A research companion to principles and standards for school mathematics* (pp. 289–303). Reston, VA: National Council of Teachers of Mathematics.

## F. Williams / December 28, 2015

Great article. I think it is equally important to teach our students to look at their work, identify errors, correct it and articulate a strategy that they need to remember to avoid making the error in the future. There are great activities for this on http://www.matherroranalysis.com .

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