The “Insertion” Error in Solving Linear EquationsKosze Lee and Jon R. Star, Michigan State University

Table 1: Transcripts of Adam’s interview Solve 4(x + 3) = 8x Adam Oh, ok I got to change. That would be (writes down 4x +12 =8x) INT Ok. And then what did you decide to do with it at this point? Adam Um, The MOVE step? INT Ok. What did you decide to move? Adam the "8". So then that would be (writes down 8 8) INT So um, if you going to MOVE the "8x", what is that going to look like in terms of

what, what you have to do. So can you tell me aloud? Adam you MOVE the "4x"? I mean then you would subtract each~ from the 4x. INT Ok, so then you would, minus 8x over here(points to "8" on the left) and what

would you do on this side? (points to "8" on the right) Adam Minus 8x over there. INT Ok, so then remember you start out with what you have and then you minus the

"8x". So write down what you have and minus the 8x on that line. Adam Ok, I would (writes down 8x after the "8" on the left and 4x after the "8" on the

right) it's 4x and you have the 8x, and the 8x (writes down "x" beside both "8" to become "8x").

INT See, um, if you look at what Dr Star did, like he took away,um, he decided to MOVE the "2x". So he took away the "2x" here(points to the "-2x" on the left side of an equation) and he took away the 2x here (points to the "-2x" on the other side of the equation.)

Adam Oh, so I did the wrong, and this (points to 8x - 4x) should be over here (points to 8x - 8x)

INT Yeah. Adam Oh, ok. (write the line "8x - 4x 8x - 8x)

IntroductionThis proposed research investigates a

particular phenomenon that occurred during a

study of students’ flexibility in solving linear

equations (Star, 2004).

Method153 6th graders participated in five hours

(over five days) of algebra problem solving. In the

first hour, the students were given a pretest and a

brief lesson on four different steps that could be

used to solve algebraic equations (adding to both

sides, multiplying on both sides, distributing, and

combining like terms). Students then spent three

hours solving a series of unfamiliar linear

equations with minimal facilitation. 23 students

(randomly selected from all participants) were

interviewed while working individually with a

tutor/interviewer. On the last day of the project,

students completed a post-test.

ResultsAnalyses of students’ work made apparent an

interesting type of error, named “insertion”, in 12

(7.8%) students’ of which three (given the

pseudonyms as Adam, Bryan and Cindy) were

interviewed. The insertion error was evident when

3x = 6x + 6 became 6x  3x = 6x 6x + 6.

Similarly, 2(x + 5) = 4(x + 5) became 2 – 2(x + 5)

= 4 – 2(x + 5). In another case, 2(x + 1) = 10

became 2(x + 1 – 1) = 10 – 1. Interestingly, this

type of errors has not previously been reported

nor classified in the literature on linear equation

solving (e.g., Matz, 1980; Payne & Squibb, 1990).

Out of the many proposed classifications of

students’ rule-based errors in computational or

algebraic problems (Matz, 1980; Payne & Squibb,

1990; Sleeman, 1984), Ben-Zeev’s (1998)

classification is the most relevant here. Its

context of solving unfamiliar problems is very

similar to the context of the present research. In

this framework, the errors are classified into two

major types: critic-related failures and inductive

failures.

Contact Information

Jon R. Star, jonstar@msu.edu; Kosze Lee leeko@msu.edu.

College of Education, Michigan State University, East Lansing,

Michigan, 48824. This poster can be downloaded at

www.msu.edu/~jonstar.

ReferencesBen-Zeev, T. (1998). Rational errors and the mathematical

mind. Review of General Psychology, 2(4), 366-383.

Matz, M. (1980). Towards a computational theory of

algebraic competence. Journal of Mathematical Behavior,

3(1), 93-166.

Payne, S. J., & Squibb, H. R. (1990). Algebra mal-rules and

cognitive accounts of error. Cognitive Science, 14(3), 445-

481.

Star, J. R. (2004). The development of flexible procedural

knowledge of equation solving. Paper presented at the

American Educational Research Association, San Diego, CA.

Critic-related failures are due to the students’

failure to signal a violation of a rule while

inductive failures are due to student’s over-

generalization or over-specialization of

conceptual interpretations or surface-structural

features of worked examples. Here the latter,

“syntactic induction”, is useful in our analysis.

The interview transcripts of three students

suggest that they have over-generalized the

procedure of subtracting the same term on both

sides in order to eliminate a term of a linear

equation. As a result, two erroneous procedures

are created – one that violates the subtraction

law by inserting “TERM –” to both sides, and the

other that violates the distributive law by

inserting “ – TERM” in between a coefficient p and

its associated term (x + n) or inside the

parenthesis. The former is seen in Adam’s

transcript (Table 1) while Cindy (Table 2) and

Bryan (Figure 2) makes the latter error. However,

they stopped making these errors after they were

made aware of the violation of such rules.

ConclusionsThe data analysis thus proposes to include the

following into Ben-Zeev’s classification: 1)

another critic-based failure whereby prior rules

can be suppressed by the desired effect of a new

procedure, and 2) errors which are generated by

the confluence of over-generalized rules and

critic-based failures even though this may be rare

in the case of the “insertion” error.

Table 2 Transcripts of Cindy’s interview Solve 2(x + 1) = 10 Cindy (Writes down "2(x+1-1) = 10-1"; followed by "2(x+0) = 9")

INT

Um, if you choose the MOVE, one of the things about the MOVE, is that, um, you can't MOVE, um, anything, you,, you can't MOVE something from a parenthesis.

Cindy Oh, ok. Um. (long pause) Um, I am not really sure what to do then. INT Ok.

Cindy Because I've already used the EXPAND as the first step, so that's all that I know that reallly like seems to make sense.

INT So what are the ones that you have done previously, is there any other that you might want to try?

Cindy

Well COMBINE won't work. EXPAND I can't do. MOVE woulld not work. Mm hmm. (shakes her pen) um, hmm. Maybe with the MOVE, can I subtract the two? (circle the "2" in "2(x+0) =9")?

INT

Actually, the "2" is connected to this not by addition or subtraction and usually you use the MOVE when there is, when you can add or subtract it. And "2" is multiplied to this, you don't, you don't add or subtract it so you couldn't MOVE the "2"..

Cindy What? INT You, you couldn't MOVE the "2". Cindy yeah.

Adam’s work that exhibits “insertion” error for the second time (“-3x” is a later correction)

Bryan’s work that exhibits both types of “insertion” errors that violate the distributive law

Figure 1 Figure 2