**Contents**

Theme

Common Core State Standards

Introduction

Symbolic Representation

Interviews with students

Mathematical Issues

Teaching Strategies

References

Your Experience

# Theme

# Common Core State Standards

7.EE.4, A-SSE

# Introduction

When students work with problems with figures in algebra they need to understand the role of variables in a new way. In algebra we are most interested in the situation in which letters are assigned to indicate the measurement of physical characteristics of shapes. For instance, in the case of perimeter, the letter represents the length of a side. Researchers have examined the development of students’ understanding of the concept of perimeter and area with identified lengths and have found some unique struggles for students as they engage in these types of problems.[i]

# Symbolic Representation

Problem |
Response |

1. Find the perimeter: |
• hhhht, 4ht, or 5ht (27%) Correct (57 %)* - A correct verbal description of how to get the answer but no symbolic representation (14%). - with the unsimplified h+h+h+h+t (29%) • hhhht or 4ht (20%) Correct (68%)** |

2. Part of this figure is not drawn. There are |
• a number between 32 and 42 (18%) correctly with 2n (38%) The rest provided no answer.** • A number between 32 and 42 (25%) correctly with 2n (9%) - 33% gave a correct verbal description of how to get an answer (such as “Two times however many sides there are”) but were unable to give a symbolic representation - The rest provided no answer* |

3. Find the area of the figure. |
• p X a + m (36%) • amXp or pam (25%) • No algebraic answer (21%) • Correct, p(a + m)(7%)* |

4. Find the area of the figure. |
• 5e2, e10, 10e, or e + 10 (42%) 41% of students thought that • Correct, 5(e + 2) (7%)* |

*Percent of 13 year old students in Booth’s (1984) study with that answer.

**Percent of 14 year old students in Kuchemann’s (1981) study with that answer.

# Interviews with students

(I: Interviewer, S: Student)

**Problem #1**

I: | How did you get hhhht? |

S: | I just added them all up. |

I: |
I see you couldn’t get an answer. Do you know how to find it for |

S: |
Counted all the numbers up. But I can't here because it's got no numbers. I do not know what “h” means and what “t” is. |

I: |
Suppose I tell you that “h” just means some number and “t” means a number, but it's a game and you do not know what the numbers are. Could you tell me anything about how you'd find the perimeter, what you'd do to the “h's” and “t” to get the perimeter? |

S: |
I suppose I'd have to measure them or something. |

I: | And then what? |

S: | Count them up. |

**Problem #2**

I: | How did you get 12 for your answer? |

S1: | I just added them all up. |

S2: |
There’s a letter there (“ |

I: | What does the letter mean? |

S2: | It's telling you how many stages. |

I: | Right. Can you write anything for the perimeter? |

S2: | Times it by that number. |

I: |
Could you write something down, to say how much that is? |

S2: |
What, shall I write down what I would do? (writes “if |

I: |
Now can you write that without using words, using math instead? |

S2: |
What, how’d you mean, like 2 times |

I: | Yes. Okay. |

S2: |
(writes “2 X |

I: | Fine |

S2: |
What, is that all it was? Why didn’t you say so, I thought you wanted the answer. |

I: | Do you mean a particular number answer? |

S2: | Yes! |

I: |
Well, is there a particular number answer? |

S2: | No! Not unless you know what n is. |

I: |
Well, then, how could you give me a particular number answer? |

S2: | Well, you can't, but I didn't know you only had to put that. |

**Problem #3**

I: | How would you do this problem? |

S: |
I suppose I could put it down like “am” times “p” (writes am X p) |

I: | What's that 'am' part? |

S: | That is the joining together of the two lengths. |

I: | How ould you get that joining to gether, what would you do? |

S: |
First I'd have to measure what the length of “a” and “m” is, and then I'd add them together. |

**Problem #4**

I: | Why did you answer "5e2"? |

S1: |
Well, I just put all the numbers and letters together. |

I: | Why did you do that? |

S1: | I didn't know what to do about the "e". |

I: | Why did you answer "10e"? |

S2: | I just took length times width and I took 2 times 5 is 10 and I just put 10e whatever e is for the variable e. |

I: |
Suppose that |

S2: |
I'd put 30 because 4 and 2 is 6 and 6 times 5 is 30 |

I: |
Does that agree with what you would give if you gave that answer with c equal to 4? Forget the figure. What would that (points to 10c) be if c was equal to 4? |

S2: | It would be 40 |

I: |
So one way you get 30 and the other way you get 40. |

S2: | Yeah |

# Mathematical Issues

Students who are successful with these types of problems are ones who have ** Acceptance of a Lack of Closure**. That is, they understand that an answer can include a variable and need not be evaluated to a single number. Those students who struggle with this sense of “incompleteness” can be at two levels. In the first level, students may not accept any answer with a letter, so will evaluate the problem into a number at all costs, putting a number in for the variable based on some rationale (see

**). For instance, in problem #2, students gave number answers between 32 and 42 or in problem #1 students might simply give a value to**

*Letter Evaluated**h*. Their sense of “complete” is that the final answer must be a number.

In the second level, students may be comfortable with a variable in the answer, but not comfortable with having an operation remaining, such as the answer “4*ht*” in problem #1 or 10*e *in problem #4. Their sense of “complete” is that all operations must be finished in order to get a final answer. At this second level of understanding, it would be important to differentiate between students who are simply collecting the letters (see ** Letter Used as an Object**) rather than adding lengths of sides to find the perimeter. In the first case, students do not associate “

*h*” with a specific length of the side but consider it a label or name for the side. For example, in the interview for problem #1, the student seems to simply be collecting things (for example, answering “4ht” or “hhhht”). Even students who respond with the correct answer “4h + t” may similarly be collecting objects. It is not necessarily clear whether they understand or do not understand the concept of perimeter as the addition of unknown lengths. They may simply be putting all the letters together in a “pile”.

On the other hand, students who answer “4ht” or “hhhht” for problem #1 may understand the concept of perimeter and the variables representing the lengths of the sides but struggle with the algebraic representation (see ** Algebraic Notation**). The student who answers 4

*ht*may intend 4

*h*+

*t*and simply does not know the convention for notation. In the interview for problem #1, the student understands how to get the perimeter with numbers, but struggles with variables representing numbers. Further questioning is necessary to reveal whether a student is regarding the letter as the unknown measurement of the side. When learning addition, students often associate length with numbers as they combine blocks, etc. For instance, a student combines a length of 2 and a length of 3 to get a length of 5. This may lead to thinking that a length of 2 added to a length of

*x*results in a length of 2

*x*, or in problem #3 writing “

*am*” as the total of the two sides.

*Algebraic Notation**may be the primary struggle that students are having with these types of problems rather than the concepts of area or perimeter.*

Problems three and four focus on finding the area. There are four possible reasons for students’ answers. The student:

1) interpreted the letter as a "thing" which could be merely collected up with the numbers (an interpretation or "input" error)

2) did not know how to interpret the letter *or* did not know how to operate with it and so performed the numerical calculation and wrote down the letter afterwards (a process or “input” error)

3) knew that “*e*” represented a number expressing part of the length of the base, but thought that area meant multiplying everything together (a process error)

4) interpreted the letter correctly, applied the correct method, but recorded the “answer” to “*e* added to 2” as “*e*2” or “2*e*” (an “output” or representation error).

And so, errors in thinking may occur at one of several points in the process of solving the problem. Students may have misread the question or part of the problem, In spite of a correct interpretation of the problem, students may have used an incorrect method in solving the problem. They may understand the problem but have had difficulty putting the answer into a format that is understood by mathematicians. Considering an input-process-output model of students’ engagement with problems may help gain a clearer picture of the points at which students' understanding of the problem breaks down.

Problem two is a different style of perimeter problem that demands an understanding of variables used for the purpose of generalization (see ** Letter Used as a General Number**). In order to deal with the unknown situation, they may simply substitute a number or complete the figure in their minds. With the first student interviewed for problem two, the student simply ignored the letters and incompleteness of the problem. The student simply counted the number of sides rather than consider the figure as going on "

*n*" times, getting an answer “32”. Other students physically or mentally continued the shape, closed the figure, and then counted, getting an answer such as “42”. These students may have lacked the ability to generalize or simply did not consider it as an option. Students may have difficulty visualizing a figure in which there are many more sides than what they can see.

The interview with the second student for problem two demonstrates that students are learning not only the mathematical procedures but also the mathematical expectations. Similar to our earlier discussion of algebraic representation with perimeter problems, the student here may simply need to learn what is ‘okay’ in mathematics and mathematical conventions. A comprehensive understanding of perimeter problems with variables involves an understanding that it is acceptable to have an expression as an answer, the appropriate mathematical notation for that answer, and a sense that the variable represents the unknown length of the side, or, in the case of problem two, the unknown number of sides.

# Teaching Strategies

- Have students explain their thinking so you can determine their understanding of variable as collecting things
*or*understanding that variables represent values. - Differentiate between students that are struggling with the concept of perimeter or area versus those that have conceptual understanding but lack the understanding of mathematical notation or conventions and those that struggle with what variables represent.
- If students have difficult with letters in perimeter or area problems consider walking through a problem with just numbers to help them get the overall understanding.

# References

[i] See Barrett, et al., 2006

Barrett, J. E., Clements, D. H., Klanderman, D., Pennisi, S.-J., & Polaki, M. V. (2006). Students' Coordination of Geometric Reasoning and Measuring Strategies on a Fixed Perimeter Task: Developing Mathematical Understanding of Linear Measurement. Journal for Research in Mathematics Education, 37(3), 187-221.

Booth, L. (1984). Algebra: Children's Strategies and Errors. Windsor, UK: NFER-Nelson.

Kuchemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children's Understanding of Mathematics (pp. 102-119): London: John Murray.

MacGregor, M., & Stacey, K. (1993). Cognitive Models Underlying Students' Formulation of Simple Linear Equations. Journal for Research in Mathematics Education, 24(3), 217-232.

MacGregor, M., & Stacey, K. (1997). Students' Understanding of Algebraic Notation: 11-15. Educational Studies in Mathematics, 33(1), 1-19.

# Your Experience