**Contents**

Theme

Common Core State Standards

Introduction

Symbolic Representation

Interviews with students

Mathematical Issues

Teaching Strategies

References

Your Experience

# Theme

Equations

# Common Core State Standards

6.EE.2b, 6.EE.5, 6.EE.6, 6.EE.7, 7.EE.4a, 8.EE.7, A-CED.1

# Introduction

There are foundational concepts that students must understand in order to be able to solve equations and simplify expressions. When numbers in the expression or equation are negative, additional consideration must be given. Equations with negatives pose some common problems for students. Solving equations often involves subtracting the same quantity from both sides to simplify the equation. However, this can lead to subtracting a negative from a negative, which confuses students. Likewise, multiplication by a negative can lead to the same kinds of issues.

In addition, there are other difficulties that arise with negatives in equations that are separate from those arithmetic difficulties. Equations are full of other symbols, and it is easy for a student's eye to skip over minus signs while trying to focus on "more important" things like variables. Furthermore, it is not always clear to a student what is an appropriate way to eliminate an unwanted term or factor from an equation when that term or factor includes a minus sign.

# Symbolic Representation

Problem |
Response |

i) 12 - x = 7
ii) 4 – x = 5 iii) -4 – x = 10 iv) -x = 7 |
Students can do (i), but the others are beyond reasoning unless they are grounded in integer computation. |

2 – 3x + 6 = 2x + 18 -2x -2x 2 – 1x + 6 = 18 |
The student has detached the minus sign preceding the 3x. That type of difficulty –identified by Herscovics and Linchevski (1991)—produced the error. |

3x -4 = x+9 -4 -4 3x = x + 5 |
In order to cancel a negative term, some students use subtraction. |

1) Adding a number to its additive inverse: -3 + 3= 6 2) Adding a positive number to a negative number -8+ 3= 5 or 11 or -11 3) Subtracting a positive number from a negative number -3 - 8 = -5, 5, or 11 4) Subtracting a positive number from a smaller positive number 3 - 8= 5 5) Subtracting a positive number from 0 0 - 8 = 8 or 0 |
1) The student has ignored the negative sign.
2) The student thinks addition should give a positive number or a number looking bigger than the addends. 3) The student thinks that the smaller number should be subtracted from the larger number. Or, the student has ignored the sign. 4) The student has ignored the order of terms.
5) The student has ignored the zero assuming it does nothing. |

# Interviews with students

(I: Interviewer, S: Student)

I: | What is a negative number? |

S: |
Possible Correct: Smaller than zero, smaller than all positive numbers, to the left of zero, zero minus something, subtracting a larger number from a smaller number Incorrect: a little, there is no such number, I don’t know, zero |

I: | -8 -3=? What does this expression mean? What does your answer represent? There are different ways of thinking to reach an answer to a question like this. Try to describe your way of thinking. |

S: |
Possible Correct: -11, you are taking away a number from a negative number and you will get a “more” negative number. This could represent that you owe money to someone and you borrow more from the same person, so now you owe more money to this same person. Incorrect: -5; you are taking a positive number away from a negative number so now it is closer to positive number. |

# Mathematical Issues

It is easier for children to learn to work with positive numbers because children can map their understanding of operations and properties to physical materials. However, developing a similar way of understanding negative numbers is harder because negative amounts do not appear physically in the child’s world.

To help students develop their understanding of negative numbers, teachers can use manipulatives (like integer chips), number lines, and several different metaphors (like money/debt, temperature, time, elevators). These ideas will help students to bridge the gap between the meaning of negative numbers and everyday world. However, manipulatives, number lines and metaphors should be used carefully so that they do not become “unnatural” experiences for students. In other words, activities using manipulatives, number lines and metaphors should be about mathematical ideas to be developed, not the procedures to be followed to use them.

Research indicates that students’ understanding of negative numbers improves when they are aware of the limitations of the models (manipulatives, number lines and metaphors). For example, a money/debt model could be used to develop the meaning of a negative number. A debt of $7 can be represented by -7. But teachers must make sure that students distinguish the ‘-’ in front of the 7 from the subtraction operation. For example, if you owe $2 to your father and $3 to your mother, then your total debt could be found by computing (-2) + (-3) = -5 which means you owe $5. It is possible that teachers represent this situation by -2-3; which is arithmetically correct but creates confusion between the negative sign and the sign for the subtraction operation. This shortcut could confuse students when they try to develop the meaning of the negative sign and the subtraction sign.

The money/debt metaphor can confuse students in problems like (-8) - (-2). Some students tend to translate this expression incorrectly as “You owe $8 and then you owe another $2.” This ignores the fact that the subtraction sign in the middle indicates that you are taking away from your debt, which does not make sense in children’s world.

Having a mental number line is also often considered a component of number sense that helps to build an understanding of negative numbers. Internalization of negative numbers is the stage when a person becomes skillful in performing subtractions. Understanding of the numerical system and relative size of the numbers, including the number zero, is an important part of having a mental number line. Understanding of absolute value is very powerful in this aspect.

Why was –x = 7 the most difficult of the introductory equations? Students consider –x statically just like -4. They treat them as the same. They could not conceive that –x = 7 could give x = -7 because to them, the right-hand side of the original equation would have had to have been -7. Students need to distinguish between the x and the minus sign. Thinking of the negative sign as signaling the opposite of x could help with relational reasoning.

Students could deal with 6x = 7 more easily than -5x = 10. They did not recognize that -5 and x were being multiplied. The negative sign persuades students that there is a subtraction of -5 that needs to be reversed with addition rather than a multiplication by -5. It is essential that students are grounded in variables and notation before solving equations.

# Teaching Strategies

Using different models for negative numbers and knowing the pros and cons of each model will help students to understand negative numbers and their operations. Teachers should alternate among models to compensate for the imperfection of each model; however, such alternations should be done by pointing out the issues in the model.

Some commonly used models are:

(i) Debt or owing: We need to be careful as we use this model because not all operations can be modeled naturally in this way. For example, subtracting a negative number from a negative number can become problematic for some children. Instead of the debt or owing model for negative numbers and operations with them, we can use the story of a postman delivering checks or bills to an address and the person at the address keeps track of the bills and checks as they are delivered. In this story line subtracting a negative number from a negative number can be conveyed by delivering a bill to the wrong address and postman comes back to take away the wrong bill.

(ii) Two color integer chips(sometimes called Equilibrium): One color represents positive numbers, the other color represents the negative numbers. With this model most of the operations are modeled naturally; however, for subtracting negatives, pairs of negative and positive chips need to be generated, and this is somewhat artificial. For example, for 4-(-3) four black chips are displayed then 3 pairs of black and white chips are formed to display zeros, then three white chips are taken away to display the value of the operation, 7, with seven black chips.

(iii) Elevators: Numbers are used to represent both position (the third floor) and the action (going up three floors). So it might be hard for students to grasp these two meaning of the number. However, this is a model that aligns well with the number line.

(iv) Time: A scale from B.C. to A.D. could be used to model numbers. However, time is already a hard concept for students to grasp, and adding and subtracting dates does not make much sense.

(v) Temperature: This model may be beneficial to use if children already have a conceptual understanding of temperature. However, if they don’t grasp the temperature idea it might be hard for them to understand what it means to add or subtract temperatures. This is another model that aligns well with the number line for those students who do understand temperature.

(vi) Number line model for operating with negative numbers: In this model addition and subtraction operations indicate which direction you face; for addition you face positive numbers and for subtraction you face negative numbers. The sign of the number indicates how you move on the number line, if you have positive number then you move forward in the direction you are facing and if you have a negative number you move backward in the direction you are facing. For example, for (-7) + (-2) you start at -7 on the number line and since you are adding you face positive numbers. Since you have -2, you move backwards 2 units facing positive numbers and land on -9. This model helps students to operate with negative numbers systematically. However, it is possible for students memorize the steps to produce the answers without really understanding the meaning behind the operation. Thus, providing an alternative model to develop the meaning of negative number is highly suggested.

Other considerations when working with equations that contain negative terms:

● Relational ("the same as") thinking can help students solve the questions posed in i-iv (see the Teaching Strategies section of Algebraic Relations: One and two step equations (*x* on one side only) for a description of relational versus operational thinking).

● Asking the students to justify their steps forces them to do only steps that they can explain.

● Students should be expected to defend the validity of their solutions for confidence improvement.

● Common errors should be discussed at the appropriate point, both in lessons and in textbooks, especially at the introductory level because it prevents development of bad habits and misconceptions on the part of the student.

● Interviewing students rather than just looking at written work will help uncover misconceptions.

# References

Altiparmak, K., & Ozdogan, E. (2008). A Study on the Teaching of the Concept of Negative Numbers. International Journal of Mathematical Education in Science and Technology, 41(1), 31-47.

Asghari, A. (2009). Experiencing equivalence but organizing order. Educational Studies in Mathematics, 71(3), 219-234.

Borba, R. N., T. (2000, July 23-27). Are young children able to represent negative numbers? Paper presented at the 24th annual Psychology of Mathematics Education (PME) Conference, Hiroshima, Japan.

Bruno, A., & Martinon, A. (1999). The teaching of numerical extensions: the case of negative numbers. International Journal of Mathematical Education in Science & Technology, 30(6), 789-809.

Demby, A. (1997). Algebraic Procedures Used by 13-to-15-Year-Olds. Educational Studies in Mathematics, 33(1), 45-70.

Dickson, L. (1989). Equations. In K. Hart, D. C. Johnson, M. Brown, L. Dickson & R. Clarkson (Eds.), Children's Mathematical Frameworks 8-13: A Study of Classroom Teaching. Windsor, England: NFER-NELSON.

Falle, Judith “From Arithmetic to Algebra: Novice Student’s Strategies for Solving Equations” p337.

Filloy, E., Rojano, T., & Solares, A. (2010). Problems Dealing With Unknown Quantities and Two Different Levels of Representing Unknowns. [Article]. Journal for Research in Mathematics Education, 41(1), 52-80.

Fischer, M. H. (2003). Cognitive Representation of Negative Numbers. Psychological Science, 14(3),278-282.

Fischer, M. H., & Rottman, J. (2005). Do negative numbers have a place on the mental number line? Psychology Sciencd, 47(1), 22-32.

Gallardo, A. (2003). "It is Possible to Die Before Being Born". Negative Integers Subtraction: A Case Study. Paper presented at the 27th International Group for the Psychology of Mathematics Education Conference, Honolulu, HI.

Gallardo, A. (2005). The duality of zero in the transition from arithmetic to algebra. Paper presented at the The 29th Conference of the International Group for the Psychology of Mathematics Education, Melbourne.

Gallardo, A., & Hernandez, A. (2006). The zero and negativity among secondary school students. Paper presented at the 30th Conference of the International Group for the Psychology of Mathematics Education.

Gallardo, A., & Hernandez, A. (2007). Zero and negativity on the number line. Paper presented at the 31st Conference of the International Group for the Psychology of Mathematics Education.

Gates, L. (1995). Product of two negative numbers: An example of how rote learning a strategy is synonymous with learning the concept. Paper presented at the 18th Annual Mathematics Education Research Group of Australasia Conference. Salma Tahir Macquarie University <mrs_galaxy@hotmail.com> Michael Cavanagh Macquarie University <michael.cavanagh@mq.edu.au (Attorps, 2003, p. 1-2).

Godfrey, David, & Thomas, Michael O. J. (2008). Student Perspectives on Equation: The Transition from School to University. Mathematics Education Research Journal, 20(2), 71-92.

Hativa, N., & Cohen, D. (1995). Self learning of negative number concepts by lower division elementary students through solving computer-provided numerical problems. Educational Studies in Mathematics, 28(4), 401-431.

Hayes, B. (1994). Becoming more positive with negatives. Paper presented at the 17th Annual Mathematics Education Research Group of Australasia Conference.

Hayes, B. (1996). Investigating the Teaching and Learning of Negative Number Concepts and Operations. Paper presented at the 19th Annual Mathematics Education Research Group of Australasia Conference. Hefendehl-Hebeker, L. (1991). Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs. For the Learning of Mathematics, 11(1), 26-32.

Kilhamn, C. (2008). Making sense of negative numbers through metaphorical reasoning. Paper presented at the the 6th Swedish Mathematics Education Research Seminar, Stockholm, Sweden.

Kilhamn, C. (2009). The notion of number sense in relation to negative numbers. Paper presented at the 33rd conference of the International Group for the Psychology of Mathematics Education, Tessaloniki, Greece.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does Understanding the Equal Sign Matter? Evidence from Solving Equations. Journal for Research in Mathematics Education, 36(4), 297-312.

Kohn, J. B. (1978). A Physical Model for Operations with Integers: Mathematics Teacher.

Linsell, C. (2007). Solving equations: Students' algebraic thinking. New Zealand.

McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. E. (2006). Middle-SchoolEquation Soving Students' Understanding of the Equal Sign: The Books They Read Can't Help. Cognition and Instruction, 24(3), 367-385.

Peled, I., & Carraher, D. (2008). Signed numbers and algebraic thinking. In D. W. Carraher & M. L. Blanton (Eds.), Algebra in the Early Grades (pp. 303-328). New York: NCTM.

Peled, I., Mukhopadhyay, S., & Resnick, L. B. (1989). Formal and informal sources of mental models for negative numbers. Paper presented at the 13th International Conference for the Psychology of Mathematics Education, Paris, France.

Pirie, S. E. B. and M. Lyndon (1997). "The Equation, the Whole Equation and Nothing but the Equation! One Approach to the Teaching of Linear Equations." Educational Studies in Mathematics 34(2): 159-181

Ponce, G. (2007). It's all in the cards: Adding and subtracting integers. Mathematics Teaching in the Middle School, 13(1), 10-17. Prather, R., & Alibali, M. (2008). Understanding and using principles of arithmetic operations involving negative numbers. Cognitive Science, 32, 445-457.

Rossi, P. S. (2008). "An Uncommon Approach to a Common Algebraic Error." PRIMUS 18(6): 554-558.

Schwarz, B. B., Kohn, A. S., & Resnick, L. B. (1993). Positives about Negatives: A Case Study of an Intermediate Model for Signed Numbers. The Journal of the Learning Sciences, 3(1), 37-92.

Vlassis, J. (2004). Making Sense of the Minus Sign or Becoming Flexible in "Negativity". Learning and Instruction, 14(5), 469-484.

# Your Experience

- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?