Using the Jar Model to Improve Students’ Understanding of Operations on Integers

1. Using the Jar Model to Improve Students’ Understanding of Operations on Integers Madihah Khalid Bny Rosmah Hj Badarudin Presented by John Suffolk Brunei Darussalam

2. Introduction • Based on experiences of teaching integers in secondary government schools, it was found that many students were facing difficulties in understanding the topic of integers • Many secondary mathematics teachers in Brunei Darussalam have expressed their concern over students’ poor performances on integers. Understanding integers is considered basic for progression to other topics. • Students tend to get confused with the signs and operations on integers although teachers had attempted to explain about it several times. • The number line model that is popular among the teachers is also confusing to the students. • Rules given make matters worse.

3. Literature Review • Pupils in Brunei start learning about integers at Primary 6 and ‘integers’ is again taught in Form 1 and 4. Most teachers use number line to explain addition and subtraction of integers as it is recommended. • Teachers find it easier to teach rules than to teach for meaning, hoping students’ understanding will develop as they operate successfully with the relatively ‘simple rules’. • Students on the other hand find it difficult to establish the rules for themselves; therefore they just rely on remembering instead of understanding. • All these led to rote learning where students only know how to solve the problems of integers but do not understand why it happens in such a way. • Many students know how to apply the method blindly and mechanically without any awareness of the significance of the answer

4. Study by Hart et al (1981) found that when students are faced with an expression like +8 – –6 many of them use the rule to work out the appropriate sign and then operate with it (in this case adding 8 and 6) ignoring the starting point. This works in some cases but not in others (such as –2 ––5, where students would give 7 as the answer). • Hayes (1999) found that slight misapplications of the rules, such as applying ‘two negatives make a positive’ to –4 + –2 to get +6, are common. We found this kind of mistake to be very common among the students in the study. • From our point of view, the difficulty with understanding negative times a negative is because it’s something that students do not do in their everyday lives. The operation of subtraction, especially subtracting a negative, is difficult for students to make sense of.

5. According to Kuchemann (1981) understanding does not necessarily flow from the use of number line because it too is an abstracted representation of an abstract idea. • Shifting along number line was criticized by Hayes, (1999) because it has the potential to cause confusion between operations and integers especially in secondary school students because the calculation involves a lot of signs and operations. • Kuchemann (1981) found that the number line is not a good representational model for working with integer operations, except for addition. A discrete model (e.g. where the positive elements can cancel the negative elements) is preferred because it has documented success with students and it is more consistent with the actions involved.

6. The Study • This paper reports part of a bigger study where students were pre-tested, taught operations of integers during intervention for four weeks and then post-tested. Students were already taught integers in February of the same year. • Research Questions: √What pre-existing knowledge do the students generally have about integers? √To what extent does the strategy used in the intervention enhance the students’ performances on operations with integers?

7. Methodology • The present study is an exploratory study which used a multiple perspectives research design. A combination of qualitative and quantitative methods were used to gather data. The sample of the study consisted of 149 Form 1 (grade 7) students in one government secondary school in Brunei. The results of this study presented in this paper were obtained from the analysis of the following data: √ Document analysis; √ Analysis of performance data from pencil-and paper pre-test; √ Interview data analysis – both teachers and students; √ Analysis of performance data from pencil-and paper post-test;

8. Result Table 1: Pre and Post-Test Mean Total Score, Standard Deviation, and t-test Results for Students from each of the five classes invloved in the study (*p < .05)n

9. Student-explanation: Minus and minus become plus (pointing negative sign of 2 and 6). Using number line starts at minus 2 then move forward 6 times. Answer is 4. Move forward because it is plus (pointing the addition operation). Student-explanation: Negative 2 put in the jar. Add 4 pairs of positive and negative. Then remove the entire negative. Student-explanation: Negative 2 and negative 6 combine in the jar become negative 8. No cancel because all negative Student-explanation: Minus and minus become plus (pointing the negative sign in front of 2 and the subtraction operations) become 2 + (– 6). Minus and plus become minus. Using number line starts at minus 2, turn back 6 steps become minus 8. Students’ solutions (from pre and post test)

10. Student-explanation: Draw 2 negative and 6 positive. Positive and negative become 0. The answer is 4 Student-explanation: Minus and minus become plus. Minus 2 plus 6 cannot find the answer. I don’t know what to do. Student’s explanation: Draw 2 negative and 6 negative. There is no positive and negative to become 0. Just take all the negative become negative 8. Student-explanation: 2 plus 6 is 8. Minus and minus become plus

11. Student-explanation: Take the minus sign in front. 2 plus 6 is 8. Answer, minus 8. Negative because minus and plus becomes minus Student-explanation: Draw 2 negative and 6 positive then 2 pairs of positive and negative become 0. Student-explanation: Draw 2 negative and 6 positive then 2 pairs of positive and negative become 0. Student-explanation: Minus and minus become plus (pointing the negative sign of 2 and 6). 2 minus 6 is minus 4

12. Student-explanation: Minus and minus become plus become –2 + 6. Start at minus 2, move backward stop at minus 8. Move backward because of minus sign of 2 Student-explanation: Take the minus sign in front. 2 plus 6 is 8. Answer, minus 8. Negative because minus and plus becomes minus Student-explanation: Using the jar model draw 2 negative and 6 positive. Positive and negative become 0. Student-explanation: Using the jar model draw 2 negative. In the jar not enough negative to be removed add 4 pairs of positive and negative. Remove the entire negative leaves 4 positives Student-explanation: Draw 2 negative and 6 positive then 2 pairs of positive and negative become 0. Student-explanation: Using number line start at minus 2 move backward 6 steps. Move backward coz minus and plus become minus (pointing the negative sign of 2 and the operation of addition).

13. Student explanation: Err. . Minus and plus (pointing the negative sign of 2 and addition operation) become minus. Put the sign in front. 2 plus 6 is 8. Answer is minus 8. Student explanation: 6 is bigger than 2, take the minus sign in front. Minus and minus (pointing the negative sign of 2 and the subtraction operation) become plus. 2 plus 6 is 8. Answer, minus 8. Student-explanation: Draw 2 negative; add 4 pairs of positive and negative because not enough negatives. Remove 6 negatives because of the minus. Student-explanation: Draw 2 negative in the jar then add 6 negative in the jar become negative 8 Student Explanation: 6 is bigger than 2. Put the sign (minus) in front. Minus and minus become plus (pointing the negative sign of 2 and 6). 6 plus 2 is 8. Since the minus is in front the answer is minus 8. Student-explanation: Draw 2 negative and 6 positive. Positive and negative cancel become 0.

14. Student explanation: Em... 6 is bigger than 2, take the sign in front. Minus and plus (pointing the negative sign of 2 and addition operation) become minus. Student explanation: Minus and minus become plus (pointing the negative sign of 6 and 2). 6 minus 2 is 4. Student-explanation: Draw 2 negative and 6 negative to get negative 8 Student-explanation: Draw 6 negative then take away 2 negative Student explanation: Using number line start at minus 6 move forward 2 steps. Start at negative six maybe because it is bigger Student-explanation: Draw negative 2 and negative 6. Add become negative 8.

15. Discussion • Students got mixed-up with operation and signs when there are subtraction and negative signs in a problem. To avoid confusion, teachers can try to use certain words for operations (like subtract, take away etc) and refer to the signs as positive and negative (not plus and minus). • Some students’ errors are a mixture of many types of error. They take the sign of the larger number and put it in front and multiplied the sign of the numbers to get the operations. They apply multiplication rule everytime they saw two signs. They also take the sign of the larger number and then subtract or add the rest. • Students generally do better in multiplication and division of integers compared to addition and subtraction involving negative integers. This can be due confusion because there are too signs of the same kind in a problem involving subtraction. It can also be due to multiplication rule is easier to remember?

16. Data from the interviews revealed that some students thought that multiplication or division and -ve sign become only multiplication or division. They were thinking that the -ve sign will disappear. If they have two -ve signs in a problem on multiplication or division they only take one -ve sign and ignore the other. • Teachers should avoid starting lessons with rules because this will hinder understanding. Instead, make pupils understand and help them come up with their own rules later. • Misunderstanding in signs and operation also led students to use distributive law incorrectly in solving problems related to integers.

17. Conclusion • Relatively few students displayed understanding of integers. They tend to memorise “rules” given by their teachers. Knowledge memorised tends to be registered only in the short-term memory, and is likely to be quickly forgotten. • Students had little idea why they write what they had written. On many occasions they said that they were using methods they had been taught in class. All the students seemed to be mainly interested in getting right answers. For them, understanding mathematics meant the same thing as getting correct answers. • Most of the interviewees gave the impressions that they could not cope with the number of rules that they had learnt for integers. They struggled to remember all those different rules, and even if they could remember rules, they were often not sure which rule applied to which task. Often an interviewee would say that they had forgotten what they had learned before, or they were confused because there were “too many rules to remember.” • Teachers should try to teach for understanding and not resort to short cuts so that they can get good marks in examinations. In a long run, teaching for understanding saves time.

18. From the study carried out, we can confidently say that the jar model is less confusing than the number line model and created better understanding in the students compared to the rules and analogies that teachers are fond of using before. • Jar method works best for addition and subtraction of integers. • The model does not explain situations when a negative number is multiplied by a negative number and when a positive number is divided by a negative number well. Other explanation like counting outside the jar is needed to explain this phenomena and this can confuse students.