Preservice teachers' number sense.

Many calls for reform in school mathematics emphasize number sense (Australian Education Council, 1991; Cockroft, 1982; Japanese Ministry of Education, 1989; National Council of Teachers of Mathematics, 1989; 2000; Swedish Ministry of Education and Science, 1994). But what is number sense? According to Case (1998), "Number sense is difficult to define, but easy to recognize. Students with good number sense can move seamlessly between the real world of quantities and the mathematical world of numbers and numerical expressions. They can invent their own procedures ... represent the same number in multiple ways.... recognize benchmark numbers and number patterns.... and gross numerical errors...." (p. 1).

I am using number sense to refer to "the general understanding of number and operations, along with the ability and inclination to use this understanding in flexible ways to make mathematical judgments and to develop useful and efficient strategies for managing numerical situations" (Reys & Reys, McIntosh, Emanuelsson & Johansson, and Der, 1999, p. 61).

Most research on number sense has, focused on "average" children's number sense (e.g. Reys & Reys et al., 1999; Turner, 1996; Yang, 2002), while some studies have focused on the number sense of children with learning disabilities (e.g. Gersten & Chard, 1999; Griffin, Case, and Siegler, 1994). However, studies on teachers' number sense (e.g. Ma, 1999; Menon, 1999) have been limited, and hence this study serves to augment knowledge about teachers' number sense, specifically preservice teachers' number sense.

Participants and Methodology

This study involved 142 preservice teachers from four groups of mathematics methods classes/sections I taught. These students were in a teaching credential program, which included 8 credit hours of mathematics education, leading to a K-8 teaching certification. Since these preservice teachers were going to teach mathematics that can be considered foundational to students in K-8, I believed it would be beneficial for me to find out these preservice teachers' mathematical competence, specifically their own number sense.

Many of them (almost 90%) had not taken algebra and geometry in high school. But all of them had taken mandatory mathematics content courses for teachers (that included algebra and geometry), taught by the mathematics department faculty of the university. (In this university, the math methods courses are taught by the college of education faculty, but the math content courses are taught by math department faculty.)

In spite of their taking, and passing, the mandatory university math content courses, results from an informal survey revealed that about 85% of them still lacked confidence in teaching middle school mathematics, mainly because they were unsure of their ability to successfully do mathematics content at that level.

On the first day of their mathematics education class, these 142 preservice students were given a 10 item, multiple choice paper and pencil test on number sense. For each item, they were given space on the test paper itself, to give written explanations for their answers. They were given 25 minutes to complete the test.

The 10 items were selected from those used in previous studies on number sense (e. g. Menon, 1999; Reys & Reys et al., 1999). The components of number sense (also based on previous studies), together with the relevant item number assigned to test them, are given next:

1. To make mathematical judgments (J), for example, by determining appropriateness and sufficiency of information--item #s 1 & 2.

2. To develop useful and efficient strategies for managing numerical situations (E), for example, by using estimation and number relationship--item #s 3 to 5.

3. The general understanding of number and operations (U), especially those related to fractions and decimals--item #s 6 to 10.

Note that, for the purposes of this study, a participant is considered to exhibit number sense, if the participant can not only select the correct response, but also give a correct solution. By correct solution, I mean an explanation that demonstrates explicit use of the relevant component of number sense for items related to that component. In other words, if no explicit use of an efficient strategy such as, say, estimation, is demonstrated in the explanation for an item requiring efficient estimation, it will be inferred that number sense might be lacking. Further examples of the difference between correct response and correct solution are given in the next section.

Results and discussion

The results of the study are summarized in 4 Tables (Tables 1, 2, 3, & 4). Table 1 shows the percent of correct responses and correct solutions, by item. Note that by "correct response," I mean that the correct response was selected, and that by "correct solution," I mean that the correct response was selected, and a correct explanation, based on number sense components, was given. For example, if 3 were chosen for item #1, together with an explanation, such as "We cannot have 2-1/4 taxis, as there can only be a whole taxi, not a part of it, like, 1/4 of a taxi, so we need 3 taxis," that demonstrates appropriateness of information, then it would be scored as "correct response," and also as "correct solution." If, on the other hand, 8000 were chosen for item #3, that would be scored a "correct response," but if the explanation showed, say, 30 years X 365 days = 10950 days [approximately equal to] 8000 days, then this would be scored "incorrect solution," as the number sense component, using estimation as an efficient strategy, was not used to get an initial estimation of, say, 400 days, for 365 days, so that 30 X 365 [approximately equal to] 30 X 400 = 12000 [approximately equal to] 8000. In short, "correct solution" means "correct response" plus "correct explanation, using number sense component."

Table 2 shows the percent of preservice teachers getting exactly 0 to 10 correct responses for the number sense test. Table 3 shows the distribution of different multiple choice responses according to test items. Table 4 shows the percent of correct responses and correct solutions according to components of number sense. Reference is made to these tables, and the relevant questions/items, in the discussion that follows.

Discussion of results for item #s 1 & 2.

1. A taxicab is only allowed 4 passengers in one taxi. How many taxis would be needed to take 9 passengers? Circle your answer. Explain why you chose that answer.

Draw pictures if you need to.

A. 2

B. 2-1/4

C. 3

D. Not enough information given

Explanation:

2. A cab driver picks up 4 passengers. The passenger's ages are 9, 11, 14 and 34.

What is the age of the cab driver? Circle your answer. Explain why you chose that answer. Draw pictures if you need to.

A. 20

B. 48

C. 68

D. Not enough information given

Explanation:

From Table 1, for items #1 and #2, more than 90% of the preservice teachers gave the correct response and correct solution, indicating they could make mathematical judgments (J), by determining appropriateness and sufficiency of information. In other words, they did not just blindly do the computation, but were aware of appropriateness of answers in given contexts, and realized when insufficient information was given.

Discussion of results for item #3.

3. About how many days old are you? Circle your answer. Explain why you chose that answer. Draw pictures if you need to.

A. 800

B. 8000

C. 80000

D. 800000

Explanation:

As for item #3, 74% got the correct response. This meant that almost a fourth of the preservice teachers were either unable to link the number of days to their age, could not multiply a two digit number (their age) with a three digit number (the number of days in a year) correctly, or were unsure of large numbers. Among the 74% who had it correct, only about 20% used estimation: most multiplied the age with 365, and then rounded up/down the answer, rather than estimating (365 as 400) first, and multiplying next. Hence, we might infer that since about 15% (i.e. 20% of 74%) gave a correct solution, only these 15% used mental arithmetic or other efficient strategies for managing this numerical situation.

Discussion of results for item #s 4 & 5.

4. Given that 48 + 37 = 85, what is the answer to 49 + 36? Circle your answer. Explain why you chose that answer. Draw pictures if you need to.

A. 85

B. 86

C. 715

D. 4126

Explanation:

5. Given that 38 + 47 = 85, what is the answer to 85 - 47? Circle your answer. Explain why you chose that answer. Draw pictures if you need to.

A. 38

B. 42

C. 132

D. 817

Explanation:

For item #s 4 & 5, all had correct responses. However, about 75% of them used actual computation to arrive at the answer, as shown by their explanation. They did not show explicit use of the relationship between the numbers and operations involved to arrive at the answer efficiently and rapidly. While it is true that these were rather trivial computations, and the preservice teachers might have overlooked the relationships between the numbers and operations, we might conjecture that this might be a case of using inefficient strategies for managing numerical situations, and an over-reliance on algorithms. However, a more definitive inference might have been possible, if larger numbers (say, 4 or 5 digit numbers) had been used.

Discussion of results for item #6.

6. How many different fractions are there between 2/5 and 3/5? Circle your answer. Explain why you chose that answer. Draw pictures if you need to.

A. None

B. One

C. A few

D. Lots

Explanation:

For item #6, 52% gave the correct response and the correct solution. This means that almost a half of the preservice teachers believed that there were no common fractions between the two given ones. Most of those who chose "none" explained that there was no whole number between 2 and 3. Such a response evidenced a poor concept of fractions, and a negative transfer of learning from whole numbers to fractions.

Discussion of results for item #7.

7. Which is the largest fraction? Circle your answer. Explain why you chose that answer. Draw pictures if you need to.

A. 4568/4569

B. 4569/4570

C. 499/500

D. 500/501

Explanation:

For item #7, 26% responded correctly. This item seemed very difficult for the preservice teachers, possibly because the fractions had three to four digits in both the numerators and denominators. But almost all those who gave a correct response, did not give a correct solution, as shown by their explanation. For example, they gave explanations such as "4569 and 4570 are the largest numbers here, so that fraction has to be the largest" or that "4569 is the largest numerator, so the fraction with the largest numerator has to be the largest." (This is similar to the reasoning of students who say that by "canceling" the 6 in the numerator and the denominator of the fraction 16/64, the fraction "reduces/simplifies" to 1/4: correct answer, but wrong reasoning!)

Discussion of results for item #8.

8. How many different decimal numbers are there between 1.52 and 1.53? Circle your answer. Explain why you chose that answer. Draw pictures if you need to.

A. None

B. One

C. A few

D. Lots

Explanation:

For item #8, 59% chose the correct response and gave the correct solution. The explanation generally was that "there are lots of numbers between 52 and 53." A comparison of the results from item #s 6 & 8 seems to imply that decimal fraction computation is easier than common fraction computation.

Discussion of results for item #9.

9. Without calculating the exact answer, circle the best estimate for 292 X 0.96 Explain why you chose that answer. Draw pictures if you need to.

A. slightly more than 292

B. slightly less than 292

C. 292

D. Cannot tell without calculating it

Explanation:

For item #9, 64% chose the correct answer, and gave a correct explanation (such as "292 X 1 is 292, but 0.96 is less than 1, so, 292 X 0.96 should be slightly less than 292"). That is, 64% gave the correct response and the correct solution to item #9. Those who got it wrong explained that multiplication would give a larger product. Such an explanation implies that these preservice teachers did not realize that sometimes multiplying (by a decimal fraction less than 1) could result in a product that was smaller than the original multiplicand.

Discussion of results for item #10.

10. If a "broken" calculator displays 6858 as the answer to 15.24 X 4.5, where should you place the decimal point in the answer? Circle your answer. Explain why you chose that answer. Draw pictures if you need to.

A. 6.858

B. 68.58

C. 685.8

D. 0.6858

Explanation:

For item #10, 23% got the correct response. Among these, about 25% used a correct explanation based on estimation, such as "15 X 4 = 60; so the product should be 68.58." That is, there were 23% correct responses and about 6% (i.e. 25% of 23%) correct solutions. The remaining 75% actually computed 15.24 X 4.5, and realized there was a zero at the extreme right of their product. Then they applied the rule of counting the number of decimal places. Those who got it wrong just used the rule of "moving the decimal places." Once again, there seemed an over-reliance on algorithms and rules, with little or no reliance on estimation and making sense.

Table 2 shows that the preservice teachers had between 3 and 10 correct responses out of a maximum possible 10 correct responses. Given that these 10 items needed very little computation, it is troubling that only about 13% had at least 9 items correct, and about 39% had 8 items or more correct. In other words, about 61% of the preservice teachers had 7 or less items correct from the 10-item total--and we are talking about correct responses, not correct solutions.

From Table 3, we can see that for item #3, on the age of the preservice teacher in days, almost a fourth of them chose the incorrect response C, 80,000 days as their age. Their explanation was that the other numbers were either too small or too large. In other words, they guessed the answer, and did not seem to comprehend that 80,000 days would be more than 200 years!

For item #6, a little over a fourth of them, chose the incorrect response A. They believed that there were no fractions between 2/5 and 3/5, mainly because they thought there were no numbers between 2 and 3.

For item #7, over a half (53%) of the preservice teachers chose the incorrect response C, stating either that they guessed or because a three-digit fraction (in the numerator and denominator) would be larger than one with four digits, as four digits would imply smaller parts. When questioned about the choice of C (499/500) as opposed to D (500/501), they stated that since the denominator 500 is less than the denominator 501, the former fraction would be larger.

For item #8, slightly less than a fourth (23%) of the preservice teachers chose the incorrect response A. Their reason was that there could be no numbers between 2 and 3.

For item #9, the incorrect responses A and D had about the same percent (16% and 14%, respectively) of preservice teachers attracted to them. The former was chosen by those who thought "multiplication will make the answer larger." The latter was chosen by those who felt that only a computation would permit an answer, and that they could not estimate because the numbers were "too large (too many digits)."

For item #10, about three-fourths of the preservice teachers chose the incorrect response A. They explained that the rule was to "count the total number of places after the decimal point."

From Table 4, it can be seen that for the 3 components of number sense, the first one, that of making appropriate mathematical judgments, J, was about 93%, for both correct responses and correct solutions. In other words, it would seem that a majority of these preservice teachers were able to make appropriate mathematical judgments, and that component of number sense seems reasonably strong among them.

The second component of number sense, that of using efficient strategies for numerical situations, E, was about 91% for correct responses, but about 41% for correct solutions. (The 41% is computed as shown next: 2 out of the three items under E, items #4 & 5 were correctly obtained by NOT using relationship by about 75% of the preservice teachers, but by straight computation. From Table 1, then, the total number of correct solutions for item #3 would be 105, that for item #4 & 5 would be 71 [being 25% of 284], giving a total of 176 correct solutions out of a possible 426. This would result in about 41% correct solutions for category E.) Given my earlier caveat about how number sense in this study is going to be judged by the use of correct solutions, (as opposed to correct responses only), we might infer that this component of number sense, E, needs to be further studied and strengthened.

The third component of number sense, the general understanding of number and operations, U, was about 45%, for both correct responses and correct solutions. From this, we might infer that about 45% of the preservice teachers seem to have a reasonable understanding of numbers and operations, but about 55% might need to strengthen this component of number sense.

Hence, it seems that only in category J (for making mathematical judgments) are these preservice teachers confident. While this study has no data on whether such proficiency/confidence could be an artifact of maturation rather than number sense per se, it would be interesting to study this aspect further. Also, given the operational use of number sense here, as exhibiting explicit explanation, using the components of number sense, or demonstrating correct solutions (as opposed to correct responses), it might be inferred that only about half of these preservice teachers a) could use efficient strategies for managing numerical situations and b) have a reasonable understanding of numbers and operations.

Possible implications

While the items may be too few or not standardized for reliability, the results, based on the way judgment about number sense was defined operationally here, are still a cause for concern. For, if these future K-8 math teachers seem to rely on learned procedures, without the profound understanding of fundamental mathematics suggested by Ma (1999), as shown by some of their explanations to the number sense test items, how well equipped will they be to teach conceptually?

One way to begin to help preservice teachers have a "deeper conceptual understanding of mathematics," (Olson & Berk, 2001, p. 306) might be for mathematics educators to first identify any weakness in number sense and conceptual understanding of mathematics, by assessing their number sense. Math educators can also make preservice teachers aware of a) the disadvantages of learning math procedurally, and b) the empowering nature of making sense of mathematics. However, to have a lasting impact, far reaching programmatic and systemic changes would have to be implemented to increase the pedagogical content knowledge of preservice teachers (such as decisions on what math content they have to learn, and what, if anything, can be done about the implied separation of math content and pedagogy by the common practice of the content being taught by the math department and the pedagogy by the education department). How we do this, and whether our efforts produce any changes to the way math is taught to K-8 students remain a challenge.

Table 1 Item number, number of correct responses and correct solutions,
and percent of correct responses and solutions by item

Item #  # correct  % correct  # correct  % correct
        responses  responses  solutions  solutions

 1      133         94        133         94
 2      132         93        132         93
 3      105         74         21         15
 4      142        100         36         25
 5      142        100         36         25
 6       74         52         74         52
 7       37         26          0          0
 8       84         59         84         59
 9       91         64         91         64
10       33         23          8          6

Note: "Correct response" = correct response selected
"Correct solution" = correct response selected, plus correct
explanation, using components of number sense

Table 2 Nmber of items selected correctly from the 10 item test,
according to number and percent of respondents selecting correct
responses

# of items correct  # correct responses  % correct responses

          0                  0                    0
          1                  0                    0
          2                  0                    0
          3                  2                    1
          4                  7                    5
          5                 18                   13
          6                 32                   23
          7                 28                   20
          8                 37                   26
          9                 14                   10
         10                  4                    3

Table 3 Item number, number of respondents choosing each response (with
key marked *), and percent choosing each response

Item #   Choice A  Choice B  Choice C  Choice D
         #      %  #      %  #      %  #      %

 1         0    0     6   4  *133  94     3   2
 2         0    0     5   4     5   4  *132  93
 3         0    0  *105   7    35  25     2   1
 4      *142  100     0   0     0   0     0   0
 5      *142  100     0   0     0   0     0   0
 6        38   27    16  11    14  10   *74  52
 7        17   12   *37  26    76  54    12   9
 8        33   23    21  15     4   3   *84  59
 9        22   16   *91  64     9   6    20  14
10       108   76   *33  23     0   0     1   1

Table 4 Percentage of responses and solutions according to components of
number sense

                       # of correct  Maximum #    % of correct
Number sense           responses/    of possible  responses/
component     Item #   solutions     responses    solutions

     J        1 & 2    265/265       284          93/93
     E        3 to 5   389/176       426          91/41
     U        6 to 10  319/319       710          45/45

J: To make mathematical judgments.
E: To develop useful and efficient strategies for managing numerical
situations.
U: The general understanding of number and operations.

REFERENCES

Australian Education Council. (1991). A national statement on mathematics for Australian schools. A joint project of the States, Territories and the Commonwealth of Australia, Australian Education Council and the Curriculum Corporation.

Case, R. (1998, April). A psychological model of number sense and its development. Paper presented at the annual meeting of the American Educational Research Association, San Diego.

Cockcroft, W. H. (1982). Mathematics Counts. London: Her Majesty's Stationery Office.

Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of Special Education, 44, 18-28.

Griffin, S. A., Case, R., & Siegler, R. S. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.). Classroom lessons: Integrating cognitive theory and classroom practice (pp. 25-49). Cambridge, MA: MIT Press.

Japanese Ministry of Education. (1989). Curriculum of mathematics for the elementary school. Tokyo: Printing Bureau.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

Menon, R. (1999). Preservice teachers' number sense. Paper presented at the 49th Annual Conference of the Ohio Council of Teachers of Mathematics (OCTM), Dayton, Ohio.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Olson, M., & Berk, D. (2001). Two mathematicians' perspectives on standards: Interview with Judith Roitman and Alfred Manaster. School Science and Mathematics, 101(6), 305-309.

Reys, R., & Reys, B., McIntosh, A., Emanuelsson G., & Johansson, B., and Der, C. Y. (1999). Assessing number sense of students in Australia, Sweden, Taiwan, and the United States. School Science and Mathematics, 99(2), 61-70.

Swedish Ministry of Education and Science. (1994). Syllabi for the compulsory school. Stockholm: Author.

Turner, B. (1996). A sixth grade teacher's beliefs about her students' number sense. Unpublished paper, Department of Curriculum and Instruction, University of Missouri.

Yang, D. C. (2002). Teaching and learning number sense: One successful process oriented activity with sixth grade students in Taiwan. School Science and Mathematics, 102(4), 152-157.

Ramakrishnan Menon

California State University, Los Angeles