Understanding decimals: Kevin Moloney and Kaye Stacey find out why our students are getting tied up in knots about decimals.

Moloney, Kevin ; Stacey, Kaye

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Although students first learn about decimal notation in primary schools, it is well known that secondary students in many countries, including Australia, do not have an adequate knowledge of the concepts involved. Even some students who can calculate with decimals do not understand the comparative sizes of the numbers involved. Understanding decimal notation is an important part of basic numeracy. Our society makes widespread use of metric measurement for scientific and everyday purposes. Computers and calculators use decimal digital displays, so making sense of input and output decimal numbers is essential. In this article, we will demonstrate some of the ways in which students think about decimal notation and how this changes as students get older. Our testing at one school, which seems to us to be quite a typical Australian high school, showed that about a quarter of students still had important misconceptions in Year 10. We present a simple test that teachers can use to help them diagnose the mistakes that their students are making and we discuss some of the help that is appropriate. In this article 'decimals' will refer to 'decimal fractions' and 'fractions' will refer to 'common fractions'.

Understanding decimals

Understanding decimals is a multi-dimensional task. Students need to coordinate place value concepts with aspects of whole number and fraction knowledge. Making the transition to understanding decimals relies on having a thorough understanding of previous concepts fully integrated with new information.

In this paper, we are concerned with whether students know the relative sizes of decimals well enough to order sets of decimals correctly. For example, can students place in increasing order the decimals 3.214, 3.09 and 3.8? Only a few students have trouble ordering decimals when the integer parts are different, so all the examples we use here will have the same integer part. Research studies in France, Israel and the United States (cited in the references to this article) have established that there are two main patterns of erroneous thinking, called the whole number rule and the fraction rule. In all of the studies, including our Australian study (Moloney, 1994), significant proportions of students were seen to be following these erroneous rules consistently. Students construct these rules for themselves when they are trying to assimilate new knowledge about decimals with what they already know about whole numbers, place value and fractions.

Students' ideas about decimals

Whole number rule. When asked to order decimals, some students systematically choose the number with most digits after the decimal point as the largest. These students are said to be following the whole number rule. They would say 4.125 is greater than 4. 7 because the whole number 125 is larger than 7. The decimal point is recognized but only as a separator, and the decimal portion is often read as a whole number, e.g. 'four point one hundred and twenty five', rather than the conventional four point one two five'. These students have effectively constructed a number line such as the one shown on the left of Figure 1. Researchers have observed that whole number rule students often have a weak understanding of whole number place value. Even with whole numbers they simply reason that longer means bigger, without understanding why.

Fraction rule

Some students consistently choose the number with least decimal places as the larger. This is called the fraction rule. These students would say that 2.3 is greater than 2.67. They reason that the 3 is tenths, the 67 is hundredths and tenths are larger than hundredths. Alternatively, some may confuse 0.3 with 1/3 and 0.67 with 1/67, giving the same result for a different reason. In effect, these students have constructed one of the number lines as shown on the right of Figure l. In showing the 'number lines' in Figure l, we are not implying that students are sufficiently explicitly aware of all the logical consequences of their thinking that they could offer such a number line. We also know that students will often have other special knowledge (e.g. that 0.5 is a half) that cuts across the application of their rules for uncommon numbers.

Fraction rule students recognize that digits after the decimal point relate to fractions but they do not connect the size of the parts and the number of parts. For example, when asked to write 5/5 in decimal form, some fraction rule students wrote 3.4, 0.3 or 0.34. Are these errors common in Australia? We carried out two studies to gather data on the ideas about decimals held by Australian students. Firstly we tested two classes of secondary school students twice, a year apart. and looked at how their ideas changed over the year. Secondly we tested classes of students from Year 4 to Year 10 and looked at how the numbers of students making each type of error varied. The studies were carried out in middle-class, coeducational Catholic schools in Melbourne. Changes in a year of schooling We tested 26 Year 7 students and 24 Year 9 students in rnid-1992 and again one year later, when they were in Years 8 and 10. The test we used is shown in Figure 5. One third (8) of the Year 7 students and one half (12) of the Year 10 students were 'experts': they got nearly all of the questions correct. We found that four of the Year 7 students and two of the Year 10 students showed the whole number misconception. A third (9) of the Year 7 students and one quarter (6) of the Year 10 students showed the fraction rule misconception.

The test was measuring change over time without any formal intervention other than normal teaching. It was very disappointing to see that there was only a small change in understanding of decimal notation during the course of a complete school year. Only six of the fifty students changed categories. Five moved into the expert category two from the fraction rule category and three previously uncategorised. Thirteen of the fifteen students retained their fraction rule misconception. All six whole number rule students stayed in their category.

This study demonstrated that there is a significant problem with misconceptions about decimals, at least at these secondary schools. It is of great concern that half of the students at Year 10, and a larger percentage at other year levels, did not compare decimals correctly. Their teachers had previously been unaware of the extent of this problem. It is also very surprising that only six out of 50 students showed any movement between categories over a full year of schooling. This provides evidence that students were responding to the items in a way which reflected a stable thought pattern which teaching during the year did not alter.

The Year 4 to Year 10 study

In the second study, we tested 379 students in Years 4 to 10, two mixed-ability classes from each year level. Line graphs illustrating changes in the percentage of students with expert, whole number and fraction rule usage from Years 4 to 10 are given in Figures 2, 3 and 4. The percentage of students classified as expert rises reasonably steadily (considering the small number of classes sampled), but was still only at 73% by Year 10 (see Figure 2). A large proportion of the younger students demonstrated the whole number rule misconception (see Figure 3) but this reduced in the secondary school. There was a slow decrease in the fraction rule misconception but, as Figure 4 shows, it remained prominent in higher year levels (20% in Year 10 in this study).

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This misconception in particular has the potential to remain with students into adulthood unless it is challenged.

It is interesting to note in the graphs that the Year 4 students had better understanding of decimals than the Year 5 students. As there are only two classes at each year level, we cannot conclude that this would usually happen. One of the Year 4 teachers attributed it to the fact that the Year 4 students had commenced their study of decimals just prior to the testing whereas the Year 5 students had not had any decimal work for some months prior to the testing. It might be that for Year 5, the simplistic whole number rule reasserted itself in the absence of instruction and practice.

Testing students' understanding

The test we used is a simple one which teachers may like to use with their own students. We adapted it from Resnick et al (1989). The test is given in Figure 5 and the answers that students in each category will give are shown in Figure 6. Students who are thinking about decimals with the whole number rule will tend to get the first five questions wrong, the second five wrong and the last five correct. Allocate a student to this category if at least four out of the five answers match this prediction. Fraction rule students will probably be correct for the first two groups of five questions and incorrect for the last five. Again use a four out of five criterion for making the classification. Students classified as experts will get at least four out of five correct in each group of five questions. (There is another small category of students who will get the first five incorrect, the second five correct and the last five correct. These students know that a zero in the tenths place makes a decimal small, but otherwise are like the whole number rule students. For more information, see Resnick et al. (1989).) It is better if the fifteen items in the test are mixed randomly rather than given in the three blocks of five as shown in Figure 5 although this makes categorizing students a little slower.

Using the results

When students' misconceptions about decimals have been classified, teachers may be able to target their teaching to particular groups of students. From our results, it is clear that students are able to carry out the standard tasks of arithmetic at school without having their understanding of the fundamental meanings of decimal notation challenged. They can carry out routine procedures without developing a real sense of the size of the numbers and without indicating to their teachers the depth of their misconceptions. Students must be continually challenged as to the reasonableness of any answers obtained: "Does this answer make sense?" It is important that students are not able to mask misunderstanding of decimals by carrying out routine procedures such as adding zeros to the end of decimals to compare them; e.g. to compare 0.3 with 0.12, change 0.3 to 0.30 and then compare the whole numbers 30 and 12. We advocate doing plenty of calculations (by hand and by calculator) which use decimals of varying lengths in the one question. If students only operate on decimals with one decimal place, say for months, they need never learn how decimals with different numbers of decimal places compare.

Whole number students may need quite basic assistance with ideas of place value for whole number. Teachers will need to discuss the reasons for referring to a decimal such as 0.29 as 'nought point two nine' rather than 'nought point twenty nine'. Counting (e.g. by 0.1) with the support of a calculator's constant addition facility can reinforce the idea that 0.9 is followed by 1.0 (and not 0.10, 'nought point ten').

Positioning numbers on a number line is useful for students with either misconception. Facility with relationships based on place value are essential. For example, it is critical that students learn that 0. 7 is equivalent to 0. 70 and why it is so.

Fraction rule students especially need to learn to consider both the magnitude of the parts of a fraction (e.g., tenths for 0.4) and the number of parts (e.g., 4 for 0.4) to appreciate the size of a decimal. A variety of concrete aids (even colouring in squares on a 10 x 10 grid) is useful here. Converting from fractions to decimals and vice versa, by calculator or otherwise, can be useful, provided students look at the answers rather than simply working them out! We hope to give further ideas for teaching in a future article when we have done further experimentation.

Conclusion

The overall finding of this study, that a very significant proportion of Year 10 students can not reliably decide which of a pair of decimals is the larger, is disturbing. Although this study was conducted in only two schools, we know of no reason why these schools would have been different to most Australian high schools. In our metric, calculator world an understanding of decimal notation is a high priority for all students. Teachers who use the test are invited to send student responses to Kaye Stacey at the University of Melbourne, along with a brief description of the recent teaching about decimals that students have experienced. The results of retesting over time are especially interesting to us.

References

Moloney, K. (1994). The evolution of concepts of decimals in primary and secondary students. Unpublished M. Ed. thesis: University of Melbourne.

Nesher, P. & Peled, I. (1986). Shifts in reasoning. Educational Studies in Mathematics, 17, 67-79.

Resnick, L., Nesher, P., Leonard, F., Magone, M., Omanson S. & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1). 8-27.

Sackur-Grlsvard, C. & Leonard, F. (1985). Intermediate cognitive organizations in the process of learning a mathematical concept: The order of positive decimal numbers. Cognition and Instruction, 2(2), 157-174.

Figure 1. Student's ideas of the order of decimals
between 1 and 2.

2              2           2               2
[up arrow]     .           1.9             1.1
               .           1.8             1.2
               .                           1.3
               1.1002      . tenths        1.4
               1.1001      .               1.5
               1.1         1.1             .
               1.999       1.99            .
               .           1.98            .
               .           .               1.9
               1.101       .               1.1
               1.1         .               1.11
               1.99        1.91            1.12
               .           1.9             .
               .           1.89            .
               .           . hundreths     .
[down arrow]   1.21        .               .
               1.2         1.02            1.2
               1.19        1.01            1.21
               .           1.999           .
               .           .               .
               .           . thousandths   .
               1.12        .               1.99
               1.11        1.002           1.1
               1.1         1.001           1.101
               1.9         1.9999          .
               1.8         1.9998          .
               1.7         .               .
               .           .               1;.999
               .           1.0001          1.1
               .           1.99999         1.10001
               1.3         .               .
               1.2         .               .
               1           1               1

               Order of    Order of decimals for
               decimals    a fraction rule student
               for a       (two possibilities)
               whole
               number
               student

Figure 5. Test of understanding of decimal notation.

Name: Class:         Date:

On each line below there is a pair of decimal
numbers. Put a ring around the larger one of
the pair. This sample is done for you: 6.8 (6.9)

(i)      4.8         4.63
(ii)     0.4         0.36
(iii)    0.100       0.35
(iv)     0.75        0.8
(v)      0.37        0.216
(vi)     4.08        4.7
(vii)    2.621       2.0687986
(viii)   3.72        3.037
(ix)     0.038       0.2
(x)      8.0525738   8.514
(xi)     4.4502      4.45
(xii)    0.457       0.4
(xiii)   17.353      17.35
(xiv)    8.24563     8.245
(xv)     5.62        5.736

Figure 6. Responses given by students in the
major misconception categories.

Number pair           Whole number

4.8        4.63       4.63       x
0.4        0.36       0.36       x
0.100      0.35       0.100      x
0.75       0.8        0.75       x
0.37       0.216      0.216      x
4.08       4.7        4.08       x
2.621      2.068798   2.068798   x
3.72       3.073      3.073      x
0.038      0.2        0.038      x
8.052573   8.514      8.052573   x
4.4502     4.45       4.4502     [check]
0.457      0.4        0.457      [check]
17.353     17.35      17.353     [check]
8.2453     8.245      8.2453     [check]
5.62       5.736      5.736      [check]

Number pairFraction rule     Expert rule

4.8        4.8     [check]   4.8      [check]
0.4        0.4     [check]   0.4      [check]
0.100      0.35    [check]   0.35     [check]
0.75       0.8     [check]   0.8      [check]
0.37       0.37    [check]   0.37     [check]
4.08       4.7     [check]   4.7      [check]
2.621      2.621   [check]   2.621    [check]
3.72       3.27    [check]   3.27     [check]
0.038      0.2     [check]   0.2      [check]
8.052573   8.514   [check]   8.514    [check]
4.4502     4.45    x         4.4502   [check]
0.457      0.4     x         0.457    [check]
17.353     17.35   x         17.353   [check]
8.2453     8.245   x         8.2453   [check]
5.62       5.62    x         5.736    [check]