This study explored the effects of an instructional intervention using technology on students’ understanding of fractions on the number line.

Fractions are foundational for more demanding concepts and applications later (Common Core State Standards, 2010).  Previous work claims the difficulty exists because of  different interpretations and uses of fractions, including ratio, quotient, measurement, and part-whole (Bezuk & Bieck, 1993). Others claim the challenge of understanding fractions is the result of interference from prior knowledge of whole numbers (Saxe, Shannon, & Chinn, 2007; Pearn & Stevens, 2007).

Based on the work of Seigler, et.al. (2011), we approached fraction instruction with an emphasis on numbers as magnitudes using the number line as the sole representation.  An Android application featuring lengths and estimation was created to use with touchscreen tablets. The number line is a continuous and extendable representation, as opposed to pie charts and area models, which has been found to be helpful and supportive in learning fractions (Bright, Behr, Post, & Wachsmuth, 1988, Pearn & Stephens, 2007; Saxe etc, 2007). Further, the Common Core State Standards call for students in Grade 3 to “understand” and “represent fraction as a number on number line” (Common Core State Standards, 2010).

Although the number line seems a promising way to unify whole number and fractions instruction, provide an extendable representation, and integrate visual and symbolic notation, students still have difficulties.  Students need to learn to pay attention to the unit, attend to intervals created by equipartitions, and match the symbolic notation to the representation.  (Bright etc, 1988; Pearn & Stephens, 2007; Saxe etc,2007; Drake, 2007; Mitchell & Horne, 2008).  This study, as well as subsequent associated studies, aims to shed light on both the challenges and affordances associated with teaching and learning using the number line as the preferred representation for fractions. This is critical work if the Common Core State Standards will soon drive curriculum, instruction, and assessment.

The two research questions we explored in this study are:

1. What is the effect of a brief instructional intervention on students’ understanding of fractions?
2. What effects does technology have on student learning fractions using the number line, particularly as it relates to their paper-pencil work with number lines?

Methods

We tested 81 sixth graders (55.6% female) enrolled in a regular mathematics class in a middle school in a small school district located within a large urban area. Using a pre-post-control design students were randomly assigned to treatment (N = 41) or control (N=40) by receiving a tablet with either the application we created or a math game application downloaded from the Android Market. Students used the app for 20 minutes per day for three non-consecutive days over two weeks. The assessment consisted of items grouped into the following subscales: naming and locating fractions on the number line, using other representations for fraction quantities, lengths, comparing and operating with fractions, and a set of items assessing student understanding of the density of rational numbers.  Both quantitative and qualitative analyses were conducted on the assessments, including coding of correct and incorrect, as well as coding strategies for several items.  As validation for the qualitative analysis, we used interviews with students solving the assessment items both before and after interaction with the number line app.

Results

We found no overall difference in gains between the control and treatment group.  However, on the subscale of “naming” we found that students in the treatment group improved significantly as compared to the control group (F (1, 79) = 2.417, p <.10)   We interpret these results with caution, as this could be primarily due to improvement in understanding the semiotics of the representation, and not necessarily deeper understanding of fractions concepts.

In both the treatment and control group a significant association between adding extra marks to the number line and answering correctly was found for two out of a total of five naming problems (X²(1)=18.300, p<0.05 and X²(1)=18.557, p<0.05) and six out of a total of ten locating problems (all p<0.05). Both naming problems are non-routine problems with unequal intervals, meaning that students noticed the unequal partitions and used marks to create equal intervals. Making extra marks on the locating items indicates that students were doing more than merely estimating the location of a given fraction. Although several types of extra marks were coded, including labeling other numbers on the number line, the most frequent was “tick” marks similar to the marks included in the instructional app.

In a qualitative analysis of interviews, we found that students’ understanding of the number line is quite limited, and that their prior work with fractions seems to interfere or override their work on the number line.  A striking example is a student who perfectly described his method for locating 3/4 on the interval between 0 and 1 by equipartitioning the interval using both precise markings and language.  However, when he decided where to place 3/4, he placed it very close to 1 rather than on the mark he created which would have been correct.

We also found that students do not typically associate the segments they create with their marks with lengths or magnitudes.  This leads us to the conclusion that learning the mechanics and understanding the semiotics of the number line is necessary but not sufficient for understanding fractions as magnitudes. 

The implications for instruction are important, as careful attention to instructional strategies for using the number line need to be complemented with assessments for understanding. We believe interactive technology provides a unique opportunity to explore this integration.  Further, the call for the almost exclusive use of number lines when working with fractions provides an opportunity for students to integrate fractions into their idea of “number”.  This direction for research addresses the increased use of technology for learning and assessment.  In addition, it recasts the previously described challenges of learning fractions by unifying fractions and whole numbers as representing magnitudes.

References

Bright, G. W., Behr, M. J., & Post, T. R. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education. 19(3), 215-232.

Bezuk, N. & Bieck, M. (1993). Current research on rational numbers and common fractions: summary and implications for teachers. In Owens, D.T.(Eds.), Research Ideas for the Classroom (Middle Grades Mathematics). NCTM.

Drake, M. (2007). Informal knowledge and prior learning: student strategies for identifying and locating numbers on scales. In Watson, J. & Beswick, K. (Eds). Proceedings of the 30^th Annual Conference of the Mathematics Education Research Group of Australasia. 255-264.

Mitchell, A., & Horne, M. (2008). Fraction number line tasks and additivity concept of length measurement. In Goos, M., Brown, R., & Makar, K. (Eds). Proceedings of the 31^th Annual Conference of the Mathematics Education Research Group of Australasia. 353-360.

Mathematics standards (2010). Common Core State Standards.

Pearn, C., & Stephens, M. (2007). Whole number knowledge and number lines help to

develop fraction concepts. In J. Watson & K. Beswick (Eds). Proceedings of the 30^th annual conference of the Mathematics Education Research Group of Australasia. 601-610.

Saxe, G.B., Shannon, A., & Chinn, R. (2007). Learning about fractions as points on a number line. In Martin, W.G., Strutchens, M.E., & Elliott, P.C. (Eds), The Learning of Mathematics, Sixty-ninth Yearbook. NCTM.

Siegler, R.S. et.al. An integrated theory of whol number and fractions development.  Cognitive Psychology (2011), doi: 10.1016/j.cogpsych.2011.03.001.