About time: Syntactically-guided reasoning with analog and digital clocks

ABSTRACT

Around the globe, young students are expected to learn about time, yet how is it that they themselves make sense of this topic? From a sociocultural perspective, sense-making about time emerges in relation to properties of available tools and representations, such as analog clocks or digital notation. Such interactions with the symbols and structural properties of clocks are examples of syntactically-guided reasoning, a key domain of early algebra. In this paper, I focus on how students’ syntactically-guided reasoning emerged when reasoning about non-routine time problems with different clocks, whether or not such reasoning was consistent with accepted conventions of time measure. I present three case studies of typical Grade 2 students describing time on a particular clock (analog or digital) as they solved tasks related to time identification and elapsed time. I describe how symbolic properties of clocks enabled pathways of thinking about time-related ideas in relation to that particular clock. Implications for the treatment of time in elementary mathematics are discussed.

KEYWORDS:

• Clock-reading
• student thinking
• elementary math
• representations
• time
• elapsed time
• early algebra
• syntactically-guided reasoning

Acknowledgments

I would like to thank Stephanie B. Purington, Amy L. Smith, and three anonymous reviewers for feedback on earlier versions of this manuscript. Special thanks to Michelle Eastman, Alicia C. Gonzales, and Anna M. Plant, who assisted in coding and analysis, and to the students and teachers who generously shared their time with me.

Notes

1. This convention was enabled through the development of mechanical clocks. Some cultures across history would describe the event “day” differently, for example, in relation to available sunlight.

2. I report here information related to Grade 2 only. This data set comes from a larger study that included students (N = 612) from across elementary grades (see Earnest, 2017).

3. Source: http://profiles.doe.mass.edu/search/search.aspx. Demographics reflect information reported to the Massachusetts Department of Education. Because research questions did not concern issues related to racial or gender identity, these data were not collected from individual participants in this sample.

⁴. Interviews also included 7 Hand Positioning tasks as described in Earnest & Chandler (in press) that are not a focus of the present analysis.

Interviews also included 7 Hand Positioning tasks as described in Earnest & Chandler (in press) that are not a focus of the present analysis.

5. Because research questions do not pertain to sex or gender identity and I did not ask participants to report this identity, I use they/them pronouns to refer to individual participants. Note that, when referring to previously published reports in which gender pronouns were used, I continue to use those pronouns to clearly reference those reports.

Additional information

Notes on contributors

Darrell Earnest

Darrell Earnest is an associate professor of education in the College of Education at the University of Massachusetts, Amherst. He received his Ph.D. in Cognition and Development from the University of California, Berkeley, in 2012. His research and teaching focus on the interplay of mathematical representations with learning and teaching, as well as the role of time representations in supporting time management among undergraduate students. His work has been published in Cognition & Instruction, Journal for Research in Mathematics Education, Journal of Mathematics Teacher Education, and Mathematics Education Research Journal, among other venues.