Dynamic Representations in Mathematics Learning Part 2: It’s About Time

Guest blog by Dr. Jeremy Roschelle, Digital Promise, @roschelle63

Last week, we introduced Dynamic Representations, which are time-based visualizations of mathematical concepts (see Part 1 here). To understand mathematics, student need to connect ideas. Dynamic Representations can help students to connect familiar with formal mathematical representations and can help students to connect visual and linguistic representations. We discussed why dynamic representations are beneficial and how to choose good ones. This week we will discuss how to use dynamic representations effectively, which requires integrating these tools with other elements of an effective approach to teaching and learning mathematics.

Integrating dynamic representations to support learning

No technology alone changes learning. With dynamic representations, students will not “get it” just because you put a dynamic representation in their hands or show them a movie of mathematical objects changing in time.

First off, students need good instruction as to what they are doing and why. Just giving students a dynamic representation and letting them play with it in the hope that they’ll discover what it is that they are supposed to discover won’t work. They need support and guidance in what they are going to do and are doing. This includes instruction prior to the experience and just-in-time information presentation during the experience. See for example the implementation of the 4-Component Instructional Design model in the Ten Steps to Complex Learning (Van Merriënboer & Kirschner, 2018).

Students also need an activity to do in which they will use the dynamic representation to accomplish a goal. The goal should be easy to understand, so they are not lost. For example, one well-specified goal would be “Change the slope three times, each time making the football player run faster than before. Then without touching the computer, explain to your neighbor how you could make three more.” This goal is simple and clear, and also implements questioning/prompts and peer teaching into the learning equation.

The activity should be central to conceptual foundations of a larger span of instruction — for example, several weeks of instruction. The above activity makes sense at the beginning of a curricular unit that will focus on the meaning of slope. Conversely, using dynamic representations as a fix for a student’s small slip or minor error can be a waste of time — it can take much more time to make sense of the new representation than it would take to remedy the error by more direct means like explaining it to them. A dynamic representation should be something that can be used as an ongoing reference point for a larger arc of proper mathematics instruction.

A related principle is to be judicious in picking dynamic representations because they each take time to learn. There are hundreds of permutations of ways to represent a fraction using technology. But it’s NOT a good idea to introduce them all or to let students choose among them all. A dynamic number line is a judicious choice because number lines are powerful representations that carry forward to future topics (like the axes of a graph). Pizza pies are less judicious — they might be used for the topic of fractions, but then nothing else afterwards.

A further related principle is that students will struggle if you ask them to too many connections all at once. Unfortunately, the term “multiple representations” is sometimes used as a synonym for dynamic representations and this can be misleading. Although the goal is for students to develop many connections among related mathematical ideas (as stated earlier, mathematics is highly interconnected), they should work with a small number of representations at a time (Ainsworth, 2006). An appropriate instructional design will introduce connections over time and gradually help students form a broader web of connections from activities in which they focused on just a few connections. This is what Van Merriënboer and Kirschner (2018) refer to as going from simple to complex, which is also a characteristic of Charlie Reigeluth’s elaboration theory (1999). During the larger arc of instruction, additional connections are gradually introduced, refined, and mastered.

Pedagogies that support learning with dynamic representations

Pedagogies that make the most of dynamic representations build on the points above and have a few other important elements.

First, pedagogies focus on concepts in addition to procedures. For example, in the simple activity of making slopes that correspond to faster running, the important conceptual idea is that steeper slopes indicate faster rates. The importance is NOT just to be able to correctly calculate each slope with a procedure. This plays into assessment. If a teacher’s assessments only give credit for fast, accurate procedures, then students will soon learn that developing concepts is not valued. Hence, it’s important to have formative assessments that focus on the concepts, like “draw a line that indicates a faster rate than the line shown in the graph, and complete this sentence about it: The line I drew indicates a faster rate, because in the same change in time on the x-axis, the __________ is more/less”

Second, they often move from more intuitive student engagement towards progressive formalization. Often, after a short teacher introduction or (“launch”) that orients students to the mathematical concept and task, students try an activity where they can explore how the dynamic representation works and try to make sense of it through their own active control of the representation. They need some hands-on time to engage their sense-making capabilities and to start forming some connections. Over time, teachers recognize what students have made sense of and helps them to formalize it in mathematically appropriate ways (Kapur, 2008). This might take the form of a teacher-led discussion or an activity in which students explain to each other using a specific mathematical idea. Often activities move off the screen to use traditional mathematical representations like a table on paper — but with connection to the on-screen experience of a mathematical phenomenon as an anchor for meaning. Notice that the trajectory is from engaging student reasoning towards formalization. The opposite order is more traditional (teaching a formal idea and then being given an opportunity to “apply it” to make it more “relevant”) but the traditional order is not typical of research on effective use of dynamic representations.

Progressive formalization almost always involves a teacher leading a classroom discussion — a discussion where the teacher focuses on developing coherent connections that incorporate and respect the formal mathematics (Empson et al, 2013). The expert connections in mathematics are well-organized and compact, and very few students are going to reduce a web of messy connections to a well-organized and compact set of fundamental connections on their own! Hence, the right pedagogy usually isn’t “discovery learning” — rather it is active learning with an emphasis on growing the coherence among student and expert reasoning. There is a “time for telling,” (Lobato, Clarke, & Ellis, 2005; Schwartz & Bransford, 1998) often soon after students have gotten an intuitive sense of a phenomena and realized how mathematics might help.

These discussions honor student ideas — so student ideas are NOT elicited only to shut them down (“That’s pretty good, but now I’ll tell you what an expert does.”) Instead, the focus is on the connections (“I like how you said it. Let’s see if we can connect that idea to the textbook equation we discussed.” Or “That’s pretty good, but when might it not be the case?”[1]) Students feel it makes sense when their ideas and ways of reasoning are reflected in a more expert structure (Kaput, 2008). Further, once initial coherence is built, an expert teacher will connect back to the same prior learning experience as new topics are introduced — the best way to achieve further mathematical understanding is to build on previous mathematical sense-making.

Overall, pedagogies often focus on mathematical discussions, explanation and argumentation (Knudsen et al, 2017) — talking about mathematical reasoning complements the hands-on experience of controlling a representation that changes in time. Drawing out “why” is important to a sense of understanding, as is focusing on justifying mathematical ideas. Articulating comparisons between right and wrong (or between better and lesser) strategies is important. Drawing out examples and counterexamples is important. Building on each other ideas to come up with ways of saying it that are clearer, stronger, and more general is important.

Conclusion

Representations that change in time can be a powerful tool to advance students’ conceptual understanding of mathematics, by focusing cognitive activity on coherent connections among mathematical ideas. Realizing this potential requires orchestrating a sequence of student activities, classroom discussions, assessments, and other elements of a strong instructional design around the mathematical connections made visible over time within the dynamic representation.

References

Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16, 183-198.

Empson, S. B., Greenstein, S., Maldonado, L., & Roschelle, J. (2013). Scaling up innovative mathematics in the middle grades: Case studies of “good enough” enactments. In S. Hegedus & J. Roschelle (Eds.), The SimCalc vision and contributions (pp. 251-269). Springer, Dordrecht.

Knudsen, J., Stevens, H. S., Lara-Meloy, T., Kim, H. J., Schechtman, N., & Shechtman, N. (2017). Mathematical argumentation in middle school-The what, why, and how: a step-by-step guide with activities, games, and lesson planning tools. Thousand Oaks, CA: Corwin Press.

Lobato, J., Clarke, D., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 101-136.

Ohlsson, S. (1995) Learning to do and learning to understand: a lesson and a challenge for cognitive modelling. In P. Reimann and H. Spada (Eds.), Learning in humans and machines: Towards an interdisciplinary learning science (pp. 37–62). London, UK: Pergamon.

Reigeluth, C. M. (1979). In search of a better way to organize instruction: The elaboration theory. Journal of Instructional Development, 2 (3), 8-15.

Reigeluth, C. M. (1999). The elaboration theory: Guidance for scope and sequences decisions. In R. M. Reigeluth, (Ed.), Instructional-design theories and models: A new paradigm of instructional theory, Volume II (pp. 425-454). Mahwah, NJ: Lawrence Erlbaum Associates.

Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16, 475-522.

Van Merriënboer, J. J. G., & Kirschner, P. A. (2018). Ten steps to complex learning (Third edition). New York, NY: Routledge.

[1] Stellan Ohlsson (1995) call this making use of epistemic activities or tasks.

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