Learning to Make Conceptual Connections: Analogical Reasoning in Classroom Mathematics
The Science of Learning Lab conducts studies to explore the teaching and learning of mathematics through making comparisons. Our research seeks to identify specific teaching practices that can increase the efficacy of everyday teaching activities such as leading mathematical discussions by reducing the cognitive burden on children to make connections and broad generalizations.
Recent Studies
Many of our classroom studies use a unique video-based methodology that we recommend for maintaining experimental control while testing teaching strategies, while maintaining higher ecological validity than a laboratory setting (for more detail, see Begolli & Richland, 2016). We work with a teacher to develop a mathematics lesson that includes many opportunities for deep thinking and drawing connections through making comparisons. The teacher then teaches this lesson in his or her classroom, and we videotape the lesson from multiple angles simultaneously. Then through video editing, we create experimental versions of this lesson so that we have the same audio-stream, the same lesson plan and student talk, but we can use different cameras to create manipulations of features of the lesson. These videos are then used as the experimental materials that we test in other classrooms. This means students in our studies learn from interactive videos of complex math concepts, but they may see one of two or three versions, which allows us to randomly assign students within a classroom to different versions.
When faced with the COVID-19 pandemic we moved this study online. To view one of our platforms and video materials on proportional reasoning, click here.
For instance in the below videos from one study (Begolli and Richland, 2010), we created a lesson on rate and ratio in which students shared and compared their different strategies. We then varied whether experimental participants could see all three solutions written on the board (High Support Condition) whether they could only see the solution that was being discussed (Part Visible Condition) or whether they only listened to their classmates describe their solution (Not Visible Condition). Each are realistic teaching practices used regularly by teachers, but this videotaped methodology allowed us to hold the rest of the lesson constant to test whether seeing the solutions written led to better learning. Short clips of the three versions are shown below to illustrate this technique:
When faced with the COVID-19 pandemic we moved this study online. To view one of our platforms and video materials on proportional reasoning, click here.
For instance in the below videos from one study (Begolli and Richland, 2010), we created a lesson on rate and ratio in which students shared and compared their different strategies. We then varied whether experimental participants could see all three solutions written on the board (High Support Condition) whether they could only see the solution that was being discussed (Part Visible Condition) or whether they only listened to their classmates describe their solution (Not Visible Condition). Each are realistic teaching practices used regularly by teachers, but this videotaped methodology allowed us to hold the rest of the lesson constant to test whether seeing the solutions written led to better learning. Short clips of the three versions are shown below to illustrate this technique:
All Visible: High Support |
Part Visible: Some Support |
Not Visible: Low Support |
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All solution strategies are visible
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Only the most recent solution strategy is visible
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No solution strategies are visible
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Our studies have revealed that details in the ways that teachers use visual representations and highlight similarities during comparisons can have implications for student learning. We found in the above study that making all solution strategies visible was the most effective teaching practice. We also found that simply discussing the solutions verbally led to adequate learning, though not as strong. Surprisingly we found that in this study, the part visible led to the lowest learning, perhaps because the first solution discussed was a common misconception. Seeing it first may have increased the cognitive load on learners to listen and then inhibit this solution.
In other studies, we've found that using several techniques for reducing the required cognitive resources for noticing and learning from comparisons led to the highest higher order thinking following the lesson. These include making representations visible, sing spatial alignment to highlight their relations, using hand and/or body gestures to highlight how they are similar or different, and reminding learners of relevant core knowledge they had previously.
In other studies, we've found that using several techniques for reducing the required cognitive resources for noticing and learning from comparisons led to the highest higher order thinking following the lesson. These include making representations visible, sing spatial alignment to highlight their relations, using hand and/or body gestures to highlight how they are similar or different, and reminding learners of relevant core knowledge they had previously.
Classroom Implications
This work has important implications for the ways that teachers lead mathematical discussions. Cognitive science research has well established that making comparisons and learning from higher order relationships is effortful and often fails when not constructed optimally, and this lies at the heart of what many math-talk interactions are designed to support. Thus, we draw on principles of cognition such as relational reasoning, transfer, and cognitive load theory to develop techniques for supporting student thinking that are usable and realistic within classroom lessons.
Research Funded By:
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The National Science Foundation
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The Office of Naval Research
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