Using multiple methods and representations promote conceptual understanding through comparison and contrast so students can build conceptual bridges between what may seem like two unrelated mathematical approaches. (Note: “multiple methods” does not mean the use of multiple instructional delivery methods together.) When we adjust our instruction in-the-moment to compare and contrast, we can clarify misconceptions by discussing how methods and representations attempt to express similar ideas. While the following methods are structured, they also lead students to reevaluate and reject fixed ways of approaching the task at hand. Overtime and with continual use, multiple methods and representations foster the belief that mathematics is not “I do math when my teacher tells me to” yet a true curiosity and study of patterns across the many ways they manifest. That mindset is critical in addressing misconceptions not just in class yet surmounting barriers we all encounter in our lives.
Set two different examples illustrating different solutions to the same task, one containing a misconception, and compare and contrast the methods two students used. Research supports that comparing two side-by-side solutions is more effective at improving students' problem-solving ability, procedural flexibility and conceptual understanding than showing each of these methods separately in a sequence (Star, 2013). Contrasting Cases has a firm structure: the first five steps must be implemented with fidelity while the sixth step, Take a Stand, is versatile. This way, students are led to gather a variety of information about what they’ll progressively understand as a misconception before using that information to construct viable arguments. While research for this method was conducted in Algebra courses, the method can be applied to practically any task where comparison is possible.
| Process | Sample Script |
| Heighten Awareness: Cue students to the instructional adjustment. |
“Let’s explore why we’re arriving at two different solutions by comparing and contrasting two different methods.” [Sets two worked examples side-by-side on the document camera: one with the misconception on the left side and the other with correct mathematical reasoning on the right side.] |
| Focus on Method 1: Gather information on this method alone. |
“First, focus only on Method 1 on the left. What do you notice about this method?” [Leads students to objective statements only about Method 1 like “I notice that there are three stages to the process they used” rather than judge the work.] |
| Focus on Method 2: Gather information on this next method alone. |
“Now, focus only on Method 2 on the right. What do you notice about this method?” [Again, leads students to objective statements only about Method 2 rather than judge the work or make comparative statements to Method 1.] |
| Find Similarities: Gather information solely on likeness. |
“How are the two methods similar?” [Leads students to discuss only the similarities in what they notice. If students note anything other than a similarity, they are guided back to finding similarities.] |
| Find Differences: Gather information solely on differences. |
“So, how are the two methods different?” [Leads students to discuss the differences. If students begin to share why there are differences, the teacher guides them back to just summarizing the ways they are different rather than the causes.] |
| Take a Stand: Construct a viable argument for preferred methods. |
“Which method do you prefer to use and why?” [Encourages students to use what they noticed, similarities and differences to attend to the precision of their preferences. Evaluates arguments for either method, both or neither. Focuses dialogue on the roots of the misconception be they from a use of a tool, structure of a process, repeated reasoning, use of a model, etc.] |
Source: Star, J. R., Rittle-Johnson, B., Lynch, K., Durkin, K., Gogolen, C., & Newton, K. J. (2013). The impact of a comparison curriculum in Algebra I: A randomized experiment. In Society for Research on Educational Effectiveness Spring 2013 Conference . Washington, D.C.
An analogy compares two things that are similar as well as different from one another, allowing rich connections between mathematical concepts and lived experiences. Consider a common mathematical misconception: a prime number is any number divisible by 1 and itself. (A prime number is any number with exactly two factors: 1 and itself. 1 has only one factor; it is neither prime nor composite.) To clarify this misconception, we can use the Teaching-With-Analogies Model (Glynn 2007, 53). Here’s a sample script:
| Process | Sample Script |
| Re-introduce the target concept, prime and composite numbers, to students |
“What types of numbers do we know about when factoring?”
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| Remind them of what they know about the analog concept (i.e. the thing that’s comparable), friendships | “Prime and composite numbers are like friendships.” |
| Identify relevant features of prime and composite numbers and friendships |
What do we know about both?
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| Connect (map) the similar features of prime and composite numbers and friendships |
What is similar about prime and composite numbers, and friendships?
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| Indicate the breakdown in the analogy between prime and composite numbers and friendships |
What doesn’t really fit when we compare prime and composite numbers and friendships?
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| Draw conclusions about prime and composite numbers |
I made an analogy between prime and composite numbers and friendships to help us understand what is unique about the number 1. What can we say about the number 1?
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Source: Glynn, S. (2007). The teaching-with-analogies model. Science and Children, 44(8), 53.