1. Honors Signature Assignment: Math/Ed 3250
Engaging students in Authentic Research
October 23, 2018

2. What are some difficulties prospective teachers experience in geometric thinking?
Activities:
Read & discussed literature review on geometric thinking of prospective elementary teachers
Item analysis from a pre-assessment to develop hypotheses
2 Cognitive Interviews with peers
Create a video to address the misconception/struggle
Meetings and memos after each step
Context: a mathematics course for prospective teachers that includes a focus on pedagogy and geometric thinking and data analysis.
We collectively did the item analysis
We came up with hypotheses about PT thinking based on literature review and item analysis. They wrote them up in a memo and we discussed them, then talked about activities they could do in the cognitive interviews to test these hypotheses. This was a collective effort. It included my input a good deal; I know how the order of questions, for instance, can influence student thinking. I also know the particular students that they were going to be interviewing. We had a discussion about their findings, using their notes. From there, we identified what would be the best use of their video. We meet almost weekly; whenever there is a product to discuss (notes on the literature review, notes from the cognitive interviews, etc.)

3. Item Analysis
Looking at the most common pattern of answers, what does it seem that our class is typically thinking?
This is what I mean by item analysis. We had a pre-assessment from our class. Item analysis is looking at the pattern of answers, including the wrong answers, to figure out what students are thinking. For example, students who are choosing f might just be confusing area with periemeter. However, students who are choosing e have a different problem. Students who are choosing e and f may not understand equivalent expressions.

4. Themes
When thinking about perimeter, prospective teachers tend to focus on squares and not on units of length
Some prospective teachers are at the analysis van Hiele level; not abstraction for shape classification
Some prospective teachers have accurate abstraction ideas, but do not represent them visually or in spoken language precisely.
Concept images over ride definitions
Knowledge is not necessarily integrated
We identified some themes. First, we found that some PTs are attending to the squares and not to length when finding perimeter. (Right triangle example – the base is “4 ½” the hypotenuse is ….). Now, this is particularly important because one of the students interviewed actually presented on the triangle inequality theorem. So, she should know that the hypotenuse cannot possibly be the length of the sum of the sides – that would just be 2 straight lines. She isn’t integrating her knowledge. We found a similar phenomenon with regard to solids. Everyone in the class got the question about pyramids and prisms being a subset of each other (TF) wrong. The interviewed students also claimed that cylinders and cones were prisms and pyramids, respectively, though they did say that the prisms and pyramids did not overlap. They were then given the definition but did not revise their to exclude cones and cylinders. This, despite the fact that they both knew that a polygon must have straight sides. So, their concept images are overriding the use of definitions. Relatedly, in regards to shape categorization, some students are still at the analysis level – they attend to properties – or at visualization (matching to a concept image or prototype). They do not use the definition and they do not think in terms of subsets, even after instruction. For example, the trapezoid was not identified as a trapezoid because it was on its side. “It just doesn’t look like one.” These are all things that elementary students do, too.

5. Challenges & Benefits Benefits and Good Challenges
Linking with the course: depth of fluency with course concepts
Richer understanding of misconceptions
Engagement in the real work of teachers
Affective benefits of small learning community
Challenges to overcome
Dead-ends and their affective consequences
Challenges to overcome: one of my students, in her cognitive interviews, found that it was probably the structure of the question that led to so many wrong answers, and the expectation that in math, everything is a subset of something else. Her interviewees clearly indicated that pyramids and prisms do not overlap, contrary to what they answered on the pre-test. She’s a little bummed – how do you create an intervention for that? What should she base her video on, her action? For me, I think that it’s about developing metacognition, and she could create a nice video on this topic – checking your thinking, etc. – but it’s kind of abstract and not as clear cut, and she’s feeling a little stuck.

6. Lessons learned Faculty Leadership Make Time
Hypothesis: by having students start in the middle of an inquiry cycle (data collection, rather than problem formulation or planning for data collection) they can be more productive and organized
Make Time
Weekly meetings with clear discussion topics and outcomes that are led by student work (notes on reading, notes from interviews)
Separate but related projects spur discussion and creative ideas