But instead of quizzing them on content that they've mostly had at some point, I want to know more about how they experience the problems. (I do peek at their answers, of course. Data is good.) In particular, if the answer was available by recall, by one or two steps, required more thought than that, or if they do not know how to begin the problem, or cannot answer. I feel like that moves it away from feeling like a quiz, and the results have felt more honest since I started polling for this kind of information.

Here's the assessment and results. The questions are mostly mild modifications of released items and sample practice items for the big state assessment.

The lack of comfort with unit conversion and the metric system is quite typical, as is the challenge of recalling and applying formulas. All of these students have had our course for all elementary teachers, and are typically quite sound on most of the content. It's quite striking for me that even math majors have these issues. What chance does a typical student have? A challenged student?

At the end of the content, I ask them what questions this raised for them about teaching. Bold and italics are my categorization of their responses.

Questions on teaching K-8 Geometry

**Big Teaching Questions**:

*What should the teacher be doing?*How do we find ways of teaching that creative, engaging and instructive? Where do we find the best ideas for lesson plans? How can we use the standards to help plan a fun lesson?*What should students be learning?*What exercises lead to deeper understanding? How can we be sure our students understand vs performing procedures?- How much time do you spend on a topic? How do you teach everything in a year? What do I do when one or two students don’t understand – do I continue or stop for them? How do you teach so everyone is on the same level?
- How much work should be shown on each problem by students? (Different at different grade levels?) Should it be shown or mandatory?
- How to introduce a brand new topic?
- Are students allowed to use calculators? What tools can they use on assessments?

**Students**:

- How do teachers organize material to make it easier for students?
- What is their vocabulary? How do we take that into consideration? How do we teach the language?
- How will I teach to students with different learning styles? How do we explain well enough for all students to understand?
- What do students find most engaging?
- What variety of solutions do students come up with?

**Content Specific**:

- How can I apply this stuff to life?
- How does it [all this content?] all fit together?
- What geometry concept is the most difficult for students? What strategies do we teach to solve geometry problems? How do children do these when I used later knowledge?
- How much geometry is there in the younger grades?
- Is measurement hard for students?
- How to teach formulas without just memorizing? Are there other techniques besides formulas for volume, etc? How do you break formulas down? How do you help kids memorize them?
- How do you teach conversions so students can remember?
*Quite Specific*: How do you teach to estimate? How do you teach congruence? Do students use pi=3.14 or do they need to know more? How do you find areas of arced shapes?

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