**J.F. writes:** "I am sorry to keep asking you questions over the years but...
Reflecting over the summer about my geometry class and what I can do better. This year I would like to concentrate on student’s visualization skills. We do it in class some and talk about it often but it is not well organized let alone scaffold for students to find success.

I was wondering if there are any resources that you know of that walk through a set of lessons or progressive instructions that help students become better visualizers?"

**My response to J.F.:** "First, I know of no such book/course/materials that focus on the idea of visualization within a math context....snippets yes, but no full set of focused materials. For example, I know of materials on visual illusions, constructing 3-D images, rotating/reflecting effects, etc.
Second, beware when you try to find texts, as the visualization texts on the market are usually tied to some meditation philosophy...and perhaps do not fit your needs (i.e. a mathematics context).

Third, to help form a better response (i.e. suggest materials), can you answer these questions...

- What specific weaknesses do you find in in students...i.e. is it that they cannot see important aspects within a geometric diagram...can not see 3-D image from a 2-D representation....cannot build a visualization from a written context....etc. The more specific you can be the better on my end.
- What specifically would you like students to be able to do (i.e. visualize) after completing these materials?
- Finally, you use the phrase "help students become better visualizers"...which implies that students have some visualization skills, but have weaknesses that can be overcome....any specifics?

I look forward to responding to your specifics..."

**And J.F. responded:** "I think I can give my best reponse by giving specific examples of where I find my class is divided with not many kids in the middle.
Construction of an equilateral triangle given a line segment. Students will have done some simple constructions previous (although this is not hard) so there is some basic knowledge, they are working in groups so helping each other out, they are asked to visualize an equilateral triangle and sketch it in a small box before they attempt the construction. Students crush this exercise and the rest of the year they are able to construct an equilateral triangle, no problem. Same type of set up with some scaffolding, I give students two line segments meeting at 90 degrees, and I ask them to construct a square, the wheels fall off. They can sketch a square and mark it with appropriate marks, but there seems to be no "minds eye" of what needs to be done to finish the construction. We will go through the construction and some will get on board but coming back to ones like it in the future or even this one and a high percentage of the students will struggle.

Another example, 'class picture an equilateral triangle,' (which they seem to be able to do 80-90%?) "what are the measurement of the angles? What will happen to the triangle if one of the sides suddenly gets a little bit longer? What type of triangle will it become? What will be the relationship of the angles?" I lose a good portion of my class. If I ask them to sketch it, I will pick some up but I want them to be able to "picture" it?

We work with blocks, we do quite a few constructions, we do some origami, we use 3-D shapes and prisms, and use *Rhino* as well but I don't think I am connecting the dots to help all students visualize better. I think some of it comes with the territory of studying geometry naturally but I don't have a, spine or ladder (for lack of a better term) running through my class where I really walk students through specific exercises that I "know" are helping weaker "visualizers" become better. If that makes sense. It would be nice to be able to assess as well."

So, **HELP!**...do you know of resources that fit these needs? Send their names...and I will share them with both J.F. and on this web site as well....Thanks in advance.