In order to create "Looking Back" moments in undergraduate courses, one needs some familiarity with the typical content of grade 7-12 mathematics and the student experience there. Here are some informal thoughts on how to (re-)connect with that part of the mathematics journey.

1. Personal recollection

Topics you recall studying from your personal school experiences can serve as bridges to current 7-12 mathematics and provide inspiration for meaningful “looking back” experiences for your undergraduates. (Of course, one has to be aware that curriculum topics have changed over the recent decades.) For example, although synthetic division is no longer a standard algorithm taught in schools, it could behoove undergraduates to see the method, figure out why the method works, understand the computational advantages of expressing polynomials as nested products, explore generalizations to division beyond just linear terms, and so on. This work connects to the polynomial algebra that is taught today in high schools.

2. Witnessing your children’s education

Observing your children’s mathematics puts you in direct contact with content being covered today.

3. Reading the Common Core State Standards and the MET II Report

The majority of the States follow the Common Core State Standards. See www.corestandards.org/Math/ . To get a sense of how particular standards are being implemented, look at the “tasks” offered at www.illustrativemathematics.org/.

The MET II document (chapters 3 and 6) http://www.cbmsweb.org/archive/MET2/met2.pdf has ideas on this issue too.

4. Your Undergraduate Students

Your students have very recently been working through high-school mathematics. Ask them as you go along the semester if they see any connections to ideas, examples, or work from their high-school days.

As you read and talk about middle- and high-school mathematics content, or observe it through your children and students, some features will strike you as unfamiliar. (After all, you are encountering these pieces out of context.) As you mull on these pieces and try to establish your own meaningful context for them you will, in effect, be creating a possible “looking back” experience for your undergraduate students too.

Personal Example: I was perplexed when I first read standard G.C.1 Prove all circles are similar. How are students meant to prove this? [Answer: This standard is in the context of geometric transformations, a focus of the Common Core. Via a translation we can assume the two circles are concentric and a dilation suffices to map one onto the other.] I then wondered if young students could then explain why the value of pi is the same for all circles—after all, the circumference and diameter of a circle scale in proportion under dilations. [Undergraduate Problem: How do we know dilations scale lengths of curved segments appropriately?] Further mulling led me to then wonder about the hidden assumptions of this argument. For example, for two circles on a sphere a “translation” followed by a dilation will map one onto the other, but the value of pi is not the same for all spherical circles. Resolving this paradox is fodder for a good undergraduate project.

Try looking into the history of mathematics: the development of a topic or, more novel, the development of the notation we use for it. Learn about the vinculum, the obelus, the radix, the formulation of matrix notation (why row times column?), vector notation, and so on. Such forays for you will no doubt spawn project ideas to give to your students.

Once you have it on your mind to look for middle- and high-school connections, ideas will naturally come to you. Write down the ideas as they do. They tend to slip away.

TURNING BEGINNING IDEAS INTO “LOOKING BACK” EXPERIENCES

As mathematics educators we are experienced in creating and writing good problems. The task here is no different. The key is to ask:

After working through this problem, project, or experience, will my undergraduate students be able to explain to the world this particular piece of mathematics in a simple, effective, and mathematically correct manner?

The best way to start creating examples is to just start creating them! (And when you have an example, test it out.) Feel free to

• Copy ideas you’ve already seen and tweak them to suit your style and your course.
• Create small new examples at first. (Don’t be overly ambitious. Baby steps.)
• Ask your students to create examples.
• Don’t fret if your course content doesn’t seem to mesh with the idea of “looking back.” Just don't worry about it. (Certainly don’t do anything forced).

Remember, we are not looking for problems and projects for high-school students to do. These are problems and projects for mathematically sophisticated undergraduates with the goal of helping them notice middle- and high-school connections, to experience innovative learning, and to begin to develop the skills to talk about the subtleties of grade 7-12 mathematics content with clarity and correctness.