Dan Meyer’s blog, dy/dan shares the thoughts, insecurities and efforts of a terrific young urban educator via words and remarkable videos. His blog is worthy of your attention.

Dan should me commended for making his thinking public and discussing issues rarely explored in public. A recent blog started out by wondering if Washington D.C. Chancellor Michelle Rhee’s tactics and hostility towards teachers would be fruitful in the public schools of the nation’s capitol.

Along the way, Dan asked a serious question about how to improve his students’ geometry test scores, regardless of how each of us might feel about the value or use of standardized testing.

Here is my first, albeit incomplete, set of recommendations.

Dear Dan:

First of all, I wish to share my admiration for the sincerity and courage inherent in your question.

I got my 07-08 Geometry results back yesterday and they were not acceptable. Too many kids listing along at Basic levels, not enough kids rising to Proficiency.

My question to so many commenters here: what would you have me do with that data?

Asking this question is critically important. You can’t be good at anything, much less teaching, without being reflective.

First, let’s assume that the test actually attempts to assess “geometry.” Many standardized tests give kids a score for something like “algebraic reasoning” when the test only included one question on the topic. It would also be nice if you continued to work with the same students tested. Having test results after the kids move on to another teacher is hardly useful as a corrective instrument.

Since you can’t cure poverty or the other socioeconomic and cultural obstacles experienced by your students, solutions will need to be relegated to what you do in your classroom.

One mistake frequently made when confronting such issues as your geometry scores is to assume that blame lies with either a) the teacher or b) the student. There is a third player at work here – the curriculum. Why don’t we ever challenge the assumptions underlying the curriculum?

While I realize you have a limited ability to replace or abandon the curriculum, it is equally true that doing the same thing louder will not achieve a different result.

But both of your responses dodge the question. From the perspective of someone opposed to the accountability measures of NCLB and skeptical of standardized tests, what would you have me do with the knowledge that (e.g.) four out of ten students I taught last year couldn’t find the volume of a unique swimming pool?

Why should students be able to find the volume of a swimming pool? How often do you have to do that? I never calculate unique swimming pool volume.

How many of your students have access to a swimming pool or even swim? (Oh, I know. Tests are supposed to be culturally neutral.)

It’s worth asking yourself the question Seymour Papert used to challenge my own teaching and curriculum planning. "What can they *DO* with that?"

Such a question goes well beyond matters of relevance. Knowledge is constructed as a consequence of experience. What sorts of experiences do your students have?

I’m not a Utopian. I know that you have to "teach" the kids "math." However, you may need to ensure understanding before covering the curriculum. Perhaps you can change the order of the curriculum. Perhaps you can supplement the curriculum with more imaginative texts (including trade books written by experts). Perhaps you can use Logo with kids – still for my money the richest environment for developing geometric reasoning. Perhaps you can find a way for students to be less hostile to the curriculum being shoveled in their direction. In any event, you need to take the kids from where they are and help them move forward.

You may need to change everything, just to “catch-up!”

The research of Constance Kamii and others, plus your own common sense indicates that "practicing" more pool volume problems is unlikely to help students improve their scores, or more importantly understand volume. Check out Kamii’s books here. Her videos are available from Teachers College Press.

In that spirit, here are some resources and practical ideas you might consider:

- The concrete inquiry model presented in How Big is the Moon? should be adaptable to your circumstances to great effect.
- Civil Rights hero Bob Moses’ Algebra Project may have resources to help. His book, Radical Equations: Math Literacy and Civil Rights, may be a source of inspiration to you as well.
- I LOVE LOVE LOVE LOVE an out-of-print book, entitled Build-a-Book Geometry. Get yourself a copy and try the author’s approach to kids owning and inventing geometric knowledge.
- Visit the Constructivist Consortium’s Online Bookstore for additional book recommendations.
- Use Logo (not Scratch) and find ways to integrate turtle geometry and programming into your required curriculum.
- Use activities from Harold Jacobs’ brilliant textbooks Mathematics: A Human Endeavor or Geometry: Seeing, Doing, Understanding
- Use the strategies of Think, Write, Talk proposed by Marilyn Burns.
- Use Geometer’s Sketchpad and the books and manipulatives provided by Key Curriculum Press.
- Use MicroWorlds to have kids construct their own "Sketchpad" as described on my site.
- Borrow activities from more progressive texts like "Connected Math"
- Identify real projects requiring geometry outside of the classroom.
- Plaster your classroom walls with interesting posters of geometric art like these.
- Remember that less is more!

As Papert and Harel teach us, "It’s OK to worry about what to teach Monday, as long as what you do points to what you want to do someday." Don’t get distracted by the immediacy of the curriculum or tests. I hope this helps.

Veteran educator Gary Stager, Ph.D. is co-author of Invent To Learn — Making, Tinkering, and Engineering in the Classroom, publisher at Constructing Modern Knowledge Press, and the founder of the Constructing Modern Knowledge summer institute. He led professional development in the world’s first 1:1 laptop schools thirty years ago and designed one of the oldest online graduate school programs. Gary is also the curator of The Seymour Papert archives at DailyPapert.com. Learn more about Gary here.

Makes me wish I still taught math so I could use some of these great resources.

Taking a second to echo Gary’s recommendation- Key Curriculum has some wonderful discovery activities. I use the IMP texts, but the “Discovering” Series has nice activities too.

I’d also like to point out the resources here. I especially like the “Rethinking Proof” text. I use many of the activities in the Exploring series too. If you don’t have Sketchpad, I think the lessons could be adapted to use in GeoGebra (which is free multi-platform dynamic software)

Gary,

Just a quick question… why would Scratch be inferior to standard Logo in geometric understanding?

When I first started working with Scratch, I suspected it would let the kids go too quickly through the “basic” stages of forward, right, … and that they wouldn’t learn the basics before going on to make some sort of video game, etc.

What I found in my own experience, however, was that I still ran across the geometric problems, but that they were more contextualized within the program I was working with.

I’d certainly appreciate your perspective.

Regards, Kent

Kent,

While you certainly can do turtle graphics stuff in Scratch, it’s intent is otherwise.

The lack of procedures, the size of the drawing area and the limited use of number, naming and variable makes it less rich for formal mathematics.

Plus, there is a vast library of activities available already for Logo.

IMHO,

Gary

Gary,

Very good points! I had also forgotten about the lack of string variables / methods and comments.

Have a great day,

Kent

While I didn’t have to tackle the *specifc* problem of a unique swimming pool, I certainly have had to take on volume problems in real life.

In buying rock to fill a ditch in my front yard, I needed to give them a figure in cubic feet.

I also have needed the volume of a room for working out which ceiling fan would be best.

Jason,

Wow! That’s great. You should learn to calculate volume of an irregular container since you actually need to know it.

Could you have looked the formula up on your cellphone or a chart at Home Depot?

Could you have estimated how much rock you need for your yard?

Well, it had to be an estimate because the ditch (which was left by the previous owners) was irregular. But in the end I did need to calculate the formula, otherwise I would have bought an extra $100 of rocks I wouldn’t have needed.

Someone could look up a formula as long as they knew what to look for.

(Also related: On our standardized test students don’t memorize any formulas — there’s a formula sheet. I don’t know the case in California. That doesn’t mean they know how to apply them, though.)

I feel like in many ways this notion of teaching ideas/thought processes in ways that connect personally is a no brainer. I think one of the reasons that I did not ever connect with algebra, when I was in school, is because none of my teachers could tell me how I would use it in later life.

However, I also feel like this is not entirely plausible. As a fourth grade teacher there are multitudes of standards laid out by by my state that I must cover over the course of 180 days. The notion that I might be able to creatively connect my students to each concept seems like a potential recipe for failure.

Again I should refrain that I was the student who required the real world connection in order to devote my attention. I regularly seek to foster those same connections in my own space.

The difficulty lies with several resulting issues.

1. Creativity in the classroom requires a commitment to time that may not be plausible and generally requires a deviation away from most formalized assessments.

2. Melting pot classrooms frequently present such a broad spectrum of life experience that connecting all of your students may not be feasible.

3. Ultimately we come back to that darned test. Formalized, regimented, standardized assessment does not lend itself to creative approaches for engaging curriculum.

If we are going to ask that our general public consider us “highly qualified” then we ought to be treated as such. How is it that we have come to a place where the results of one cold hard assessment have become our sole measure of a years worth of work?

Math is so wonderful, yet so inaccessible to so many, it has made me wonder if we are doing a disservice to mathematics by having all learners encounter the same mathematics at the same pace at the same age.

What if students could approach math at their own pace and out of their own curiosities?

That question inspired me to write the following post on my blog at gregorylouie.edublogs.org

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There is an issue in educational reform that is often noted but inadequately discussed.Individualized instruction and/or tutoring is probably the single most effective means of raising performance. It is easy to understand why it isn’t discussed in reform circles as individualized tutoring is seen as far too expensive to be practical.

A classroom teacher simply doesn’t have the time or resources to individualize instruction for each student. At the same time, we know that the uniqueness of individual cognitive development coupled with the impact of the proper or improper timing of instruction can make the difference between alienation from or love of learning.

What will happen when artificial intelligence advances to the point where expert cognitive tutors can facilitate individual learning?

Imagine an educational system in which each individual learner can follow their own intrinsically motivated curiosities and be guided by expert cognitive tutors. How would that unleash the creative energy of students?

Also, imagine a world in which a teacher’s passions guided their curriculum and students came to them without the physical constraints of time or place. How would that transform their teaching?

Imagine a world in which individuals self-selected their own educational paths. Would society collapse without the current one-size-fits-all extrinsic controls on what students learn? Or would society flourish as individuals blossomed to their fullest capacities?

I don’t know. But I do believe that technology is making such rapid progress that such an imagined world will be possible in my lifetime. And I believe in Thomas Friedman’s premise that in today’s flat world, whatever can be conceived will be acheived. So dear friends, in your opinion, how would this possibility change the world?I really liked this article. The part I enjoyed the most was about how blame falls on either the student or the teacher and often times the curriculum is neglected.

Antonio Perez