Year: 2009

Total 161 Posts

Asilomar #5: What Can We Do With This?

Session Title

Lights, Camera, Action! Fun And Success For All In Algebra


Allan Bellman, Supervisor of Teacher Education, UC Davis


“Why would we want to teach with digital images?” he asked and then answered, “Because it’s better to watch it than read about it.” where “it” referred to any generic textbook problem.

I had no disagreement insofar as we keep text on the table as an option for those who do find it better to read about it than watch it. Math education, however, is not suffering from a surplus of visual representations.

I’m naturally inclined to this kind of discussion. He pointed at an image on the screen and asked, “What can we do with this?” which is obv. one of my favorite questions to pose, discuss, or answer. Bellman and I answer that question differently, however.

The image was of a Volkswagen Bug and he wanted us to point out all the mathematical shapes we could see on its frame. We eventually settled on the parabola that formed its canopy. He passed out a printout of a Volkswagen Bug and a transparency of graph paper. He asked us to find the equation of that parabola.

We could position the car wherever we wanted. Some positioned it upright. Others upside-down. Some defined the origin of their coordinate system at the top of the car. Others at the bottom. We all derived our equations.

Then he put a transparent Bug on top of a TI ViewScreen panel connected to a TI-84 Plus. He put the Bug in different positions and we had to modify our parabolic equation each time to match it, which was an interesting exercise in transformations.

Then we reworked the same exercise with an advertisement ripped from a magazine that featured lines, parabolas, and sinusoids. There were TI-Navigators on every table connected to hubs which were connected to some Windows software that would graph the equations we submitted after we logged in.

I graphed y = 4, which traced a horizontal line across a rooftop, and looked smugly at my tablemates.

Bellman then put up a picture of a golf course. He draw a ball next to the ninth hole and asked us to determine the equation of the line that went through those two points. He challenged us to find a quadratic model that fit to those two points. He mentioned that we could even ask our students for an exponential model.

And this is where my purpose splits from his on using digital media in the classroom:

Neither that line, that quadratic model, nor that exponential model have anything whatsoever to do with golf. If we’re going to use an image of a golf course we need to ask a question that clarifies or has even a glancing connection to golf itself.

  1. Will she sink the putt?
  2. How far has she hit the ball off the tee?
  3. Which club should she use here?
  4. How fast is the club head moving?

By setting the background of a coordinate plane to an image of a golf course, we may engage a few more students than if we used a plain plane but I think we’ll also lose a few students on the other end who recognize the arbitrary, artificial nature of the setup. (ie. “Why not a picture of a baseball diamond?”) I worry that if we use digital media in our classrooms like this, we’ll define mathematics even more as an abstract thing rather than as a tool for explaining our own lives.

Bellman then brought up a video of a basketball player shooting a jump shot. We used Logger Pro to pull down some coordinates from the first half of the ball’s arc. Then we answered the question, “will he make the basket?” which is exactly the approach I’d like to see to digital media in the math classroom.


Some PowerPoint. A lot of modeling.


Transparencies and paper to push around and play with.


  • Someone asked him “How long does it take you to come up with all of this?” and he nodded at the difficulty of assembling all this digital material but pointed out (rightly) that you only have to develop it once.
  • “There are two kinds of video. The kind you make and the kind you get from Blockbuster.” Someone asked him how you extract a clip from a DVD and his status as a paid TI rep forced him to play it coy. I am not likewise burdened so here: Handbrake.
  • Bellman kept a running tally of the cost of his setup over the course of his presentation, beginning with a $100 video camera. Clearly, it balloons as you start deploying TI equipment and I’d like to open the floor here to a cost-benefit analysis.
  • In general, I admit to some cynicism about graphing calculators, which occupy a strange corner of the edtech market. The equipment is cumbersome to set up. (Bellman had two lackeys there from TI to help him install those hubs.) It doesn’t come with a web browser. The resolution on its black-and-white screen is worse than the cell phone in my pocket. Can anyone explain to me how graphing calculators are going to avoid getting crushed in the tightening vise between cell phones and netbooks?

Asilomar #4: Be Less Helpful

Be Less Helpful – CMC North 2009 – Dan Meyer from Dan Meyer on Vimeo.

Session Title

Be Less Helpful


Dan Meyer, High School Math Teacher / Google Curriculum Fellow


30% of my talk targeted how we teach – the subtle ways we encourage students to stop thinking. 70% targeted what we teach – the not-so-subtle ways our adopted curriculum makes our students intellectually timid and incurious.

Very little of the content will be a surprise to regular readers. However, after my O’Reilly webcast, I resolved to always add some new content or some new analysis every new time I present the material, both out of appreciation for those attending who, in their loyal readership and commenting, have done a great deal to shape these ideas, but especially because ideas, if they’re worth anything, should keep growing and changing.


Big pictures designed for conversation.


A place for people to interact with ideas using notes and drawings. Very similar to the handouts I designed for last year’s presentation, which I model in this video.


  • I was really, really sick throughout the entire thing.

Asilomar #3: How To Teach Geometry

Session Title

Don’t Just Cover Geometry, Discover Geometry


Michael Serra, Teacher / Author


Serra took the participants through his well-worn strategies for teaching Geometry, emphasizing manipulatives and induction. His students develop definitions and terms together. At whatever point they see a pattern in their experimentation (ie. “okay … all the angles in a triangle seem to add to 180°”) the students write the conjecture.

This approach requires of a teacher a rare mix of knowledge, confidence, and humility and it’s no surprise to me that operationalizing his approach into a textbook has yielded unpredictable returns. Whereas other books will cite and define the parallel line postulate, his book cites it and starts the definition but leaves the critical parts blank. They aren’t defined farther down the page. They aren’t defined in the back of the book. They’re just blank. Some teachers will simply fill in those blanks for their students who then dutifully record the conjectures in their notes. Others will have the students develop those conjectures through observation and experimentation, as intended.

Serra demonstrated the latter approach over the course of ninety minutes, developing 75% of Cartesian Geometry with nothing more than induction and patty paper.


Document camera for modeling the experiments.


Pre-printed shapes and forms. We operated on them in groups with scissors and rulers.


  • “This is all of Geometry in one class. Finish up in a day. Let’s watch movies now.” – Ian Garrovillas.
  • I wish I had something insightful here linking Serra’s nascent web presence to Cringely’s Burn, Baby, Burn, which speculates on the fate of higher education in the Internet age. Serra’s name carries a lot of currency in the business of math education. Converting that currency into imaginary Internet money and then that imaginary Internet money into real money is the necessary journey but there isn’t a road map. I don’t know what I’d do in his position.
  • Ian asked if I had ever thought about monetizing my blog, by which I think he meant sidebar ads or something. I’m so lost in that world but my creeping sense is that sidebar ads would cheapen the experience of publishing content (for me) and interacting with content (for you) in a way that would be zero-sum or quite possibly negative-sum. I know for certain, however, that writing content here has been a financially profitable experience, though that profit has come about in some strange, indirect ways that I don’t know how reproduce with any reliability.

Asilomar #2: What Do We Do With The Seniors?

Session Title

What Do We Do With The Seniors?


Robert Loew, High School Math Teacher


Loew and his colleagues wanted more options for students who finished Precalculus during their third year of high school (or earlier) but who weren’t going on to a STEM major and didn’t want to take Calculus. They came up with two.

1. Math Analysis

College prep. Approved by the University of California for a-g credit.

One semester of “Calculus Lite,” heavy on application, light on theory, including:

  • continuity and limits,
  • average/instantaneous rates of change,
  • the derivative as the rate of change at a point,
  • basic rules for differentiation, including the chain rule,
  • the meaning of extreme values,
  • the meaning and use of the first and second derivatives,
  • the integral as the cumulative effect of change / area under the curve.

One semester of “other math topics,” including:

  • management science,
  • Eulerian and Hamiltonian circuits,
  • critical path scheduling,
  • game theory and negotiation,
  • the prisoner’s dilemma (as it applies to arms negotiation),
  • fair division (as it applies to settling an estate between three heirs),
  • the time value of money,
  • models for saving and investment (as they apply to calculating the value of a stock, the greater fool theory),
  • decision analysis.

Key texts:

2. Problem Solving

Approved for elective credit. These are techniques for solving problems that aren’t neatly defined, for answering the question “what do you do when you don’t know what to do?” The course emphasizes both individual initiative and group collaboration, rewarding creativity and divergent thinking.

Chapter headings:

  • Draw a Diagram
  • Systematic Lists
  • Eliminate Possibilities
  • Matrix Logic
  • Look for Patterns

Class norms/values:

  • open ended inquiry / divergent thinking,
  • tolerance for ambiguity,
  • collaboration,
  • sustained effort,
  • many students will be uncomfortable and “may need to be filtered out.”

Key text:

I applaud this kind of curriculum design but it seems a shame to me that students who aren’t already tolerant of ambiguity or already patient in their problem solving “may need to be filtered out” of a class designed to teach tolerance of ambiguity and patience in problem solving especially since those seniors have likely been indoctrinated with those bad habits by eleven years in the very same school system.


PowerPoint. Texty. Comic Sans.


PowerPoint printouts. An interesting tactic here: he withheld the handouts until the very end and passed them out only in exchange for a completed session review slip. Seems to me to miss the point of handouts as another space to interact with ideas, but whatev.


  • A family of deer skipped across the path as I walked to this session. The grounds here are incredible.

Asilomar #1: What Do We Do With Algebra II

Session Title

Thoughts On Rationalizing Algebra In Ways That Serve Kids, Not Universities


Steven Leinwand, Principal Research Analyst, American Institutes for Research


The day before CMC-North I was trading notes with our lead counselor, just swapping stories about kids, when she mentioned a student who was at the end of her turn at the local community college. She’d be transferring to a state college to complete a liberal arts degree if it weren’t for a failing grade in Algebra II. Because she can’t yet perform long division on polynomials, she’ll have to postpone her degree in (just guessing here) linguistics a full year.

Leinwand opened his talk: “The great divider of our time is the Algebra II final exam. Algebra II squeezes off options for so many kids. Algebra II is anathema to all but the top 20% of the population. My premise: as currently implemented, high school algebra I and II are not working and not meeting either societal or student needs.”

He described the courses as “focused on increasingly obsolete and useless symbol manipulation at the expense of functions, models, applications, big ideas and statistics.”

He works with schools across North America and when he’s trying to get a feel for the tenor and rigor of their math programs, he asks for:

  • the courses they teach,
  • their course descriptions,
  • the books they use,
  • the balance of course enrollment,
  • last year’s final exams for every class.

He said they give him unrestricted access to the first four but balk at the fifth. He said, “if you want to engage people in discussion, go and get those finals.”

Leinwand asked, why are most Algebra II final exams balanced towards the verbs:

  • Simplify,
  • Solve,
  • Factor,
  • Graph.

… when math is ever so much more about being able to:

  • Find,
  • Display,
  • Represent,
  • Predict,
  • Express,
  • Model,
  • Solve,
  • Demonstrate.

Lynn Steen: As mathematics colonizes diverse fields, it develops dialects that diverge from the “King’s English” of functions, equations, definitions and theorems. These newly important dialects employ the language of search strategies, data structures, confidence intervals and decision trees.

Leinwand: “No one is saying throw out the old dialect, but what about the new dialect.”

This all came across depressingly but he ended on a hopeful note, citing several promising projects. Among them, The Opportunity Equation, which aims to:

… explore the feasibility of offering a mathematics pathway to college for secondary students that is equally rigorous to the calculus pathway and that features deeper study of statistics, data analysis, and related discrete mathematics applications, beginning with a redesigned Algebra II course.

He called the forthcoming Common Core math standards “the last, best hope” for meaningful math reform. He ended with a proposal for Algebra I and Algebra II curricula, paced at one chapter per month.

Algebra I

  1. Patterns.
  2. Equations.
  3. Linear Functional Situations.
  4. Representing Functional Situations.
  5. Direct and Indirect Variation.
  6. Data.
  7. Systems of Equations.
  8. Exponential Functions.
  9. Linear Programming.

Algebra II

  1. Review and Reinforce Big Ideas and Key Skills of Algebra I.
  2. Quadratic Functions.
  3. Polynomials and Polynomial Functions.
  4. Patterns, Series, and Recursion.
  5. Exponential and Logarithmic Functions.
  6. Rational and Radical Functions.
  7. Probability and Statistics.
  8. Optimization, Graph Theory, and Topics in Discrete Mathematics.


PowerPoint. Black text on a white field. He introduced his slides with this, “These are terrible slides coming up. You want to read PowerPoint slides that break every rule of PowerPoint these are them.”

I felt sick. Leinwand had attended my PowerPoint: Do No Harm talk last year and I could only hope he hadn’t added that disclaimer on my account. He was wrong anyway. He used his slides as conversation pieces. Doesn’t matter to me that they were monochrome.




  • There is a gentleman at the table across from me murmuring and nodding agreement at Leinwand’s every line. It would not be inappropriate to describe the atmosphere in this session as something like religious conversion.
  • New rule: “Legislators can’t require a test that they themselves don’t take and publish the results of on their websites.”
  • If you’re looking for an example from Leinwand of the “old dialect,” here’s one: rationalizing roots in the denominator of fractions. Here’s another: the conjugate in the same context. Can anyone make a case for that?
  • One of “the most honest and important documents in our business in the last five years”: the $3.1 billion budget State Superintendent Jack O’Connell submitted in response to Governor Schwarzenegger’s pressure to make Algebra I an eighth-grade standard.

BTW: Fantastic follow-up from Josh G.

All of this just highlights the real problem: universities and colleges want a gatekeeper. They want that extra way to filter admissions, because they have to do it somehow. Worse, they don’t want to be seen as the “easy” school to get into, because this lowers their respectability. (This also drives me crazy.) So they demand gatekeepers, whether or not those gateways are actually a more useful math education for their students.

BTW: I have attached Leinwand’s slidedeck here.