Month: March 2013

Total 14 Posts

Three-Act Modeling v. Textbook Modeling v. Common Core Modeling

Bill Carey:

What’s so compelling about the three-act math project isn’t that it does a better job of teaching the body of knowledge of mathematics; it’s that it reshapes the cultural practice of mathematics in a way that more closely reflects how grown-ups engage in mathematical inquiry.

That’s the goal anyway, particularly w/r/t mathematical modeling. Pick any definition of modeling you want — the IB, the Common Core, the modeling cycle, anything. They all define modeling in similar terms. Here’s the Common Core. It’s scary:

  1. identifying variables in the situation and selecting those that represent essential features,
  2. formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
  3. analyzing and performing operations on these relationships to draw conclusions,
  4. interpreting the results of the mathematics in terms of the original situation,
  5. validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable,
  6. reporting on the conclusions and the reasoning behind them.

That is a huge list of important, valuable skills. The scary part is how little our curriculum helps students develop those skills. Here’s a task from Pearson’s Algebra I text, which is pretty typical in this regard:

That brave little icon indicating the “Modeling” practice begs the question: Is this modeling? Who is doing the modeling? Try to locate each of the six parts of modeling in that textbook problem:

  • Who is identifying essential variables? Where?
  • Who is formulating the model for those variables? Where?
  • Etc.

Then do the same for any arbitrary three-act lesson plan.

The three-act structure isn’t the only worthwhile approach to modeling and it’s still a work in progress. But we should all stop pretending that including some real, physical, made-from-atoms item in a word problem does justice on its own to the complicated, exhilarating stew of skills we call “modeling.”

BTW. While you’re at it, feel free to compare the Common Core modeling standard against the Common Core modeling assessments. As you may know, there are two consortia developing the assessments. Here is an item from SBAC and an item from PARCC. They are more different than they are alike.

Featured Comment

Bowen Kerins offers up a useful analysis of the SBAC and PARCC tasks.

[3ACTS] Finals Week

Which of these drinks has the strongest caffeine concentration? Can you rank them from strongest to weakest? I couldn’t. What information would you need to know to find out?

Two options here:

One, have students talk about the information they’d need and how they’d get it. (Two questions central to mathematical modeling.) Then just give them that information.

Two, have students talk about the information they’d need and how they’d get it. Give them the names of the drinks and have them research the information themselves.

I’m not too uptight about the difference. Option two has the students practicing their Google-fu. Option one costs less time.

Here are the goods. I’m pretty sure caffeine doesn’t work the way I illustrate it in the third act, so you may want to skip that clip.

I also included the cost of each drink in case you’d like to ask students to calculate the “bang per buck” ratio as a follow-up, a cool improper fraction that reads “milligrams per ounce per dollar.”

The Soaring Promise Of Big Data In Math Education

Stephanie Simon, reporting for Reuters on inBloom and SXSWedu:

Does Johnny have trouble converting decimals to fractions? The database will have recorded that – and may have recorded as well that he finds textbooks boring, adores animation and plays baseball after school. Personalized learning software can use that data to serve up a tailor-made math lesson, perhaps an animated game that uses baseball statistics to teach decimals.

Three observations:

One, it shouldn’t cost $100 million to figure out that Johnny thinks textbooks are boring.

Two, nowhere in this scenario do we find out why Johnny struggles to convert decimals to fractions. A qualified teacher could resolve that issue in a few minutes with a conversation, a few exercises, and a follow-up assessment. The computer, meanwhile, has a red x where the row labeled “Johnny” intersects the column labeled “Converting Decimals to Fractions.” It struggles to capture conceptual nuance.

Three, “adores” protests a little too much. “Adores” represents the hopes and dreams of the educational technology industry. The purveyors of math educational technology understand that Johnny hates their lecture videos, selected response questions, and behaviorist video games. They hope they can sprinkle some metadata across those experiences — ie. Johnny likes baseball; Johnny adores animation — and transform them.

But our efforts at personalization in math education have led all of our students to the same buffet line. Every station features the same horrible gruel but at its final station you can select your preferred seasoning for that gruel. Paprika, cumin, whatever, it’s yours. It may be the same gruel for Johnny afterwards, but Johnny adores paprika.

Featured Comment:

a different Dave:

Enjoyable games/activities in general are difficult to create, especially in any quantity. Learning and teaching are complicated and personal by necessity. The combination is exceptionally difficult. [..] It’s just not realistic for this to happen on any timetable or method I’ve seen proposed.

2013 Mar 11. Michael Feldstein links up this post and wires in a comprehensive “Taxonomy of Adaptive Analytics Strategies.”

Precious moments:

First of all, the sort of surface-level analysis we can get from applying machine learning techniques to the current data we have from digital education system is insufficient to do some of the most important diagnostic work that real human teachers do.

Then there are those systems where you just run machine learning algorithms against a large data set and see what pops up. This is where we see a lot of hocus pocus and promises of fabulous gains without a lot of concrete evidence. (I’m looking at you, Knewton.)

And guess what? Nobody’s been able to prove that any particular theory of learning styles is true. I think black box advocates latch onto video as an example because it’s easy to see which resources are videos. Since doing good learning analytics is hard, we often do easy learning analytics and pretend that they are good instead.