Thursday, July 2, 2015

Math problems of the week: Common Core-inspired 3rd grade test question

From the SMARTER Balanced Assessment, 3rd Grade Mathematics Sample Items:






Extra Credit:

Compare the simple vocabulary and sentence structure of the Sample Top-Score Response with other aspects of its communicative demands, and relate this to the communication skills of 3rd graders.

Tuesday, June 30, 2015

Confusing math with math education

It strikes me that much of what is wrong with math education results from a confusion of math with math education. Is the goal to teach kids how to do math, or how to be math teachers?

Consider two tasks common to today’s math assignments but rare before Reform Math: explaining answers verbally, and explaining what’s wrong with other people’s solutions. Variants include having third graders write letters to second graders about why, say, 1/3 is bigger than 1/4, or to Jack “telling him what he did right, and what should do to fix his mistake.”

I and others have argued here and elsewhere that explaining your answers verbally is often a counterproductive waste of time that, in particular, shortchanges second language learners and students with language delays. Similar arguments apply to explaining why someone else’s answers are wrong. But, if you’re in a teacher education program training to be a math or a K6 general education teacher, then suddenly being able to provide these types of verbal explanations is absolutely essential.

Ironically, these explanation demands are especially common in elementary school, when students are least able to verbalize things clearly. Perhaps this has to do with the profile of the typical elementary school math teacher, who, in his or her teacher training program, has had to take courses in math education, but not in actual math. To some extent, however, such teachers are simply following the math curriculum that others have written and/or selected for them. So what about those most responsible for creating and selecting math curricula--the Deborah Balls and Andy Isaacs and Jo Boalers of the world? Is it possible that most of them get more training in math education than in math?

When it comes to educating K12 students, math education should primarily involve math, and not the infinite regress, as it were, that comes from educating students in math education.

Sunday, June 28, 2015

Problematizing grit, II

How hard you work on something isn’t the only effort (or grit)-related variable. Also key—and what Angela Duckworth’s questionnaire doesn’t probe—is how you direct that effort within the project. I realized that recently when, for the first time in over a decade, I decided to learn a new piano piece. Having allocated myself a mere 15-20 minute window on weekday mornings (a rare stretch of quiet solitude), I was determined to practice as efficiently as possible.

And this meant resisting all sorts of temptations that as a student I often succumbed to: the temptation not to bother working out fingerings and using them consistently; the temptation to interrupt my work on the sections I knew the least well, or found the most difficult, for the satisfaction of breezing through easier or more familiar sections; the temptation to play the piece too fast, too soon. It’s not just the distracting temptations outside a project, I realized, but also the distracting temptations within a project, that need resisting.

Directing your efforts appropriately involves brains as well as brawn. In learning a piano piece, for example, it helps to realize that muscle memory is essential, and that muscle memory will develop fastest if (1) you use consistent fingering and (2) you play slowly enough to minimize errors.

Teachers, too, can be smarter about grit. Neither should they try, vaguely, to "teach" it (e.g., by spending lots of class time on "growth mindsets"); nor should they simply give students tons of work or make them "grapple" indefinitely without guidance. Rather they should give students frequent advice and feedback about performance--and about how best to allocate their efforts.

Friday, June 26, 2015

Math problems of the week: Common Core-inspired "algebra" test problem

A problem from the "calculator section" of Algebra II  Performance Based Assessment Practice Test from PARCC (a consortium of 23 states that are devising Common Core-aligned tests).





Extra Credit:

Based on the given information, determine the ratio of algebraic to verbal challenges in this problem. Describe the steps used and explain any assumptions made. Create a model and describe the steps used to create it. Enter your answer, model, explanation, and assumptions in the space provided.

Wednesday, June 24, 2015

Problematizing grit

In her Ted Talk on “grit,” Angela Duckworth offers the following definition:

Grit is passion and perseverance for very long-term goals. Grit is having stamina. Grit is sticking with your future, day in, day out, not just for the week, not just for the month, but for years, and working really hard to make that future a reality. Grit is living life like it's a marathon, not a sprint.
All this, Duckworth finds, predicts long term success. So far so good—but (dare I say it?) hardly surprising.

What’s a lot less obvious is whether grit can be taught. Of course, this hasn’t stopped the education establishment, ever eager to focus on something other than academic instruction, from jumping to conclusions. Here, on the other hand, is Duckworth:
Every day, parents and teachers ask me, "How do I build grit in kids? What do I do to teach kids a solid work ethic? How do I keep them motivated for the long run?" The honest answer is, I don't know.
Duckworth says the best idea she’s heard is Carol Dweck’s “growth mindset”: “the belief that the ability to learn is not fixed, that it can change with your effort.” Duckworth cites Dweck’s finding that:
when kids read and learn about the brain and how it changes and grows in response to challenge, they're much more likely to persevere when they fail, because they don't believe that failure is a permanent condition.
Again, so far so good—but (dare I say it?) hardly surprising.

Plus, there’s only so far mere beliefs can get you. Indeed, the questionnaire that Duckworth uses to measure grit (and predict success) addresses how distractible you are, how fickle vs. sustained your interests are, and how hard and how diligently you work on things; not what you think about failure.

Given this, perhaps a better way to raise students’ perseverance is to provide extra incentives for hard, concentrated work. Ideally these incentives would be built into the work itself. You make sure that it’s interesting; that students get timely feedback about their progress through it; that completing it results in a satisfying final product, set of revelations, set of new skills, and/or sense of accomplishment. As far as these things go, much school work (whether because it’s busywork, easy work, group work, vaguely defined, and/or lacking in timely feedback) comes up short.

But even with some of the best types of assignments, and/or with certain types of students, there may be insufficient incentives for perseverance. In that case, as we’ve seen with J, why not resort to extrinsic incentives? For those who fail the marshmallow test, why not incentive them with marshmallows?

Monday, June 22, 2015

All about meteors or all about MEteors?

According to Michael Tscholl, a researcher at the University of Wisconsin (as reported in a recent article in Edweek):

Most students harbor fundamental misunderstandings about how forces such as gravity and acceleration operate in outer space. That's because their beliefs about physics tend to be based on their experiences in their own bodies.
Bodies on earth, Tscholl explains, need energy to keep moving; objects in space don't.

How to overcome these fundamental misunderstandings? Guess what Edweek/Tscholl propose? Is it:

1. Enhance students understanding of the concepts of friction and inertia.

2. Give students "embodied cognition" exercises in which they move their bodies around through earthly friction?

Hint: the solution proposed by Edweek/Tscholl is MEteor,
a room-size "simulation environment" that calls to mind a space-age version of the popular space-age version of the popular arcade video game Dance Dance Revolution.
Still stumped? Here's more:
In MEteor, planets and other space objects are projected on the floor and walls. The students must predict the trajectory of an object moving through space by physically moving along the path they think a meteor (projected on the floor) will travel. Laser scanning technology tracks their movements, offering real-time feedback on whether their predictions are correct. Based on that feedback, students adapt their beliefs about scientific principles, then adjust their movements to reflect what they are learning.
Final hint: it's probably reasonable to assume that these MEteor-facilitated embodied cognition exercises don't take place in outer space.

Another problem reported by Tscholl: "students are scared of symbolic representations." Given this, what do you think his solution is?

1. Give students more practice with symbolic representations and their relation to physical phenomena.

2. De-emphasize symbolic representations.

Stumped? Consider: (a) how facility with symbolic representations, and with manipulating these mathematically, is essential to doing physics, and (b) how little sense there is in anything in this article.

Saturday, June 20, 2015

You need to do some graphics to make it look like they’re flying, when they’re not really flying

I recently came across this un-facilitated, unedited, in-class assignment that J wrote for his graphic design class. Somehow, with its earnest attempt to cope with whatever the prompt was, and with his years in high school now weeks away from their conclusion, I found it quite endearing. I reproduce it here with permission from the author.

Given what I’ve written recently about Facilitated Communication, I should note that, in a sense, the author’s in-class communication is facilitated. J’s handwriting being so bad that often even he can’t read it, he regularly uses an AlphaSmart keyboard. But the keyboard remains stationary, sitting on his desk rather than on the palm of someone else’s hand; it offers no text-completion software with pop-up windows of likely next words and grammatical corrections; and no one would even consider hovering over J and supporting his wrist while he types. This is an author who feels strongly about being left alone while the creative juices flow:

There are some people who becomes a graphic designer. Like making a fictional movie, you’ll have to do some graphics on some objects. Like when Violet turned into a blueberry, people had to do some graphics since you obviously can’t inflate people into a ball. 
You have to be good at programming. Graphics require some programming. When you make a movie, you’ll want it to look real, and not make it look like it’s edited. Like when we see Violet turning to a blueberry, it looks real, and has not been edited.
You have to be good at painting to make some cartoon movies. In cars, Lightning McQueen and other cars look like they’re real, but they were actually painted. You would want to make it look real, and not look like they have been painted.  
You have to have a software to do some graphics. Photoshop is one of the software. It can edit some things out, and put some new things in. Like if you want to change some of the words, you’ll want to remove the words, and put new words in, and you’ll want to make it look like real, and has not been edited.  
So if you want a graphic designer, you need to be prepared. You want to make a movie look real, and not been edited. Like in Harry Potter movie, quidditch is obviously not real. You’ll have to make some graphics to make it look like they’re flying, when they’re not really flying.