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Month: November 2012

Total 8 Posts

Computers Are Not A Natural Medium For Doing Mathematics

By • November 5, 2012 • 42 Comments

Exhibit A:

The simplest thing, “Take a picture of one of the proofs you just wrote and email it to me.” turns into twenty minutes of troubleshooting cameras that don’t work, and picture files we can’t find in order to attach them, and how to login to your school email account.

This isn’t an exhibit of doing mathematics or of technology enabling a classroom. This is an exhibit of an entire classroom spending time and administrative capital accommodating the limitations of computers, of technology disabling a classroom.

The tools need to get out of the way. When I use the Internet to communicate these words across time and space, I’m not consciously aware of all the technologies that facilitate that communication. They are out of my way. Computers are a natural medium for communicating words. In Kate Nowak’s class, the tools are consciously in the way.

Featured Comments

Dave Major:

Over the past couple of months I’ve heard “yeah, that’s cool, but I can do the same using x, combined with y and converted using z, backing onto Dropbox” far too many times.

Paul Topping:

With plain text, we go to a computer first to type it. Many of us have noted how he hardly ever handwrite anything longer than a phone number or address these days. The same can’t be said for math notation. Some can write math using LaTeX but that is far from ideal. Even mathematicians who are LaTeX experts do not handwrite it on paper or a whiteboard. They use standard math notation.

[3ACTS] Pixel Pattern

By • November 5, 2012 • 8 Comments

You’ve heard of pile patterns? There are variations but generally you have three snapshots of a growing shape like this:

Questions follow regarding future piles, past piles, and a general form for any pile.

I wanted to know what this old classic would sound like with newer equipment. Would video add anything here, for instance? Here is the result of my tinkering:

Video adds the passage of time. I added a red bounding box to the video, which was an attempt to make the question, “Where will the pattern break through the box, and when?” perplexing to students.

I also added different colors, which allows students to track different things or ask themselves, “What color will be the first color to break through the box?” Different questions require different abstractions. If you care about total tiles, you’ll model the total. If you care about the breakout, you’ll model the width and height. Each one will require linear equations, which is nice.

Other notes:

  • The sequel asks about the “aspect ratio” of the growing shape which is a useful way to dig a little at limits.
  • Real-world math. Here again I’m thumbing my nose at our conviction that math should be real. This isn’t real in the sense we usually mean. If it interests your students, it will interest them because it asks questions that rarely get asked in a math classroom, questions from the bottom of the ladder of abstraction:
    • What questions do you have?
    • What’s your guess?
    • What would a wrong answer look like?
    • What information do you need to know?

The Smarter, Balanced Sample Items

By • November 1, 2012 • 24 Comments

The Smarter, Balanced Assessment Consortium:

Five swimmers compete in the 50-meter race. The finish time for each swimmer is shown in the video. Explain how the results of the race would change if the race used a clock that rounded to the nearest tenth.

You should take a tour through the Smarter, Balanced Assessment Consortium’s released items, make an opinion about them, and share it here. California is a member state of SBAC, one of two consortia charged with assessing the Common Core State Standards, so I’m comparing these against our current assessments. Without getting into how these assessments should be used (eg. for merit pay, teacher evaluation, etc.) they compare extremely favorably to California’s current assessment portfolio. If assessment drives instruction, these assessments should drive California’s math instruction in a positive direction.

The assessment item above uses an animation to drive down its word count and language demand. It’s followed by an expansive text field where students are asked to explain their reasoning. That stands up very well next to California’s comparable grade five assessment [pdf]:

  • Elsewhere, we find number sense prized alongside calculation (here also) which is a step in a very positive direction. (ie. Our students should know that $14.37 split between three people is between $4 and $5 but it’s a waste of our time to teach that division longhand.)
  • I’ve been assuming the assessment consortia would run roughshod over the CCSS modeling practice but on the very limited evidence of the sample items, we’re in good shape.
  • The assessments do a lot of interesting and useful things with technology. (Reducing word count, at the very least.) I only found one instance where the technology seemed to get in the way of a student’s expression of her mathematical understanding.

I can’t really make an apples-to-apples comparison between these items and California’s current assessments because California currently has nothing like this. No constructed responses. No free responses. No explanation. It’s like comparing apples to an apple-flavored Home Run pie.

Featured Comment:

Candice Frontiera:

Next thing to explore: Technology Enhanced Item Supporting Materials [zip]. [The “Movie Files” folder is extremely interesting. –dm]

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