Monthly Archives: July 2013

Mathphrase

Even after my vocab-building epiphany, I was inclined to be overly forgiving of slow vocab development. Fortunately my school has speech-language pathologists on staff to collaborate with teachers, and my SLP was quick to direct me toward more exact standards.

For example, one of our favorite vocab-building activities was Math Catchphrase (later dubbed “Mathphrase”). I laminated little cards with mathematical terms on them (some were review words, others were new for current material). Then we played Catchphrase, but instead of reading the word from the disk you pass around, students grabbed their word from a turned-over stack of cards in the middle of the table. On each corner of the table was a piece of paper with the words “Skip Zone,” where students could put the words they had to skip so we could talk about them at the end of the round. After watching us play it once, my SLP saw that the game needed an obvious external incentive for the students to use mathematical descriptions of the words instead non-mathematical ones. So I spruced up the scoring system to reward mathematical descriptions, which made a big difference. Unlike regular Catchphrase, we would reshuffle the words and play again with the same stack for the sake of repetition and reinforcement.

For her end-of-year review, a student came up with her own vocab game (“Draw, Act, or Describe”) that I’m excited to throw into the mix this coming year to prevent overusing Catchphrase.

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A Case for Mathematical Vocab-building

I knew my students had language-related learning disabilities but didn’t know how that would manifest itself in the classroom. Many of these students hear words they don’t understand so often that they don’t like to make a show of it. They let the moment pass, acting like they understand what you’re saying and assuming they’ll figure it out later if it’s important. In time I began to notice signs that my students didn’t understand certain terms, but for a while I effectively ignored their vocabulary needs. My thinking was that with as many gaps as there were in their understanding, conceptual mathematical vocabulary didn’t rank highly on the list. I mean, they knew the most common vocab terms (add, subtract, multiply, divide, distribute [on good days], equation, etc.), so I tried to express new ideas in those terms. I did use more advanced vocabulary in class, but I heard myself saying those terms quickly, self-consciously, like I know this word means nothing to you so let’s just get it over with.

Finally, I was working one-on-one with a student (a high school senior in trigonometry, one of the highest math courses taught at my school) because she didn’t understand a homework problem. We read through the problem, I tried rephrasing it, breaking it into manageable pieces, but made no headway. Then she pointed to two words and said, “I don’t know those words.” She was dyslexic, so maybe it was a symbological thing. Maybe she knew the words but was confused by their written forms, so I said the words out loud: “Radius and variable? You don’t know those words? Do they sound familiar at all, like you’ve heard them before but can’t remember what they mean?” “No. I don’t know those words.” I know for certain she’d heard them before, multiple times, but the fact is that she had no recollection of hearing them, was aware of no meaning associated with them. Radius and variable. This is a student who has taken Algebra 1, Geometry, and Algebra II, and made positive impressions on her teachers. She should know the words radius and variable.

What a wake-up call. Maybe she’d had other teachers who, realizing what a jumble her command of language was, had decided to de-emphasize vocabulary, to get her to do the math without worrying overmuch about the words associated with it. Or maybe they’d given vocabulary the normal amount of emphasis and this was what that level of emphasis resulted in for students like her.

She had taken all the prerequisite courses for trig and had been able to “do the math” so well that teachers had given strong recommendations of her ability. But here she sat without even basic tools for expressing the math she had learned to do. She could neither produce the words on her own, nor recognize them when written and pronounced for her. She could not communicate the ideas she had worked to learn, and without communication, ideas wither.

Boy, was I wrong about the ranking of vocabulary in the hierarchy of important mathematical subjects. Getting these students to “do the math” without enabling them with tools for communicating the math is nearly worthless. After this, cumulative vocab-building ceased to be a dismissable time-drain in my class and became recognized as central to the students’ learning and reviewing. It absolutely takes time away from other pursuits. It is essential.

Two Highlights and a Lowlight of Quadratics, 2012-13

  • Highlight: Having students use arms to form the shape of a parabola. Engaging large muscle groups in class gets a little extra blood flowing and promotes retention–I don’t do it as much as I’d like. One student turned the “arms parabola” into a dance move to go along with her hit new song, “I’m a Parabola!” When we got into graphing quadratics in standard form, a student stood in front of a projected Cartesian plane and others instructed him how to position himself and his arms to represent the function.
  • Highlight: The quadratic formula song.
  • Lowlight: Major struggles with simplifying the quadratic formula once they had all the values in the right places.

Goals Jotted Down

I remember drafting this post in May, but I guess it never made it out of my drafts folder. Until now, that is!

05/23/13 Here are some goals I jotted down for improving Trigonometry next year.

  1. Set better sections for student notebooks. The notebook sections I required this year were basically useless. Here’s what they should be:
    1. Notes and Classwork
    2. Homework
    3. Quizzes and Tests
    4. Journals and Projects
    5. Miscellaneous
  2. Trigonometry is not bathroom and drinking fountain time (yeah, it was a problem this year)
  3. Support the “function box” concept better*
  4. Make better intro learning activity for quadratic functions**
  5. Emphasize slope as a rate of change
  6. Contrast rate of change vs. accumulation
  7. More “discuss/work with a partner”
  8. Follow through on promise of notebook checks

7/10/13 Of course there are others, it seems hundreds more. Tweak that activity. Improve those notes. Allow them to critique their own presentations. Make sure homework assignments are worthwhile. But at the end of the school year, the bulleted eight were the goals that seemed big, essential.

*7/10/13 After coming across the illustration of inverse functions that I featured here, I’m not sure if I’ll continue to use the function box. I might.

**Number 4 is my primary curricular goal. At the end of last year I set a primary goal to improve the way I taught logarithms, with the result that this year the logarithm unit was one of the best of the whole year. Quadratics are the first non-linear functions my students study so their introduction deserves to be genuinely meaningful. Thus, it receives “primary goal” status.

Introducing Quadratics

Quadratics were the first non-linear functions my students ever encountered. From reading different blogs I had the idea that it would be cool for the non-linearity of the relationship to be a surprise that they would discover and therefore take to heart by exploring the relationship between two quadratically related variables. This plan was also influenced by my own experience in 9th grade of being utterly confounded by seeing the teacher graph a parabola for the first time. I was convinced it was wrong. When you graph a function, it has to be straight! This can’t be right.

As for the two quadratically related variables, I figured it would be good to work with a physical shape, so I went for the base-length of a rectangle with constant perimeter and the area of that shape.

The collective student response I anticipated:

“Gee, I wonder how the base length of a rectangle with constant perimeter is related to the rectangle’s area. This is a neat-o activity. Okay, now I’ll plot the ordered pairs I found and…wait a minute! Something’s weird here. My data points don’t lie in a straight line! How can this be? This is like nothing I’ve seen before! Well, since I’m working with a physical shape with which I’m quite familiar, and have proven for myself that these two variables are related in this curved way, I must accept the reality and strength of this kind of relationship. I wonder what other variables are related this way. Wowzers, this is fascinating!”

The actual collective student response:

“Why are we doing geometry? What’s the point of this? What do I do with the values in the table? How do I graph them? And then I connect the dots? Okay.”

Not even the slightest hint of interest that they had graphed a curve instead of a line.

Problems:

1) The apparent aimlessness and disconnectedness of the rectangle activity. Blame this on my thirst for surprising them with the quadratic relationship. I should have introduced the activity with something like, “These two values are related quadratically. This activity will help you explore what that means.” I hadn’t wanted to taint their discovery process with my teacher-y information, but I ended up robbing them of the sense of purpose that they needed.

2) Many students lacked automaticity for graphing a table of values. A problem indeed.

3) Many students showed no meaningful comprehension of the concepts of mathematical relationships and functions (related to problem 2). While certainly not everyone will be intrigued by the same mathematical discoveries that I was, I’m sure some of the lack of interest in the curve was caused by the meaninglessness of even line graphs to these students. If a graphed line communicates no information to you, a graphed curve is hardly different.

Improvements for next time:

1) Give an intro to the rectangle activity. Okey dokes.

2) Provide physical rectangles that allow them to adjust and measure the side lengths; allow for non-whole number side lengths. (This time through, the rectangle was only hypothetical, and they were dealing with whole numbers exclusively.)

3) Consider spending time practicing reading graphs before teaching quadratics, and moving the review of domain and range up from where it currently is (right before teaching inverse trig functions). Although a more realistic expectation would be to review domain and range before quadratics and before inverse trig functions.

4) Prep self for the potential dismay of the pre-calc teacher that my students have covered so little material because I was forever adding preparatory and enrichment activities to the material we did cover. Dismay or no, I’m convinced that it’s better for these students to understand some things than to have seen many things and understand none.