Writing in Algebra 2

One of my pushes this year is to get my Algebra II students to write math better. Last year I put “explain this” problems on a few exams and wasn’t so impressed with their responses. This year I am teaching my kids to write responses.

On their first assessment, I put a question similar to one we talked about in class:

Explain to someone who doesn’t know a lot about math why you can never find an x which would make |25x+5.1|-5=-6.

The responses were disappointing across the board. There were bits and pieces of gems, but nothing complete. Not a single student was able to construct a well-written response. Things I received included:

  • The other side of the equation is negative, leaving no possible solution to the problem.
  • You can never find x because the answer is negative and an absolute value problem with a negative after the equal sign is not possible.

So what I did was type up the following document and passed it out a few days after the assessment:

We talked about the vagueness of the responses, the use of pronouns like “it” and making references to “the other side of the equation,” and most crucial, the lack of reference in almost every solution to the original equation. How can you answer a question about an equation without even talking about the equation?

My favorite moment of the discussion this generated was when one student raised her hand and critiqued her own solution, and then said: “I wrote this and don’t even know what I meant.”

On the next assessment, without telling them I was going to do this, I threw the exact same question down. It was on. I saw my kids reread their responses after they wrote them, and really pay attention to their writing. Let me tell you: it all paid off. On this second round, most students got full marks. (On the first assessment, almost no one got full marks, or close to it, for that matter.)

Here are some random smatterings of their thoughtful answers:

  • You could never find an x to make the absolute value equation above true because you would have to subtract -5 from -6, which still gives you a negative number. |-25x+5.1|=-1. An absolute value equation cannot equal a negative number because absolute value is the distance from zero and is always positive [my correction: or zero].
  • In this absolute value equation there is no solution because any number in the absolute value has to be 0 or a positive number. And if you subtract 5 from 0 or a positive number, there is no possible way that can equal -6. So there is no solution to this equation.
  • An absolute value of anything can never be equal to a negative number, since it expresses a distance. When this equation is simplified, it becomes |-25x+5.1|=-1. If the ‘-1’ were replaced with a positive number, you could find the answer [for] x. But since it is a negative, you already know that is impossible.

I am continuing to ask them to express themselves through writing. On that same assessment where I asked them to repeat the absolute value problem, I also asked the following two questions, to which I got some really nice writups.

The following two questions build upon each other. The solution to part (a) will very much help you explain part (b).

(a) Explain why a^2a^4=a^6 without using your exponent rules. Explain it to someone so they can understand it simply!

(b) Explain why a^ma^n=a^{m+n} is true. You can assume m and n  are positive integers. Explain it to someone so they can understand it simply!

I still have to do more work with this, but I just wanted to say: it is worth it to talk with your kids about writing. One 15/20 minute conversation has already yielded great dividends for me.

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8 comments

  1. This is really good, especially your discussion of what was weak about your students’ initial answers (vague, using unclear pronouns like “it,” etc.). I can imagine this was very helpful for them. I suspect that a lot of your students really did understand why the equation had no solution, they just didn’t communicate it very well. Students need to be reminded (and taught!) that just because something is clear to them inside their own head, they still have to work at making it clear to someone else.

    (BTW, the footnote in the Scribd doc was a little unclear; I suspect you and your students understood it, but just made me wonder what “question 27” was.)

    1. The footnote reads:
      “Almost no one in the class got full credit for this problem. But that was expected because I haven’t “taught” you to write good explanations yet. (That is the reason why I made question 27 “extra credit.” Surprisingly, most all of you wrote really cogent and wonderful explanations for that problem.) The 9 solutions that I picked to share were totally random.”

      Question 27 was:

      27. Explain why the following is WRONG:
      If I have the compound inequality x>3 AND x3 AND x<=3 simply gives you all real numbers. To explain further, it is all numbers greater than 3 and all numbers less than or equal to three. All the real numbers.

  2. I think one of the important reasons to be asking students to write about math is because it reinforces the (often forgotten) fact that the math they’re learning isn’t just a random collection of facts. They may know how to manipulate formulas, but asking them to step back and think about what they’re saying brings home that this is a logical problem-solving framework we’re developing.

    That and the fact that expressing oneself clearly is a key skill in the “real world”, no matter what the subject is.

  3. Thanks for posting this. I’ve been reluctant to try to teach math writing because … I know I don’t know how to teach it. But if a class discussion of samples of the students’ own work helps, there’s no reason not to have a go at it.

  4. Thanks for the post!

    I have been working with the 9th graders I am student teaching with to try to improve their writing and reading comprehension skills in math class. The textbook is difficult to read but key to all the lessons. I did a readability test of the textbook and found that the book is between level 11 (junior in high school) to 15 (yep – junior in college) – so the reading level is much higher than these 9th graders need. Not only does the textbook require a lot of reading, all the math problems require explanations – which is good except the students don’t know how to compose them. It seems the question of “is it reading comprehension?” and “is it their writing skills?” or “is it the math concepts?” is pertinent here.

    My first try at improving writing was to outline and discuss the elements of an argument. After a quick discussion, we watched a TED video where Art Benjamin argues that statistics, not calculus, should be the pinnacle of high school math [1] and picked out the different elements of his argument. (I don’t know that I have formed an opinion on this, yet, but it made sense to the students since they have gone through one unit of algebra and one unit of statistics, thus far. So they were basically comparing the two units to form their own opinions.) I then had them write an argument of their own.

    First drafts not so good, but I plan to work on them one-on-one during independent work time. I hope it can lead to some better “explain” responses in the homework, quizzes, and tests.

    Challenge: I am not a language arts teacher! I want to emphasize having a claim and support for that claim, but I don’t know how to teach grammar and sentence structure. Should I be? Do I just go to the real Lang Arts teachers and ask them what to do? What next? Ideas?

    Love the sharing,
    Joan

    [1] http://www.ted.com/talks/lang/eng/arthur_benjamin_s_formula_for_changing_math_education.html

  5. Hey, my name is Jordan, and I’d just like to say that finding this post has been EXTREMELY helpful for me. I am currently enrolled in a grad school class called “Teaching Interdisciplinary Writing.” Right now, I’m working on a project where I have to create the syllabus and several lesson plans for an imaginary class in Writing in Algebra.
    I am an English major. The last time I took any kind of math class was back in high school, and even then, I didn’t “get it.” This project, then, is a tough one for me. But here I find you, a real life Algebra teacher, and you are teaching your students to “write math!” !!!!!! This is a huge thing for me!

    Bottom line: could I convince you to mentor me, briefly, with this class I’m creating? I would love the opportunity to hear and learn from more of your experiences in teaching writing to math students. I don’t know if you even check these pages of your blog any more, but if you do, and you are willing to be the Yoda to my Luke, my email address is jordanmccown@yahoo.com.

    Thank you!

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