Wednesday, January 16, 2008

Emergency Math

Sarah at Mathalogical has suddenly gotten her course load increased to four preps (General Math being the latest addition) with little curriculum attached, and she's asking for suggestions. I'm responding here because the comment got too long.

First, four preps without textbooks or curriculum is rough. I did that last year, am veryvery glad it's over, and wasn't proud of the results. On the positive side, it gives you exposure to a large range of typical conceptual hurdles in a short amount of time, and your toolkit will grow very quickly. You'll know a lot more about just what your students in later courses aren't getting due to your experience with this course. In order not to get too discouraged it may sometimes be necessary to remind yourself of how much you're learning when you don't get enough time to prepare what it takes to have the students learning enough, selfish and futile as that may sound. And starting this marathon now rather than in August means you can try things out knowing that you can start over again in just one semester.

The three resources I found of most use last year were
  1. I Love Math
  2. The Math Worksheet Site (this costs $20 per year), and
  3. The National Library of Virtual Manipulatives
I don't know that these are better than anything else out there, but these are the ones I returned to again and again, and where whatever did work usually came from. With only that basis for recommending the following, here's what I did:

There was no time for dreaming up a coherent curriculum with much by way of unifying themes or red threads, so in the General Math type courses I prioritized according to what skills I thought were hindering students the most in accessing more math. Some areas I focused on were
  1. Integers on the number line. The Math Worksheet Site has neat pages of number lines with addition and subtraction problems that the students solve by diagramming the problem on the number line. A large number of 10th graders could not deal with negative integers, and in most cases these number line problems helped. The very idea of associating the numerical operations of addition and subtraction with the geometrical idea of motion along a line is the Big Idea that students just have to get in place, it's much less obvious than we like to think, and missing skills in this area really holds the students back.

  2. Place value, and decimal numbers on the number line. First, placing these on the number line was a priority - though in many cases I did not succeed in teaching this. Dan Greene has great stuff on it (as you would already know) - but teaching place value just is not easy. It's awfully important, though, as the kids trip badly over this missing skill when they attempt to do more advanced stuff, so if you can do anything for them in this area, you're helping, even if it sucks up quite a bit of time. The Math Worksheet Site has lots of practice sheets for translating between Decimals, Percents and Fractions, and they're tidy and neat for what they do. As for resources for placing the numbers on the number line, the worksheets at this site aren't that satisfying. There must be animations out there that let you zoom in on a piece of the number line to study place value - but I haven't found anything great, and spent quite some time searching for it last year.

  3. Solving simple linear equations. The common student error that bothered me the most was students' insistence on subtracting the coefficient of the variable instead of dividing by it - my explanations just did not work, and they were inelegantly wordy. What did work for many students was practicing with the Algebra Scale Balance at the National Library of Virtual Manipulatives. After working on this site the incidence of that error went down very noticeably, and it's the concrete representation that does the trick - doing a verbal version of this lesson, well, good luck. For practice problems, the "Partner Problems" worksheet for equations at I Love Math is great. It has two columns of problems of increasing difficulty, and horizontally aligned problems have identical solutions, so that the students can get near immediate feedback on their solutions. The students liked that sheet, and would gladly redo it if I photocopied it onto paper of a different color (and yes, they did need the repetition).

  4. The basic operations. Many kids were more likely to settle down and do something when their assignment was a boring worksheet on practicing multi-digit multiplication, a fact that always puzzled me - my "interesting" discovery activities were much less likely to elicit absorbed concentration (they would involve reading a line or two of directions for each task - bad, bad idea :) The Math Worksheet Site has lots of practice worksheets, at various levels of difficulty, and the card game Top Deck at I Love Math (in the Middle School Folder) is a lot of fun. (Digression: The card games for practicing skills with fractions worked less well, because students tended to devise their own rules that defeated the purpose of the activity: for example, they'd agree to match denominators of different fractions rather than matching fractions for equivalence, as I wanted them to do!)

  5. Area and Perimeter. If students can just get the difference between the two, nevermind formulas for calculating anything, that helps - it was a defining moment for me that October day when I realized that the students truly were unable to distinguish the two - that was when my ideologically rigid commitment to grade level standards started to give. A hands-on activity (measure the area of your desk in terms of number of colored paper squares you need to cover it; measure the perimeter of your desk in terms of number of standardized pieces of string you need to reach around it) did some good, but only some. A worksheet from a colleague, which involved drawing rectangles on a grid that all had the same area but different perimeters, or the same perimeter but different areas, did more good. There were still plenty of students who had plenty of trouble with just counting up line segments to find a perimeter of an irregular shape, though, and - well, I don't know what to do about that.
That was a sort of braindump of what for me emerged as priorities in a general math type course for underperforming students last year, without any authoritative pretenses. Never took a math ed class (and wonder whether they'd ever deal with 10th graders enrolled in Geometry who add 5 and 3 on their fingers and don't know multiplication from addition). If you are able to post about upcoming topics a week or two ahead of time ("We're doing the Pythagorean Theorem next week - what are the students going to struggle with?") you just might avoid some of my unpleasant surprises ("Squares? Square roots? What's that?") and do something relevant to what the students actually need.


Sarah Cannon said...

Thanks H! Emergency Math is a perfect title in so many ways.

I had a longish response that got deleted. I think I'm too exhausted to rewrite it tonight. For now thank you, thank you, thank you for bringing some hope to my night.

H. said...

Sleep well. It will make you a better teacher :)

Henry Borenson, Ed.D. said...

It would be worthwhile to look at the Hands-On Equations approach for teaching elementary and middle school students how solve algebraic equations,such as 4x + 3 = 3x + 10, and how to apply the methods to solving verbal problems.

The students use pawns and number cubes to quickly set up and solve the equations. The Making Algebra Child's Play workshops,held throughout the country, are very helpful in introducing this teaching method.

A research study at shows that 4th, 6th and 8th graders are all equally successful with the first seven lessons of the program.

The above website also has a "Verbal Problem of the Week", and it presents the solution using Hands-On Equations.

Sarah Cannon said...


Thanks again. I think the way I made it through this week was by looking at people's blogs online to remember that other people in other places are facing similar challenges. It doesn't make it any better that they exist. My department is made up of two first year teachers, so it can be hard to tell what's us, what's the school, and what's normal.

I read your post about number lines back in the fall, but it makes so much more sense now than it did then. Having seen work from one of the school's better students where she had to draw a number line to subtract two-digit numbers, I'm now sold on the value of them.

Area and perimeter and any geometry in general look crucial. I suspect teachers have focused so much on the arithmetic, anything else is bonus.

Hopefully, I'll take some time this weekend to figure out what I want to do with this class. Then specific requests will be forthcoming.