Nov 4 2007
03:31 pm

RB asked ((link...)) about my views on how to bring up performance in math-related school subjects. Here you go, RB, and thanks for asking! :)

The things one must do to bring up "math" performance in middle and high school are dictated by the nature of the "math" tasks themselves.

First of all, get rid of the calculators. They are Garbage In, Garbage Out (GIGO) devices. Without a sound ability to do a problem without a calculator, a student will not be able to check his or her calculator work and will not know whether the answer received is anywhere near correct. I have seen too many engineering graduate students miss problem after problem by entering garbage (typos) into calculators, then believing the resulting answers.

Next, look at the nature of algebra (and middle school math such as fractions, etc.) as taught in school. Success in these requires the following:

1. a very short list of algebra- or middle-school-specific facts and concepts;

2. neatness (so as not to induce mistakes when moving from one step in a problem to the next);

3. the ability to resist the temptation to skip steps before one is ready--and this can be taught;

4. the willingness to explain each step in writing--and this can be taught, too;

5. the willingness to check one's work--again, teachable;

6. and "last but not least," 100% knowledge of addition, subtraction, multiplication, and division facts such as the answers to 8x7, 6+9, 17-8, 9*6, ....

Yes, I said 100%. Here's why this is crucial. Suppose a student has been getting A's in these "baby facts," earning a 90% or even a 95% average on exhaustive tests of these "little" facts. A typical algebra problem (or middle-school math problem involving fractions, long division, etc.) will contain a few to even many of these "little" facts, all of which must be correct to get the larger problem correct.

Even assuming the student knows the short list of middle-school- or algebra-specific concepts perfectly, and does the rest of items 1 through 5 above perfectly, what is the probability of that former 90% or 95% "A" student getting 7 little facts correct in a row to successfully complete some new algebra (or middle school) problem?

It is 90% times 90% times 90% times 90% times 90% times 90% times 90% = about 48%, not a very good grade. Or at 95%: 95% times 95% times 95% times 95% times 95& times 95% times 95% = about 70%, still not a very good grade. :(

Just doing those simple multiplications above required 33 "little facts" in a row (even using knowledge of powers to eliminate steps) to calculate 90% to the 7th power = 47.829697% = about 48%, and (if I counted correctly) 125 "little facts" in a row to calculate 95% to the 7th power = 69.833729609375% = about 70%.

Even the ability to correctly do 99% of the "little facts" will still cause a student to fall flat on a problem of any size, such as the multiplications in the preceding paragraph. [99% to the 33rd power is about 72%; 99% to the 125th power is about 28%.]

Nothing less than 100% mastery of the addition, multiplication, etc., tables is required to guarantee success in middle and high school math, even when the student knows and understands the higher math concepts perfectly.

In addition--no pun intended :)--since many students (and parents, and teachers!) have "math anxiety" or "test anxiety" (or both!)--in order for students to be able to work problems without freezing up or blanking or being otherwise affected by anxiety, hence missing any of the "little facts," the students must not only understand them but *must* memorize them all.

The fact that 8 times 7 = 56 should be as much a part of the student as his or her bones. "8 times 7" and "56" should be as inextricably linked in the student's mind as "sun and moon," "black and white," and "up and down" (or whatever else students reliably associate these days ("Paris" and "Hilton?"). So should "7 times 8" and "56." And "7 times 8" and "8 times 7."

So, given that students must know these "little facts" in order to have a good chance at succeeding in middle and high school math, the answer of what to do is clear.

In addition to teaching students #1 through #5 listed above, all teachers must learn these little facts! If the teacher does not have 100% mastery, students *will* know of this weakness, and will not listen when told they must learn them well.

Once the teachers lead by example, students can be taught how to memorize these facts. They can be shown how to drill in order, drill out of order, make up their own exhaustive tests, over and over, until they get them right 100% of the time.

Then the students must be shown the value of repetitive reviews, at first daily, then weekly, then monthly, until the student has solidly cemented these little facts into his or her brain.

The nice thing about this approach is that it is not only inexpensive, costing nothing in $ except pencils and paper, but also tried and true. This approach is unpleasant--reliable learning of actual facts has been out of fashion in some circles for a long time--but is based on the nature of the math itself. It works.

It is to be hoped that all the teachers already are 100% solid in these "little facts." If so, there would be no barrier to immediately proceeding with a plan that emphasizes the addition and multiplication tables, neatness, writing out all steps with explanations, and the checking of each problem.

Add in a speech about what to do when a student sees a minus sign--many people are somehow afraid of them--and performance will go up dramatically. I always make students memorize this "minus-sign-anxiety" to-do list, so when they get stuck on a test they can at least mentally calm themselves by reciting the steps:

***An Aside about Minus Signs***
[What should the student do if nervous about a minus sign?

1. DO NOT PANIC. Always good advice, whether when thinking about minus signs or when reading Douglas Adams books.

2. Stop and think about what is going on.

3. If necessary, draw a number line.

4. If necessary, rephrase the immediate problem in terms of money. People rarely confuse owing money with being owed, getting paid with paying, or having money in the bank vs. being overdrawn.

5. Lastly, always check the work.]

Students who do all of the things mentioned in this post still need to learn the relatively small number of "higher" middle school and algebra "math" concepts, but at least they have a decent chance of succeeding if given the simple yet solid grounding outlined above.

And one might be surprised at how quickly students learn the tables if shown how and work a little on them each day in an organized way.

-- OneTahiti

This looks REAL good...

OneT. Thanks for the effort to put all that in! It'll take me a while to digest it - and even then my response may be less than intelligent :-)


Thanks, RB

I will look forward to your comments. :)

I forgot to put in that I make students memorize the minus-sign to-do list as a kind of calming mantra, so I went back and added that bit.

-- OneTahiti

I think this is GREAT!

I've read it once top to bottom. I really like it. And you're right - it ain't expensive, just distasteful :-) making it unpopular.

I'm gonna read another time or two, then see if I can offer a quasi-intelligent comment or two.


RB: It's been awhile. Any comment?


I am looking forward to your comments. :)

-- OneTahiti

OneT the Math Master


A little late reading your Nov, 2007 post. But I completely agree. Your six requirements for success were strictly adhered to by my incredible high school math teacher 20+ years ago. Her name is Nancy Thompson.

We could NEVER skip a step, we had to be able to explain the steps, we had to be painfully neat, and we always had to check our work mathematically. Plus she made it FUN. We would play "Name that Theorem" modeled after "Name That Tune". Before Christmas our class would walk the hallways singing Math formulas to the tune of favorite Christmas carols. Mrs. Thompson would lead us, fashioning an aluminum foil wand and crown.

It was COOL to be smart! She made us feel "above" geekiness. On the wall in her classroom hung a huge "Round Tuit" button. Anyone who was slacking had to shamefully go "get around to it" by getting up and pushing the button. She employed so many ideas to engage her students. She was strict, disciplined and very scary. Students either loved her, or they failed. Shoot, I struggled. I almost failed one of her classes, and I still love her.

Most importantly, anyone who wished to survive her classes absolutely had to come to her as a master of their basic math facts.

Both my daughters learned math through the Saxon program. My oldest only had to be told once what 4 times 3 was, and she committed it to memory and knew it. My youngest wasn't so lucky. She needed much more time and drilling of the basic math facts. She didn't get the time she needed, too many concepts were presented too quickly for her, and she is still behind today. In fact, her inability to perform in math spilled over into her other courses in regards to her level of confidence.

We tried flash cards, coins, beads, candy corn, whatever we could find to count with her. But we were not enough. Her class studied their multiplication tables in a few weeks. When I was young we studied them for months, almost my entire third grade year. She'll be a freshman in high school this year and it breaks my heart she'll be taking basic math when her sister took Algebra II her freshman year.

I look back and I think of all the time the teachers' had my youngest in class to teach her math. All the movies they watched. All the field trips and assemblies they attended. All the "busy" homework the teachers sent home. But no single trained teacher spent the time with her to assure she knew her basic math facts. They focused on those who could move quickly forward like my oldest, and they left those like my youngest behind. Mrs. Thompson taught us for every minute she had us. If we didn't get it, she would work with us individually after her lesson until the bell rang. And I know I am one of many who still, to this day, remember her home phone number.

Emotionally, my youngest has moved on. She is compensating. She's bright and she's a networker.

But I really hate that she will never experience those grueling but rewarding times of working a complicated math problem one step at a time until that defining "awwww' moment at the end when she could proudly say to herself "I got it, I did it!"

Instead, she just muddles through. My oldest is a problem solver.

Math prepares one to problem solve in life. That's what my high school math teacher told me. And I never forgot that.

I know there's always Quick Books, that's what I use now. But it still makes me sad.

Hopefully our schools will get back to the basics in Math. I agree with you 100% OneT.

Thanks for the feedback, Grasshopper


Thanks for the feedback! I read your post with pleasure. I am hardly a math "master"--math is a huge field!--but appreciate your comments. Your memories of your high school math teacher Nancy Thompson are wonderful.

The talk elsewhere on RoaneViews about the need for folks to offer ideas for helping their school reminded me of another post ((link...)). I find myself wishing I could read your comments on that too. :)

-- OneTahiti

Tribute to Ms. Thompson

I loved your post! Can I make a suggestion? Send that to Nancy. It would mean a lot to her. Both she and Johnny are well into their twilight years but still active as they can be. Their address is the same as it was 20 odd years ago

As for the Saxon Math Program. It was pitched as the answer to all math
problms by the publisher and the state adopted it.They form a Textbook Committee where the teachers have input but it is a token offering.


A # burned in many minds! Right?

Real teachers those Thompsons!

Good community leaders too!

Grasshopper--it is not too late


It is not too late for your youngest! She has all summer to learn those facts. Twice-daily drill combined with repeated testing, re-drill of the missed facts, then more testing, should help.

-- OneTahiti

GH, I had the Thompson Treatment, too...

I'll never forget plane geometry. First thing she (tried) to teach us was that geometry wasn't about math, it was about learning to think things through in logical sequence and the use of applied logic to problem solving. That was WAY more important than the Pythagorean (or any other specific) theorem. They were only illustrations of her point. And I well remember several stories that remind me of her meticulous adherence to the right way to "prove" a theorem. It didn't matter AT ALL if you got the right answer if you didn't get there by the right path.

And I think (if I'm "hearing" right) that's a big part of what OneT's trying to say, as well.

NMT, as well as Physics/Chemistry/Biology teacher JMT, taught us that we couldn't hope to get to the sexy, esoteric, fun parts of the stuff if we didn't learn to put the building blocks in the foundation in the right order and with the right cement. Again, I think that's what OneT is saying. Ya gotta fight your way through the stuff that seems like drudgery, INCLUDING sheer memory work, before you can really have fun with the fun stuff.

In addition to the basics of "gazintas" and "timeses", that includes basic algebra concepts like FOIL for multiplying the terms of 2 binomials, etc. Without those just coming out of the brain reflexively, it will be much harder to win the battle.


So, this is the delayed response...

... to OneT's original posting of this Math Mastery thesis.

Having had to teach other things - thankfully not math - that ABSOLUTELY REQUIRE that the building blocks of the foundation be laid in the right order and using the right cement, I think I can wholeheartedly endorse the concept as explained by OneT.

Although I had the benefit of good basic stuff as I have already related in my response to Grasshopper about Nancy Thompson, that process of NAILING basics first has to continue with more complex types of math as well. Unfortunately, I ran across some really sorry-ass math instructors in college, and it has messed me up to this day as to where I am versus where I should be. I had one fellow, who was exceptionally bright, that taught an absolutely engaging and interesting course on the history and philosophy of math. The only problem was - that wasn't the course I was taking. I was trying to learn about limits and functions and all that stuff. His course was interesting. But useless. He didn't know how to get the stuff across that was so far at the bottom of his own understanding pile. Even said he graded by the stair-step method: throw the tests down from the top of a stairway. The heaviest ones landed nearest the top and they had the most on 'em, so they got the best grades. What a jerk!

Anyway - I think what's here in OneT's thesis really makes sense. Hats off, OneT!


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