(long) discussion re early algebra, homeschool program

In a message dated 11/15/2001 12:54:51 AM Eastern Standard Time, [email protected] writes:

Subj:Preparing for Algebra
Date:11/15/2001 12:54:51 AM Eastern Standard Time
From: [email protected] (Jean Hancock Eye)
To: [email protected]

Victoria,

I’ve gotten such good info from your posts on LD Online, I was hoping I
could get some advice from you. We are homeschooling and my older son
(10yo) is in 4th grade now. He is a little bit ahead in math and really
wants to move ahead quickly and start algebra. That’s fine with me as long
as he gets a solid foundation in the basics. It looks to me like math texts
from 5th grade through pre-algebra all cover the same things- fractions,
decimals, percent,integers, exponents, a little geometry and graphing, and
some more work with variables in pre-algebra. All are important, but I
hardly think we need to spend 3 or 4 years to cover the material. He’s
already been introduced to these topics, and I honestly think that over the
next year or so he’ll pretty well master them.

Would I be crazy to try to start Algebra I in 6th grade if in fact he
continues to make good progress and has gotten down the basics? I don’t
want to use one of the “play around with algebra without really learning
anything” programs that seem to have become popular in recent years, but
real Algebra I- probably Paul Foerster’s text. I don’t mind going at a
slower pace, but I don’t want to use a watered down curriculum. Any thing
in particular that I want to be sure we cover in order to prepare him for
Algebra?

I’d appreciate any input whenever you get the time. Thanks.

Jean Eye

Subj: Re: Preparing for Algebra
Date: 11/16/2001 1:31:30 AM Eastern Standard Time
From: VictoriaHal
To: [email protected]
CC: VictoriaHal

Best question I’ve had in years.
Important issues:
(1) There really is a lot of important stuff to be covered in the Grades 5 through 8 curriculum. It tends to get lost in the math-as-trivia presentation, but it is there. Get a series of books that you trust, and *at least* work through the chapter summaries and/or self-tests at the end of each section (This is a way of going faster while being sure not to omit anything). If he hits a snag, slow down and do the whole section, but if he’s OK, go on.You can make this an even more organized and effective presentation by going through all the books topic by topic — first all the multiplication reviews, then all the division reviews, then all the fraction reviews, then all the area reviews, then all the geometry concept reviews, then all the graphing reviews, then the pre-algebra reviews. At that point you will have enough work and a systematic curriculum for one to two school years, and your son will really benefit from being finally allowed to see the big picture. Hopefully, if you really have a decent set of texts, problem-solving will be an integral part of each unit; if not, use the problem-solving sections of the text and/or other resources and make absolutely sure that you include problem-solving applied to each skill (After all, if you can’t use it to solve a real-world problem, why learn it anyway?)
(2) You’re absolutely right; most of the algebra stuff being sold commercially, especially the “teach your baby algebra” type, is absolute trash and to be avoided at all costs. Better not to be exposed to algebra than to be taught all sorts of erroneous ideas that then have to be untaught. This ALSO applies to the stuff in most texts that purports to introduce algebra in elementary grades — it’s well-intentioned but we know where that path leads.
(3) Mental maturity is a very important issue here. I made this mistake with my own daughter. Have you read any Piaget? Piaget is NOT to be taken as gospel, he has many odd ideas, but his general plot of mental growth is a good outline and helps clarify some important pedagogical issues. The elementary school child is described as being in the stage of “concrete operations” — able to solve problems and perform basic logical arguments, IF a concrete model or visualization is available. (A lot of the so-called algebra programs try to provide a concrete model, with notably little success in the long run because they are shooting themselves in the foot). As the child enters physical adolescence, the stage of “formal operations” also develops — IF taught (many people never reach it, and the present-day educational system has in large areas given up on trying to help them reach it, or adopted counter-productove strategies). A person who has reached the formal operational stage is able to perform generalized logical operations and can hold a theoretical argument about abstractions such as “justice” etc. Think about arguing a question of right or wrong with a ten-year-old versus a sixteen-year-old — there will be a qualitative difference: the ten-year-old will constantly go back to examples and particular cases while the sixteen-year-old will be able to discuss the general philosophical issues. OK, so algebra IS a system of formalized logic. One of the reasons WHY we teach it is to help the student reach the stage of being able to abstract, to discuss general terms rather than particular cases. This is why concrete models for algebra are counter-productive — sure you produce *answers*, but who cares? They’re in the back of the book. You don’t produce *thinking skills* that way. Exception — some bright kids make the leap on their own. But in general, to teach a thinking skills or logic class by providing ready-made recipes is shooting yourself in the foot.
Anyway, as I said I made this error and tried to introduce algebra before my daughter had become able to perform formal or abstract operations. Luckily she has a distinct character of her own and she told me she wasn’t ready to do it and that was final. Three or four years later she took to it like a duck to water and learned half the course in the car on the way to a placement test for a higher-level class. She had the thinking skills, and just needed the language at that point. With your son at age ten, he is borderline — just at the beginning of being able to do abstract logic. A good student at twelve or thirteen can do basic algebra — this is the usual age in European schools. American schools used to wait until fourteen-fifteen (beginning Grade 9) but when the school system was actually working, the kids were prepared at this age, had number sense and problem-solving down pat, and were really able to absorb all of basic algebra in one year by this age. The present American system teaches it in dribs and drabs, a chapter here and an extra exercise there, starting at age ten or so in Grade 5 and trying to teach algebra twice or three times by using “preparation for algebra” and “pre-algebra” at ages thirteen and fourteen in junior high, and then in the latest generation of “algebra” books, watering it down by doing all sorts of games with calculators and data processing and not even mentioning equations until the second semester. As you have noticed, this is not particularly successful.
OK, back to your son, I woud not do much algebra solving if any at his age. What usually happens is teaching guesswork, a habit that then has to be untaught later. On the other hand, he can very succesfully learn to use formulas, such as areas of rectangles, triangles, circles, volumes of blocks and pyramids and cones and spheres; interest rates; distance = rate*time; and so on. He can also learn to draw graphs of all sorts and interpret data from graphs, a very useful skill in real life and all sorts of careers. (I don’t like to see this stuff take up the first semester of Algebra 1, because that is TOO LATE. These are important skills that need to be learned, and junior high is the best place to learn them for a reasonably able student.) This allows him to get used to letters standing in for numbers without asking of him logical operations he can’t do yet.
(4) When you do get to equations and problem-solving in a couple of years:
*Get a text that has lots and lots of problems, pages and pages of them, and very few pages of lists of equations. Know anybody who sits and solves equations either for fun or a job? But lots of people solve puzzles for fun and use algebra in scientific and technical jobs. Also avoid 1970’s new math texts that have pages and pages of verbiage in between actually doing math — there’s a big difference between doing a skill and talking about doing it.
*Avoid “pick the number up here and move it there”. That’s what you do on a flannel board in kindergarten. Stress the idea of the equation as a balance and of doing the same thing on both sides. This becomes more and more important later, not less, as the math gets more complex and you have to have a plan of attack.
*Avoid quick tricks in general. You are teaching a logical problem-solving method. Any memorized quick trick detracts from your main message of logic.
* Stay as general as possible. A text that always uses the unknown as letter x and always has it on the left because this is “easier” has again missed the entire point. The idea is to think abstractly, to realize that any letter can be chosen to stand in for an unknown, and it can be in any position in an equation. Do lots of problem-solving and choose meaningfuil letters, eg “Let M = Mary’s age” and so on.*
* Take time and do it right the first time. Yes, write out those steps. The easy problems where you can guess and not write anything down are there as stepping-stones to the real stuff. If you only do the easy readiness problems and skip the real meat in the rest of the exercises, the things where you do have to write things down and think hard, you’ve cheated yourself out of the real value of the subject. If you skip steps and hurry, sure you can fill up a page of answers — but again, so what? They’re in the back of the book anyway. It’s the *problem-solving approach* that is the subject and that will stay with you when school is out.
* If you do it right the first time, understand the meaning of the problem, and write out the steps, guess what? You’ll end up being faster than the guessers anyway. There’s nothing more useless than a fast mistake. And on the same line, teach yourself to use pen instead of pencil — this is a class in doing math, not erasing it.

Write me again — I’ll answer as I have time.

Do you mind if I post this for others to read, your question and my answer?

Subj: Re: Preparing for Algebra
Date: 11/18/2001 8:08:52 PM Eastern Standard Time
From: [email protected] (Jean Hancock Eye)
To: [email protected]

Victoria,

Thanks for your wonderful reply- this really helps me clarify some of my
thoughts. We are more or less using the topic by topic approach you
suggest, taking the time to really master a given topic before going on.
That’s one of the reasons he’s getting ahead- once you really know how to
use the multiplication algorithm, you can zip through all the
multiplication chapters- same for division, fractions, decimals, etc. It
takes time to figure it out at first, but then you can roll through a lot
of material pretty quickly. Also, we’ve done a lot of work with problem
solving this year- I’m trying to get him to move away from “Do I know THE
way to solve THIS problem?” and think instead “What’s ONE way that I can
this problem? Is there an easier way?”

I know a little about Piaget’s work, and one of my concerns has been that
no matter how good my son is at all the arithmetical operations needed for
algebra, if he hasn’t reached the “formal operations” stage it won’t make
sense. Solving equations is fairly straightforward, but setting up an
equation to mathematically express the relationships described in a problem
is another matter. Now and then I see glimmers of this sort of abstract
reasoning skill, but for the most part, he’s still in the “concrete
operations” stage. I like your idea to spend lots of time working with
formulas, basic geometry and graphs and before we start Algebra I. As long
as I call it pre-algebra, I think he’ll be satisfied, and he’ll just have
to trust my judgement on this one.

One question about using concrete materials for algebra. You wrote…

OK, so algebra IS a system of formalized logic. One of the reasons WHY we
teach it is to help the student reach the stage of being able to abstract,
to discuss general terms rather than particular cases. This is why concrete
models for algebra are counter-productive — sure you produce *answers*,
but who cares? They’re in the back of the book. You don’t produce *thinking
skills* that way.

It makes sense that using models just to find answers is
counter-productive, but what about starting with a concrete model, then
showing how equations are used to represent the model, then moving on to
using equations to represent more abstract problems? Is this a reasonable
approach?

Of course you may post your reply along with my original question. Thanks
again for your help- time to print this out and put it in my math notebook!

Jean

Subj: Re: Preparing for Algebra
Date: 11/21/2001 2:08:13 AM Eastern Standard Time
From: VictoriaHal
To: [email protected]

Yes, do use a concrete model to start algebra — the scale, equation as a balance, is almost required to develop a real sense of the meaning of the equation operations. But this is step 1, Chapter 1; after that, it is mostly dropped and referred to only in passing when introducing a new operation to do on the equation.

What I am arguing against is buying a large, complicated, multi-part, brightly-coloured plastic set of gizmos (which will be quickly broken and lost, this keeping the company’s sales high) — such things are a big deal in the marketing of “math” programs right now, and as I was saying, are counter-productive as well as expensive.

If you find a *good* pre-algebra book, it will be full of work with formulas and graphs and measurements and business math including formulas and tables and graphs, and problem-solving; a *good* program will not try to teach algebra in a hurry just for bragging rights. Two ways to get a good program: search new texts on the Web and from publishers (examine Singapore math, which I haven’t seen but which I have heard good things about); OR haunt used-book stores — I have a number of old junior-high texts which have excellent work in them. I prefer pre-1955 and the occasional selected post-1990; the period roughly 1956 to 1990 was the period that school math went off-track. A book of this sort will say Grade 8 or junior high or junior math on the cover, and it will be obvious to your son that he is not being held back in baby stuff.

Good luck and feel free to ask again. My time available to answer varies, but I’ll be around (email addrsss may change — watch for a note).