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).
ONLY if it replaces thinking -- but that happens too often
I agree with you wholeheartedly about the problem of filing “math class” in one brain segment and “reality” totally separate. At that point little can be done. I often tell both students and colleagues that we would be better off teaching Chinese Poetry than math class as it is now — at least they would come out with some learning and culture.
My issue is not with the use of concrete per se — I am one of the most concrete thinkers and teachers around — but with the use of concrete INSTEAD of teaching abstraction, problem-solving, thinking skills, and so on.
After teaching for a while, you see fads come and go. Whenever somebody comes up to me with their new wonderful game that will cure botts, glanders, and the hives, I always ask “remember teaching machines?” (ask for explanation later if you’re not old enough to have seen this one …)
The fact is that nothing can replace those three dirty little four-letter words, time and hard work. Hard work and time on the part of both teacher and student. Hey, I LIKE algebra, and I solve puzzles for fun, but it didn’t all come to me in a blinding flash — I had to wrestle some problems to the ground first.
The latest fad that I’m worried about is the programs that claim to teach algebra to ten-year-olds through playing games with plastic shapes. This hits several favourite American fad-loves right on the nail — my kid can do it at ten where your old-fashioned school has to wait to age fourteen, so we’re smarter than you are, as well as more modern; I can do it with games and fun, fun, fun, not the horrible old boring blackboard stuff you old fogies use; and I have a cute kit full of bright-coloured toys that I can buy that will do all the work for me, and you just have dull old books and have to write stuff down and waste time and energy. The only favourite fad item that isn’t in the programs I’ve read about is playing with computers and calculators, but no doubt that’s in the next version.
A lot of these programs are like urban legends — everyone’s heard of a great program that works miracles, but it’s somebody else’s in some other town. Meanwhile, I keep asking, “OK, you have this great algebra program; hand me the class of kids who have done Algebra 1 when they’re 10, and we can have great fun with geometry this year now that they’re 11.” But darnit, nobody can produce the class full of kids.
Yes, the occasional kid is gifted and advances faster than most — and will advance faster in almost any program, so that proves nothing — but nobody can show the group of successes. Well, if they can’t be found, I guess they’re mythical.
This takes a load off the rest of us — we only have to keep up with real standards, not mythical ones.
Re: (long) discussion re early algebra, homeschool
Wow, Victoria,
That is the best explanation of Algebra that I have ever heard in my life! I’m quite sure that I did not understand it’s value well until now.
I think with that kind of knowledge, it would be critical to know what math programs currently produced fit your criteria. Do you know anything about Saxon Math? I have looked at the Singapore math site in the past, but was not able to reach any conclusions from the site.
I am not currently homeschooling as I am hopeful that my child will be provided the remediation she needs through the school and by me. But having the knowledge of which programs and curricular material to use is the essential factor. My child is in a fairly new charter school and they are very open to trying innovative and successful programs. I just have to do the research and present the case for particular materials. (So far we do not have math problems as she is only in first grade. She has an auditory processing disorder which has resulted in a language delay, and she is beginning to have some difficulty in reading and spelling. Listening comprehension is poor. I’d appreciate it if you have the time to comment on my question about essential programs on the Teaching Reading board.) All that said, I am preparing myself with the knowledge needed to homeschool if necessary.
Janis
Re: ONLY if it replaces thinking -- but that happens too oft
Hi Victoria & Sue,
Here goes my age showing again. I went to school in Maryland and was a victim of “new math” in ‘62 what a disaster. It took years to get over it.
Math is math is math and some people find it easier than others, it was never easy for me and took hard work. I am glad there are people like you teaching who know playing games won’t get it.
Re: programs
I am not personally familiar with all the newer programs — so much stuff is published that it’s impractically physically and financially to collect or even read everything.
From what has been written on this board and the Teaching Math board, I think that Singapore Math may be a very good program.
Remember — any program is only as good as its implementation. You have to take the time, do the physicalmodelling experiments, draw the diagrams, and all that, to get any good out of any program; and the better the program and the more real-world and physical things are brought in, the more work you have to do (handing out and checking off a multiple-choice worksheet is so easy — and that is the problem with most programs).
Also remember to make haste slowly. It’s a lot faster to take your time and do things right once than to hurry and do them over and over and over again repeating the same errors.
And finally, remember that no book and no program is complete. If kids could learn just from books and computer screens, all the teachers in the country would be out of jobs. Instead, there’s more and more need for special ed teachers and tutors — because the automatic systems can’t do the job. The interaction and discussion of problem-solving are vital parts of learning to really use math.
… for many learners, it is very helpful to see a concrete version of what is going to be represented symbolically. Unfortunately, most of ‘em just stop there and get an answer based on concrete — but one good way to teach that formal operations thing instead of just hoping it happens is to start with the concrete and introduce the abstract generalizations and then practice stretching out the abstract ideas. I’ve learned to refer back to the concrete more often than I *think* should be necessary and have found in the long run that okay, some of these kiddos seem stuck in concrete, but most of them *do* move away from it. Of course, one difference between me & lots of math teachers is that I’ll be watching like a hawk for the kinds of shortcuts that won’t hold up later, and use the concrete stuff to challenge an oversimplified way of doing things.
My opinion’s out on the funky packages of complicated manipulatives — I’m going to do some exploring with them. I suspect that *some* learners will see them as a real bridge to the symbolic — but I could be wrong.
I’m with you all the way on teaching the ideas instead of fifty-seven different procedures. It’s a real *bear* trying to unteach this to older college students who are smart enough to memorize fifty different procedures but have developed a near phobia about thinking about what the stuff means. But this is where I think *more* concrete examples help- these students tend to have concrete and symbolic filed in totally different places in their brains. “Stuff for math class” has its own file drawer and they’ll come up consistently with stuff that flies in the face of the good sense they have otherwise. When I plowed through research on it, it seems this happens at around fifth grade — this ever-widening gulf between “what you do to solve a math problem in math class” and how you think about numbers and amounts when you see them.
The whole advantage goes to the kiddo who’s advanced enough to keep making the connections because his/her number sense is going to keep developing. Math in class will be easy — and s/he’ll get lots of added practice because s/he’ll be observing how it works in the real world, as if it were a language being practiced to fluency. The other people might do okay in math class (for a while) but they may never see the connection between fractions and ratios in a math book, and fractions and rations in the real world — and it really hurts ‘em both ways.
… and there are folks who do solve equations for fun…