American Readers: What *is* PreAlgebra?

[elmerRex]

What is prealgebra supposed to be?

I can not figure this out.

 

There is a page here, but...how is this different from arithmetic or algebra? It makes no sense to me.

[maize]

It isn't different.

 

Prealgebra really is nothing on it's own, it is just a final year of consolidating arithmetic skills and possibly introducing basic algebraic concepts.

[kiana]

What Maize said.

 

Pre-algebra originally was a high school course intended for students who were not ready for algebra. Later middle schools also started calling the last course before algebra pre-algebra.

 

In whatever curriculum you are using, pre-algebra will be the last year before Algebra. For example, in Saxon, 8/7 and algebra 1/2 are both pre-algebra (because they both review all arithmetic in preparation for entry to algebra).

[RootAnn]

The previous posters have all answered your question well, I think. I wonder, however, what a pre-algebra course could be...

 

I took pre-algebra way back when (after its invention but perhaps before it had been well defined?). My course really wasn't the solidification-of-arithmetic-skills-plus-some-algebraic content that it is now. Mine was more of a exploration of topics that I hadn't seen plus allowing us to experience the beauty of mathematics above just arithmetic. Looking back, it seems kinda Math Count-ish without the actual competition aspect to any of it. My teacher didn't use a text at all. He just made his own exploration sheets for us in a blue ink (not sure what this method of copying was called - ditto machine?). We were encouraged to figure out the answers to word problems, pictorial representations (esp. ones that weren't to scale - aGGH!), and do a lot of problem solving. There was definitely some geometry and trig, but I hadn't had any of those topics before and I'm not sure how much typical teaching he did. 

 

It awakened my awe of mathematics and my love of numbers. It also set most of us up for a great year of algebra the next year as we'd already had experience with thinking through problems that didn't have just numbers we had to plug-and-chug (which is what my math background before that had been).

[mom2bee]

Pre-algebra is a made up course for helping students who have been failed in the teaching of arithmetic and logic and don't have the supporting skills to be ready for algebra.

 

Harsh, but for most students who are developmentally and computationally ready, it's unnecessary. And it didn't used to be part of the scope and sequence of education either publically or privately. It was added to address a need, and that need was mistaken for slower courses instead of better teacher training in how to explain and impart mathematical knowledge and reasoning.

 

:iagree: You said that so much better than I could have!

Please read Arctic Mamas post twice. First the whole post, then only the bolded parts.

[Dmmetler]

It lets kids who aren't quite ready for algebra have a book with "algebra" on the cover. For my DD, who had basic math down, but needed more time learning how to transfer problems from a book and do more difficult problem sets, "Pre-Algebra" let her consolidate those skills and grow up a bit, so when she hit truly new content in Algebra, she already had those EF skills down. If she'd been doing it at 12-13 vs at 7-8, we could have probably skipped that step without issue.

 

 

[Dmmetler]

The previous posters have all answered your question well, I think. I wonder, however, what a pre-algebra course could be...

 

I took pre-algebra way back when (after its invention but perhaps before it had been well defined?). My course really wasn't the solidification-of-arithmetic-skills-plus-some-algebraic content that it is now. Mine was more of a exploration of topics that I hadn't seen plus allowing us to experience the beauty of mathematics above just arithmetic. Looking back, it seems kinda Math Count-ish without the actual competition aspect to any of it. My teacher didn't use a text at all. He just made his own exploration sheets for us in a blue ink (not sure what this method of copying was called - ditto machine?). We were encouraged to figure out the answers to word problems, pictorial representations (esp. ones that weren't to scale - aGGH!), and do a lot of problem solving. There was definitely some geometry and trig, but I hadn't had any of those topics before and I'm not sure how much typical teaching he did. 

 

It awakened my awe of mathematics and my love of numbers. It also set most of us up for a great year of algebra the next year as we'd already had experience with thinking through problems that didn't have just numbers we had to plug-and-chug (which is what my math background before that had been).

 

This is why I like AOPS PA. I know not everyone does, but it's basically a course in competition math using basic arithmetic and in starting to learn basic proof skills using basic arithmetic. There's a lot developmentally packed into that book.

 

 

[Haiku]

I don't have a negative view of pre-algebra. In going through Dolciani with my dd12, what I take from it is that pre-algebra allows students to put their computational skills to work on more complex problems and synthesizes what has previously been taught. I feel that Dolciani is a "capstone" of sorts before my dd begins algebra. I'm a bit baffled by the derision toward pre-algebra.

[RootAnn]

This is why I like AOPS PA. I know not everyone does, but it's basically a course in competition math using basic arithmetic and in starting to learn basic proof skills using basic arithmetic. There's a lot developmentally packed into that book.

 

I know. I thought about saying that AoPS Pre-A is different than a lot of other pre-a courses, but some kids (including many of mine) couldn't handle that book. There are some other books/resources that have bits & pieces of the idea in my post in them including Zaccaro and maybe AoPS's contest math books?

 

I finally understood Base 10 when I encountered other Base systems in my pre-A year. (It was a year of epiphanies since that was also the year I started to understand English grammar through the grammar taught in my French class.) 

[Alessandra]

I don't have a negative view of pre-algebra. In going through Dolciani with my dd12, what I take from it is that pre-algebra allows students to put their computational skills to work on more complex problems and synthesizes what has previously been taught. I feel that Dolciani is a "capstone" of sorts before my dd begins algebra. I'm a bit baffled by the derision toward pre-algebra.

I agree. You can go online and look at AOPS to see what topics are covered.

 

I think the ease/difficulty of the algebra course will determine how challenging the prealgebra course is, in a given curriculum sequence.

[wapiti]

What "prealgebra" (or "pre-algebra") means depends entirely on the context.  Like any other math, the designation of "prealgebra," by itself, says nothing about the challenge/depth level of the material.

 

For the most part, as PPs have described, usually it can be boiled down to the year prior to algebra when arithmetic skills are consolidated *and used in combination* and some basic algebra is introduced.  One might simply call it the final year of middle school math.

 

In some courses/schools, "prealgebra" topics often covered by some other single course/text might be spread over more than one year - say, 7th/8th or 6th/7th/8th, typically designated only by grade level, without the "prealgebra" name.

 

In some schools/texts, "prealgebra" might be a remedial course for high school students, though note that presently, "prealgebra" is not typically considered worthy of high school credit.  High school level math starts with algebra 1.

 

Some of the Doliciani editions with "pre-algebra" in the title date back to the early 1970s, and no one would call those remedial.  In the 80s, IIRC, Dolciani authored two or three different challenge/depth levels for prealgebra.  (As for AoPS Prealgebra, that text is in a class by itself  :001_wub: )

 

ETA, topics often not covered until a "prealgebra" level include arithmetic with exponents, square roots (simplifying, approximating, arithmetic with, fractional exponents), and the Pythagorean theorem.

 

Note also that the Common Core, if it sticks around, may mess a bit with what is considered the standard sequence, around the 8th grade and algebra levels.

[mom2bee]

... I'm a bit baffled by the derision toward pre-algebra.

My post wasn't meant to show derision. Its just...the origin of "PreAlgebra" is a bit goofy to me. Yes, there are definitely worthwhile and valuable books out there that have PreAlgebra on the cover, but most of the prealgebra books (especially those being made today) are not worthwhile books. If K-6 math was taught better, there really would NOT be a widespread need for "prealgebra" texts that are just rehashing fractions, decimals, ratios and NOT expanding problem solving skills.

 

If a book has value, it has value. Regardless of what is on the cover, but if a book does NOT have value, then it just does not have value.

[mathwonk]

Please forgive any perceived derision.  I think this is a result of the conditions under which some of us encountered various topics, and has nothing to do with the value of the material itself.  I am so old that in my day there was nothing but arithmetic, algebra, geometry (plane and solid), trig in high school, then calculus in college.  So to me precalculus seemed like a made up subject introduced for students who had not mastered algebra, (coordinate) geometry, and trig.

 

Then I saw some books called "college algebra", and I thought those were a fake too, another course made up to remediate college students who had not learned high school algebra.  It turned out in some cases these were really old books written before some of those topics were taught in high school, or maybe before everyone took them there?

 

Then in the upper reaches of undergraduate school there began to be classes on intro to analysis, apparently for people who were not ready for just plain analysis.  (Some elementary high school books also use the word analysis, quite incorrectly by mathematicians' standards, for precalculus topics.)  Later we introduced courses in college called "introduction to proof" for students who had not absorbed logic and proof techniques in the natural sequence of courses on geometry, calculus, etc....

 

I had almost never heard of pre algebra until now, but logically speaking it should be arithmetic in my simple world  By the previous posts, when well done, it serves its student population with whatever needs exist at an introductory and preparatory, and sometimes exploratory level.

 

So some of us are answering just what topics are likely included in a book with this on the title, some of us are addressing what topics should precede algebra, and some of us are trying to explain why this name exists, I think.  Forgive me if this analysis (!) is unhelpful.

[wapiti]

If K-6 math was taught better, there really would NOT be a widespread need for "prealgebra" texts that are just rehashing fractions, decimals, ratios and NOT expanding problem solving skills.

 

I agree with this.  The real value of prealgebra, from my perspective, is taking the already-mastered tools and combining them in new ways for problem solving.  Using the tools within a bigger picture/context leads to a greater depth of understanding.

[kiana]

I can't really agree that it's unnecessary. So many programs have named their last year before algebra as pre-algebra that if you skip it, you haven't really completed the arithmetic scope and sequence.

 

Where it's unnecessary is if you do a program such as CLE (where the last course is still named Mathematics 8) and then think that since you haven't done something labeled "pre-algebra" you still need to do a textbook called that.

 

Just follow the scope and sequence of the program you're using. If they have pre-algebra before algebra, do it. If they have math 7 before algebra, do it. If you're changing programs, do a placement test. If you're not sure whether everything is covered, ask here.

[Arcadia]

What is prealgebra supposed to be?

I can not figure this out.

As a foreigner, prealgebra here to me is like the end of elementary school math exam. Whatever topics you didn't get a perfect score for you review and retest.

[mathwonk]

In that sense, I guess ideally a pre algebra course would be prerequisites for algebra, i.e. if you know this material then you are ready for algebra.  That would explain why it might contain both arithmetic and some elementary algebra.  Precalculus could ideally be taken in the same vein.  The realist in me immediately recalls however that in actuality our precalculus students at university seem to have  had a somewhat mediocre success rate in calculus afterwards.

 

Historically, it does seem that such courses are created in response to an observed need, i.e. our students are not doing well in X, so we create a pre-X course to try to help out.  it does not mean this succeeds, but that is the goal.  In college we had such a course to prepare students for proof oriented classes like abstract algebra and analysis.  So when I taught it, it covered logic and proof and some of the early parts of the abstract algebra course.

 

Then I realized that my students were not learning any analysis concepts, such as rigorous limits, so I considered I should have covered some of the easier parts of analysis as well, but there was never enough time for everything.

 

The most difficult challenge was always to get students to realize that each statement is only true under certain conditions.  For instance, if a product XY of integers is divisible by an integer n, then is it true that n also divides one of the factors X or Y?  This is true if n is prime, but not usually otherwise, and I never figured out how to make this sink in, even though I called it the "prime divisibility property".

[mathwonk]

Another odd thing about a "pre-X" course, from the teacher's viewpoint, is that it becomes judged by the results of the next class, i.e. the X class, rather than the present class.  Thus the pre-X teacher will be criticized even if he/she covers well every topic on the syllabus, if someone teaching the next class complains that the entering students do not know everything they need.

 

Of course I was also prone to complain that my entering students had been ill served by their previous classes, all the way back to grade school, if they failed to know what I wanted to assume.  Eventually I began to make every class somewhat self contained, or at least include a quick review of previous material I wanted to use.

 

I always tried to learn from Jesus' parable of the sower, in regard to prerequisites, i.e. sometimes even the message from the best teachers fall on infertile soil, but one keeps trying.

[Momling]

I disagree with the negative pre-algebra view. It's just a title for a book or course. Like most math classes, it includes a bit of review, a deepening of concepts and applications, an introduction to new areas, and a preview of what's to come. We did a few years of what might be called "Pre Algebra" or "Middle School Math" or "The Tricky Bits of Arithmetic with an Introduction to Algebra, Geometry, Trigonometry, and Statistics, etc...". Anyway, it's just a name for the math that typically gets taught in US schools around age 12 and 13 or so; it's not inherently good or bad.

[Arcadia]

Another odd thing about a "pre-X" course, from the teacher's viewpoint, is that it becomes judged by the results of the next class, i.e. the X class, rather than the present class. Thus the pre-X teacher will be criticized even if he/she covers well every topic on the syllabus, if someone teaching the next class complains that the entering students do not know everything they need.

Summer brain drain is real. Some schools give math review homework packets for summer to try to help lessen the brain drain.

 

And of course like you said, even if the teacher did teach, it can fall on deaf ears.

[ElizabethB]

The 1970 Dolciani Pre-Algebra is a work of art!

 

It starts with a few chapters on set theory. Then it incorporates set theory and foundational math properties to prove each topic it teaches. I did well in math and understood it, but did not truly understand it as a coherent whole until I read through my Dolciani Pre-Algebra book. It is definitely not just a rehashing of elementary math if well taught.

 

(I took through Calc 3, Diff Eq, Linear Algebra, college level stats, and also took Ops Research.)

[sweetpea3829]

Oh my lam....math is not my strongest subject, though I enjoy it very much.  

 

After reading this thread, I'm left wondering...how would you define Algebra?  Trigonometry?  Calculus?

 

I took all of these classes, remember very little from them (especially Trig and Calculus) and never really gave much thought to what the math itself was.

 

I'm rethinking a lot of it now, because my son is math-gifted and...let's face it....he's going to enter a realm of mathematics that I do not quite understand.  

[kiana]

Oh my lam....math is not my strongest subject, though I enjoy it very much.  

 

After reading this thread, I'm left wondering...how would you define Algebra?  Trigonometry?  Calculus?

 

I took all of these classes, remember very little from them (especially Trig and Calculus) and never really gave much thought to what the math itself was.

 

I'm rethinking a lot of it now, because my son is math-gifted and...let's face it....he's going to enter a realm of mathematics that I do not quite understand.  

Many people do this.

 

Calculus is the mathematics dealing with changing quantities. When you take a derivative, it's an instantaneous rate of change. When you integrate, you're dealing with the accumulation of quantities. Here's a website I like with a "what is calculus and why do we care" explanation: http://www-math.mit.edu/~djk/calculus_beginners/chapter01/section02.html

 

Trigonometry literally means "measurement of trigons" (triangles). Originally it really applied only to angles within triangles, but later it was expanded to angles larger than 180 degrees, and later still to real numbers. As functions of real numbers, trigonometric functions are very useful for studying periodic phenomena -- this is because every 360 degrees (2pi radians) the values of the basic functions repeat.

 

Algebra is a lot more difficult to define because it is studied at many different levels. There is high school algebra/college algebra (really the same thing) which is more about setting up equations and solving them for unknown quantities. But, in a broader and more abstract sense, it is really about studying number systems and operations. For elementary algebra we restrict ourselves to the real numbers or possibly the complex numbers, but there are many other systems that we study in an advanced algebra class for math majors (seen in university catalogs as abstract algebra or modern algebra).

[Arcadia]

I took all of these classes, remember very little from them (especially Trig and Calculus) and never really gave much thought to what the math itself was.

Trig & Calculus are easier to forgot if you don't need it for daily life. I could remember my Calculus even now only because of all the "torture" in engineering maths. Trigonometry was used in physics/engineering mechanics and in electronics three phase diagrams.

 

My older has already exceeded me in C&P on the combinatorics part. Luckily AoPS does a decent job for independent learning.

 

My math was integrated so I don't have Calculus as a subject but I have differentiation and integration from 9th grade so Calculus was spread out.