Response Please

[David Chandler]

As some of you know I produce curriculum materials for homeschoolers (see http://mathwithoutborders.com).  I currently have completed Algebra 1 through Precalculus and am doing Calculus this year.

When parents ask me for recommendations for Prealgebra I don't know what to tell them.  My (candid) personal feeling is Prealgebra is not a "thing."  It is a holding pattern for students who have finished their arithmetic skills but are not mature enough to start Algebra 1.  I don't know of anything that is covered in Prealgebra that wasn't already covered in K-6 or won't be covered better in the high school curriculum.  My tendency is to have students who have mastered all their arithmetic skills and can do word problems to start Algebra 1, and take two years, if necessary, speeding up if appropriate.  I have made up an arithmetic skills checklist and have parents go over this with their kids to decide if there are missing elements in their backgrounds.  (See attached.)

Are any of you in love with Prealgebra?  Do you have alternate responses you would give on this question?  Any omissions on my checklist you would add?  (I sad down one afternoon and did a "brain dump" to come up with the list.)

--David Chandler

[Paul Libbrecht]

Looking at the curriculum in, say, Germany or France, you'd find such topics as value-tables (e.g. linear proportions), and mappings. I don't see these there.

paul

[David Chandler]

I'm not sure I recognize the term "value tables."  Do you mean reading and interpolating tables?  Like old time trig tables or log tables?

--David Chandler

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[Paul Libbrecht]

On 17 sept. 2014, at 09:24, David Chandler <david...@gmail.com> wrote:

I'm not sure I recognize the term "value tables."  Do you mean reading and interpolating tables?  Like old time trig tables or log tables?

;-)

I mean like the representations of such indeed.

Where you have an input and an output (often with several outputs).

This is commonly used when noting results of an experiment.

This is an effective representation of an idea of a function.

This is very commonly used as "tableaux de proportionalité" to show linear proportions.

And… isn't this a similar representation used with spreadsheets?

paul

[Christian Baune]

Paul,

in databases, we call these "value tables" "relations" and we use set theory to massage them :-)

All of these "value tables" model a domain (most of the time, a business model).

So, "value tables" are still in use, they are only a bit hidden from the end user view.

Should one visit a website, shop online etc. At the end, at least one read is made somewhere in a "value table".

Christian

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[Maria Droujkova]

On Wed, Sep 17, 2014 at 2:15 AM, David Chandler <da...@mathwithoutborders.com> wrote:

Are any of you in love with Prealgebra?  Do you have alternate responses you would give on this question?  Any omissions on my checklist you would add?  (I sad down one afternoon and did a "brain dump" to come up with the list.)

--David Chandler

Pre-algebra or pre-calculus as holding pens are not for me. But here is something totally else to consider while making checklists: early algebra. Which I do love!

The term applies to intuitive, informal activities you can do with young kids, even toddlers and babies. Typically, these activities sidestep arithmetic prerequisite checklists, and can be done in parallel to mastering arithmetic.

For example, function machine games, visual explorations of symmetry groups, exponentiation via building fractals, "solving for X" with subitizing or approximate number sense in storytelling... We have lots and lots of examples in the Moebius Noodles project: http://www.moebiusnoodles.com/ And this year, we are working on publishing several more early algebra books by different authors. 

 

Cheers,
Dr. Maria Droujkova

moebiusnoodles.com
919-388-1721
-- .- - ....
 

[Sue]

David, do you participate on Living Math Forum (a Yahoo group)? They (over 5000 members, mostly homeschooling moms) discuss questions related to this in depth. If I remember correctly, Julie Brennan (the host) and others have found AOPS's Pre-algebra text immensely valuable.

I think lots of kids finish learning the basics needed for algebra before they're developmentally ready for the full-fledged algebraic thinking, and need some curriculum to keep them moving forward at that stage.

Warmly,
Sue

[John Mason]

I am intrigued:

Jean Schmittau and Davidov seem to show that very sophisticated
algebraic thinking is available very young.

I was asked recently by secondary teachers here in Calgary, how to
respond to the question "why do I need to know about factoring
quadratics?". It seems to me that algebra is about learning to work
with the general, the as-yet-unspecified, the symbolic, and as such is a
vital component of citizenship, since participating in society involves
handling of all sorts of symbol systems. I tried them on "customers want
a number; entrepreneurs need algebra (to state and analyse policy)"; but
they wanted to think about it.

Surely algebraic thinking starts in the womb, and is needed in order to
learn arithmetic (eg. Dave Hewitt)?

JohnM
Killam Visiting Scholar
Calgary

[Joseph Austin]

What’s the difference between Freshman College Algebra and Senior HS Calculus?  4 years of your life!

Kids should learn math as soon as they want, not be forced to wait until the teachers are ready!

I agree with “early” algebra.  I remember trying to learn percentage in 4th or 5th grade (make that 4th AND 5th grade),

and having to memorize 3 formulas and decide which to use when. When I got to algebra, it seemed it would have been so much simpler if only I had known about rearranging formulas.

  

One key skill that I believe must be included (but in my experience often is not) is setting up formulas from word problems.

Again, I think word problems would have been so much easier if I could have just set up the formulas as phrased and rearranged them algebraically. As it is, I had to do that anyway, in my head, without actually having been taught a procedure for doing so.

I had hated word problems. I think now the reason is, I had never been taught how to do them.  I suspect the reason I was not taught its that the teacher probably didn’t know herself.  But a friend of my mother was a substitute math teacher, and she specialized in teaching word problems—that’s what she did whenever she subbed, because the students never knew how to do them.

[David Chandler]

Hey...That's a great use of sub time.  If a "real math" sub could go in with a large collection of word problems in various contexts, set aside the nominal curriculum, and teach problem solving, that would be a fantastic use of what is too often wasted time.  One of the best experiences I had, at the start of a school year in a new school, was when I just polled my Algebra 2 class on what they were confused about in their prior background.  We spent a couple of days going over topics of their choosing and establishing a common foundation in a class where the spread in abilities and preparation is often quite wide.

By the way, I don't consider "Precalculus" in the same boat as "Prealgebra."  It's a time to deepen skills and introduce some of the many topics that get shortchanged along the way.  I think the primary reason students have difficulty in Calculus is they really do need to know pretty much everything they have been taught up to that point.  So many new ideas are introduced in Algebra 2, Precalculus gives them a chance to achieve fluency.

The same attitude probably applies to Prealgebra, but I haven't seen it handled well so far.  I have ordered the AOPS text as suggested.  The title looks promising!

--David Chandler

[Maria Droujkova]

On Wed, Sep 17, 2014 at 1:09 PM, David Chandler <david...@gmail.com> wrote:

 I think the primary reason students have difficulty in Calculus is they really do need to know pretty much everything they have been taught up to that point.  

While this might be true about many courses, everything John Mason wrote about early algebra applies to early calculus. Let me quote John:

"Jean Schmittau and Davidov seem to show that very sophisticated algebraic thinking is available very young.

I was asked recently by secondary teachers here in Calgary, how to respond to the question "why do I need to know about factoring quadratics?".  It seems to me that algebra is about learning to work with the general, the as-yet-unspecified, the symbolic, and as such is a vital component of citizenship, since participating in society involves handling of all sorts of symbol systems. I tried them on "customers want a number; entrepreneurs need algebra (to state and analyse policy)"; but they wanted to think about it.

Surely algebraic thinking starts in the womb, and is needed in order to learn arithmetic (eg. Dave Hewitt)?"

I just put up the recording of our Monday event in the Math Future series, where Melissa Kibbe shared her cognitive psychology research on algebraic thinking with kids ages 4 to 6: http://www.moebiusnoodles.com/2014/09/cognitive-psychology-of-kids-ages-4-6-learning-algebra/

This year, I've been experimenting with early calculus, as well. Here is my interview for The Atlantic that mentions it: http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/

I do think prerequisite-happy course design contributes to the mass failure of calculus students in conventional courses.

Cheers,
Dr. Maria Droujkova

moebiusnoodles.com
919-388-1721

.-- .... -.--

 

[David Chandler]

I am fully in favor of experiences with "early algebra" and "early calculus."  My concern is that the rush to calculus I see among some high school students is a misguided focus on acceleration rather than depth.  There is a lot more math to explore at the pre calculus level (or "other than calculus" level) than can be crammed into the Algebra 2 curriculum.  I would rather spend more time exploring trig, logs, and exponential functions, analytic geometry using vectors, topics like matrix transformations, formal and informal logic, deepened problem solving experience, and how about some computer fluency somewhere along the line, rather than the headlong rush for the holy grail of slopes, areas, and limits.  In my experience with math outside the classroom, including physics and engineering related projects, it is these other topics that account for  90%+ of the actual mathematics that comes into play.  Having an extra year of math after Algebra 2 before Calculus gives space for this deepening and broadening.

--David Chandler

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[John Mason]

Hear Hear, David!

I agree... depth is much to be preferred to acceleration!

JohnM

[kirby urner]

On Thu, Sep 18, 2014 at 7:32 AM, David Chandler <david...@gmail.com> wrote:

I am fully in favor of experiences with "early algebra" and "early calculus."  My concern is that the rush to calculus I see among some high school students is a misguided focus on acceleration rather than depth.  There is a lot more math to explore at the pre calculus level (or "other than calculus" level) than can be crammed into the Algebra 2 curriculum.  I would rather spend more time exploring trig, logs, and exponential functions, analytic geometry using vectors, topics like matrix transformations, formal and informal logic, deepened problem solving experience, and how about some computer fluency somewhere along the line, rather than the headlong rush for the holy grail of slopes, areas, and limits.  In my experience with math outside the classroom, including physics and engineering related projects, it is these other topics that account for  90%+ of the actual mathematics that comes into play.  Having an extra year of math after Algebra 2 before Calculus gives space for this deepening and broadening.

--David Chandler

Yes, I'm in sympathy with this view.  We miss too much in the rush to Delta Calculus, as I've taken to calling it. [1][2]

Where we introduce the greatest common divisor (GCD) is a place to slow down more, as if gcd(a,b) == 1 then a, b are relatively prime, what some call "strangers" (no factors in common).  The whole prime versus composite number distinction comes in.

The strangers < N, of N, are its totatives (including 1 itself) and totient(N) is how many totatives N has.  If in is prime, its totient is N-1 as all numbers less than it are strangers.[3]

All this stuff is highly comprehensible without a lot of prior math, if taken slowly, plus here's a place to introduce Euclid's Method for the GCD, identified by Knuth and others of paradigmatic of what we mean by "method" in the sense of "algorithm".   A little programming comes in.

However K-12 happily bleeps over Euclid's Method for finding gcd(a,b) in favor of factoring a,b into primes and intersecting the two sets for a GCD product.  It's not either/or and shouldn't be.

Anyway, a pet peeve. 

All that totient stuff comes home to roost when we get to public key cryptography and the RSA algorithm in particular. 

Euler's Theorem and Fermat's Little (not Last) Theorem then get front burner treatment.

But hardly ever do we follow this spiral in K-12, because of the rush to do Riemann Sums, as if that's all the math you'll ever need or care about. [4]

Kirby

Notes:

[1]  http://mybizmo.blogspot.com/2014/08/delta-calculus.html
[2]  http://mathforum.org/kb/thread.jspa?threadID=2649411
[3]  http://mathworld.wolfram.com/TotientFunction.html
[4]  https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/definite_integrals/v/riemann-sums-and-integrals