http://www.aps.org/publications/apsnews/200910/backpage.cfm

I'd like to hear your reactions to this article.

I've run into this exact same thing with my DS14. It's memorization without any clear intent for understanding.

I tutored math in college. I agree with this author.

Most people will struggle with Calculus even if they are very bright. It takes a certain level of maturity to handle it and a good ability to solve problems methodically. It also takes a thorough grounding in Algebra and Trig. I always started with teaching how to solve problems using a method, then went into the nuts and bolts. I've seen D students make A's and I've seen a lot of others continue to fail.

The answer to the issue is threefold.

1. A lot of people do not belong in a technical field of study at the collegiate level. They need other technical outlets that are not primarily intellectual.
2. Many teachers are not up to the task. Most retired engineers can teach math with their eyes closed, but no amount of instruction will make most teachers' college grads able to teach it.
3. Kids should be grouped by ability based upon standardized tests. This way material can be presented in a way that is appropriate.

The fact of 1-3 means the good teachers ability to impact the right students is diluted by pushing too many kids on the wrong study track. And then most students do not get the right exposure at the right time.

Ouch. That article sparked painful memories of my first attempt at college-level calculus. I was woefully unprepared & dropped the class rather quickly.

I guess that nonsense in my senior year of sitting in the back of a classroom with a pre-calculus book to figure things out on my own with a small group of similarly ignored students didn't help me one whit.

The author's comment about maturity resonated as well. I took another crack at college a handful of years later and absolutely smoked calculus (relatively speaking). I'd had no formal math since my first attempt, so could credit nothing other than being a little bit older.

The author's three main complaints are discussed beautifully:
1. Confusing difficulty with rigor.
2. Mistaking process for understanding.
3. Teaching concepts that are developmentally inappropriate.

Looking back, I think our teachers in HS stumbled the most @ #2. There was a whole lot of memorization, but understanding? Not so much. I'm one who thrives on knowing all the why's & wherefore's -- once I get those under my belt I can run in any direction.

Determining what exactly constitutes "developmentally appropriate" for DS is going to be a challenge -- I can see that already.

Thanks for sharing this article!

I think this is an interesting article and it makes a lot of good points, but it skims over a very real problem: most adults are not developmentally ready for calculus. A currriculum path that prepares students for a science major will not work for most students. As long as we try to find a one-size-fits-all approach, we are going to fail. This article advocates a developmentally appropriate curriculum. Developmentally appropriate for whom? If we design a curriculum that is appropriate for most students it will fail to adequately prepare those with math talent.

It is true that some people confuse rigor with difficulty, but the real problem illustrated by the author's anecdotes is that the teachers were not capable of understanding the material themselves! How does he expect bright students to be prepared for calculus if they can't do rate problems in eighth grade? Or understand what an identity is? These concepts are developmentally appropriate for eighth graders on the track for scientific careers.

Yes, the focus on testing is counterproductive. But it exists because we perceive that there is a real deficiency in the ability of teachers to actually teach math! To fix this problem, we need to pay math teachers enough to buy competence for our public schools. And we need to provide developmentally appropriate curricula for different ability groups.

Of course you can't instruct students by assigning material that they aren't prepared for. I don't think that necessarily means that the material is inappropriate, but that the lack of preparation is systemic. Ganem doesn't address that. He seems to excuse teachers from knowing the material by saying that he himself found it difficult.

Articles like this are used as support for those who want to dumb down the curriculum. I don't think that's the right approach. I think we need to stop expecting to prepare everyone for calculus. I don't think that's a realistic expectation.

My impression of this editorial was similar to Cathy�s crystallized response.

Imagine the �take away� many educators would find and later mention to parents seeking math acceleration for their gifted child. Dr. Ganem should have included the �gifted caveat� as was later added by the author of �The Hurried Child� after his book had been cited as reason to retain countless kids with age peers.

I found myself bristled when reading his analogy about developmentally inappropriate skills since each of my kids walked (cruised) at six months and sight-read a fair amount at two years.

I was nodding my head in agreement at the beginning of the article as he described the problems of students not being prepared for college level math. By the end, I was bristling like delbows when I realized this would probably just be fodder to accuse parents of "pushing" children into developmentally inappropriate math.

DD7 is on a different developmental path as Grinity described so well in another post and I copied below. To me it is developmentally inappropriate to focus on the speed of her math drills. Her basic addition and subtraction speed is fast enough so it's not a hindrance as she tackles 2 digit multiplication and long division. Isn't that the whole point of math drills and not just speed for speed's sake? Grrr!

I have some further questions for the science/engineering experts here.

What is the appropriate level to introduce calculators for use in classroom math and should their use be allowed without restriction from that point forward?

I ask because my kids attend different schools with different approaches (neither could use one in their elementary/middle school math courses). My daughter was required to have a graphing calculator beginning with her freshman advanced algebra (1 and � of 2) class. My son still hasn�t been allowed a calculator in math class (high school honors geometry), including for the ITBS and other standardized tests.

Which approach is better and why? Thanks.

Well now you've got me trying to figure out what the three or four things are! LOL... If you count "factoring" as one thing (although it covers several chapters when you go into the detail of linear vs. quadratic and simple quadratic vs. completing the square and/or the quadratic equation), that's a start... then maybe "isolating a variable" could include all the rules of manipulating equations and situations where one equation will do vs. when you need two or three...

Hey, that covers most of it right there! Maybe there are only two things you need...

I just this morning started algebra with DS8. I have been dragging my feet on it, reviewing the basics because I feared he wasn't ready. But he has been acting out, and I realized that his behavior looked a lot like it looked in 1st grade when he was underchallenged. I made the switch and his behavior this morning was the polar opposite of how he has been most of the school year this year.

That's a long way of saying that I am reading this thread with interest because I'm personally trying to find that balance right now! Tips are welcome!

That helps, Dottie. Thanks. I have been fretting over this for a while.

You know, it's funny how most of our problems seem to occur because he's ready for something and I'm not!

Oh right graphing! If only I had access to that "brilliant" friend's email outbox.....

I agree with the first statement and have been horrified by the math phobia I've seen from some teachers. While retired engineers could make good math teachers for students whom math is intuitively obvious, it can be a pretty bad fit for struggling math students.

It would be better to have teachers like John Mighton who initially struggled with math but overcame through perseverance. He developed the JUMP (Junior Undiscovered Math Prodigies) program and wrote The Myth of Ability. I expected to disagree with much of the book based on the title. It was not what I expected and he asserts young children are capable of understanding advanced mathematics, but doesn't ask students who have fallen behind to struggle with open-ended problems without guidance.

The majority of the book is a step by step logical approach for teaching mathematics. Dottie, it may be a good resource for the girl you are tutoring.

Your book suggestion does sound interesting, Inky.

I�ll admit to being a weak math student (thought to be underachieving due to high math reasoning scores in K-12. In college, my psychology professor guessed I must be math LD after some IQ testing). In any case, the only time I understood math well was on the few occasions, usually the night before a test, when my parents asked the engineer neighbor to work with me for an hour or so.

I wish every school could waive the certification requirement to allow at least one retired engineer on staff for the math dept.

Yeah, I�ve seen that before at a public elementary school, primarily SAH Mothers, who held technical degrees and either volunteered or worked part time in the school being referred to as �paraprofessionals� by the teaching staff.

Hi Guys

I have to agree with what Cathy wrote on page one of this thread. Majority of people will never be developmentaly ready for Calculus. Yet we push it, push it like crazy.
From what I am seeing right now it is becoming a norm, not a high standard to take Calculus in HS. It will not lead to anything good.
Even on this site, and I know that majority of you will disagree with me , I am seeing this huge push to do more complex math early on. Why? I have been for years advocating problem solving.
I have one very high ability math kid and another that is right where she should be, being a bright kid. I so regret starting my, then 5th grader - 10 year old, in Algebra 1. It has lead to problems that I was not able to forsee then.
My now 7th grader is in Algebra 1 with majority of other 7 graders. I think that the brilliance of my first kid has actually done disservice to the school! Now most 7th graders take Algebra and really, really strugle with it! A good friend of mine who teaches math at the parochial high school told me a few years back that she is not going to put any freshman higher than Algebra 2. Now, a few years later she is forced to placed them in pre-calc! Sometimes a good number of freshman, just because they have finished Algebra 2 in elementary school.
Now, and do not think that I am elitist, I firmly believe that a kid like DS happens one in a 1000 or less (probably way less). But the parents, stupid parents, see that there was this kid at school that just wheezed through math and they want their kid to follow the path. Last year, when my DS was an 8th grader, there were two kids in his class that doubled up on math. In elementary school! Just because their parents thought that they can't be that far behind Ghost in math!
So to those that still believe that walking at 6 months or reading at 1 year means that one can do Calculus in elementary school - good luck How old was Einstein when he started talking
Problem solve!

About retired engineers teaching math.
My husband, an electrical engineer, showed me this article ones. It was by the retired CEO or another O of Lockheed. The guy did have original degree in engineering. He is or was a presidential advisor etc, etc. He wanted to give something back to the community and asked to be a part time math teacher at his local school. He was told he lacked credentials (certification?)
Princton took him instead as a guest lecturer...

Raising my hand too!
P.S. I'm not sure I could go back to work for another bureaucracy though. It may be better to stick with being a volunteer.

Two thumbs up!

^^ thanks

Yes, I was schooled in Europe but it was many years ago and it is different now. What I said above comes from observing the behavior of US schools/parents over the last few years. Not from my past schooling experience.

Then again, Ania, I've been doing my darndest to slow DS8 down in math. I've not wanted to let him move past pre-algebra this year. Serious foot-dragging on my part...

But he started acting up. Bored. Cranky. All his usual signs of underchallenge. I can't stay put, no matter how much I want to.

My plan is to move slowly through a simple version of algebra (the "Key to" series) and then have him take the AoPS version next year. I'm hoping that in this way I can keep him challenged, plug any gaps in "the basics" for him (I'm not seeing many), and still slow him down at least a little. (Not as much as I would like!)

But I don't see a way to keep him from algebra right now. It's where we are.

Its a lot faster to do computations in your head for many numbers rather than use a calculator. Its also a lot easier to check work if you know numbers well. There are the basic number things like multiplication tables and division - memorization of up to 20 x 20 or 99 x 99 is not a bad goal. Beyond that, knowing the log tables for base 10 and e will allow you to do division and multiplication for much larger number sets. Knowing sine ( and by implication - cosine ) by tenths of a radian is a good target as well. Knowing the numerical methods to derive other numbers is a good target, too. Knowing primes between 1 and 1000. I could go on. You just never know where you will find a number that you know..

When to use calculators? I say from the very beginning. But with the caveat that you should be able to walk away from the calculator and still be able to do the numbers to a good approximation quickly as a rough estimate at least. Calculators will have to be used if there are a lot of numbers to process - ie stats or a very long problem.

Can most adults do calculus? The answer is no. By calculus I mean a word problem involving rates that must be solved using integration. One needs to know Geometry, Analytical Geometry, and Algebra fairly well to even set up and solve most problems. And one must be able to hold all this information in one's mind and decide from the list of stuff what to use and how to use it.

Calculus is just one small branch of Mathematics, too. Most mathematicians consider Calculus to be trivial.

What is developmentally appropriate? If someone is good at the basic prerequisites for that subject, then they are ready for the subject. Some kids are ready for Algebra when they are nine and others when they are 14 - and others when they are 18. Some - never. Once one has Algebra down and Geometry down, then Analytical Geometry comes next, then Calculus. Mathematically mature students could jump Analytical Geometry and do Logic then go into Real Analysis then back track to Calculus.

Two of our retired PEs teach remedial math and Calculus as insructors at our local Community College. They get pretty high marks. So based on this, I'd have to disagree.

Just one other thing - if some has a very high verbal ability, then they will do very well at math. They just need the exposure. The real mathy stuff like proofs may be more accessible to them than the highly visual stuff in Calculus. Just something to think about.

http://www.megsss.org/index.php?option=com_content&task=view&id=31&Itemid=62

Here is the curriculum for an organization established in 1978 by mathematicians, scientists and engineers to challenge and train (only) the most highly able math students beginning when they are as young as 4th grade. The organization�s mission is to identify and instruct the most promising future scientists. At least in the St. Louis area, far more than 1 in 1000 kids qualify for the lessons based on their results after a battery of math achievement, math aptitude, logic and reasoning tests. Many graduates of this three-year program do test into pre-calc as freshmen when they enter either public or private high schools and continue on to prestigious universities.

I wish I had known about it earlier for my son.

Kristen, I had thought this might be a good home school curriculum for your son. I don�t know if you can buy the books anywhere though.

Interesting that you mention this as one of the initial qualifiers for the program mentioned above is reading scores at the 95th percentile.

That's good to hear. Don't get me wrong, it's critical that the teacher get math but it's even better if he/she gets why some students don't get math. Unfortunately, many of the engineers I know don't get why other people don't get math.

For example, I'd be in class following along in lecture with one of my engineering professors. All of sudden he'd stop deriving the equation and just say "the rest is obvious." Well, it usually wasn't obvious to me.

It's the same with grammar though, so don't feel too guilty. It can be hard for English-y types to explain why something is incorrect or inelegantly written. It just is. I think that's just the nature of being more-or-less a natural at something.

delbows, thanks for the link! I really appreciate your thinking of me! I'm looking at it now...

I owe you one!

Hi Ania,
It's late and perhaps I'm misreading your post but I'm not understanding why you regret early AlgI for your DS? Is he having problems now or is it b/c of the effect it had on the school in placing other kids in early AlgebraI who don't belong there?

^^ CFK is right on the dot. It has to do mostly with logistics.
And you really do not know where you are going to be until you are there, unfortunately.
To be more specific - University is far away from us, classes have to be taken in the evening as freshman and sophomores have to have a full load of classes at school, which really means doubling up on workload. Having U classes in the evening takes away some extracurriculars. He doubled up on science this year and even though he is doing physics, not normally taken by freshman, it is still not calc based, which turns to be really tedious for him. Again - do not get me wrong - this is honors physics and he will most likely switch to AP Physics in a couple of weeks, and he is learning tons of new things, but doing it based on calc would have been so much easier for DS!
What can you do not to rush ahead? Spend 2 years on Geometry! Geometry is not really taught in american schools. One year of basics. Never returning to it again. Kids do not know how to write proofs. Even the smart ones do not know! So if you kid tells you that he knows or understands something intuitevily tell him to prove it. Prove it in two or three different ways!
That should cure any boredom...

I just read this thread with great interest. I have a technical/math background as well and am homeschooling an up and coming 3rd grader. I have a BS in math, took grad level math, and have never had a graphing calculator! I guess that would make me mostly in the anti-calculator camp.

I do think the article has some valid points for the ND child. Of course, YMMV when you're dealing with a HG+ child. I do think in the average classroom pre-algebra operations should be quite automatic by the time you're getting to algebra and that conceptual understanding should be there.

We are using Singapore at home. We'll come to the end of the elementary Singapore series this year some time and then we'll have some tough decisions to make! I have been introducing pieces of algebra as we go along, but am really dragging my feet getting to "real" algebra. I guess it's the traditional math student in me that wants him to have the very solid math grounding before we get there.

Also, sign me up to teach math in a few years! I would love it. I love math too.

This is why many rigorous math programs now require a year of Logic or its equivalent before starting Real Analysis. There is not a lot of difference between reading proofs and reading essays. At least, if the essays are logically written. This is why I believe that high verbal ability lends itself well to math achievement.

I really like this! It'd be great to have the option to compress the elementary math and then use that time to expand time spent on things like geometry, problem solving, and logic.

Inky, that's to a great extent why we're homeschooling. I still feel like we're wandering in the dark sometimes, but for my deep-but-not-fast kid, doing harder math and doing it for longer makes sense.

Anyone have a way to do that within a bricks-and-mortar school? I'm curious about other solutions that use a similar approach.

Austin - I read the very same thing on another website re: high verbal ability and math achievement. Because my son had always seem mathy (although not to the level of many here), I had expected his PRI to be greater than VCI on WISCIV. His VCI blew his PRI out of the water. He grasps algebraic concepts easily and makes those leaps of thought which makes me go "hhhmmm." But give him a page of computation and he's under the table. ha ha ha. It took reading "Math Talent" to get over my thoughts that math-talented kids are fast at computation, can hold a ton of numbers in their head...I know they often are but they don't have to be.

Dazey

Finding a good teacher. My dd9 has been very fortunate this year to get a teacher who gets her and LOGs. A couple of weeks into the school year he noticed that her rate of learning was faster than that of the other kids in his class (7/8th graders). He decided that she could cover the whole years' material in a couple of months, so he is letting her go "as fast as she wants but as slow as she needs" (meaning she doesn't have to slow down to wait for the other students but if she has problem with a concept she can take all the time she needs). He decided that bumping her another year would not solve the problem as she would be in the same situation in a couple of months. Instead, he is giving her and another kid individualized work within the classroom. He is allowing them to go deeper into the subject and spending a lot of time on problem solving. My dd has been accelerated in math since 1sth grade (3 years so far), but she always had to stick with the class until now, which meant a lot of waiting (although she had a great 2nd great teacher who was wonderful about teaching them advanced concepts). Now for the first time, she is allowed to go at her own pace. This year she is really enjoying math. Her school teaches up to calculus AB in middle school, so we may have some of the same problems Ania is having, but we will have to cross that bridge when we get there. With any luck we will find a HS that offers advanced math work (Exeter seems to be the favorite right now). I really don't want her to attend college courses until she is in college....

Bianca, funny you metioned Exeter Check my other posts.

kcab - my "not mathy" DD 12 (who is still the best student in her Algebra 1 class) is starting an AOPS class this week. I will let you know how it goes...keeping my fingers crossed...and I am very thankful for their very generous withdrawal policy

Hey... funny you BOTH mention Exeter!

I haven't (yet?) regretted the rush to Algebra, but then a) we're homeschooling, so logistics isn't a problem and b) once we got to algebra we had so much more we could "wallow" in that we haven't kept up the pace and I'm saving as much regular math as possible for as long as possible.

We generally alternate years between regular math courses and interesting side things (Algebra/ Statistics/ Geometry/ Number Theory? that's next year...) in an effort to keep from hitting every single topic on the standard scope and sequence before high school...... But one of my backup plans is boarding school. Exeter looks fascinating for many reasons... but especially that they seem to have a really high "ceiling" on their math possibilities.

Here are some ideas from Developing Math talent (Davidson recommends this book)

accelerate students who have mastered computation, Include open-ended problems and projects that challenge able students to keep their interest in the subject.

enrichment topics:
Fractals and chaos
the Pythagorean theorem and Triples
Fibonacci Numbers
Finite differences
Pascal's triangle
Golden triangles
magic Squares
Other numerical systems (Babylonian, Egyptian, Chinese, Mayan, Roman, ect.
Pi
Topology
Pendulum and other applied physical problems
Transformational geometry
Combinatorics
Graph theory
Computer programming and robotics, including Logo and Lego- logo

Hope this helps. I don't even know what most of this means but I have in mind to pace my DS by giving him some programming to do at school to replace some of the repetions in work he doesn't need. I'm hoping I can make this work.

Feeling dumb, but what's Exeter? Where is it?

If all of you are excited about it, it sounds like something good! I trust your judgement!

Thanks for the recs, CFK and onthegomom. We've already done several of the enrichment topics, but not most of them. And the book is new to me. Good stuff!

This helps!

Wow! and this too:

Exeter offers a free education to any admitted student whose family income is $75,000 or less

I don't know if this will be helpful or not (bear in mind that musicians only need to be able to count to four!, so I am probably very far from having much to offer on the math front), but a book we've been reading lately sprang to mind as I read through this thread: Moses Richardson's "Fundamentals of Mathematics" (Macmillan, 1941; OOP, but there are editions up through 1973, so lots of copies around). It's a textbook Richardson developed to teach the one-year compulsory math course for humanities majors at his university; while he presupposes previous exposure to elementary algebra and geometry, "almost no accurate recollection of the details of these subjects is prerequisite for this book" (vi). He thinks mathematics has a great deal to offer students who have little need for technical skill in the field, and his objectives here are to give the student:

"(1) An appreciation of the natural origin and evolutionary growth of the basic mathematical ideas from antiquity to the present;
(2) A critical logical attitude, and a wholesome respect for correct reasoning, precise definitions, and clear grasp of underlying assumptions;
(3) An understanding of the role of mathematics as one of the major branches of human endeavor, and its relations with other branches of the accumulated wisdom of the human race;
(4) A discussion of some of the simpler important problems of pure mathematics and its applications, including some which often come to the attention of the educated layman and cause him needless confusion;
(5) An understanding of the nature and practical importance of postulational thinking" (v-vi).

In service of these objectives, he examines several topics (number theory, algebra, logarithms, impossibilities, analytic geometry, functions, limits and the calculus, trigonometric functions, probability and statistics, finite and transfinite cardinal numbers, euclidean and non-euclidean geometry), using the sense of mathematics "as the totality of logical (hypothetico-deductive) systems and their applications" as the organising principle/unifying theme throughout. He points out that while some of the book looks "hard," "the early fundamental paragraphs of a so-called 'advanced' subject, presented at the proper level are often easier to grasp as well as more important and more interesting than the later technically complicated paragraphs of what has traditionally passed for an 'elementary' subject" (ix, vi).

Now obviously, our collective kids are not first-year liberal arts majors, but it seemed to me on reading this thread that this kind of approach might for some people be both very useful and rather intriguing in regard to our kids and their unusual needs. Like many others here, we are bumping up against algebra early, and this struck me as one possible means of slowing things down, as well as laying a pretty thorough grounding in a real understanding of mathematics; especially for homeschoolers operating outside of the system, anyway, I thought I'd toss this out there as an idea. I worry a lot about math, since Harpo seems pretty good at it, and (standard academic dodge) it's not really my area.

I could be completely out to lunch about this book, though! Anyway, so far, both Harpo and I are enjoying this book a lot (I've only had it since Canadian Thanksgiving, though, so we're only a few chapters in at this point).

Hope that helps somebody (or at least wasn't completely irrelevant)--

peace
minnie

It interested me enough to track down a copy. Thanks!
http://www.amazon.com/Fundamentals-Mathematics-Moses-Richardson-Leonard/dp/0023996900

^^we have this book, I think?
DH used it with younger kids when training for Math Olympiad.
I think we even got it from Math Olympiad side???

DD starts tonight. I will report back.

First class is over. It went extremely well for her! I was /still am actually shocked!
I always judge how my kids understand the material by the way they can respond in the classroom. DD had over 10 published responses and right after class she jumped into assigned homework.
So far very well

I found this, and thought it might interest some (I don't know anything about it, but thought it looked kind of keen):

Problemoids Math Challenge Program (Bill McCandliss, Albert Watson, Royal Fireworks Press, in 3 volumes).

http://www.rfwp.com/series42.htm#119

There are some other math resources on that site that look interesting, too.

peace
minnie

^^ like I have said previously - great return policy!
I like your location - middle of the mess...

(Joining this thread late.)

The approach to teaching maths in the French system is quite different from what I've seen in US schools.

French math classes move somewhat more quickly than US classes, but not wildly so (eg, they start doing multiplication seriously in late-ish second grade). They also go more deeply and seem to make a greater effort at ensuring that students understand concepts.

For example, the idea of tens is taught by showing boxes with up to ten balls in them. Once a box gets filled, you close it and it becomes one ten instead of ten ones. This idea feeds nicely into regrouping: you take a box of tens out of the tens area, move it to the ones area, open it, and turn it into ten ones. Does that make sense?

I know that US schools teach regrouping, but I've never seen an American example that's as crystal clear as the French method. Plus the French method is used all over France and no one wastes/duplicates effort on designing their own methods.

I don't know about mathematics at the higher levels. The most I can say is that my husband worked for a French company for a while and he always talked about the exceptional math skills of the French employees. And I think that math education in other countries is even better (Russia? Finland?).

We noticed big differences between the French and US approaches when our DS was still in 1st grade.

HTH,


Val

Wow MON! Of course each kid is different, but I'm thinking - keep public for good math, and send them to CTY online during the year or over the summer to the summercamp for the writing. Who wants to pay for private and then pay more for 'enrichment?'

Wink
Grinity

Regarding logistics and university. Don't universities in the US offer distance education?