kitchen table math, the sequel: help desk - precalculus

Tuesday, May 24, 2011

help desk - precalculus

This year's math course has been a disaster. Beyond disaster, actually. Where math is concerned, this year has been epically* bad.

Some kind of emergency repair has to happen this summer - but what?

Any suggestions?

I could conceivably sign C. up for an algebra 2/precalculus course at the local community college - but are the teachers there going to be any better? At this point, we desperately need an actual math teacher: a person who can teach math to a student who doesn't teach math to himself.

We could do ALEKS, but ALEKS is super-slow and overwhelmingly procedural; the 1 1/2 courses I took on ALEKS taught me one disembodied procedure after another. At this point disembodied procedures might be better than nothing -- but then again to the extent C. learned anything this year he learned disembodied procedures.

I could insist that the two of us work through Foerster (I own the teacher's edition).

We could work our way through Saxon.

I could advertise for a math teacher or check out the various tutoring companies....

We also have to do serious SAT prep (though that's not going to be onerous & time-consuming).

I'm thinking this wasn't the summer to sign up for the precision teaching institute at Morningside.

* My neighbor's son, who is a terrific writer, is constantly inventing words he says ought to exist, and having watched him do this a few times, I think he's right. Epic is an excellent word, and its non-existent cousin epically is the word I need today.


Jo in OKC said...

More options:
* Work through an Art of Problem Solving Algebra book. Look through the table of contents to find the appropriate one.

* Thinkwell. Ed Burger's an awesome teacher and all the videos are of him. There's not an unending supply of problems, though (it doesn't algorithmically generate more problems on the same topic, just with different numbers).

* EPGY through Stanford.

* Johns Hopkins CTY uses something other than EPGY for their advanced math classes. Not sure what it is.

Some people like ChalkDust or Math-U-See. I don't have any experience with them.

Anonymous said...

There is also a DVD that goes along with the Foerster book with video lessons.

Jean said...

I will say that the two math courses I took at junior colleges were the best two I ever took. (I did college algebra and trig.) But it's really going to vary a lot, of course.

Maybe hire a tutor? Have you got a local university with a math department? You could find a math major maybe? Or if you know any adults in math-heavy occupations--my husband is an excellent tutor in higher math...

Catherine Johnson said...

Thank you!

(And keep it coming.)

I've emailed the tutor we have now, who I think is fantastic; looks like he'll be able to teach an algebra 2 course this summer to 4 of us (assuming that's how many decide to do it).

He only wants to do algebra 2, though, not precalculus.

I don't know the difference ---

I have the Art of Problem Solving Algebra 1 book -- I'll look at the Algebra 2 TOC & have C. take a look, too.

I've gotten very tired of the 'discovery' organization of the AoPS counting book, though. VERY tired. Am finding it tedious constantly having to construct a hierarchy of the subject & its concepts & techniques.

That might not matter for algebra 1 & 2, though, where we already know a fair amount of the material & something about the structure of the subject....

Catherine Johnson said...

Is there a good assessment I could have C. take?

Maybe the COMPASS test?


Anonymous said...

Bummer about this :-(

Do you know why this year's math (I'm assuming it was Algebra 2) was a disaster? Specifically, did C. have the pre-reqs mastered and then flailed at the new stuff *OR* did he struggle because the pre-reqs were rocky?

-Mark Roulo

Anonymous said...


Even though you don't seem to want to use ALEKS for learning, you could use ALEKS for assessment only.

It would give you an idea of which concepts have not yet been mastered.

ChemProf said...

While I'm not a huge fan of, this seems like a place where it can be useful. Can you look up the possible instructors at the community college? If you use it carefully (since you don't want a class with a high "easiness" score), that kind of rating site could help you avoid a disaster.

Catherine Johnson said...

This is probably better off-list, but it's a teacher problem.

The teacher is ill and is probably cognitively impaired (that's what it looks like from email). He can't be fired (we've been told this directly), and the school did not help us when we asked to have C. taken out of the class last fall.

That was another epic fail moment: last fall I was teaching AROUND THE CLOCK and dealing with my father's sudden death; I didn't have the wherewithal to fight a get-my-kid-out-of-the-class battle, and Ed didn't pick up the slack on that one.

The kinds of things that are going on are gobsmacking.

The teacher gives the kids 0s on quizzes when they are out sick; he just gave everyone 0s because they were taking AP exams and so weren't in his class.

The other day he told them all they're bad at math & said they belong in special ed. This is what goes on in class time.

When they do badly - and there is a huge amount of doing-badly in the class - he berates them. He makes no adjustment to his teaching at all, and his teaching consists of simply writing up the examples from the book on the blackboard.

He has quizzes and tests where the highest grade is a C, and the only kid who gets that is the gifted kid who doesn't need teaching. Half the class fails tests, and when that happens he tells them "You don't do the work."

He also tests them on material he hasn't taught.

It's been a nightmare, and it's gotten fantastically bad here at the end where the class is covering material C. has never seen before.

Neither Ed nor I realized how much of the material C. already knew going into the class. He got through the first quarter very well because he'd already learned the material. Then he got through the second quarter pretty well for the same reason - though Ed realized then that we were headed for disaster.

Basically, C. has had an entire algebra 2/precalculus course without a teacher.

A friend's son told his mom he's been teaching himself the material out of the textbook --- and the text they use is not a good book for that.

Now they've started Stewart's calculus & it's incomprehensible as a self-teaching text.

The tutor we hired has also discovered a number of grading errors, so we have NO idea what C's real grade actually is (true of the other boy we know).

There's damage all over the place, not just to C. Kids we know are being kept out of courses they need to take on grounds of their grades in this course.

The thing is, precalculus & calculus tutors are impossible to find. We've been able to locate only one, and he wasn't effective.

I don't know what we could have done this year other than fight like the devil to get him out of the course last fall -- and we had no assurance of succeeding in that.

Catherine Johnson said...

While I'm not a huge fan of, this seems like a place where it can be useful. Can you look up the possible instructors at the community college?

Good idea!

I also have an acquaintance whose cousin teaches at the community college. I met him a few years ago & he said he could vet teachers for me. So I may need to get to that (assuming we're not already too late to sign up for summer classes...which we may be.)

Catherine Johnson said...

Even though you don't seem to want to use ALEKS for learning, you could use ALEKS for assessment only.

I was wondering about that myself---their assessment seems pretty good.

SteveH said...

I'm not sure what "disaster" means. If you are talking about a remedial problem for material already covered, then a one-on-one tutorial approach would work best. If you are all covering new material (pretty much), then a group lesson might work well.

With my son, I can cover a lot of ground very quickly. I know what difficult variations he might not have seen and whether he can approach a problem from different directions.

There is also the question of how you will decide what level you need to get to and how you are going to determine success. With my son, I was just following the textbooks he had, which were decent. However, now I have the issue of getting him prepared for SAT math, which is a different thing, especially the need for speed. Of course, SAT only goes so far. When you get to pre-calc and calc, you're on your own to determine whether you are prepared enough. I don't even think AMC/12 works well for that. There is nothing like a good textbook followed carefully.

I don't like AoPS as the initial introduction to new material, although I haven't seen all of their publications. I think their general problem solving books will be good for my son now as more of a second, more rigorous, pass of the material.

In general, I would advocate an emphasis on getting really good at algebra; backwards and forwards. This will make calculus pretty easy. My son is wrapping up his course in Algebra II and I can see him getting there. I can give him complicated "simplify" problems, and he doesn't make mistakes.

Anonymous said...

A course called precalculus draws its topics from three areas:

1. Algebra 2 + Trig (solving polynomial equations, working with logs and exponents, trigonometry)
2. The beginning of a calculus class (limits)
3. Grab-bag of topics that don't fit in elsewhere (polar coordinates, DeMoivre's Theorem, proof by induction, summation notation, etc.)

In order of priority, #1 is most important. Lacking the skills of #1 will lead to a calculus disaster.

#2 depends on the calculus class next in the sequence: Do they expect students to remember limits? Or will they expect forgetting and reteaching?

Not knowing the items of #3 will lead to comments about "kids these days" not knowing as much math as we did in the past. This is unlikely to lead to problems, especially in college, as most Kids These Days don't know these things, so no one expects them anymore.

SteveH said...

"The other day he told them all they're bad at math & said they belong in special ed. This is what goes on in class time."

Yikes! This sounds like a teacher my son had in 7th grade. Sarcasm and putdowns were used for class control. He also told the class the low math scores some received on the state test and then lied about it.

A good tutor can make all of the difference, but it sounds like that's tough to find.

Anonymous said...

I'm surprised that Algebra 2 is being mixed with pre-Calc. Usually, pre-calc is some combination of Trig and "College Algebra." The entire set-up sounds weird, in addition to be useless teacher. Sounds like the school (is this the Jesuit school?) should provide a summer class for all the students, to catch them up.

Catherine Johnson said...

In general, I would advocate an emphasis on getting really good at algebra; backwards and forwards.

What about trig?

Catherine Johnson said...

I'm surprised that Algebra 2 is being mixed with pre-Calc. Usually, pre-calc is some combination of Trig and "College Algebra."

What is "College Algebra," anyway?

Catherine Johnson said...

I'm surprised that Algebra 2 is being mixed with pre-Calc."

Here's what I think is happening (based on what our school has told us & what the tutor has told us).

I think a lot of schools are giving kids longer to take calculus by including some calculus topics in a junior-year algebra 2-type course.

In fact, C. took algebra 2; this year is 'precalculus.'

Catherine Johnson said...

This is the moment where kids' future math learning is ended if the parents can't reteach.

I can't reteach.

I don't think a weekly tutor could have fixed this situation.

We need a math teacher.

Catherine Johnson said...

This sounds like a teacher my son had in 7th grade. Sarcasm and putdowns were used for class control.

Apparently, there was quite a lot of negative treatment of students in the past. Today we have that along with cognitive impairment.

I'm not sure the school is aware of the cognitive impairment issue. They know he's sick; they know that he's gotten worse.

It's not clear that they know this illness brings cognitive problems with it.

I wasn't, either, until I started looking into it.

At the moment, I'm talking to the tutor about reteaching algebra 2 this summer - Steve's advice about getting algebra 2 rock solid sounds right to me.

The tutor says the kids have to have algebra down cold -- though he thinks you need to have trig down cold, too.

Is that right?

What do you need to know to take calculus?

Catherine Johnson said...

anonymous -

you're right

the school should offer a summer course to make up for this

Anonymous said...

To leave no stone un-turned...what about Kumon? Their material goes up to Calculus.

Ben Calvin

Anonymous said...

To succeed in calculus, not only do you have to know algebra backwards & forwards, but you really need to have facility with trig. Many of the "tricks" for integration are based on trig functions, so if calculus is the first time you're seeing trig - well, good luck with that.

Catherine Johnson said...

Anonymous - THANK YOU!!!!

You guys are a HUGE help -- I can't tell you how important this is.

I'm far outside any capacity to make judgments based on my own 'background knowledge' --- I don't have background knowledge.

Catherine Johnson said...

Hi Ben!

Great to see you!

(Hear from you...)

Kumon is a GREAT idea - but it's going to be a tough sell.

I think I better take it up with C - I could at least get him to go to Kumon with me and take the placement test.

The one bright spot in all this is that he has SO MUCH MORE basic algebra & geometry than he did...we do SAT math sections together, and I'm shocked by how much he knows (and often by how much he understands).

I would say that he's had so much repetition in his math studies (spiral!) that he's actually managed to get pretty solid in algebra 1 and in **some** of algebra 2. I have no way of knowing how much ---

Hainish said...

Catherine, you do need to know algebra forwards and backwards.

You do need some trig, but it's not relied on so heavily. Also, you/C should understand volumes of solids and maybe unit-circle concept.

And...I always thought college algebra was algebra for kids who didn't take more than 2 years of math in high school.

Anonymous said...

"College algebra" is remedial algebra in college for students who did not do well enough in algebra 2 in high school.

Algebra 2 and precalculus vary a bit from place to place. Generally precalc should include a little of each of trigonometry, analytic geometry, linear algebra, and complex numbers, as well as some (but not too much) review of algebra 2.

My son is loving the Art of Problem-Solving precalculus class, but I can't recommend it for a student who has trouble self-teaching. The approach there is to give almost no easy problems. Ten problems a week that take half an hour to an hour each is more the style. It is great for the kids who don't need drill, but not at all suitable for those who need careful scaffolding to get up to speed.

The Foerster book is a good one, and working through that with a tutor would provide a solid algebra 2 background (not much of the other topics of precalc, though).

Ben Calvin said...

I think the Kumon placement test itself would be worth doing to get a quick idea of where the gaps are.

My son (5th grade) started Kumon this year. The placement test was very revealing. His current teacher was doing a fine job teaching fractions.

But neither his 4th grade teacher or curriculum had emphasized computation. His ITBS percentile ranking in computation went from the high 90s to the 50s .

We were planning on enrolling him this summer, but when I caught him guessing on multi-digit multiplication we knew it was time.

In terms of motivation....I am a firm believer in bribery in general and money in particular. My son will get cash for getting back on the Principle's Honor Roll. He's also motivated to get ahead of his class so that by next year he will already be exposed to the material before he gets it in school.

Perhaps (in addition to bribery) you can pitch is as being the only additional math you're going to ask him to do. Say if we do this (and it works) this will directly address the issue. The nice thing about Kumon is it's just math. It's very straightforward.

Anonymous said...

College Algebra is not necessarily remedial. At my high school, it was part of a junior year course for students who had aced Algebra 2 in 9th grade, and done well in Geometry in 10th. The content was definitely all of the algebra beyond Algebra 2 that you would need to be ready for Calc in your senior year (this was back when that was as quickly as you could progress through the Math curriculum. It included the very beginnings of Calc.

Catherine Johnson said...

I'm still confused about "College Algebra" (not anyone's fault - )

What do you think ACT means by "College Algebra"?

When they send college readiness scores to high schools, the math they're talking about is "College Algebra."

I wonder if they define the term anywhere on their website.

Catherine Johnson said...

In terms of motivation....I am a firm believer in bribery in general and money in particular. My son will get cash for getting back on the Principle's Honor Roll.

I agree!

Also agree that just taking the placement test would be a good idea.

The tutor is here; he's basically stunned at our situation.

He says the two kids he's tutoring aren't even close to being ready to take calculus - which is what they're both supposed to take next year.

He teaches algebra 2 at a public high school & says that C. and the other kids can't do his tests, which they ought to be "laughing" at.

Basically, C. and the other boy would not be able to pass the Regents exam for algebra 2.

That's another thought: I should give him the Regents exam.

Catherine Johnson said...

The tutor also says there are numerous topics that weren't covered at all -- and some topics in trig that were covered had maybe a week or two of time devoted to them.

Crimson Wife said...

I have heard really high praise for Courtney James as an online math tutor.

palisadesk said...

I'm thinking this wasn't the summer to sign up for the precision teaching institute at Morningside.

Why don't you contact Kent Johnson and/or Joanne Robbins (principal) at Morningside and see if they have anything appropriate for C.? Then you can both go!

They guarantee their results. The guarantee is more modest for the summer session but they might be able to get C. pretty much up to speed in the time available, using accelerated learning techniques.

No harm asking....

Glen said...

Catherine, calculus is all about functions. (I'll ignore the difference between function and expression here, because the book and chapter titles you need will use the term "functions" for all of it.)

In calculus, you learn to find more advanced properties of more difficult functions than you did in algebra. You learn how to figure out rates of change of a varying function (I give you a function showing a dropped ball's POSITION at any time "t", and you have to use that function to derive another function for the ball's SPEED -- rate of change of position -- at any time "t"). You learn how to take a function showing variable rate of change and use it to produce another function showing how the results accumulate. Or you are given a function for how a value changes over time, and you have to figure out when the value will be at its maximum, or minimum, or will touch zero. Or you are given a function for the rate of change of one thing (eg, rate of flow of water filling a spherical tank in liters/minute) and you have to produce a function for a related change (eg, rate of increase in depth of water in cm/min when tank passes depth "d".)

To prepare for this, you should be as familiar as possible with the major classes of functions and their properties. There aren't many major categories: linear, polynomial, trig, log, and power are probably all you need prior to calculus. If I missed one, you'll find it if you browse several textbooks' presentations of functions.

You want C. to know the basic "behavior" of these functions. I'll take a simple polynomial, y=x^2 as an example. What does its graph look like? If you change it to y=x^2-2, what happens to the curve (same shape, slides down by 2). If you put the -2 in a different place, y=(x-2)^2, what happens (same shape but slides right by 2). If you change to y=-x^2, what happens? (and know your order of operations; I don't mean y=(-x)^2. The shape turns upside down). Or y=(1/2)x^2 (shape widens).

And so on with y=Ax^2+Bx+C, and y=Ax^3+Bx^2+Cx+D and on to higher "orders".

And same for trig functions. What does y=sin(x) look like? What about y=sin(x+pi)? Or sin^2(x)? Or sin(x^2)? And so on through the major categories.

Think of a variation and try to imagine what it would look like ("what about y=(x-2)^2 - 2?") before verifying it with a graphing calculator.

Become very familiar with these functions. You should be able to see in your head that y=-x^2+2 can never be greater than 2, or that y=sin^2(x) can never be greater than one or less than zero. Know them like your little friends.

Glen said...


And you need to be very sharp at manipulating all of these categories of functions algebraically. You should know that y=x^2-5x+6 and y=(x-2)(x-3) are the same thing and be able to derive one from the other easily. You should immediately know that the log of x^2 is twice the log of x, or tan(x) is the same as sin(x)/cos(x), or how to find the cosine of something if you know the sine.

Also, review the formulae (think of them now as functions, too) for circumference, area, and volume of various shapes. Play with them a bit. You know the area of a circle with radius r? Good. Suppose the radius is 2r. What's the radius now? Or r-2? Or the area of an equilateral triangle with side length 2x? Practice playing with these variations.

Calculus will give you power tools to work with these functions, almost all of which use the notion of infinity, and these shouldn't be too hard as long as your ability to manipulate the functions algebraically (no bogus operations), and to have an image in your mind of what the result should be, are sharp.

It sounds like a huge task, but there aren't many MAJOR categories of functions to get familiar with, and it will give your algebra a good workout to play with them.

Have the behaviors of these functions in your head, not in your graphing calculator, and be quick and accurate with algebra, and you'll have nothing to fear from calculus.

Glen said...

As for how you make sure that C. knows the above things, I wish I could give you a certain answer, but if it comes down to it, you can even do this yourself if you know which areas to focus on. The trickiness of AoPS or AMC contest prep or SAT prep aren't really features of a standard calculus class. Books that give you plenty of straightforward algebraic manipulation practice (think fluency in manipulating, rearranging, factoring, etc., not trickiness), chapters that have you do lots of similar work with trig, log, and power functions, chapters that have you explore how various functions look under various circumstances, and exercises where you create functions to describe situations ("a spherical tank of radius r is filled to a depth d; what is the diameter of the surface of the water?" or the "how high off the ground does a ladder of length L touch the side of a house if the base is pulled out a distance x from the house?"--these are just geometry, not calculus) are what you need.

If you can find someone to teach these things, good. If not, you'll still be okay if you just look in books for practice with these things.

Anne Dwyer said...


College Algebra is a class that is taught at community colleges (in Mich in my experience.) The universities teach pre calc. There is not much difference between the two classes. These are considered 'college' level classes and transfer. I have taught college algebra and would be happy to give you a summary of the topics that I taught.

At the community college level, the classes that match up (somewhat) with high school are elementary algebra, intermediate algebra and college algebra. At our community college, there is a geometry course, but it is rarely taught.

There is a lot of overlap between these classes. So, for example, intermediate algebra is a repeat of elementary algebra, but each topic has more depth and a little more material. However, calc is not like that. There is (practically) no review at the beginning of calc. We start with limits and everything is new.

Taking a class at the community college is a good idea but....
you've already missed the beginning of a full summer semester. (Again, this is in Michigan.) You could take a 12 week semester, but I wouldn't recommend a 10 week or 7.5 week semester. Even with a 15 week semester, things go FAST!!! You have to be ready from day 1. If you fall behind, there is really almost no way to catch up.

As for, in the summer, most of your instructors will be adjuncts. They won't have very many ratings. (I have 5 and I have been teaching since 2007...and yes, I look.)

Many colleges use one of two books to teach calc: the Stewart book and the Larson book. I am teaching from the Larson book this summer. There is an extensive algebra and trig review geared towards what you need to know for calculus. If you decide to use a tutor, I would use this review as a guide on what to cover.

Good luck and contact me if you need anything.

Allison said...

I commented on the other thread about what content you should try to ensure mastery of, but I just want to add one thing here:

my precalc class 20some years ago was just as bad, just as procedural, and just as completely unmemorable. I cannot even tell you what topics we covered, and I knew at the time that I had no idea what topics we were covering. It was completely unclear to me and everyone else in the class what we were learning in that year. Our calc teacher the following year was apparently well prepared for our ignorance, or maybe she just taught well, and I did extremely well in AP calc anyway (though truthfully, I was desperately unprepared for higher math because of gaps like those I had in precalc.)

My point is to NOT PANIC. The course is usually a hodgepodge of bad procedural stuff anyway. Concentrate on functions and graphs, trig, and limits, and you'll be okay.

Allison said...

Here's the book I'd buy:

at a magnificent price of under $15 new.

Anonymous said...

Have you tried the free courses at ?

There are a lot of video tutorials going all the way through college math. And he can work through the quizzes.

SteveH said...

"What about trig?"

All I can say is that for me, algebra really didn't "click" until I finished Algebra II in high school. Before then, I could approach problems two different ways and get two different answers. It might have been a sign issue, an exponent issue, or simply not being careful enough. This was in spite of getting A's in math.

I remember a teacher once simplified an equation on the board to show that 1 = 2, and we had to find the mistake. I am trying to teach my son what these issues are. I think they come from the difficulty in applying simple rules to complex, rational equations. These are issues that will cause problems in everything else you do. You might think you have difficulty with trig, but it may really be an algebra issue.

I didn't find trig or calculus to be difficult, but I distinctly remember the semi-lost feeling I had when I couldn't do algebra correctly all of the time. This is a hump that many never get over. It makes everything else much more difficult than it really is.

Back when I taught college algebra (an oxymoron), I would have students point an arrow at the equals sign. Then I would have them put a box around each term and include the leading sign in the box. Then I would tell them to put a '+' sign between the boxes. The next step was to change all terms to rational expressions. If there was no dividing line, I would have them draw one in and put a 1 in the denominator.

Next, if the leading sign was a minus, I would tell them to put that in the numerator as a (-1) factor. (Time had to be spent justifying why that could be done.) Then, I would have them put parentheses around all of the factors in each numerator and denominator. (There were lots of mistakes here.) If a factor didn't have an exponent, I would tell them to put in a 1.

They didn't like this sort of anti-simplify process, but I told them that this is how they should see the equation. They can then determine what can be moved around and why. They can move any factor up or down across the dividing line just by changing the sign of the exponent. They can swap the position of any factor. You can say a*b = b*a over and over, but you have to see what that means in a complex rational expression. You have to know what 'a' and 'b' are.

In this form, they can also easily find the least common denominator. If one contains something like:


then students should easily "see" the factors.




I think this relates to the simplify discussion we had in another thread. Students should not automatically think of simplify when they look at an expression or solve when they look at an equation. They need to know what can or can't be done. They have to "see" the terms, factors, exponents, and missing denominators.

Anonymous said...


I haven't read all the posts yet, but have you looked at the Life of Fred books yet?

Mr. Schmidt has a way with explaining things.

Anonymous said...

Here is a list of higher math resources that you may be interested in:

Kinetic Books and VideoTexts also offer online courses in higher math

Dr Callahan offers a Calculus course (College Level, First Year):

Karen Graves

Catherine Johnson said...

oh my gosh - Allison - I have that book! It's fantastic! (Have just read a bit.)

I was planning to give C. a copy as a going-away-to-college present.

I've also been planning to write posts about it ----

Glad to get your endorsement - !

Catherine Johnson said...

Thanks for reminding me about Life of Fred -- I remember looking at those a while back --

Will revisit!

Catherine Johnson said...

algebra really didn't "click" until I finished Algebra II in high school. Before then, I could approach problems two different ways and get two different answers.

When I interviewed Philip Keller for a newsletter Debbie S. and I are starting, he said that there's a group of students who can reliably set up an algebraic equation -- but cannot reliably solve it.

I was really intrigued by that.

Catherine Johnson said...

Students should not automatically think of simplify when they look at an expression or solve when they look at an equation.

I wish someone had told me this a long time ago!

AND: this is a HUGE issue in SAT math, which has an entire genre of problems you solve by NOT simplifying as far as you can go.

(i.e. you substitute a value for x+y instead of just for x)

When you've spent your entire math education simplifying expressions & equations, it's quite difficult to break set.

Catherine Johnson said...

Interestingly, as I practice SAT math sections, I'm starting just to 'see' what level of simplification I need to do .... can't explain it any better than that.

It's not exactly conscious; it's 'cognitive unconscious' learning.

I have got to figure out how to write a post about 'basal ganglia' learning one of these days.

The basal ganglia is pretty much IT when it comes to a lot of what we've all been talking about here on ktm for lo these many years.

Catherine Johnson said...

My point is to NOT PANIC.

Thank you!

I needed that.

Anonymous said...

"My point is to NOT PANIC.

Thank you!

I needed that."

You have clearly misplaced your towel.

-Mark Roulo

Anonymous said...

Math U See might be a great option. It's sequential.

LexAequitas said...

"epically" is a word.

Anonymous said...

I recommend taking a look at Lial's Precalculus. Lial's books are excellent for remediation.

Catherine Johnson said...


epically is a word!

and rightly so, may I add

Catherine Johnson said...

Crimson Wife -- thank you SO much for the reference - I had never heard of Courtney James.