select another subject area

Entries from HighSchoolMath

CompareAndContrastPart4 10 Oct 2006 - 01:54 CatherineJohnson

thank you: Elizabeth Carson, Co-Founder NYC HOLD

update: I forgot to mention that this chart comes from an article circulated on the NY Math Forum list, The Curricular Smorgasbord by Williamson M. Evers & Paul Clopton. (pdf file)

CompareAndContrast
CompareAndContrastPart2
CompareAndContrastPart3
CompareAndContrastPart5
CompareAndContrastPart6
CompareAndContrastPart7
MathInSalinaKansas

keywords: the f word the f-word bibliography greatest hits

TitlesOfConstructivistMathCurricula 19 Jul 2005 - 01:46 CatherineJohnson

Jo Anne Cobasko has taken the time to construct a complete list of NCTM standards based math programs.

update: Department of Corrections

This list is David Klein's handiwork, not Jo Anne's.

Thank you, David! (For everything you do.)

All of us should keep this handy, because none of these programs ever calls itself constructivist, and schools don't seem to advertise this piece of information, either.

When I first raised the issue of TRAILBLAZERS being a constructivist curriculum with a teacher on the textbook selection committee, she looked at me blankly. I got a number of those blank looks before I discovered that everyone in the school knows what the word constructivism means, and knows what a constructivist curriculum is.

The reason I know this is that I finally read the original committee report, which states explicitly that the new curricula must have a constructivist approach with modeling. I was a little behind the curve there.

Elementary school

Everyday Mathematics (K-6)
TERC's Investigations in Number, Data, and Space (K-5)
Math Trailblazers (TIMS) (K-5)

Middle school

Connected Mathematics (6-8)
Mathematics in Context (5-8)
MathScape: Seeing and Thinking Mathematically (6-8)
MATHThematics (STEM) (6-8)
Pathways to Algebra and Geometry (MMAP) (6-7, or 7-8)

High school

Contemporary Mathematics in Context (Core-Plus Mathematics Project) (9-12)
Interactive Mathematics Program (9-12)
MATH Connections: A Secondary Mathematics Core Curriculum (9-11)
Mathematics: Modeling Our World (ARISE) (9-12)
SIMMS Integrated Mathematics: A Modeling Approach Using Technology (9-12)

Programs explicitly denounced by over 220 Mathematicians and Scientists:

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)
Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

printable page

Thanks, Jo Anne, for taking the time to do this!

key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

MathOlympiad2005 22 Jul 2005 - 21:45 CarolynJohnston

Check out the problems from the 46th Math Olympiad competition in Mexico (held last week).

The competition is held over two days. Each day, the kids are given 3 problems, and 4-1/2 hours to work them.

I haven't even dared look at them yet -- I'm afraid I'll be lost for days.

(ht: Charlie Martin).

MathOlympiadProblem
HappyJulyFourth (Moise & Downs)

DimensionalAnalysis 25 Jul 2005 - 20:05 CarolynJohnston

DanK brought up dimensional analysis in this thread, and it's such a useful idea that I thought we should have a thread to explain what it is, and talk about it and its possible uses in math education.

Here's a very simple example, where dimensional analysis can help you get the right answer.

Suppose a man drives 60 miles in 50 minutes. How fast is he driving?

There are two answers a kid is likely to come up with: the first (and correct) one is 60/50, but a kid might very well come up with 50/60 and not notice he's made a mistake.

Here's how dimensional analysis could help this student get the right answer: he knows he wants a rate for an answer; distance per unit of time. If he thinks of the 60 as '60 miles', and the 50 as '50 minutes', then his two choices are:

(60 miles)/(50 minutes) = 60/50 miles/minute

(50 minutes)/(60 miles) = 50/60 minutes/mile.

This gives him more context to help him choose the right answer. Miles per minute are units that make sense for this answer: minutes per mile don't.

In addition, dimensional analysis is the tool to use to make unit changes. If the question requires the answer to be given in miles per hour, then 60/50 is not the right answer, because the units are miles per minute. How to do the conversion to miles per hour?

As with converting fractions to have common denominators, the trick is to multiply the answer by 'one'. In this case, the conversion factor will be (60 minutes)/(1 hour). (You see why this is really 'one'?)

Thus the answer in miles per hour is:

(60 miles)/(50 minutes) x (60 minutes)/(1 hour).

Notice that (60 minutes/1 hour) is actually 1, expressed in different units in the numerator and denominator!

Now for the trick. Move the units around a little, just as though they were numbers in fractions being multiplied, and you get

(60 miles/1 hour) x (60 minutes/50 minutes).

Now the minutes cancel in that second term, and you are left with 60/50 (otherwise known as 6/5) as a dimensionless number. (A dimensionless number is a number without any units attached. For example, all ratios are dimensionless).

So the answer is: 60 miles/hour x 6/5, or 72 miles/hour.

There's even more that you can do with dimensional analysis. As Dan points out, it's a very handy concept, but hardly any math text uses it to the fullest extent they could.

At the undergrad level, it's something engineers and scientists learn explicitly. They have to know it in order to make unit conversions. I was a graduate student when I learned it in a geochemistry (i.e., thermodynamics) class; I had already had a complete undergraduate math education. I taught that whole class of geochemists how to do differential calculus; in return, they taught me dimensional analysis, and I think I got the better end of the deal.

So: when are kids ready to learn, and to start using, dimensional analysis?

Manipulating dimensions is a lot like manipulating fractions, and largely uses the same skills. You can't add dimensioned quantities, for example, unless the dimensions are the same: for example:

x miles/hour + y meters/minute = x+y miles/hour

doesn't make any sense unless you first convert the y term to miles/hour. Identical units can cancel (as the first example showed, when I canceled minutes in the numerator and denominator). So right about the age Ben and Christopher are now -- tennish or elevenish -- is about the earliest kids could really start using it, and it's also about the time that math texts stop emphasizing units (as DanK pointed out).

Plus, if the parents don't know it, how can they teach it?

Once again, it's the internet to the rescue.

WichitaBoyOnMath 31 Jul 2005 - 22:15 CatherineJohnson

We have an embarrassment of riches! At least 2 great comments from WichitaBoy, and Ed sat down and wrote out his constructivism-as-psychoanalysis thoughts, too.

Here's one of WichitaBoy's observations:

"Writing is organizing." Now there's a great thought I can take to the bank.
Here's one back for you: professional mathematics is organizing. You have vague thoughts, you notice a vague pattern, and you try to organize your thoughts, to nail down the pattern, to really clarify what's going on beneath the hood. When you've nailed it completely, when you understand with perfect perspicacity the essence of the pattern, then you've got a proof of a new theorem. If you've really organized it, you've got a theorem that goes in the "Book of God".

There's this, too, in response to my saying that what is brilliant about Saxon Math--the structure--is largely invisible:

Read Confucius or Socrates. The ideal teacher should be able to fade into the background like the Cheshire cat. And so with the ideal textbook.

HighSchoolAlgebraTexts 02 Aug 2005 - 15:49 CatherineJohnson

Temple and I are writing an op-ed about American high schools, and I just came cross a treasure trove of PowerPoint slides filled with Horror Statistics, so naturally I had to stop dead in my tracks and get one posted on ktm......

This is a case where PowerPoint has a distinct advantage when it comes to conveying the Bad News.

The whole entire key to conveying bad news on PowerPoint is:

one piece of bad news per slide

or, alternatively,

don't bury the bad news inside a bunch of other junk

source: PowerPoint presentation on U.S. high schools at U.S. Department of Education

EasyMathIsHarder 02 Aug 2005 - 22:22 CatherineJohnson

Another slide from the Department of Ed.

Unfortunately, they don't have the lecture notes up along with the slides, but I think this is self-explanatory. Assuming I'm reading the slide correctly, it tells us that for all but the lowest quarter of students, 'hard' math is easier than 'easy' math.

In other words, the top 75% of students get better grades in college prep math than they do in 'low-level' math.

This is one of those cool findings that inspires me to search for terrific, high-level material for Christopher.....but I'm afraid the reasons for this phenomena may be that the college prep kids have better teachers. The report includes numerous slides showing that the poorest teachers are assigned to the lowest level classes, and that the quality of teacher makes a huge difference in children's achievement. (I'll drop those slides in soon.)

Still, I wouldn't rule out the possibility that 'real' math is more learnable than stripped-down, pretend math.

update

This slide, and a number of others in the presentation, is based on a study of 3000 high schools done by the Southern Regional Education Board, Middle Grades to High School: Mending a Weak Link.

This research brief is based on an SREB study of nearly 3,100 students from 44 middle grades schools and 38 high schools. It shows that ninth-graders in higher-level courses have a lower failure rate than students with similar characteristics in lower-level courses. The report offers specific actions that schools can take to improve student achievement.

The finding that the same level of student will do better in college prep courses than in non-college prep courses wasn't limited to math. It was true across the board.

from the SREB report (pdf file):

Take 100 ninth-graders with similar characteristics and test scores in the eighth grade. Place 50 in higher-level ninth-grade courses. Place 50 in lower-level courses. What happens? If you said fewer students fail in the higher- level courses, you are correct. Please read on.
The Southern Regional Education Board conducted a follow-up study of nearly 3,100 students from 44 middle schools and 38 high schools and found:
Ninth-graders who are placed in higher-level courses have a lower failure rate than students with similar characteristics who are placed in lower-level courses.
This fact begs the question:
Why do we continue to place large numbers of students in lower-level courses where they have little or no chance of gaining the skills and knowledge they need to succeed?
Here is what we know …
Our studies suggest that students who are assigned to higher-level, more challenging work are more successful in high school.
We also know that about one in five students in SREB's network of middle grades schools fails at least one course in the ninth grade, and about 10 percent do not earn enough credits to stay on track for graduation with their classmates.
Clearly, raising the achievement of high school students requires three actions:
1. Students must be challenged to perform at high levels.
2. Students must be prepared before they enter ninth grade to meet these challenges.
3. Students must be given the extra help and extra time they need to succeed.

Key Findings

Many students who expect to go to college are not taking the necessary courses in high school.

Some schools enroll many more students in college-preparatory courses than others. The difference is not explained by differences in students or demographics.

Enrollment in more demanding courses does not result in more failures. In fact, the evidence suggests that challenging content results in lower failure rates. It appears that many students in all kinds of schools can handle more challenging intellectual assignments than schools are willing to give them.

Taking algebra or pre-algebra in the middle grades leads to enrollment in higher-level mathematics courses in high school and does not increase failure rates.

Middle grades schools that successfully prepare students for college-preparatory courses in ninth grade provide extra help and link students with an adult mentor. Successful schools come in many sizes, and their students vary by ethnicity and socioeconomic status.

Teachers matter enormously; middle grades students who have teachers as advisers are more likely to have educational goals and plans for high school.

There are simple steps that middle grades and high schools can take to make sure almost all students will be successful in college-preparatory courses.

Now that I've had a chance to look at the report, I think we're seeing confirmation that people rise to expectations.

I notice, too, that this report does not find that differences in college-prep placement can be explained by 'differences in students or demographics.' I'm inclined to believe this, given my own experience here in Irvington. Last year we had, I believe, 40% of 6th graders enrolled in pre-algebra; next year this figure will be subtantially lower.

Reducing the number of students in accelerated math was a plainly stated objective of the middle school administration and math faculty.

We're talking about a super-affluent suburban district spending $18,000 per pupil.

Meanwhile 80% of 8th graders at the KIPP Academy, in the Bronx, pass Regents A. Compared to 40% of kids here.

I continue to find this utterly shocking.

MoreOnAlgebraInEighthGrade 02 Aug 2005 - 22:45 CatherineJohnson

More from Middle Grades to High School: Mending a Weak Link (pdf file)

A comparison of our eighth- and ninth-grade data reveals three middle grades experiences associated with students who take and succeed in higher-level courses in grade nine.
These experiences are:

studying “something called algebra” in the middle grades;

reading a great number of books in grade eight; and

expecting to graduate from college.

Studying “something called algebra”

Across all schools, 62 percent of the students who said they had a course with “algebra” in its title during the middle grades were enrolled in college-preparatory mathematics in ninth-grade. Eighty-five percent of these students earned a “C” or above. High enrollment schools enrolled 82 percent of students who had algebra in the middle grades in college-preparatory mathematics courses. They had virtually the same success rates as schools with lower enrollment rates. Clearly, students who begin algebra earlier are more likely to succeed in an accelerated mathematics curriculum if high schools choose to enroll them inthis curriculum.

I love this.

You can just feel how much fun it is trying to drag information out of young teenagers for the purposes of a Major Report.

Yeah, I studied something that said algebra. I think.

HotMath 14 Aug 2005 - 15:09 CatherineJohnson

Thanks to Dan K, I've found a fantastic resource:

Hotmath.com

[Hotmath provides] explained solutions to the odd-numbered homework problems from most of the popular secondary math textbooks used in California. Thus, teachers can now assign practice problems for homework where teacher-prepared, explained solutions are instantly available, and can mix in even-numbered problems for challenges. Students who do not need to see the worked solutions needn't bother, and students who might abuse the availability of worked solutions will be tested on the even problems.

Here is a sample worked-out problem: algebra problem

And here are the 2 critical paragraphs from the Hotmath 'white paper'. I've begun to come across these studies elsewhere, and I'm inclined to trust these summaries, in part because this discussion jibes with my own experience re-learning maths:

Providing students with worked out examples of math problems has been found to be more effective than simply assigning the same problems for the students to work out on their own. In one experiment (Carroll, 1994), 40 high school students were instructed in how to solve linear equations. In an “acquisition phase” the students were divided into two groups and their instruction differed in the following way: in the “conventional learning” group, students were assigned 44 unsolved problems to work out (in the classroom and at home homework), and in the “worked examples” group students were provided with the same problems, but half of the problems were accompanied by correct solutions. After completion of the assigned problems, both groups were tested on 12 related problems, 10 of which were very similar to the linear equations presented in the acquisition phase, and 2 of which were word problems, used to test whether students could transfer and extend their knowledge to a new context. No worked out examples were available during the test. The test results revealed that students in the “worked examples” group outperformed students in the “conventional learning” group on both types of the test problems. A second experiment, employed a similar methodology but focused on “low achieving” students (students with a history of failure in mathematics, and students identified as learning disabled). Here, the data revealed that students in the “worked examples” group required less acquisition time, needed less direct instruction, made fewer errors, and made fewer types of errors than students in the “conventional learning” group.
Related research (Pass & Van Merrienboer, 1994) sheds light on the cognitive underpinnings of the effects described above. In this study, 60 college-aged students were instructed in geometry concepts. As in the Carroll experiments, students were assigned un-worked problems to solve or worked out examples to review (unlike the Carroll study, the “worked examples” group was assigned no un-worked problems to solve). In this study, the researchers manipulated the nature of the problems presented to the students: within each group, some students received problems which were all similar to each other, while others received a more varied problem set. Furthermore, the researchers measured the “cognitive load” experienced by the students. This research revealed that while students in the worked examples group completed their work more quickly, they perceived the work as less demanding and displayed better transfer performance at test. The effect was most pronounced for the students given highly-variable problems. The researchers suggest that the reduced cognitive load associated with the worked examples enabled students to “take advantage of” the variability in problems by using the available cognitive resources to process the underlying similarity in the problems (i.e., the mathematical concepts being taught), and to integrate the current problem with existing knowledge (Linn, 2000).

The site covers Prentice Hall Pre-Algebra, the book Christopher will be using in the fall, so I'm going to subscribe. Cost is $49 for 12 months.

I think it's going to be fantastic for Christopher to have an answer source that isn't His Mother.

Especially since it looks like I'm going to have to start some heavy-duty Writing Instruction this year. (That's another story.)

cognitive load

This is going to be an important term for me. It perfectly captures what it is we're trying to do when we push our kids to practice to the point of automaticity.

We're trying to reduce cognitive load.

update

I've just re-read Dan's original post, and I don't see a reference to hotmath. hmmm. Maybe one of the sites he mentioned pointed me to hotmath. In any case, I'm recommending hotmath, not Dan. (He'll let us know what he thinks, I'm sure.)

StatisticsHelp 19 Aug 2005 - 17:51 CatherineJohnson

I mentioned that Temple and I are writing an op-ed on U.S. high schools.......and I'm stumped by a statistical issue.

What does it mean to say that multivariate analysis shows that a certain factor is highly predictive of a particular outcome, while another factor is less predictive?

What does this form of analysis imply about causality, if anything?

I ask because of an apparently highly influential government report published in 1999: Adelman, C. (1999). Answers in the Tool Box: Academic intensity, attendance patterns, and bachelor’s degree attainment. Washington, D.C.: U.S. Department of Education.

This study finds an extremely high correlation between rigor of high school curriculum and students' eventual completion of a college degree--far higher than the correlations with the factors we're used to hearing about, such as parents' level of educational attainment, socioeconomic status, and race (and substantially higher than high school GPA and SAT scores).

The report itself, which I've barely skimmed, as well as other accounts of it, seem to imply that the relationship is causal. It's not that being smart and motivated in the first place causes a student to take a more rigorous high school curriculum and attend and complete college, but that the more rigorous high school curriculum sets him or her up to succeed in college.

from the American Educator:

Academically well-prepared students are likely to graduate from college regardless of their social background. Unprepared students of all backgrounds are not likely to do so.
The graph below breaks students into quintiles based on their level of academic preparation and their socioeconomic status (SES). As you can see, among the lowest SES students, a bachelor’s degree was earned by 62 percent of those who were well prepared, but only 21 percent of those who were not. Similarly, among the highest SES students, 86 percent of those who were well prepared--but only 13 percent of those who were not--earned a bachelor’s degree.

Percentage of students who graduated from a four-year college by socioeconomic status (SES) and academic preparation.

key words: rigorous high school curriculum predicts graduation from college
how can you tell whether A caused B?
low birth weight paradox
how good are our best students?
statistics and law

HighSchoolMathDebate 09 Aug 2005 - 21:56 CarolynJohnston

Here's another good post at Tall Dark and Mysterious -- a discussion of a laundry list of topics that kids ought to be learning in high school math, in order to prepare them for college math.

The laundry list is good, but the comments are fascinating.

Here's a rather heart-rending comment, number 24:

As a high school math teacher, i can assure you that each of the prerequisites you propose is taught ( or, at least, presented ) to every student, college bound or not, in my district starting in the eighth grade. I teach the same material to ninth, tenth, eleventh and twelfth graders when I'm not trying to teach them material they should have learned in grade school.
My district increased its math requirement this year from two years to three. So now, when a student shows up in a college math course, she will have had three years of math beyond the elementary prerequisites you have proposed.
And still... you will find your classes include students without the least of skills.

On a more positive note, here is comment 37:

As a high school math teacher, I have to say that your list is very good. In fact it is covered in the exam that all of our students must pass in the state of Texas in order to graduate.
The only way that colleges can be sure that high schools are teaching what needs to be taught is to hold kids accountable. And that means standardized tests.
One of the good things from the No Child Left Behind initiative here in the US is the requirement that all states have in place an assessment to show whether the kids have mastered what needs to be mastered. The test we give is given in 4 separate parts for each of the 4 content areas, and if the students don't pass all 4, they don't get a diploma! The math test covers the curriculum of Algebra I and Geometry - the minimum courses that all high school students should take. In order to receive a college bound diploma, the student must have also taken Algebra II.
As soon as these tests are implemented nationwide, I think the colleges will see a student body that is a lot better prepared.

(Unfortunately, commenter 37 then goes off on a rant about how colleges need to move into the 21st century and realize that calculators are here to stay).

And then, someone named Charles Williams hits the nail flat on the head. Many people will never need algebra, and other people will need a lot more, in order to work in technical fields. The problem is that we don't know which 11-year-olds are which, and we don't have the national stomach to try to separate them at that age; nor should we. Here's his comment:

Perhaps 80% of our students will not use algebra in their careers. Nonetheless, a student who wants to master a technical field must be in a rigorous math program from the middle school on. The minute we give up teaching a middle school student fractions and hand him a calculator so he can do computations, technical careers are no longer an option for him. Now do we have the will to track college prep students starting in the 6th grade so that the others will not be burdened with the unpleasantries of algebra? Or will we insist that all students graduate with what they need to succeed at college in technical fields. Unwilling to face facts and make hard choices, we muddle through in a dishonest way. We prepare all students for college on paper but not in reality. We do de facto tracking while denying that this is happening. We require more and more high level courses from our students and then water them down.

I can't tell whether he is implying that we ought to be making hard decisions, and cutting kids out of possible future choices in careers. I hope not; I don't think that's necessary (as the successes of other countries show). But if we are going to force unwilling kids to do the work that's really required to succeed in college math, we should be pushing those kids all the way, and recognizing that we can expect a lot of pushback from them.

MathForumArchivedNewsletters 14 Aug 2005 - 01:37 CatherineJohnson

I've just been alerted to a terrific resource, the Math Forum Newsletter.

They have an article about Kitchen Table Math in the latest issue! (Although so far I haven't been able to find it.....I don't think....)

Sigh.

However, I have managed to attach and display the logo they sent me!

GretaFrobieterOnUsingAlgebra 19 Aug 2005 - 17:21 CatherineJohnson

Another terrific comment in one of the threads, from Greta Frohbieter (Welcome, Greta!)

What about learning algebra 1) for the sake of becoming knowledgeable, 2) as a tool for understanding the universe, and 3) because it CAN be an enjoyable subject for the vast majority of students when taught properly? This whole discussion is based on the premise that algebra (and higher math in general) is taught and learned in preparation for certain careers, and while there is obviously some of truth in that, I think we do the subject and our students a disservice by focusing solely on the vocational aspect of math.
How many of us directly use our knowledge of ancient history or chemistry or poetry in our jobs? Probably no more than use algebra, and yet these subjects are not constantly required to defend thier existence in the curriculum. Can anyone enlighten me as to why the "utility standard" is applied to mathematics by both students and educators so much more often than it is to any other subject?
By the way, as a civil engineer in the aerospace industry, I used math from all of my courses through differential calculus. (I don't remember using integrals, but it's possible that I did.) Now, as a middle school math teacher I use algebra constantly!

I think I mentioned in a post at some point that when I had to sit down and think about it (to answer an online quiz), algebra was either my first favorite or second favorite subject in high school (after Spanish).

I don't know that algebra was taught especially well in my high school; probably it wasn't.

But it was fun. (And it wasn't supposed to be fun; it wasn't gimmicked up or jokey.)

It was fun.

Heck, it's fun now. I can't even finish knitting a baby sweater these days, because it's more fun to lounge around on the bed doing RUSSIAN MATH.

update

I get HUGE, VAST quantities of grief from Christopher about this.

ARE YOU DOING RUSSIAN MATH ON MY BIRTHDAY???????????

ARE YOU BRINGING RUSSIAN MATH TO THE SWIMMING POOL?????????

YOU'RE OBSESSED WITH MATH!!!!!!!

Things of that nature.

UniversalAlgebra 17 Aug 2005 - 00:25 CatherineJohnson

This is incredible!

A ktm guest left a link to an online course that is actually called Universal Algebra.

I'm going to add this link to the Favorite Math Supplements for Kids page (on the sidebar) so people can find it.

By the way, anyone who has resources to add to the list should let Carolyn or me know.

DepartmentOfCorrections 25 Aug 2005 - 16:42 CatherineJohnson

I completely & totally misunderstood Carolyn's comment about geometry being a predictor of math talent!

Boy, trying to put the English language together with math is not easy. (This reminds me of another topic.....more later)

Here's Bernie:

I think a couple of things are getting mixed up here. Mathematicians tend to view the results of the standard course in 10th grade (probably pushed down to a lower grade these days) Euclidean geometry as a good indicator of talent as a pure mathematician. This has nothing to do with geometry per se and everything to do with logic. Pure mathematics is accomplished through stringing together known abstractions in a logical way to produce new truths. That is exactly what is done in high school geometry.
But there are many professional mathematicians who can't even draw a 3-dimensional cube comfortably.
Also, there are many different aspects to what is called "geometry". There are, for example, algebraic geometry, Euclidean geometry, non-Euclidean geometries, topology, analysis in many forms. These are all conceptually different abstractions from our ordinary perceptions and different mathematicians may or may not have any talent in any of these separate fields.
Still, I think working out perimeter problems as given above is good practice all around.

metacognition & math homework

I've been wondering--I think I've mentioned this before--whether mathematically talented kids get lost in the crowd more often than verbally talented kids.

I bring this up based on a sample of 5 kids, 2 of them what I would call 'word' kids and 3 of them probably 'math' kids.

(Whether or not anyone should trust my judgment on this, I don't know. Can a middle-aged adult 'tell' that a particular kid is a 'math kid'???)

I know two of these 'math kids' personally, and have worked directly with them. Both of them are--and this I will stand by--simply far more mathematically talented than anyone other than their parents has noticed. To me it seems as if they've been kind of invisible; they've been kids who were seen by the school as having some problems early on, or as needing a multisensory teaching approach, or as maybe being slightly at risk for LD.....the point is, I haven't seen their files & I don't want to, but these two boys have not been seen as possessing mathematical talent.

I think they do possess mathematical talent.

I'm thinking that math talent, unless it's extraordinary, doesn't 'read' well. Math talent doesn't announce itself to adults the way word talent does.

Verbal talent isn't a sure ticket to Teacher's Pet-hood, of course. If a child is verbally talented and hyperactive, or verbally talented and 'dark,' that's not going to win hearts and minds at school, either. But a calm, nice-mannered verbal kid like Christopher--or like E., the little girl who was 'class brain' last year--will 'jump out' at the grownups.

This is all speculation; I don't know what to think.

I just know that I've been working with a couple of boys who, I believe, have a lot more math talent than anyone realizes.

And I don't think I've seen the same thing with verbally talented kids.

back shortly

Oops--I realize I haven't said a word about 'metacognition & math homework,' and now I've got to run.

Back in a bit.

does math talent 'show'?

Back again.

I remembered a story on my way home.

Back when I was at Dartmouth, I was introduced to a guy one day who I took to be a bit of a dufus. Not so bright.

This was unusual for me, because I don't usually stand around evaluating other people's intelligence. It's rude. And yet, with this guy, I did.

Two weeks later I found out he was our class valedictorian.

I don't know whether he was a math guy, but the story is relevant anyway, because this is what I've been wondering about: do kids whose main talent is math 'read' less smart than kids whose main talent is verbal?

Or--a variant on this question--is it hard for adults to perceive math talent in elementary school kids?

I ask because I've started working with one of Christopher's friends who, to me, seems obviously Good At Math.

He's Good At Math, and yet he's not doing well in math; his scores aren't great, and he's in the Phase 2/3 track. (I keep mentioning 'Phase 4,' but this fall the school will have 3 tracks, not 4. Christopher will be in Phase 3, and his friend will be in Phase 2.)

When I gave him the Saxon placement test, he sailed through it, except he couldn't do any fraction or decimal problems at all. And he didn't remember ever having been taught fractions, which can't possibly be the case.

After getting advice from ktm readers, I decided to skip Saxon and go directly to the fraction & decimal lessons in PRIMARY MATHEMATICS, which I'm having Christopher do along with him.

Those have gone great.

It's obvious that this boy has been taught fractions, because he learns the material instantly, and can apply it instantly to the workbook exercises.

Also because he says things like, 'Well I would say 1 2/5 is less than 1 3/7 because it's closer to 1.'

So this is a case where he was taught a subject in math, understood the subject, and then had no ability to recall or retrieve it at all.

This has led me to wonder about a bunch of things.

If teachers perceived this boy to be talented in math, mightn't that help? I'm guessing that a Talented In Math designation would cause teachers to think there's something wrong when his scores are low, something wrong that needs to be figured out and fixed. (I have no idea what his teachers or the school think, btw. I'm speculating.)

Good math teaching means constant formative assessment. This is why I mentioned 'metacognition' earlier. From the teacher's standpoint, metacognition means finding out what a student knows & doesn't know (as well as what he knows that 'isn't so.') That doesn't seem to have happened here. Until the moment I gave this boy the Saxon test, I don't think anyone had said to his mom, 'Your son doesn't know fractions & decimals.'

Parents are going to have to be in charge of knowing what their kids know & don't know.

parents & grade school math

I keep coming back to the idea that the best thing for a parent of a grade school child to do is to re-learn math right along with his child.

buy a copy of the texbook (if there's a textbook)
buy a copy of the consumables
buy a copy of the teacher's guide
do your child's homework the same time he or she does the homework

There's probably an easier way, but since I don't know what it is, this is my way. I'm going to be working through the Prentice Hall Pre-Algebra book this year. (If it's better for me to do the problems on my own time, before Christopher comes home, that's fine. My goal isn't to crowd him, but to learn & re-learn math well enough that I'll be able to help with questions. With a little kid, doing the problems at the same time might be nice.)

If you're like me, and you didn't learn math well to begin with, then re-learning math gives you power. It's that simple. You can see how an elementary math curriculum is put together, you can see whether your own school's curriculum is acceptable, and you have a shot at seeing whether your child is actually learning what he needs to learn.

the book parents needs

People who think about math ed spend a lot of time talking about understanding math &/or being able to do math.

We should probably spend more time just talking about how to remember math.

Remembering math is hard. Here's Ingrid Wickelgren, Wayne Wickelgren's daughter:

The brain has trouble with math, not so much because there are so many facts to learn, but, surprisingly, because the facts are so similar. For example, the fact 3 + 5 = 8 is not so different from 3 + 6 = 9. They both contain 3, they both contain +, and they both contain single-digit numbers. To a child, the facts overlap in the brain, creating a blur that confuses them and makes it difficult remembering any single answer. It's like static on the radio, which often occurs when other stations or electrical impulses interfere with a radio station's music or speech. When the child sees 3 + 5 = --, all the arithmetic facts involving 3 and 5 get activated in the mind, and the wrong answers create interference for the right answer.
In every area of math, a relatively small number of basic concepts are used to express a large number of more advanced concepts or facts. But as each basic concept activates many other facts in the brain, the facts interfere with one another.

I believe this Without Question.

Christopher's friend has forgotten what he learned. He's also forgotten that he ever did learn fractions in the first place. (It's starting to come back to him a little now. He'll say, 'I think maybe they did teach that.')

Parents probably don't need to sit down and re-learn math with their kids......That's just my desperation measure. (A desperation measure that also happens to be fun.)

But parents obviously have to do something, because kids are getting lost in the schools. Even in good schools with good teachers.

So, for the time being, my guess is that what parents must do is to make sure their kids are overlearning math.

Unfortunately, most of us don't just naturally know which particular math factoids & concepts have to be practiced to the point of overlearning.

The book I wish I could buy would tell me exactly which math facts & concepts a child must overlearn.

And it would give me a coherent set of one- or two-minute worksheets my child could do 5 days a week every week until he leaves home for college.

tutoring advice

CalculusInHighSchool 23 Jan 2006 - 16:36 CatherineJohnson

From an interesting page by this title at Math Forum:

What causes students to experience difficulty once they are taking calculus?

Many aspects of algebra and pre-calculus are self-contained. A student's success will not necessarily depend upon his or her proficiency with material in a previous chapter. Calculus, on the other hand, is entirely new material that builds upon itself continuously.

Calculus students cannot always assimilate the material quickly enough.

Calculus students can fall behind and find it difficult to catch up.

Some students have difficulty with the large number of word/application problems.

Some students do not possess the cognitive flexibility to switch strategies that can be required to solve specific calculus problems.

Some calculus students have difficulty with active working memory and consequently with manipulating all aspects of a problem without getting lost.

Some calculus students do not remember all of the necessary material, especially formulas, from algebra and pre-calculus.

Calculus students are challenged to learn inverse operations in close succession; consequently, they can confuse one type of a problem with its exact opposite.

This list is especially interesting in terms of what cognitive science tells us about learning & remembering.

I mentioned that I need a book that tells me exactly what formulas, concepts, math facts, etc. a person has to know cold in order to take the next course up.

This list reinforces my feeling that we need such a book, along with a workbook that would structure an ongoing sequence of practice.

I think the RUSSIAN MATH book does an amazing job of fending off the last item on the list.

One last note: I suspect this list is probably a good rundown of why students of any age in any level of math have trouble learning it.

cram school

I had an interesting experience yesterday that illustrated the importance of a student having time for math to 'sink in.'

I was trying to teach Christopher the Saxon 8/7 lesson about subtraction of fractions with borrowing (or regrouping).

He couldn't do it at all.

Then his friends Drew & Marc, who've been in Phase 4 since the beginning, came over. Both of them could not only subtract fractions using regrouping, they could do a darn good job explaining what they were doing & why.

They told me they'd learned fraction subtraction with regrouping in 4th grade, back when Christopher was learning basically nothing.

Then they learned it again last year, in 5th grade. I know this, because Christopher & Drew were in the same class by the time Mrs. Woeckener taught the subject.

The difference between Drew & Marc, who've had a year and a half to know what fraction subtraction with regrouping is & how it works, and Christopher, who's had a huge amount of Intensive Math Intervention but didn't learn this topic in the 4th grade, was stark.

He did pick it up almost immediately, after Drew & Marc showed him how to do it. ('Drew and Marc are better teachers than you!') So that's something.

But this is extremely new & fragile knowledge in his head. Drew & Marc have a far sturdier base on which to build.

ExtendedResponse 08 Nov 2005 - 22:52 CatherineJohnson

My sister-in-law, a fantastic teacher in central Illinois, says the Big New thing in math is extended response. She's going to fill me in when she finds out what it is.

In the meantime, I found this page of released extended response items on the ISAT.

my extended response to extended response

OK, my initial reaction to extended response is: I'm against it.

Actually, make that mixed. My initial response is mixed.

Here's one of 2 released 2004 extended response gr5 items:

A company makes a wall calendar each year. The company sells ad space
around the calendar to local businesses. The cost of ad space is based on
the number of square units each ad contains. The company charges $40.00
for Ad Space D. Using this information:

Draw an Ad Space that costs exactly $60 in the gridded space on page 10 of
the answer document.

And here's the illustration:

I like this problem, although wiser heads here at ktm may give me reasons why I shouldn't, in which case I'll revise my opinion.

I like it because it's visual & spatial as well as 'numerical' (if that's the right word), and because I've found Christopher to be very challenged by any problem that asks him to combine numerical thinking or problem-solving with spatial 'thinking' or problem solving. And of course I love the Singapore bar models, and this problem reminds me of them.

I also like it because it has 2 steps: you have to figure out how much each square costs & then you have to figure out how many squares $60 would buy.

I like the open-endedness of this particular problem, too. A child could simply count the number of squares in Ad Space D (40) and then divide 40 dollars by 40 squares to get $1/square. Or he or she could notice that Ad Space D is a standard multiplication array, and multiply 4 by 10 to get 40. I'm sure a lot of kids would start out counting & then notice, mid-stream, that they could have arrived at their answer more efficiently by multiplying instead. Which is good. A little Math Object Lesson buried inside a story problem.

I like that!

Last but not least, I kind of like the fact that each square turns out to cost exactly one dollar. I don't know why. It reminds me of a genre of problems in Russian Math, in which you go through all kinds of elaborate, painstaking calculations only to end up with an answer of ONE. Or maybe TWO. Or, when things get really fancy, ONE HALF.

Interestingly, I'm finding, as I work my way through RUSSIAN MATH, that I'm becoming quite attached to the number one. Every time it crops up as an answer I think: I should have seen that coming. An answer of one always seems like a flag, a sign that there was an easier, more elegant way to do whatever it was I was doing.....but I missed it.

Russian Math has all kinds of 'surprise answers,' and I think a surprise answer in the middle of an ISAT could be slightly.....fun?

An answer of one is like a little joke.

What I don't like...

...is the injunction to Explain in words how you got your answer and why you took the steps you did to solve the problem.

That is a terrible, terrible idea for a test.

It's a good thing to do on homework once in awhile, or in the classroom. RUSSIAN MATH asks students to write out explanations, although it doesn't ask students to explain how they did a problem. It asks them to restate the definitions & explanations given in the lesson.

Items like these can't possible be graded well on tests. They are far too time-consuming, and graders will end up scoring on length or number of explanations given. When you have items like these teachers are going to end up devoting all kinds of class time to writing extended responses, as Susan H says is already happening. We're looking at a massive waste of teachers' and students' time.

Last but not least, I'd bet the ranch you learn nothing from the verbal explanation that you didn't already learn by looking at the student's work.

Being able to produce a fluent, intelligible verbal explanation of a mathematical solution is almost certainly important for math teachers.

It's not important for the rest of us.

I really don't like this one

The number of fifth-grade students going to the museum is greater than 30
but less than 50. Each student will have a partner on the bus. At the
museum, each tour group will have exactly 6 students.

How many students are going to the museum?

Show all your work. Explain in words how you got your answer and why
you took the steps you did to solve the problem.

Unless 5th graders in Illinois are doing a lot of prime factor problems, I don't see any reason to include an item like this one on a timed assessment.

First of all, no one should have to be doing discovery ON A TEST.

And second, this problem has two answers (36 & 42, right?), but the wording implies that it has just one answer, and that one answer is findable.

I am DISCOVERING the fact that I don't think red herrings belong in math classes. Certainly not in elementary school math classes.

What is the point? You are teaching children to distrust the English language at the precise moment they're learning grammar & composition. An unreliable narrator in a work of fiction can be a terrific device.

But an unreliable questioner in an examination is just wrong.

I'm against it.

update: I forgot 48!

sigh

(thank you, Dan K)

extended response in 8th grade

Here's the 2004 released 8th grade item:

Peter sold pumpkins from his farm. He sold jumbo pumpkins for $9.00
each, and he sold regular pumpkins for $4.00 each. Peter sold 80 pumpkins
and collected $395.00.

How many jumbo pumpkins and regular pumpkins did he sell?

Show all your work. Explain in words how you got your answer and why
you took the steps you did to solve the problem.

The problem is fine, assuming these kids have actually been taught some algebra.

If they haven't, this is a discovery problem on a timed assessment, and I'm against it.

So, assuming they've learned how to set up & solve equations with unknowns, the problem is good. IMO.

The demand that the student explain each step in words is not.

Russian Math rocks

Instead of writing about Russian Math, I should be downstairs (at the kitchen table!) actually doing some Russian Math.

So I think I'll sign off.

But tomorrow I'll give some examples of what a proper extended response item should be.

A proper extended response item should be a RUSSIAN MATH EXTENDED RESPONSE ITEM.

update: scoring rubric for extended response

'Student Friendly' Mathematics Scoring Rubric

Assuming I'm reading this correctly (I feel a little distrustful), students must get all computations correct in order to earn the highest possible score of 4. They can earn a score of 3 with minor mistakes in computation, which I feel is fair, though others may disagree.

What I reject absolutely is the explanation section:

I write what I did and why I did it.

If I use a drawing, I can explain all of it in writing.

This is wrong. I don't believe a 4 should depend upon being able to supply an explanation in any case. But here you have a child who can explain why he or she did what she did in a drawing, which is no mean feat (and I'm in a position to know) and even that isn't enough.

Pace Anne, you'll notice that it's not OK for a child to explain what he/she has done by offering a mathematical demonstration, as the teachers in Liping Ma's book do. Anne's right about that; it struck me, too. Over and over again, when Liping Ma asks a Chinese teacher why he/she teaches an idea a certain way, the teacher responds by writing out a proof-like mathematical demonstration. That's what makes the book incredibly difficult (and incredibly valuable) to read for most of us; the teachers don't translate math into words, and neither does Ma.

For Chinese teachers, math is math.

This drops you to a 3:

I write mostly about what I did.

I write a little about why I did it.

If I use a drawing, I can explain most of it in writing.

A couple of years ago the head of our school board sent out an email explaining the adoption of TRAILBLAZERS that included this line (from memory): In recent years math has become language-based.

I think that would come as a surprise to actual mathematicians.

extended response problem from IL state test
extended response problem 1
extended response problem 2
extended response problem 6
extended response problems 7, 8, 9
direct instruction & the rigor conundrum
Dan's daughter reacts to extended response problem
defensive teaching of Singapore bar models
open-ended problems in math ed
problems that teach - "Action Math"
email to the principal

CalculusBookRecommendationNeeded 15 Sep 2005 - 17:27 CatherineJohnson

A lot of good stuff in the comments I want to get pulled up front, but since I have to go into the city today, there's no time at the moment.

I'll just get this posted, from Anne:

Speaking of good books, does anyone have a recommendation for a good calculus book? I still have mine from college, but I have a hard time understanding the proofs. It seems like there are steps missing.
In the book Countdown about the math Olympiad, the author mentions that one of the mathematicians who constructs the problems is Russian. This mathematician says that his calculus book was only 100 pages long, but it was excellent and had no extraneous information. I wonder if anyone has translated a Russian calculus book.

I've been wondering the same thing, and ktm needs a recommendation to post as well.

So if you've got suggestions, please let us know.

I have two, potentially.

Calculus Made Easy by Sylvanus P. Thompson (and updated/revised by Martin Gardner) This is a classic (always a good sign), and people rave about it. I don't know whether it has proofs, or whether the idea is to give people conceptual understanding without formal proofs.

Also, believe it or not, the University of Chicago School Mathematics Project, the same folks who are responsible for EVERYDAY MATH, had a longrunning project translating foreign math textbooks into English. I'm not sure I can track down what's happened to the list; it seems to have moved to the American Mathematical Society, but I can't find it there at the moment.

I know I did once track it down...so I assume it's still findable.

If someone else comes across it before I do, could you post the link?

Thanks.

translation from the Russian

Calculus of Variations by I.M. Gelfand & S.V. Fromin

Is this the one?

update

Bernie & others say the Gelfand book is an advanced text. (I didn't have time to read the blurb yesterday.)

CalculusTexbookRecommendations 11 Oct 2005 - 13:52 CatherineJohnson

I love Amazon reviewers. Half my life is based on these folks.

Amazon reviewers of calculus textbooks, I've just discovered, are a different animal. Not fully domesticated, I'd say. So it's gonna take me a while to cess them out.

Here's one fellow I'll probably add to my pantheon:

If you are a serious student of Calculus, go get Anton's Calculus. I am a Math teacher in Malaysia and a long time user of Anton's Calculus since his 3rd edition. I teach Calculus the traditonal way because in my country we are still new to the computers. Prof Anton has written books in his previous editions in a lively and refreshing manner that I could read his book again and again without getting bored. I may be old-fashioned, but as a fan of Anton, reading his latest 6th editions is such a delight, and only recently I have just learned how to make use of software like Maple, I could see Anton's Calculus paving my way into new explorations, as his new book says, Calculus: A New Horizon indeed. Buy Anton's Calculus, I am sure you will not regret.

update

It's nice to see college kids are also learning nothing:

In Calculus I, I was taught using computer programs how to solve Calculus problems but never actually learned Calculus. This put me in a tough spot when I had to start Calculus II and didn't know what I was doing. In this course we weren't allowed to use calculators and everything I learned in Calculus I became useless. Fortunately, I came across this book and I was able to teach myself Calculus in a matter of days. I also tried several other Calculus supplements and the only one I can recommend is "How To Ace Calculus" and its sequel for anyone taking Calc II & III. Whether you're dumping a fortune into an education on brushing up on some old math this book is the only supplement you need.

this guy is hilarious

I also spend a huge amount of time cruising Amazon's listmanias. Here's one called So you'd like to... Learn Calculus and Analysis, And Really Understand It! by one Billy Smorgasbord, a resident, it seems, of Oxnard and Antarctica.

There is an bothersome and fairly intimidating phenonemon which is widespread among mathematics teaching and textbooks. For want of a better term, we might call it "Mathematical Macho". Now, when in the grip of this mysterious phenomenon, it seems that people get the idea that it is necessary that a deep subject like mathematics be really difficult to learn, and that there should be an effect of "weeding out the weaker students" alongside that of actually teaching the stuff.
To be fair, I should mention that, over the years, I have observed an impressive number of attempts (whether or not these were made wholly in earnest will be left to the reader) by numberless (pun somewhat intended) and often quite well-esteemed authors and, even, a whole venerable organization (this called the Mathematical Association of America), to make the subject more palatable, and perhaps even interesting, to a wider audience than yet before.
Nope, sorry, fellas. Thus far things just haven't worked out all that well.
Yup, I've seen 'em come and go, alright. Witness the sometimes abysmally constructed explanations in "Calculus Made Simple" by Silvanius Thompson, the scarifying "rigorous" language purveyed by most MAA textbooks, the quite awful wording and quite annoying imbedding of mathematical syntax within text to be found in Boas' celebrated "A Primer of Real Functions", the spotty development in Schey's "Div, Grad, and All That", et cetera. We won't even go into that astonishing and original artfulness (arguably for the delectation of brilliant student and scholarly peer, not for the now-terrified beginning reader) made of the subject in Apostol's highly-regarded two-volume masterpiece.

Billy's list, to my untrained eye, seems pretty useful, and thus far Amazon reviewers mostly second his opinions. However, his listmania on Yup, You Really Can Increase Your Intelligence opens with a book by Robert Sternberg, a red flag for me. Years ago I read a popular book on intelligence by Robert Sternberg many years ago that I thought was pretty dumb.

Plus which, until I'm persuaded otherwise, I'm rejecting out of hand Billy's opinion of Calculus Made Easy. Any book that's been continuously in print for over a hundred years gets on my short list.

(I own the book, and the introduction alone is worth the price of purchase. Haven't tackled the calculus yet.)

So Billy's on probationary status.

Billy's guide says that 23 of 26 people found this guide helpful. Read 13,887 times

I'm going to have to start paying attention to how many people read listmanias.

f6b7124128a002be707b5010.M.jpg Billy?

URLs for listmania & 'so you'd like to guides'

Top So you'd like to guides

Top Listmania lists

So you'd like to... Learn Quantum Mechanics Via Worked Problems and Solutions!

point of comparison:

35 of 35 people found this guide helpful. Read 4,060 times.

while I'm on the subject

Newt Gingrich has 14 pages of book reviews on Amazon.

I bought a book on Saving the Giant Panda he recommended. Very cool pictures.

No calculus recommendations, as I recall.

how not to title your So you'd like to guide

So you'd like to throw your writing career out the window

9 of 11 people found this guide helpful. Read 317 times

CalculusRecommendations 22 Dec 2005 - 16:49 CatherineJohnson

OK, I've collected a handful of recommendations.

Michael Spivak

First, check out the Comments thread on calculus books.

Here's one interesting comment:

Michael Spivak's books are good, as is Tom Apostol's Calculus. Personally, I prefer Spivak. They are both Americans by the way. G.H. Hardy's A Course of Pure Mathematics, and Richard Courant's Differential and Integral Calculus are both classics which are very good, but probably not for everyone. Those are all longer than 100 pages. If you are looking for brevity then you can try out Dan Bernstein's(another American) "Calculus for mathematicians" which is only 12 pages. Find it here: More Mathematics .
None of these books are typical of what you will find in the modern science/engineering calculus courses. If you want something along those lines, then I'd recommend Salas, Hille, and Etgen's Calculus: One and Several Variables.
Fomin and Gelfand's book considers calculus of variations as opposed to calculus of real variables(i.e. "standard" calculus). It's a good book, but probably not what you are looking for.

People love Spivak.

oops. Just clicked on 'See all 60 customer reviews.' Some people love him, some hate him.

Here's Apostol.

Purcell, Varberg & Rigdon

I've asked both David Klein & Barry Garelick for recommendations.

Here is Klein:

I'm not up on calculus texts. I use a standard book (one of many) along with others at CSU Northridge called, CALCULUS WITH ANALYTIC GEOMETRY, 8th ed., by Purcell, Varberg, and Rigdon. It has its faults, but isn't bad. The theory part is good, but it needs more medium level difficulty problems and more graphing examples (without calculator assistance). [One Amazon reviewer loathes it; the other likes.]
Worth avoiding in my opinion is the so-called "Harvard Calculus" books:
Calculus Reform—For the $Millions by David Klein and Jerry Rosen (you'll have to register to open this pdf file, but registration is free)
WHAT IS WRONG WITH HARVARD CALCULUS? by Jerry Rosen and David Klein

Subsequent editions have remedied the worst of the deficiencies, but I would still avoid it.

(I should add that I think Carolyn somewhat liked reform calculus. She's in transit at the moment, but when she chimes in, I'll either edit out this comment, or add hers as needed.....)

Ivan Niven

Barry's first suggestion, which comes from Dick Askey of the University of Wisconsin, is Calculus: An Introductory Approach by Ivan Niven.

I'm sorry to say I've bought the one and only used copy available at Amazon, but there are 2 copies available at Alibris.

Niven wrote his book in 1961, before graphing calculators.

Lipman Bers

Another recommendation from Barry:

Calculus by Lipman Bers, which I ordered the minute I read this Amazon review:

I had come across this book in the university library.
Before that I had been getting excellent marks in Calculus by mechanically going thru the rules in my mind. This book changed all that and gave me a proper perspective on the discipline.
The explanations are clear and this book is eminently suitable for self study.
Recommend this book whole-heartedly at least for the first and second years of calculus.
This was about twenty-five years ago ! But it's just as relevant now.

Barry says Bers was a Russian mathematician emigre to the U.S. who was familiar with Russian textbooks.

Thomas's Calculus

Another possibility might be an early edition of Calculus by George B. Thomas, now in its 11th edition.

Barry says Thomas's Calculus was a college staple for years, and is not easy.

I'm having trouble finding out when Thomas died, so I have no idea which editions of Thomas' Calculus were revised after he was gone.

.....oh, here's a clue, in an Amazon reader review:

I've used both Stewart's Calculus and Thomas'. Interestingly, Thomas has been writing calculus books for a LONG time and i've picked up several editions in the used book stores, because from the first time i bought a Thomas calc book back in Jr. High for my own self interest, i was a fan of his style.
His style is that of the old-school American text book authors who wrote in a clear, concise manner of English, using tangible and visual examples. Those old writers still thought of much of the material as novel, and were appealing to a more agrarian society of students.. especially the young and booming field of engineers. This is lacking in today's texts. The only drawback is that some old texts are much too impersonal and use the passive voice for everything, which can make them very difficult to read at times.
Thomas' recent editions (at least - i can not recall for the 60's era editions) are not only formally clear, but easy to understand and read. Here are the ways in which Thomas' book beats Stewart's book....
[snip]
Thomas' book is in fact probably the best calculus textbook around. I've looked at many many of them, and fraknly, none of them are this complete and well developed... The funny thing is, Thomas' book was one of the best decades ago. It has only gotten more exhaustive and more mature!

This reminds me of Carolyn's post about the early books in a field, Don't teach in a monotone

Thomas has 5-star & 1-star reviews. Very mixed.

James Stewart

Lastly, Barry reports that James Stewart's texts, which teach graphing calculators, are being used a great deal. Barry says Stewart's books are 'fairly good.'

The two big ones seem to be:

Calculus : Concepts and Contexts (with CD-ROM, Make the Grade, and InfoTrac)

and

Calculus

Mixed reviews, expensive as the dickens.

off-topic: Arnold Kling

I just found all of Arnold Kling's Amazon reviews....

'the calculus page'

No idea if this is worthwhile: calculus.org: THE CALCULUS PAGE

BernieOnCalculus 14 Sep 2005 - 22:37 CatherineJohnson

First off, I've become very wary of Amazon's reader reviews ever since I realized that they remove negative comments in order to boost the ratings of the books. That's not kosher. [Catherine speaking: I posted 2 5-star reviews on Amazon that have disappeared, so I'm not sure Amazon has a systematic policy against negative reviews....]

Ok, what's the big deal about Calculus? Why are there thousands of Calculus books and none of them any good?

The reason is that the subject is simultaneously too big and too deep. And there's really no good way to split it up into manageable digestible pieces.

If you want to understand a computer, say, you can split it into pieces (power, case, motherboard, plug-in cards) which are you can then study and understand separately. But with Calculus, learning the subject is more like approaching a huge ship in the fog. At first you don't have any idea what is there. Then a few points become clear, but they are disconnected and make no sense. Then a few structures show themselves, and gradually, very gradually, the whole thing starts to come together. It takes much more energy and much more determination to carry through with such a program than with simpler subjects. So most people don't carry through with it, and it becomes a filter, a flunk-out class.

Linear algebra is a much more useful subject which is amenable to being broken into manageable chunks, and perhaps for this reason it doesn't carry the same mystique as Calculus.

Let's lay out what Calculus is in order to make this clear. It consists of two new operations called "differentiation" and "integration"--roughly analogous to subtracting and adding--both of which are based on a totally new view of the world, called "limits". Limits are a pretty deep concept, much deeper than is generally supposed or understood by most people taking Calculus. In fact, I would venture to say that most people taking Calculus never really grasp limits and, as a result, end up more confused and resentful about mathematics than when they started. Moreover, limits cannot be tackled until one has already achieved a certain mastery of both algebra and geometry, for they entail a melding of these two subjects. Both subjects must have been learned down to the "have it at my fingertips" level before limits will start to make sense.

To be perfectly honest, the problem is even worse than that, because I think it's fair to say that in some sense the human race doesn't really understand Calculus yet. This is because, although there is complete agreement on what basic Calculus is and how to use it, there is still sharp disagreement on what the logical underpinnings of it should be. It's really kind of like Quantum Mechanics in this regard, and that makes it quite unlike all the other kinds of mathematics young students have ever seen, which is all cut and dried.

So, to take the larger view once more, Calculus has three aspects which the student must master more or less simultaneously: 1) the mechanics of integration and differentiation and limits, 2) a philosophical understanding of limits, 3) the thing we discussed yesterday--an understanding of the underlying meaning of the formalism of Calculus in terms of real-world problems. Because there is so much interconnected stuff to learn, the connection between formalism and real-world meaning is even more tenuous, and must be held in even greater abeyance, than is the case with standard school mathematics. The student must suspend disbelief for a much longer period than ever before. Which means that there are inevitably many more Calculus students who get left by the wayside than occurs in elementary mathematics.

It is generally accepted among mathematicians that the hardest part of learning Calculus is 2), the philosophical part, and therefore the teaching of Calculus is usually broken into two subjects, taught to two different groups. "Mechanical Calculus" (high-school Calculus) is taught to students who are deemed too hopeless to ever really learn it deeply. Almost all standard Calculus taught to freshmen college students is of this kind. The students are only taught the basic formulas for differentiation and integration and some of the applications are shoved down their throat. Limits are hand-waved and never really explained, and most students don't realize there's a problem. They're just left with a vague feeling of uneasiness. If they're engineering students, then they are drilled on the applications for another 3 or 4 years, so that they become quite good at them, without worrying too much about what it all means. It works, why worry about it?

For students believed to be budding mathematicians, the whole subject is taught, with an emphasis on the meaning of limits and being able to deeply understand the logical underpinnings of the whole enterprise, i.e., to do proofs. Applications are only lightly touched upon. That's the audience Apostol's book is written for. That's a completely inappropriate book for almost all people.

The mechanics of Calculus, i.e., the basic formulas for integrating and differentiating, aren't really that big a deal except for one fly in the ointment. They are operations applied to functions rather than operations applied to numbers, which is all that the students have ever seen before. So even here there is a philosophical hurdle, because it's hard for people to think of functions as objects. We are used to thinking of functions as the "verbs" of mathematics, not the "nouns", so operating on them seems very strange and most young students probably never really grok it. It's yet another philosophical nut to chew on before one can really understand what one is doing with Calculus. It takes time for that fact to sink in.

The single most important obstacle precluding most students from mastery of Calculus is that they don't really have any idea what functions are when they start Calculus. And that's usually because they don't have a firm grasp of algebra. This, however, is a solvable problem. I personally would reorganize the curriculum so that a year is spent just messing with functions before Calculus is tackled.

But of course that runs headlong into the problem that people in high school and college--unlike students in elementary school--have very little desire to suspend disbelief: if they can't see an immediate payoff for what they are learning right now, they don't want to learn it. This leads to a quandary for the teachers/professors, namely, in order to motivate the students they have to tell them the applications. But in order to do the applications, the students need the full machinery of differentiation and integration. This leads inexorably to the continual cycle of Calculus "reform" which changes textbooks every couple of years, seeking to do the undoable by squeezing in years of difficult philosophical struggle and mechanical practice into far too short a time period.

There's also the problem that many of today's soccer mothers and fathers want to push their children into Calculus as quickly as possible in order to put another feather in their own cap, so they have no tolerance for an extra year "wasted" on learning functions. But that's a subject for a different thread.

WilliamKSmithCalculus 16 Sep 2005 - 12:16 CatherineJohnson

Here's another recommendation from Barry Garelick:

Calculus with Analytic Geometry by William K. Smith (also available at Amazon)

I've already ordered my copy.

Have I mentioned I'm planning to take calculus?

Well, I am. I'm planning to take calculus.

But first I have to 're-take' algebra & geometry. Then trig, which I've never studied.

You folks here at ktm are helping me so much. Even though I'm a writer, I can't locate the words to describe what you've given me. The reason I can't 'locate the words,' of course, is that I don't actually know what I'm learning from ktm. I study & absorb what people say, but then forget the source of my new knowledge once it's been assimilated into my store of old knowledge. I'm left with the hazy feeling that 'I'm learning a huge amount from the Commenters at ktm.'

So I'm going to start taking notes. God is in the details.

thank you!

integers! integers!

So Christopher's math class started integers on Monday—a topic he knows virtually nothing about—and he's having a test tomorrow. He is way not prepared, so I'm busy today writing an Integer Lesson. Probably won't be posting much (though I may have a couple of things from Barry.)

I'm taking a moment to make one more plug for Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, though.

I could probably add & subtract integers in my sleep. (Though I did have to do some review last year when I first re-encountered the topic, which I take as a sign that my knowledge was more procedural than conceptual.)

But last night, after working with Christopher for awhile, who was semi-lost (I don't think he could pass a test at this point) the Math Fog rolled in.

This is the good thing about working with people who know less math than you do. Concepts and procedures you thought you understood turn out to be not quite so clear. I assume that's what Bernie meant when he said the other day that he'd realized there were aspects of reciprocals he hadn't thought about (if I've got that wrong, Bernie, I'll change it!) Carolyn has said something similar at times. I'll be asking her about some elementary concept that, for her, is as simple as breathing in and out, and suddenly she'll see why Ben--or anyone else--might get confused.

lost in translation

This is another one of those constructivist insights that's been lost in translation.

For me, and I think for most teachers & writers, teaching or writing about a subject always forces you to understand it far better than you did.

Radical constructivists conclude from this that children should explain all of their answers in words.

I'm pretty sure that's wrong, because math is not language. Math is math. A child who can explain his answer by showing the mathematical steps he took to find it has produced a proper mathematical explanation as far as I'm concerned. (Russian Math & the Chinese teachers in Liping Ma all offer mathematical explanations & demonstrations.)

But what really bothers me about the 'explain your answer in words' business is that it puts the onus on the child to teach himself. The teacher doesn't have to work and fight and struggle to find the right words; the child does. I know that's wrong.

While I'm on the subject, why don't I just go ahead and take umbrage at the suggestion that a child is capable of explaining math in words?

Writing is hard. Writing well is extremely hard. Finding the words to explain any mathematical concept well is a vast and ambitious undertaking in itself, not a toss-off in the middle of a homework assignment or state assessment. (I'm seriously against the extended response (pdf file)requirement that's taken over IL state rubrics. At least, for the time being I am. [update 5-14-06 sorry, link no longer works])

back to Russian Math

I shouldn't be putting words in people's mouths, so if I've misunderstood Bernie or Carolyn I'll issue a CORRECTION.

In the meantime, why don't I just return to quoting myself.

It's true for me that when I work with a child for awhile, I realize I don't understand things as well as I thought (or hoped).

After Christopher went to bed, I got out Mathematics 6 and turned to the section on adding & subtracting integers.

The first thing that struck me was the fact that this topic appears at the very end of the book. Prentice Hall Pre -Algebra^* opens with integers, and I question that. I question it not based on any profound grasp of pre-algebra as a coherent whole. I question it on grounds that Nurk & Telgmaa are geniuses, and they put adding & subtracting integers last, not first.

I'm sure they have their reasons. (I intend to figure out what their reasons were.)

Reading through Nurk & Telgmaa's discussion, I realized why I was confused. I think I realized why Christopher was confused, too. I hope so.

We were both, I believe, stumbling over this type of problem:

5 - (-7) = ?

Both Saxon Math 8/7 & Russian Math teach addition & subtraction of integers using the number line. Saxon's lessons were particularly strong, I thought.

But when I tried to untangle myself by resorting to the number line, I got stuck.

Start at zero, move five to the right, then.......then what?

What was my next move? My very next move, without renaming or re-expressing - (- 7) as + 7 ?

I was stuck.

Reading through Mathematics 6 I realized that the problem is something Wayne Wickelgren & his daughter Ingrid have raised: the same letter or sign has been made to stand for two different things.

There are two 'minus signs' in 5 - (-7). One means 'opposite,' and the other means 'subtract.'

One means 'perform an operation' and the other doesn't (I don't think. Is 'taking the opposite of a number' considered an operation? I don't know.)

In any case, for both Christopher and me, 'subtract' and 'take the opposite of' are two different things.

Mathematics 6 has a formal demonstration of the fact that:

5 - 7 = 5 + ( -7 )

This is something I think I figured out on my own many, many years ago. I've been using it ever since to de-confuse myself when dealing with long lines of integers to add & subtract. At some point, if I'm getting confused about whether I can or can't use the commutative or associative properties, I just turn the whole thing into addition.

Reading Mathematics 6 I realized that's what needed to happen with 5 - ( -7):

5 - ( - 7) = 5 + [ - ( - 7) ]

Voila!

Christopher and I both understand that 'the opposite of the opposite' is the number you started with originally; the opposite of the opposite of 7 is 7. (This wasn't an especially hard idea for Christopher, but the number line really nails it down.)

Once you convert '5 minus negative 7' to '5 + the opposite of the opposite of 7' it's in a form Christopher understands, and can do.

AND it's in a form you can perform on the number line, if you like or just want to check.

5 - ( - 7) =

5 + [ - ( - 7) ] =

5 + [ 7 ] =

5 + 7

Once you've converted a 'double negative' subtraction problem into addition, you no longer have an anomaly, The One Subtraction Problem That Cannot Be Done On A Number Line.

We'll see how it goes. This morning I had Christopher quickly rewrite 12 subtraction problems as addition problems. (I haven't explained to him why a subtraction problem can be rewritten as an addition problem, and I don't know whether I'll get to that today. I haven't closely studied Mathematics 6's presentation to see whether I can introduce it 18 hours before the test.

Fortunately, Ed had already introduced the idea that 'subtraction is addition' last night, when he used the addition-of-debt-to-debt (a concept that is not foreign to our household) to show Christopher that:

- 7 - 7 = - 14

I think he had a lesson in Saxon on subtracting a positive from a negative being the same thing as adding a negative to a negative, so he probably had some knowledge to build on before Ed gave him the add-one-debt-to-another example.

It's the minus-minus issue that's throwing him.

I hope.

one last thing

Looking at this, it strikes me I'm also going to have to create some problems that I ask Christopher to 'simplify'—'simplify' defined broadly as 'write it in the simplest possible correct way that will allow you to recognize what the computation is and do it.'

For instance:

-7 + 5

He probably needs some practice rewriting this as 5 - 7.

I'll see.

I'm also going to try to put together an incredibly simple 2 - 1 type problem that he can always solve quickly when he gets jumbled up. Something like this:

1 - ( - 1) = 2

-1 -1 = -2

-1 - ( - 1 ) = 0

He hasn't learned the Polya line about how 'For each complicated problem you can't do, there is a simple problem you also can't do.' I realize it's not clear that you can explicitly teach problem solving, but I'm going to have to try. He's got to learn the strategy of creating a super-simple version of a hard problem in order to see how to deal with the hard problem SOON.

^*new title: Prentice Hall Mathematics: Explorations & Applications

keywords: subtraction negative minus absolute value subtraction is addition integers extended response

MathLessonsPage 21 Sep 2005 - 15:48 CatherineJohnson

I've started to get the Math Lessons page pulled together. I'm sure I've forgotten posts that should be indexed there, so if you know of any, let me know. (Any lessons you especially like from other people's sites, like MathandText, for instance, should also be added.)

There's a link to 'Math Lessons' on the sidebar.

CalculusWorksheets 21 Sep 2005 - 20:52 CatherineJohnson

Central Lakes College has calculus worksheets, too.

Here's one. (pdf file)

Unfortunately, they've posted a link to a set of calculus notes they characterize as ++great++, but they're off-limits to me.

Yahoo's list of math links

Yahoo math links

self-instructional mathematics tutorials

This site, self-instructional math materials, looks interesting:

The following mathematics tutorials development as part of the project, Increasing Students Success: Addressing Prerequisite Mathematics Assumptions in Introductory Non-mathematics Courses, funded by The Fund for the Improvement of Postsecondary Education. (project #P116B60125)
Various introductory courses at six universities have been selected for this project. One goal is to provide self-instructional mathematics tutorials for individuals who may need review of certain topics. This self instructional approach will:

let you move at your own pace.

provide you with additional review (if necessary).

let you know how well you are doing.

Currently the none interactive versions have been developed. While some do not have a lot of graphics, the review materials 3, 4, and 5 are fairly graphic intensive and may take a few minutes to load. Interactive versions are currently being developed and will be added to this site at a later date.

keywords: Yahoo math links calulus worksheets self instruction self teaching teach yourself

OnlineMathResources 22 Sep 2005 - 22:30 CatherineJohnson

I came across all kinds of interesting-looking math web sites last night while looking for:

integers worksheets
downloadable number line worksheets

I didn't find either of the things I wanted (and almost spent $29.95 to join some teacher site linked to by FunBrain just to be able to printout their number line sheet...).

But I found all of these:

AAA Math (resources listed by grade thru gr8)
also has a potentially interesting page called World Education Levels. Unfortunately, I can't tell what 'world education levels' are without spending a lot more time on the site than I want to spend. LOTS of online quizzes that are corrected by the site, and they seem to be selling a software program on arithmetic.

Computer Lab: Math, Numbers, and Counting HUGE survey of 163 different online math sites for kids. Terrific collection.
Item number 163: Disaster Math for Kids
From FEMA! Interactive word problems graded by the FEMA site.

the aforementioned FunBrain Math Baseball is a classic.

FunBrain's teacher site, the page that almost sold me a $30 sheet of number lines. Has articles on behaviormanagement in the classroom that look good.

Harcourt School Publishers' number line express Blecch. But maybe little kids would enjoy it. There's a talking lion railroad engineer.

MacTutor History of Mathematics

Math Cats how-to for teachers Definitely worth looking at.

math clip art! possibly for autistic kids (I was on a major clip art tear a few years ago, when Andrew was in his PECS genius phase...)

Maths Dictionary (British?)

Mathsurf teacher's site word problems from Pearson Scott Foresman. If you're looking for story problems with multiple answers, this is the spot. Possibly (probably?) a good site to visit for problems your child may encounter in constructivist math courses -- worthwhile problems, as far as I can tell on cursory inspection.

Mathsurf telling time worksheet (to print)

NCTM Illuminations Looks decent. Lots of interactive 'tools': affine recurrence plotter and--Yay! nets!! I've been looking for nets!

National Library of Virtual Manipulatives for Interactives Mathematics I've visited this site several times before. It's pretty good, as I recall. (If I'm remembering correctly, the pan balance problems are kind of cute & potentially useful.)

Room 108 Looks decent. You can create online Mad Minute pages (must be answered & graded online)

odd & even numbers possibly good for autistic kids? this site speaks the directions, although I don't think the directions are also written out in words. But any time an autistic child can hear the same words spoken by the same recorded voice it's a good thing, I believe. Site is simple and graphically compelling. Has a HUGE cursor (also great for autistic kids.)

Primary Games good for autism? I have a feeling this might work with Andrew at some point in the near future. Very simple, has ONE moving image--'Squigly,' a little worm inside one of 10 apples who pops out of his apple and then disappears back inside every couple of seconds. The child has to tell which a

Navigate KTM

Kitchen Table Math

KTM User Pages

Service Groups

Parent Groups

Personal Pages

Blogs

Special lists

Help

Entries from HighSchoolMath

update: Department of Corrections

Elementary school

Middle school

High school

Programs explicitly denounced by over 220 Mathematicians and Scientists:

one piece of bad news per slide

don't bury the bad news inside a bunch of other junk

update

from the SREB report (pdf file):

Key Findings

Studying “something called algebra”

cognitive load

update

update

metacognition & math homework

back shortly

does math talent 'show'?

parents & grade school math

the book parents needs

cram school

my extended response to extended response

What I don't like...

I really don't like this one

update: I forgot 48!

extended response in 8th grade

Russian Math rocks

update: scoring rubric for extended response

translation from the Russian

update

update

this guy is hilarious

URLs for listmania & 'so you'd like to guides'

So you'd like to... Learn Quantum Mechanics Via Worked Problems and Solutions!

while I'm on the subject

how not to title your So you'd like to guide

Michael Spivak

Purcell, Varberg & Rigdon

Ivan Niven

Lipman Bers

Thomas's Calculus

James Stewart

off-topic: Arnold Kling

'the calculus page'

integers! integers!

lost in translation

back to Russian Math

5 - (-7) = ?

5 - 7 = 5 + ( -7 )

5 - ( - 7) = 5 + [ - ( - 7) ]

5 - ( - 7) =

5 + [ - ( - 7) ] =

5 + [ 7 ] =

5 + 7

- 7 - 7 = - 14

one last thing

-7 + 5

1 - ( - 1) = 2

-1 -1 = -2

-1 - ( - 1 ) = 0

Yahoo's list of math links

self-instructional mathematics tutorials