03 Aug 2005 - 16:40

Hotmath.com

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.)

Back to main page.

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Catherine, I take it Prentice Hall prealgebra is different from the Prentice Hall Math Course 1,2,3 series that Ben is using?

-- CarolynJohnston - 03 Aug 2005

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Oh!

I thought he was using pre-algebra!

Hmm.

I think they're different.....

-- CatherineJohnson - 03 Aug 2005

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I just looked, and Prentice-Hall 1, 2, & 3 are there, too.

What's the difference?

I'm going to see if I can find out.

Meanwhile, I have to get the op-ed to Temple TONIGHT and my 'upstairs computer' is crashing repeatedly.....I'm tearing my hair out.

-- CatherineJohnson - 03 Aug 2005

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Is yours by Boyd & Branch?

-- CatherineJohnson - 03 Aug 2005

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Yippee. My Mac seems to be crashing. This is a nightmare.

My Prentice Hall is Pre-Algebra by Davison, Landau, McCracken, Thompson

-- CatherineJohnson - 03 Aug 2005

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Here's what I see for Prentice-Hall Math Course 1 TOC:

Course 1
Charles,Branch-Boyd,Illingworth,Mills,Reeves
Copyright: 2004 Publisher: Prentice Hall
Grades: 6-8

1. Decimals
2. Algebra: Patterns and Variables
3. Number Theory and Fractions
4. Adding and Subtracting Fractions
5. Multiplying and Dividing Fractions
6. Ratios, Proportions, and Percents
7. Data and Graphs
8. Tools of Geometry
9. Geometry and Measurement
10. Algebra: Integers
11. Exploring Probablilty
12. Algebra: Equations and Inequalities

-- CatherineJohnson - 03 Aug 2005

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Here's the one I have (this is a revised edition; we're using the old one, I assume, but they don't look hugely different):

Pre-Algebra
Johnson, Kennedy, Thompson, Charles, Bass, Bellman, Handlin, Davison, Landau, McCracken, Bragg (Davison, Landau, McCraken & Thompson are authors on mine)
Copyright: 2004 Publisher: Prentice Hall
Grades: 8-12

Appropriate for a wide range of student abilities. Works for both the middle school and high school students preparing for success in algebra.

TOC
1. Algebraic Expressions and Integers
2. Solving One-Step Equations and Inequalities
3. Decimals and Equations
4. Factors, Fractions, and Exponents
5. Operations with Fractions
6. Ratios, Proportions, and Percents
7. Solving Equations and Inequalities
8. Linear Functions and Graphing
9. Spatial Thinking (this is called 'Algebra in Geometry and measurement' in my edition)
10. Area and Volume
11. Right Triangles in Algebra
12. Data Analysis and Probability
13. Nonlinear Functions and Polynomials

-- CatherineJohnson - 03 Aug 2005

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The one whose TOC you've listed above FIRST is the one that Ben and I are currently working from. It's nominally the 6th grade text. He'll be using the 7th grade (Course 2) text in the fall. I don't know a thing about the pre-algebra text, if it's different.

Is the pre-algebra text as hideously busy as the Math Course X series?

The pre-algebra TOC looks very intense. But -- I think you have told me that this is the course where Irvington actively tries to eat children and spit out their little bones, correct?

-- CarolynJohnston - 03 Aug 2005

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OK, so now I can't post comments, either?

-- CatherineJohnson - 03 Aug 2005

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No.

That one went through.

I'm in computer hell.

-- CatherineJohnson - 03 Aug 2005

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OK, now I'm remembering about Prentice Hall pre-algebra.

It's supposed to be an 8th grade book.

I'm going to find the mathematicallycorrect review.....

-- CatherineJohnson - 03 Aug 2005

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Sigh.

They hate Saxon 8/7, which is what I'm using, and they like Saxon Algebra 1/2, which is what I'm not using......

http://www.mathematicallycorrect.com/books7y.htm

-- CatherineJohnson - 03 Aug 2005

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Not using, and don't own.

-- CatherineJohnson - 03 Aug 2005

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Ooo, We did Glencoe Pre-Algebra last year. Good to know they liked it. Yay. It had a lot of online stuff that I liked. Parent/Student pages to practice for each chapter.

I wonder what they don't like about Saxon 8/7. Do they feel that way about all of them? I guess I'll have to look around. I haven't been to this site in a while.

-- SusanS - 03 Aug 2005

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Mathematically Correct has some disagreement amongst reviewers...

Saxon 8/7 is a bit of a controversial book as far as I can tell; the lady who loves Saxon over at lewrockwell (I've forgotten her name at the moment) says it's a 'remedial' book and she never uses it.

It's way too easy for a 7th grade book; I suspect that's the problem.

Christopher's going into 6th grade, and he's only done one Saxon book, so I think it's good for us, though I'm supplementing with one bar model a day.

My neighbor bought Algebra 1/2, which seemed incredibly 'procedural' to me; there didn't seem to be any explanations at all.

otoh, I hadn't been working with math textbooks for long when I saw her copy, so it's possible that the problem sets are where the magic lies.

-- CatherineJohnson - 03 Aug 2005

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I'm glad to know about the Glencoe site.

Did your school use Glencoe?

Or did you choose it?

-- CatherineJohnson - 03 Aug 2005

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That was their text at the junior high for pre-algebra for high math 6th graders. I think the other class is regular 6th grade math. I believe (but am not sure) that Trailblazers was or will be the text, (but I could be wrong, it might just be 4th, 5th grade. Everyone tells me different.) We had a good teacher to boot. She was very aware that my son was a 4th grader so she communicated with me alot about how he was doing. She was also a real "math" person, so it was almost like they communicated in code when I was around. It was a neat sight to see a teacher that really got him.

Teacher at a time...

I loved the online stuff. They had extra example, quizzes, parent help, etc.

The 7th grade Algebra text is the Heath, Algebra I--An Integraged Approach (McDougal^? Littell.)I need to go find out about that one.

Both texts are very slick and colorful.

-- SusanS - 03 Aug 2005

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Trailblazers stops at 5th grade.

Wow--he was in 4th taking Glencoe Pre-Algebra!

Neat.

-- CatherineJohnson - 04 Aug 2005

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It was a little difficult at first since there was no differentiating for him due to his age.

Officially, I believe they call it "single subject acceleration." It is really critical that there is major parent support and teacher "understanding" involved, especially if there is a school switch. I realize that now.

One of the complaints of the NCTM standards is that they don't seem to effectively deal with gifted issues. At first, they said nothing about it, then later they dealt with it by pushing "enrichment" only of the text. They are against acceleration even though most experts agree that in certain cases it is a perfectly legitimate and appropriate response.

From what I can tell, most teachers believe that when a child truly understands the underlying concept of math, then they are ready to learn it officially. This is pretty standard approach for gifted kids as well, particularly in math.

I do worry that NCTM will eventually convince the powers that be that it isn't necessary allow a child to move forward at a quicker rate than the others. It does require more care from the schools to make acceleration (or grade skipping)work. It would be easier to just believe that it isn't necessary (or that it is "harmful"),and just leave the kid in the classroom.

-- SusanS - 04 Aug 2005

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My observation is that grade and subject acceleration only happened in Ben's school when it was virtually unavoidable. On such occasions they did support it -- only for math courses -- I don't know whether they were gracious about it.

I do think the teacher in the middle school where the kids had their math classes (which will, by the way, be Ben's school next year) was very good about supporting the younger kids she worked with.

-- CarolynJohnston - 04 Aug 2005

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I don't think it ever happens in Irvington, under any circumstances.

It's crazy.

This year they were pulling out a tiny group of math whiz kids for enrichment. When I asked one of them what he was doing, it was tessellation-type stuff.

That might be great; I just couldn't tell.

It's hard 'interviewing' a child, but I was also teaching my Singapore Math class in the same room they were using for enrichment. Looking at what was up on the board, I couldn't see that the kids were being moved ahead in the curriculum.

Now that Dan has created his dimensional dominoes, I could also imagine that a cool thing to do would be to teach topics the rest of the kids won't have time to get to, because they need the time to master the regular curriculum. But I didn't have the sense that the enrichment class was systematically doing anything like that, either.

I talked to my neighbor, who has an M.A. in math & has been teaching & tutoring forever, about these issues. She said that the natural-born math kids learn unbelievably quickly. It's as if they 'inhale' the books; they take one (or two) looks at a math text, and they've got it.

I certainly got the impression that to her acceleration in the old-fashioned sense was obviously the way to go.

My sense from 'reading around' is that the constructivists are actively opposed to acceleration.

-- CatherineJohnson - 14 Aug 2005

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Perhaps the reasoning behind being anti-acceleration is the same as Montessori's; that a kid who is ready for acceleration will simply 'discover more' (or 'construct more knowledge') than other kids, and therefore shouldn't need actual acceleration.

I actually think all the tesselation stuff in math curricula these days is a boondoggle. It's a bit of a mathematical dead end; nothing builds on it, and it builds on nothing, and in fact the benefit of studying it isn't too clear (in other words, it's not obvious that it does what it's intended to do, which I think is 'develop geometrical thinking').

I'll bet it's being used as a sort of holding pattern -- something you can use to 'enrich' kids without teaching them something that might actually accelerate them.

-- CarolynJohnston - 14 Aug 2005