Furthermore, if you keep adding the digits, you will keep getting 9. Allow me to illustrate. Let's take 99,999,999,999, which is obviously divisible by 9. Now, when I add up the digits of this, I get 11*9 = 99. So I add up the digits of 99 and get 18. So then I add up the digits of 18 and get 9!

This is very common and my math majors do it in proof-based courses. I always prepare solutions for them so they can see the "easy" proof after they have struggled through it. (Their own correct but very long proofs get full credit, of course).

It is because we are in base ten. So every time we add 9, we are adding (10-1), so we add one to the tens place and subtract one from the ones place. The same thing will happen when you skip count by 7s in base 8. Sorry, it won't let me line up the numbers for a nice chart, but I've listed the same numbers in each row. Base 10: 7 14 21 28 35 42 49 56 63 70 77 84 91 98 Base 8: 7 16 25 34 43 52 61 70 77 106 115 124 133 142

It is a way to check for arithmetical errors that is much, much quicker than redoing the calculation. With as much technological assistance as we have now, it is of little use, but if I were adding pages of numbers in an accounts book and did not have a calculator, it would be an extremely good idea to use this for a check.

Since we're on the topic, here are some other cool divisibility rules. If you add up the digits of a number and the sum is divisible by three, the number is divisible by 3. If you add up the digits of a number and the sum is divisible by nine, the number is divisible by 9. If you add and subtract the digits in an alternating manner, and the sum is divisible by 11, the number is divisible by 11. An example of the last: Consider 143418. The alternating sum is 1-4+3-4+1-8 = 5-16 = -11, which is divisible by 11. Therefore 143418 is divisible by 11.

It is not necessary. It used to be very necessary for people who were doing accounting without a calculator or adding machine and is still useful for checking if you do a lot of adding of numbers without a calculator.

Liping Ma's book "knowing and teaching elementary mathematics" has multiple examples of flawed explanations from teachers. The four main areas are subtraction with regrouping, multidigit multiplication, division by fractions, and exploring relationships between perimeter and area. The fourth part was really seeing how they explored things they hadn't thought of before. The book also explains why these ideas are mathematically problematic. Here is an example: A teacher is teaching subtraction with regrouping, and explains "But you don't have enough ones to subtract, so you go to his neighbor here who has plenty". On the surface, this is a reasonable explanation. But it reinforces two incorrect ideas that students consistently have: 1) If you want something, you just put it there. This is also seen in fractions, where students frequently will simply add a denominator without doing anything to the numerator, because they do not understand that when establishing a common denominator they are really multiplying by 1. It is also seen in equations, where a student will be told to manipulate an expression, and will simply start squaring it, multiplying it by something, et cetera. 2) These are two different numbers. In reality, they are different parts of the same number, and this must be repeatedly emphasized. ETA: Haha Jackie and I were cross-posting, but this is a really good book.

Or honestly, put them all in TT and let the teacher facilitate by making sure they stay on-task. That would let them pace independently as well so that you don't have the slow kids frustrated by the quick ones and vice versa. Move towards group classes once you get to more qualified teachers in (hopefully) middle school and definitely high school. I shouldn't be so negative. And I really would much rather have small-group instruction with qualified teachers, preferably math specialists, starting in kindergarten. But I just don't see it happening any time soon on a wide scale, although I have some really outstanding elementary education majors in my precalculus class this semester. The other really sad thing is that it only takes one terrible teacher to undo years of good math instruction.

Very much so. Some students learn better by learning the concepts first, and some learn better by consistent practice with the algorithm. In a program designed for students in public schools, *both* should be provided. Constructivist math may work very well if the teacher actually understands the math that he or she is teaching. But when a student is aided by a teacher who only understands the math algorithmically if at all, and peers who also do not understand it at all, it is the blind leading the blind and nothing is learned.

You can check the AOPS pre-algebra post-test. http://www.artofproblemsolving.com/Store/products/prealgebra/posttest.pdf Some people who have very young children who are not yet ready for the frustration of AOPS have done Jacobs Algebra (includes necessary pre-algebra topics in the beginning of the book) and/or Jousting Armadillos (a newer program based off Jacobs). Jacobs can also be spread over more than a year.

I would really try to get the 'W' at this point and take it again later. If he can't do that, I would try to retake it whether he makes the "C" or not.

This is a big problem with Everyday Math. Some teachers do teach it well (and it's not a TERRIBLE program if drill is included) but many people have been given the impression that drill is unnecessary at best and damaging at worst if the understanding is there. This is codswallop. It is horsefeathers. It is complete and utter male bovine excrement. I do believe that drill is rather pointless if the understanding is not there, because the subject will be forgotten as soon as the drilling is stopped. And certainly some students need more drill than others. But drill and understanding MUST go hand in hand.

This is why dividing math into conceptual/traditional is really a bit of a red herring. Most programs DO teach concepts, at least to some extent. They may not assign problems that require the understanding of these concepts to the extent of some others, but they teach them. Similarly, most programs DO drill. It's just that some drill more than others. Trying to make them fit into neat little boxes is an exercise in futility -- they are really more on a continuum.

There are some examples of 'traditional' non-conceptual math teaching in Liping Ma's book. For example, where the teachers were discussing the traditional method of multiplying, to multiply 123 x 456. They were asked how they would help a student who was forgetting to move the partial sums over. Many teachers (but more of the US teachers than Chinese) indicated that they would just have the student put something there as a 'placeholder' to remind them to move the sums over. Some of them even specifically indicated that they would use something other than 0's (like asterisks) because 'those aren't 0's, those are just placeholders'. I can't remember what order they were multiplying them in, so I picked one. For example, they would expect to see something like (dots added for spacing): .....123 ....x456 ______ .......738 .....605.* ...492.*.* _________ and then the sum at the bottom. This is a completely crazy answer to give students. The zeroes are there because what we are doing is multiplying 123x6, 123x50, and 123x400, and then adding them all up. Any sort of conceptual program (including, btw, many of the pre-New-Math programs -- being old/traditional does NOT make a program non-conceptual) is going to give an explanation that involves place value in some way, shape, or form, that is not just 'memorize the algorithm because I told you to'. It is perfectly possible to teach a traditional program conceptually or to teach a conceptual program in a non-conceptual manner. This is where the teacher variability comes in. Some children will also intuit the rules just from being taught the algorithms. Some programs also stretch students a lot more than others in the *application* of concepts that they have learned. The problems in (for example) AOPS, Singapore, the starred problems in some programs, etc., will expect students to go further using the concepts they have learned than in some other programs I have used. Since most students entering college will NOT have this preparation, it is perfectly possible to enter and do just fine at most universities without this preparation. However, one of the biggest issues with students who want to major in these areas is that, although they are fine at computation, they possess little understanding of what or why they are computing -- even the majors. Trying to get students to make connections across their discipline and even outside of their discipline is a huge topic right now, and it's one reason the 'introduction to proofs/transition to advanced mathematics' course tends to be such a bloodbath. It's also why a fair number of math majors change to a different major and a math minor after introduction to proofs -- because they found out that math really wasn't just about computing and finding the right answer.

BTW, in addendum to what I said above, here's how I would *like* to assign grades, if I had only self-motivated and mature students, and also students who could be trusted not to cheat on coursework. In computationally based classes such as high school algebra, trigonometry, calculus, etc. -- One final exam assessing all work for the semester. Samples would be available to students so they could see what types of questions to expect. In theoretic/proof-based classes, one final problem set (essentially, a take-home final) with a week (or two) to work on it, assessing important proofs through the class. In both cases, homework would be assigned regularly and solutions posted after a week, but it simply would not be collected and graded. I would of course be available to the students for consultation throughout the class, but it would be their responsibility to come and see me to ask whether their answer (which differed from the key) was still valid, or how close to correct their answer was.

There are two big reasons, imo, to include daily work in the grade. 1) To manipulate students who otherwise wouldn't do it into doing it. Many students will not do homework if it is not worth points, and then fail the exams. Making it worth part of the grade pushes them to do it. 2) When the problems are legitimately too difficult to be done in an exam situation, but you still want to assess the student on how well they've done on these very difficult problems. This would rarely apply in high school math classes. Both of these (but especially the first) are much more relevant to someone who is teaching a class than to someone who is teaching one student.

I also agree with Brenda about Geometry. I find Lial's, Stewart's, and Cohen's precalculus textbooks well-explained and readable. Here's what I'd do: 1) Identify any holes. For this I would get an intermediate algebra textbook (Lial's is widely available used. Bittinger's and Martin-Gay are other popular choices at community colleges, and so should also be widely available used) and test through, using the chapter tests. I wouldn't worry about (using the table of contents from Lial here) systems of linear equations other than two-variable systems, inverse/exponential functions, conic sections/nonlinear systems, and sequences and series. These will be taught in precalculus and so it isn't necessary to go back and cover them. Looking at the Lial's TOC here: http://www.pearsonhighered.com/educator/product/Intermediate-Algebra-11E/9780321715418.page, I would do the chapter tests for 1, 2, 3, 5, 6, 7, 8, and 9. I would also do a few homework problems from 4.1. If he is weak on any topics in these chapters I would do homework problems related to them. Student solutions manuals for these are also widely available and give solutions to the odd problems. 2) Pick a standard precalculus program and work through it. If you want the video instruction Chalkdust is solid.

If you've already backed up and restarted once I'd look for a different curriculum.

:leaving:

I'd be more inclined to finish up and then test through the review in the Algebra book. To add to what I said earlier: The review in the algebra book is intended to review all of pre-algebra for students who have not been studying during the summer. Not finishing the pre-algebra (if there ARE unlearned bits, which is most likely at the end of the book) may cause her to switch from reviewing to learning too early, while she still knows 80% of what's going on. That is why I'd rather finish the book and move quickly through the review. When she gets to algebra, I'd have her take the tests until she gets under 85%, and then move back to the lessons that are covered on that test, and I'd also make sure she knows this, so that she'll be putting her best effort into avoiding arithmetical errors so she doesn't get switched into material she finds boring.

I think calc/stats would be most useful. Some programming ability would be handy but it doesn't necessarily need to be a specific course.

I don't think a kid needs to be getting B's to be challenged. But I do think we could use a different grading scale than this 90/80/70/60 ... I heard at some school districts it's even as high as 96% for an A. Quite honestly, I think that if 96% is an A, there are not enough challenging problems for the better students. I would prefer a kid to be getting 80-90% on more challenging problems.

The shape is not enough, and if I saw this (the shape with nothing labeled) it would not receive credit. How much should be labeled depends on the specific function, but I would usually expect local maxima/minima (if they exist, and if the student has the tools to find them -- for example, the vertex of a parabola), asymptotes, and intercepts to be labeled at a bare minimum. Any monkey can put a function into a graphing calculator and copy it down, but this will not develop the ability to understand that, for example, if f(x) has a point at (0, 0), then f(x-5) + 3 has a point at (5, 3). For the absolute value, does he understand problems such as |x| = 8? If he does not understand those, I would start there before moving to |x-2|=8, as the second type are quite difficult to understand if the first type are not understood. For the degree, I would go back and briefly review how to tell the degree of monomials. I'd suspect that he only has problems with the ones with more than one variable. If you can do monomials, polynomials are easy (you can just write the degree of each term underneath it and look for the highest (including ties) number if you are a little shaky on monomials). VERY brief review here -- also a video lesson, but quite honestly I think he'll be able to grok it with a brief explanation and probably respond with 'Oh yeah duh, I knew that'. http://www.mathplanet.com/education/algebra-1/factoring-and-polynomials/monomials-and-polynomials

Have you considered something like ALEKS?

Yep. An option, btw, could be (if he knows where he's going) to take college algebra/math for liberal arts/stats as dual enrollment (whichever one will transfer to where he's going) as a senior. He will have his math credit already finished at college while it is fresh in his mind.