Part of the problem is that your math program seems to be teaching that math is following steps. That is a hugely confusing way to learn math! Instead, she needs to learn that math is common sense: read a problem (and take a deep breath, panic, relax, and then read it again) figure out what basically it is asking (ie, adding means put the two chunks together) then apply her real-life understanding to the situation For example, check out the following blog post to see how very complicated the step-by-step approach to mixed numbers really is, and one way to make the problem easier: Subtracting Mixed Numbers: A Cry for Help

James Tanton's free G'Day Math courses provide a wonderful introduction to completing the square.

If you want to measure area, you need to use a flat shape. Think about what measuring means: comparing this thing we are interested in to a similar thing that we've agreed to use as a unit. To measure length, you need to compare it to some other length, to see whether they mark off the same amount of distance. To measure volume, you compare it to some other volume, to see whether they hold the same amount of air/water/ether/whatever. To measure area, you compare your shape to some other flat shape to see whether they cover the same amount of flat space. Now, what will be a convenient unit for counting off the amount of flat space something covers? Our unit should be: symmetrical, so it doesn't matter which way we lay it down space-filling, so we don't miss some part of the area we want to cover easily imagined or constructed, so we don't have to put too much effort into making a measurement The first condition limits us to circles or regular polygons. The second condition eliminates the circle and limits us to polygons that can tessellate. Of these, the square is by far the easiest to work with. These considerations do not depend on the system of measurement you are using (inches, cm, or cubits), but only on the nature of flat shapes. I don't know of any civilization that developed any other way of measuring flat space, and I doubt there ever was one. Even triangles, which seem like they should be a simpler shape than squares (having only three sides instead of four) are much more difficult to use as a measuring system, because it is harder to create fractions of triangles, and you know that measurements do not always come out in nice, even numbers. Edited to add: I've read somewhere that Descartes did not originally insist that the axes of his coordinate geometry system be perpendicular. We are used to working with perpendicular axes, but everything would work just as well with a slant-wise system. (Try it and see!) In that case, I suppose we could measure area in rhombuses. I think it was the ease of working with squares that pushed history into using the perpendicular coordinate system we have today.

When we switched to our 30-minute limit, we didn't try to finish the same amount of work in a day. I cut back quite a bit on what we were doing in our normal textbook (by skipping the easy problems and the topics I know she's mastered). I will allow a little bit of leeway if we're in the middle of a problem when we hit our time. We almost always finish the problem. When we do more than 30 minutes of math in a day, it is usually something completely different -- online stuff, or a living math book, or logic puzzles, or whatever.

There are several things that increase my rate of glitchy mistakes. You might compare them to your son's experience to see if some minor change might make a difference for him. I make more mistakes when I'm tired, fewer when I'm fresh. This is why many people try to do math first thing in the morning. We are not morning people, but my dd and I do limit our math time to about 30 minutes, because we've found that her performance (and emotional response) plummets with longer lessons. If we need more time than that, we make sure that we at least take a break. I make fewer mistakes when I'm interested in what I'm working on, more when I'm bored or hate what I am doing. A student who only wants to get the lesson over with will tend to make a lot of mistakes. I make about the same number of mistakes if I'm working orally, buddy-style, but I tend to catch them more quickly because of our interaction. When my daughter stares at me with a raised eyebrow (or when I do it to her), that's our signal to stop and rethink whatever we just said. I make more mistakes when I'm under stress. I don't know your son's age, but puberty definitely counts as "under stress" enough to make a mess of mathematical thinking. While mental glitches can hit me even on a simple problem, I make more mistakes on complicated, multi-step problems. There are so many more chances to make a mistake, and my mind is often rushing ahead to the next step and not paying sufficient attention to the current one. And finally, I make more mistakes when I'm feeling overwhelmed by something I really don't understand, or when I'm trying to remember a lot of rules that haven't become automatic. (Think of a new driver.) This last case is a problem that definitely needs attention. A student who is making mistakes for this reason needs help, review, time to let the new knowledge coalesce. He may need to redo the lesson from a different angle, with new explanations that might fit better into his overall understanding. Perhaps he needs to go back to earlier topics to firm up the foundational concepts before coming back to the idea that flummoxed him.

Several assorted thoughts: First, consider for a moment the emotional atmosphere you have described. Nobody can really learn anything in that atmosphere, except to hate math! Math Mammoth is a fine program, especially when you do it buddy-style, but no math program could teach through those emotions. If you think your daughter will enjoy Miquon, that sounds like a great change! Second, Miquon is an excellent program that gets into relatively advanced topics. A third (or even fourth) grader doing Miquon is not in ANY way "behind" in math! Third, did you know that up to 80% of what is taught in math textbooks each year is just a repeat of the year before? If you take a year for Miquon and then decide to go back to a different 4th grade math program, that repetition will help you "catch up" on any areas you've missed. Fourth, even if your daughter learned absolutely NO math for the next three years (and how likely is that?), you could teach all of elementary arithmetic in less than one year in middle school. Don't worry. Your daughter will do fine! Just keep trying things until you find a program you both can enjoy learning together.

On my blog, I have a wide variety of math games that use playing cards or dice. Click here for a quick list of my favorites.

If he loves workbooks, Miquon is a good choice, since it offers pages that can be filled out creatively. Thus it fosters true mathematical thinking, whereas the average elementary workbook encourages rule-following and answer-getting. See this video for an example of Miquon math. A counting rope is easy to make and fun to play with, and it helps build mental math skills with little numbers (which then apply to bigger numbers as the child grows). My favorite way to do math with a young child doesn't use workbooks, but relies on conversation (an adult-child discussion builds all sorts of logical and social skills without effort). And the Moebius Noodles book offers activities to build mathematical thinking in all your children.

You might also enjoy Jo Boaler's summer "How to Learn Math" course at Stanford, which starts in two weeks: Summer School for Parents, Teachers: How to Learn Math

You've gotten some fantastic suggestions so far! I especially like the one about having your daughter make up math problems. That was always the backbone of my early elementary math program, and it really helps build a strong foundation. Here are a few additional ideas (mostly from my blog) that you may find helpful: Kick up the fun factor while exploring advanced mathematical ideas with Moebius Noodles Try some hands-on living math explorations like what Lula's kids have done, or try some of Malke's creative math ideas Play around with math games And whenever you go back to a math curriculum, make it more daughter-friendly by doing the work buddy-style

Have you looked at JUMP Math? It tries hard to break down and teach all the small steps than many math programs take for granted. They offer all sorts of free samples, so you can get a good idea of whether you would like it.

First, glitchy mistakes like you describe above are NORMAL. He will never stop making them, no matter how good he is at math. This is why engineers always check each other's work before releasing a project. The mind is not a computer, and when it is focusing attention on one thing (especially on a new concept, like in math class), then it can't spare much attention for everything else (like those pesky +/- signs, or the math facts it is so sure it mastered long ago). Result: mental glitch. Second, AoPS pre-algebra is a very algebraic pre-algebra, getting to the heart of the concepts and teaching for true understanding. And it pushes the student to think really hard. [Example from chapter 5: "Find the value of c such that x=2 is a solution to the equation x/c=3."] My very-good-at-math daughter is a little more than half-way through it (and will have started 9th grade before we finish), and I do not feel that it is AT ALL beneath her level. There is really no need to rush through math, and I think the current push for algebra in 8th grade does more harm than good in most cases. Here is our method of working through the book, to avoid the boredom factor: We do not do every problem, but rather focus on (1) checking and deepening her understanding with the gray-box lesson problems, which have mostly been review so far, and then (2) working only the word problems, starred problems, and challenge exercise sets (unless we find a topic she's forgotten and needs more practice on). Believe me, the challenge problems will not be boring. Some days, one problem will fill up most of our 30-minute lesson time. But if the idea of taking another pre-algebra is truly messing with your son's head, you might want to check out James Tanton's G'day Math as a supplement. My daughter and I have been working through those lessons (slowly, at our normal laid-back pace), and she is loving it. The lessons cover quadratic equations, but in a visual way that is quite accessible. Your son may not be able to do all of the levels, but even doing a little bit of quadratic math should make him realize he's not really "behind" -- at least, I know that my daughter always feels encouraged when she gets the chance to "jump ahead" and explore advanced topics.

The contrast between your husband and your daughter in interesting to me in light of recently re-reading Jo Boaler's book What's Math Got to Do with It? Her experience has been that on average, guys seem more willing to just go with "here's the rule, do it" while girls are more likely to want to know why. She thinks that's one reason so many girls give up on math. Has your daughter looked at the samples of the AoPS books? You can print out sample lessons for each of the books. They really do a very good job of getting at the "why" of math, and the solution manuals are very thorough. Those might give her enough feedback on their own. But even better, the AoPS Forum is a great place to ask questions!

Sometimes, as others have posted, a gentle reminder is enough to drag the concept back out of the dust-bunnies of memory. Other times, I find that the student answered "I don't know" out of habit, because it was easier than actually thinking about the question. And because he'd really rather be doing something else. And still other times, the child really didn't understand the topic when we went through it before. No matter how hard I try to teach the concepts, some kids just want to be "answer-getters." They don't want to do the hard work of understanding, they just want to memorize some steps and crank through the worksheet, to get it over with. In all of these cases, what helps me the most is conversation. We always talk about our math. I ask questions like, "What do you think? What do you remember? Can you explain the question to me? What are they asking for?" And, whether the child's answer is right or wrong, I try not to give that away, but just ask, "How did you figure it out?" Having to explain his reasoning helps to make the child's understanding firm, because the act of putting his thoughts into words forces him to rethink and clarify them. This is one reason why I enjoy doing math buddy-style with my daughter, even as she is going into high school, rather than just giving her an assignment to go work through on her own.

This sounds so much like my daughter! :iagree:

Perhaps, though I wonder what you mean by the word "automaticity". The ability to crank through a page of workbook problems or a quiz or test, following the steps of an algorithm? The secret to that is not "understanding" but the ability to turn off one's mind and just follow directions. Algorithms (the mathematical recipes for performing calculation) are wonderful things, in that they provide an efficient way to perform mindless, rote calculation. As one mathematician said: "It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them." (Alfred North Whitehead) Civilization advances that way, but I don't believe that mathematical understanding does. A student learns relational understanding by thinking and discussing a lot of problems -- and especially by working the same problem several times in as many different ways as possible. Relational understanding takes a long time to build, but it takes a much different type of practice than the "crank it out" practice that builds automaticity. So I don't think there's anything unusual about a student who thinks relationally in general still needing a lot of practice to build automaticity on some particular method. The two things are entirely different. I would begin to worry, however, if the student was beginning to think that producing automatic answers is what it means to be "good at math." Oh, and I would also worry if a student whom I thought had developed a relational understanding of a topic was not able to reason out the answer to a problem in that topic. A student with relational understanding may forget the steps of an algorithm, but if he knows what the problem is about, he should be able to think his way through it somehow, assuming we teachers have the patience not to cause stress by reacting with dismay when he fails to show automaticity. Somewhat off topic, but I found a fascinating description of the difference between instrumental and relational understanding, though the author doesn't use those words -- and he's talking about science, not math: Richard Feynman on education in Brazil

Another factor to consider when we think about learning math is that there are different uses of the words "understanding" and "mathematics" that are as confusing as what Rose described concerning "working memory". People can mean totally different and sometimes even contradictory things by these words, and if you don't know which definitions a person is using, it's impossible to make sense of what they say. Take a look at this article: Relational and Instrumental Understanding. Instrumental Understanding focuses on answer-getting. If you can manipulate symbols and regurgitate facts from memory to produce answers (and the quicker, the better), then you "understand" "mathematics". This is the kind of understanding that can be tested by machine-scored, color-the-dot tests. And children with this sort of understanding often need to review things constantly, lest they forget the procedures or techniques that they have "learned". Relational Understanding focuses on recognizing connections between ideas. If you can see why things work and analyze the interrelationships and synthesize new ideas by cross-referencing older ones, then you "understand" "mathematics". This kind of understanding can only be tested through deep conversation with a student to probe the reasoning behind the answers given. Children with this sort of understanding do not need constant review, because the information is all interconnected in their minds and thus it can be retrieved when a related topic appears. Also, because the concepts are connected, work in one area draws on and reviews other related topics, though the students may need a few reminders to refresh a concept that has been left dormant a very long time.

Thank you for the links!

It is very easy for us as parents to give the impression that the answer is the important thing in math. It is very easy for children with perfectionist tendencies to focus on answer-getting and to feel like failures and want to give up if they get a wrong answer. But the answer is the absolute least important part of a math problem! The important things are learning to see the relationships between the parts of the math situation, to think it through, to make sense of it and understand how it works. I do NOT ask my kids to memorize anything at this age. But I DO ask them to tell me how they figured things out. Over and over I ask them. How did you get that answer? Whether the answer is right or wrong doesn't matter, I still ask. How did you think about it? How did you figure it out? Why do you say that? What will you try next? This problem is trying to trick you--will you let it get away with that? How will you think it through? And we play a whole lot of games, and spend only very short sessions on workbooky stuff -- no more than 10 minutes per grade level, maximum. But we do lots of talking and we take turns making up little math problems for each other, things like that. The kids especially like to make up problems for me, to try to stump me.

Also, the number that each rod represents is NOT always the same. After the child internalizes the relationships between the rods through play, then we can use the rods as we wish to teach math concepts. We can define ANY of the rods as being "one": for instance, if the yellow rod is one, then the white rod represents 1/5, the red is 2/5, etc.

My daughter is older than yours, 14yo and in 8th grade. We started AoPS Algebra last fall, but it was too much for her, so we backed off to Prealgebra to solidify the foundations. All of it has been review so far, but a very algebraic-thinking approach to the topics, which has been good for her. My dd is a natural reader, and she could certainly do the book on her own if I forced her to (and would probably get through it more quickly that way). But she really likes working together, so that's what we do. Because it's all review for her, we tend to skip and skim. We are mostly using the text, but she occasionally watches a video just for fun. She used Alcumus for awhile, but then drifted away to other interests, since I wasn't requiring it. Here is our procedure: 1) Sit together on the couch, with the book and a whiteboard. Check the time. We go about 30-40 minutes per day, because any longer than that leads to fatigue and grumpiness. 2) She attempts the "discovery" problems at the beginning of a lesson, doing all of these herself. I add comments, if I want to. After she's done them, I skim through the lesson to see if there are any key points that I want to mention, or if the book's method was different from hers. 3) We work through the exercises buddy-style, each of us doing alternate problems and explaining out methods as we work. We check each other and rarely have to look up anything in the answer book. If she had any troubles at all with the discovery problems, then we will do the whole exercise together. Otherwise, we often skip to the most difficult of the problems. We always do the star problems, and if one of them is particularly challenging, we will both work our way through it (dueling white boards?) and then compare methods. I love to see the way she thinks these things through! 3.5) If there are story problems, we explain HOW we would approach it and get the answer, but we often skip the actual calculation. 4) End-of-chapter reviews are the same: skimming and buddy math. 5) We put in a bookmark wherever we are when our time runs out, and that's where we start the next time. Sometimes we just get a few problems done (I think our "record" was two problems in a half-hour session), while other times we whiz through several pages. We try to do AoPS math at least 3-4 times a week, except on the weeks that we skip (either due to sickness or time-intensive activities like NaNoWriMo or whatever -- I've always been laid-back about schooling). On days when we miss it, she's reading Danica McKellar's algebra and geometry books and we also started working through a series of online lessons on quadratic equations by James Tanton. The latter would be too advanced for most prealgebra students, but my daughter has a strange, eclectic background that made her well prepared for it.

I have a large variety of math games, puzzles, and activities on my blog. You may also want to read through some of the Math Teachers at Play blog carnivals, which collect fresh ideas from around the internet every month.

The definition thing is built in to the Kindle, so it will work on any book that is written in words. (Some pdf pages are scanned as images, and the Kindle can't find definitions for a picture.) As for the table of contents and other links, that depends on the formatting of the book. If it's properly formatted, then the table of contents should work, but not all books are properly formatted. For books that you read straight through, like a novel, this doesn't matter much, but if you know you will be skipping around in the text, you should try out the free sample (most places offer that) before paying for a book. Here's another idea: If you can get a text version of the book, and if you know how to insert bookmarks in Word, then you can make your own table of contents. Then use "Send to Kindle" (available free on the amazon site) to transfer the file. That has worked well for me.

I use a lot of drawing to visualize and explain math concepts. We always work with a white board at hand.

I also use "mathy" to mean someone who enjoys math and (at least sometimes) has an intuitive understanding of concepts. And I certainly agree that everyone can do well at math, that being "non-mathy" is no excuse for failing to learn the basics --- just as one doesn't need to be a "bookish" child to learn to read and write. I wrote a blog post about what it means to succeed at homeschool math, which has nothing to do with mathiness. You might enjoy it: How to Recognize a Successful Homeschool Math Program