"Math Facts" are a very abstract, rote way to approach mathematics. I much prefer to teach "Number Bonds"--the idea that numbers can be taken apart into smaller numbers and put back together to make bigger numbers: Number Bonds = Better UnderstandingAnd thinking specifically about addition and subtraction, I prefer to emphasize mental math strategies and give the kids lots of practice playing with numbers before we try memorizing. Then any memory drills become a mopping-up exercise, rather than the main battle. Here's my approach: PUFM 1.3 Addition PUFM 1.4 Subtraction

Here's a different perspective about math facts, from my blog: Learning the Math FactsAnd here's a wonderful source of insight about how to help young children learn to really think about math: Talking Math with Your Kids

I think you would find a lot of ideas and resources in Lucinda's archive: Navigating By Joy posts about MathIn particular: Living Maths Curriculum 2013-14 (planning a living math curriculum for last school year) How to Help Your Child Fall in Love with Maths (Even if They Hate It) How we do maths without a curriculum (planning for this school year) How my autodidactic 9 year old is learning maths without a curriculum (this year, with a different student) And check out the AoPS website to see a variety of excerpts from the Kitchen Table Math books: Kitchen Table Math Book 1 Kitchen Table Math Book 2 Kitchen Table Math Book 3

We did the book together and skipped much of the reading, like this: Dd did as many of the gray problems at the beginning of a section as she could, on a lap-size whiteboard, narrating as she did it or explaining when I asked a question. For the ones she got right, we skimmed the lesson to see if they offered any new or interesting insights. Or if I would have done it a different way, we might compare methods. For the ones she missed (or hadn't seen that type of problem before), we read the book's explanation more closely. I generally either read aloud sections or paraphrased them---didn't make her read. When we were sure she understood the lesson problems, we did the exercise sections together, Buddy Math style. Because we were checking each other as we went, we only rarely needed to look up the solutions. We continued this through the prealgebra book and about a third of the way into the algebra, then she decided she was ready to work on her own.

Math Doodling Making abstract math visual: Math doodles let us see and experiment with a wide range of mathematical structures — and even to feel them, if we include hands-on 3D doodles in clay or other media. Links include art projects, geometry constructions, and physical models to explore. Math Doodling

Khan Academy is attractive because it is free and easy to assign. BUT It's not written by a teacher, or a curriculum designer, or anyone with experience to know what sort of trouble kids have understanding math concepts. Behind every good math curriculum there's a lot of invisible knowledge: understanding of how kids learn, the typical misunderstandings they tend to develop, and how to nip those mistakes in the bud. Mr. Khan doesn't have that knowledge, and his lessons show it. Here is an example of what I mean: Open letter to Sal Khan (About Teaching Decimals) Khan also tends to focus on procedures (at least in the videos I've seen) without building up a foundation of understanding of why those procedures work. This puts a significant strain on memory, and students who think of math as a series of procedures to memorize tend to get overloaded as they approach algebra. Everything starts to mix together in their minds, making it harder to recover the rules from the dust bunnies of memory. For these reason, Khan Academy works much better as a supplement than as your primary math program.

If you have internet access, you could do a blogging class. Our co-op kids had a lot of fun with that. We've also done recreational mathematics (logic puzzles and games), which was popular and requires very little equipment. Or if the kids enjoy drawing, how about geometric constructions? I've seen a lot of cool Waldorf geometry art.

It doesn't matter, in the sense that finding a common denominator first will not change the value of his answer. On the other hand, it does matter because it introduces additional, unneeded steps in his problem, which means there are more opportunities for a careless error or mental glitch to affect the answer. And it also matters because sometime between now and when he gets to algebra, he needs to learn to handle fractions efficiently. Algebra fractions are too complex to handle like this. It sounds to me like he has tried to get by with just memorizing rules and following them, without understanding what is behind to rule. I'm sure your math program has tried to teach conceptual understanding, and you sound like you've tried to explain things until you're sick of it. But children are naturally short-term thinkers (they haven't lived long enough to learn perspective), and rote memory seems like a quick way to get through the lesson and escape to whatever they'd rather be doing, and it sounds like your son has tried to take that "easy" way out. That is, it's easy until he collects so many rules that they start to get jumbled in his head! Here is a suggestion that might help, or might just see more confusing. I hope it helps... You might try giving your son a "rule" that can help him wrap his brain around the conceptual difference between addition and multiplication, so that perhaps he will be able to see why addition (and its twin, subtraction) need a common denominator but multiplication (and its twin, division) do not. Here is the rule: Addition means "and." Multiplication means "of."When we add fractions, we want to find the total of this AND that. (1/2)+(1/3) means we have half a pizza and a third of a pizza (or whatever model you prefer), and we want to know how much total pizza that makes---how close are we to having one whole pizza? It doesn't help just to count up all the pieces, if they are different sizes. We have two pieces, sure, but we still don't have any idea how much total pizza is there. So we have to make the pieces the same size (common denominator) in order to count up and get a useful answer. But when we multiply, we are finding what is this much OF that. Multiplication always works that way. 2x3 means to find how much is 2 of the 3s, which comes out to 6. If instead, we needed (1/2)x3, we would find 1/2 of 3, which is 1 1/2. And what would we do to find 1/2 of 1/3 or (1/2)x(1/3)? Forget rules and just reason it through---how would you do it? What is half of a third? Does changing the fractions to (3/6)x(2/6) make it easier to figure out? For fractions that are more complicated than halves and thirds, it helps to use a rectangle model, like this. The idea is still to find how much is this fraction OF that amount, but the picture really helps me see what that means. I think of it as a "pan of brownies" model---you have part of a pan of brownies left, which is one of the fractions, and you are trying to figure out how much some fraction of that amount will be.

Ooo, that would be so much fun!

You might try asking a librarian and borrow the books before buying. I know a lot of people like the MathStart books, but I personally dislike many of them and would hate to find out something like that *after* buying a whole series. The series I do like are mostly out of print, I think. The Young Math Books series is probably the most similar to the Read and Find Out science books.

You've gotten a lot of good advice already. Here are a few additional, assorted comments... On learning to teach Singapore math: The most important thing is to switch your brain from a worldview that school math is all about getting right answers and instead see math as all about ideas and the relationships between them. I have a (short, unfinished) series of posts on my blog that might help you see the big picture: http://letsplaymath.net/tag/pufm/ On division: I don't see anything wrong with asking how many 5s "are in" 15, as long as the student understands what you mean. But I do suggest varying the words and asking it several ways: How many fives does it take to make 15? If you split 15 up into groups of 5, how many groups would you have? How many fives add up to 15? If you shared 15 things evenly between five people, how much would they each get? How many 5s would you need to have the same amount as 15? If you count by fives to 15, how many steps does it take? etc. Different words create slightly different mental images for your daughter, and the more ways she can think of division, the better her understanding will be. On a younger child catching up: This happens frequently, because different children have different strengths. (It happened twice in my family alone, affecting four of my five kids.) If it is too frustrating for the older child, I suggest putting them in different math programs, even if that means more work for you as the teacher. Emotions can cause mental blocks, and math is hard enough without that additional burden. The older daughter may have to fight her tendency to competitiveness, but you have a choice about where she faces the battle. On being "behind": Some of the families on this forum can seem very intimidating! But it sounds to me like your older daughter is at an appropriate place for her age, so I would not hurry her ahead IF that's going to cause stress. [And those reviews *will* get harder!] When I had kids in Singapore Primary Math (the older series of books), I usually had them working one semester or even one year "behind" their age/grade level, even the math-intuitive kids. That fit our laid-back approach to schooling, and it hasn't hurt them at all. After all, level 6 of Singapore math, at least in the older books, is pre-algebra. I was never in a rush to get to algebra, so we spent plenty of time doing other things along the way.

Adding up is a legitimate and age-appropriate way to solve subtraction problems. Please don't make your son feel bad about using it. Nothing good will come of fighting his natural intuition about numbers! When he gets stuck on a larger problem, give him plenty of time to think and maybe a white board with colored markers to use as "scratch paper." My kids get mad at me (for good reason!) if I break their chain of thought while they are figuring out a problem. After you have waited as long as you can bear, ask him if he wants a hint. If he says yes, THEN is when you can point out the link between adding up and subtraction---not as if subtraction is how he has to do it, but as "This is how I like to think about this sort of problem."

Your son's method sounds like a fine strategy to me, at that age. He uses the facts he is confident about to figure out the things he doesn't know. Bravo! I have a series on my blog that you might find helpful for tips on mental math and arithmetic strategies. It's based on the Elementary Mathematics for Teachers textbook, but I tried to write in such a way that anyone could understand, without having to have that book or the Singapore math books: Homeschooling Math with Profound Understanding (PUFM) SeriesI also have a series on using bar diagram models to solve word problems: Word Problems from LiteratureIf we are talking about the Singapore Primary Math series (the only one I've used), then personally, I think you could work with just the HIG and the textbooks, if you are doing other stuff also and thus not wanting a full curriculum. Of course, the workbooks have practice problems, and some people like to get extra workbooks for even more practice, but the textbooks (done orally with mental math, or on a white board as the problems get bigger) are the heart of the program. I would MUCH rather omit the workbook than the textbook. Other Singapore math series might be different, I suppose.

We spent a lot of time with word problems, NOT solving them but just setting up the diagrams. Ds thought he was "getting off easy" because he didn't have to do the calculation, but what I wanted was for him to show me he could think the problem through. It also helps to go way back to the easy problems and work through them systematically, gradually increasing the difficulty and talking about how to represent each problem with diagrams. Your son might enjoy my Word Problems from Literature series for a review that walks through problems from 2nd to 5th grade levels;

The most important step in solving any percent problem is to figure out what quantity is being treated as the basis, the whole thing that is 100%. What makes this problem especially difficult is that the "whole" changes from one part of the story to the next. The discount is a percentage of the regular price (the cost to the customer), but the profit or loss is a percentage of what the item cost the store. For more help in sorting out percent problems: Percents: The Search for 100%

As an easy, portable, fun way to fidget with numbers. Counting scores in games, recognizing number bonds, early addition and subtraction, a support for mental math skills... If you need more specific ideas, here's a free download that might help: Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics

Creative? Take turns making up math stories for each other to solve. Don't try to memorize the facts, but just use them over and over and over and over and over in contexts that have meaning to your child. These don't have to be real world stories---Star Wars, or knights and dragons, or dinosaurs, or whatever he's interested in will help him focus.

Abeka may expect that, I don't know, since I haven't used their 1st grade program. But no matter what program you use, keep in mind that the curriculum is your tool, not your master. You know your children much better than the textbook author does. If you have a bad feeling about something, it's wise to pay attention to that. In my opinion, children should not be pushed to memorize facts, but should be allowed (and encouraged) to use manipulatives as much as they want through at least 3rd grade. Not that you can't work on facts, but that you shouldn't stress it. Abstract, numbers-only math does not stick in most children's minds, but hands-on stuff makes the numbers "real." A Counting Rope is an easy-to-make manipulative that kids love. As much as possible, practice the math facts through playing games, like Tens Concentration. You can modify this for smaller numbers by just leaving out all the cards above the number you want to practice---i.e., for the 6 family facts, leave out 7+up. And stories! Whenever your kids stumble over a calculation, either make it into a little story for them or encourage them to make up one for themselves. Put your kids right into the stories: "[One child's name] had six cookies on a plate, but [another child's name] came by and ate four of them. Now how many cookies are there?" Children's minds naturally think in stories, not in abstractions like "6" or "4." In fact, you can turn the make-up-a-story idea into a game that will help your kids build a firm foundation of understanding numbers. I explain it in this blog post: Tell Me a (Math) Story. Also, with children this age, I encourage you to check out the activities at Moebius Noodles for fun ways to give them a well-rounded approach to mathematics. It's MUCH more than just numbers.

I have a series of posts on my blog to help parents (and students) learn to use bar model diagrams. You can start here and go through the series in order: Elementary Problem Solving: Review

Even in algebra, you can just emphasize thinking about the meaning of the fractions. For example: 2/(x - 3) = 1/5 "Well, the equal sign means these two fractions name the same amount, but they sure don't look like it! How can we make them look more similar? Perhaps we could rename them into equivalent fractions that have a common denominator..." 10/5(x - 3) = (x - 3)/5(x - 3) "Hmm. If the fractions are equal, and the denominators are the same, that means the numerators must really be the same, too. Right? They are just in disguise..." 10 = (x - 3) "And now we can solve it..."

I would like to suggest an answer to the big question in your first post: "Why does he hate it so much??" I think he tried his best to explain it to you (see the red section above), but he didn't know the right words to describe the problem. You see, up until level 2A, the Singapore math program is fairly intuitive to a student who thinks as your son does. His mind has a naturally mathematical approach, seeing and taking advantage of connections between math topics in order to solve problems. For example, he resents being told to look at a word problem as "subtraction" because to him, it is clearly a "missing addend" addition problem---and he is exactly right! There is no inherent difference between subtraction and addition (although most students don't learn this until they get to prealgebra). The underlying mathematical structure is identical. But in Singapore math level 2A, the focus changes from this intuitive approach to mathematics, which clicked so well with the way your son thinks, to an algorithmic approach---that is, to teaching the standard pencil-and-paper procedures for addition and subtraction, with renaming (what we used to call "carrying" and "borrowing"). The focus changes from intuitive understanding (which your son was good at) to an emphasis on memorizing abstract (disconnected from real-world objects or physical manipulatives) math facts and following the abstract rules for working with abstract numbers. In my opinion, this change in emphasis is developmentally inappropriate for many second-graders who are not quite ready to think abstractly, with numbers alone. Of course, our children do need to learn how to do multi-digit addition and subtraction, but there is no reason that they need to do it in the first part of second grade. Your son's reaction is a good sign that he is not developmentally ready to make that leap. To your son---or at least, to my son who hit the same mental road block in the subtraction section of level 2A---when you bring up the term "subtraction," this feels exactly like you are telling him, "Put your brain on the shelf and do the math MY way, following my rules, because I said so." If my explanation is true, what can you do? As you mentioned in your last post, a good temporary solution is to set aside the book and play some math games. I have several on my blog that you might want to try. But if this is a developmental thing, as I suspect, you may need to wait several months before you attempt to re-introduce the subtraction algorithm. (A long wait like that is what worked wonders for my son.) And of course, you don't want to just neglect math all that time. So I suggest you take a wandering exploration through all sorts of non-number mathematics---there really is a lot of mathematics out there your son can learn without having to memorize and follow abstract rules. For instance, get the Moebius Noodles book and explore symmetry and fractals and functions. These activities are accessible to younger students, too, so the whole family can play and learn math together. Or just skip ahead to the measurement chapters of your Singapore 2A book (after a short break to let his emotional reaction calm down), and then try multiplication and division. Even fractions (in level 2B) were ever so much much easier and more intuitive for my son at this age than subtraction.

What Saxon is calling "unit multipliers" is what I have always heard called "conversion factors." By any name, they will be extremely important in chemistry. Which means that she doesn't need to worry too much about them right this moment, but that she will want to master them sometime between now and then. I have an article on my blog that you might find helpful: How Old Are You, in Nanoseconds?

If your son is enjoying ABeka, let him keep using it. But when your girls react so strongly, that is a warning, like when your hand hurts if you get it too near a flame: change course because there's danger ahead. Several ideas to keep your girls from getting burned: Definitely check out Education Unboxed. Excellent stuff there! Also check out Moebius Noodles for rich ideas to deepen all three kids' understanding of math. Pick up a few living math books the next time you visit the library. Read Talking Math with Your Kids to learn how to draw out the math in everyday life. And explore my blog about homeschooling math. This post is a good place to start: Tell Me a (Math) Story.

One suggestion: Give her the experience of success more than failure, if you can. That is, if arithmetic is her stumbling block, DON'T make it the focus of your math. For instance, check out the Moebius Noodles blog and book for a wide variety of deep mathematics that is accessible with minimal arithmetic. Or incorporate a lot of living math library books. Don't wait until she gets "the fundamentals down first" before letting her play freely with the kind of math she's good at. You can still keep working on arithmetic, too, but let it be a small part of the math you do. For example, here is one math activity with plenty of richness, but no numbers required. Click the image to go see a bigger, readable version at flickr [Copyright note: Moebius Noodles materials like this image are Creative Commons Attribution-ShareAlike 2.0 Generic (CC BY-SA 2.0)]:

I second the idea of having HIM explain the problems and procedures to you. It is natural for kids to think short-term: "What do I need to do to get through today's exercise so I can go do something fun?" They grab any trick that might help them get through the lesson, and then dismiss it from their minds. They are children, and all children are short-term thinkers---they haven't been around long enough to learn perspective. But in order to explain the procedure, he has to process it in his mind, think about it, and put it into his own words. The more you can get him to do that, the more likely it is that the concepts will stick with him. You might try doing this with the tests, too. Get a white board, if you have one, or a big sheet of scratch paper, and ask him the questions orally, and let him explain whatever he remembers, as if he is teaching it to you. If he gets stuck (there is a lot to remember in 5th grade math!), you can try to ask questions that get him thinking but do not give away the answer. If he's anything like my kids, I think you'll find that he knows a lot more than his test performance shows.