Mental math is applied common sense. Don't worry too much about the steps, but focus on how to understand the numbers. See if you can compare them to "friendly" numbers that are easier to work with, and then make the appropriate adjustment to get the actual answer required. I describe several ways to approach a problem in my blog post Mental Math: Addition, but the same principles apply to subtraction, multiplication, etc.

Try teaching any other division model that comes up as a tweak on the "equal groups" model she already knows. For instance: Arrays: The "equal groups" are the number of items per row. Sharing: The "equal groups" are the amount per person, but usually we know the number of people and are trying to find the equal group. Subtraction: Like sharing, the "equal group" is still the amount per person. But this time, we know what the amount is, but we don't know how many people we can serve. We share out (and subtract) our equal groups until we run out of candy (or whatever we are sharing), and the answer is how many groups we made. Inverse of multiplication: Multiplication is an "equal groups" operation, too. In multiplication, we know the number and size of the groups, and we are trying to find the whole amount of stuff. In division, we start out knowing the whole amount of stuff, and we are trying to find either the number of groups or the amount of stuff in each group. Fractions: Did you know that your "equal group" doesn't have to be a whole thing? Or that you can share out just part of a group? For instance, If we have two candy bars and three kids, each "equal group" of candy shared will be 2/3 of a bar. [2 bars ÷ 3 people = 2/3 bar per person] Or, if we insist on giving each person 4/5 of a bar, the third person will only get a fraction of his share---in this case, he would get half of the "equal group." [2 bars ÷ 4/5 bar per share = 2 1/2 shares] And when you get ready for long division, you can still use the equal groups model: The Cookie Factory Guide to Long Division.

I have quite a bit about multiplication on my blog. Your friend might find these helpful: A game for learning the models behind multiplication multiplication bingo (uses the rectangle model) math war times tac toe online practice target number contig tips for learning the times tables

You can play Subtraction War: Each get a deck of cards, remove the face cards, then flip up two and subtract the numbers. Highest difference wins. Advanced Subtraction War offers a bit of strategy: Remove the tens, too. Then turn up three cards. Choose two of the to make a two-digit number, and then subtract the third card. More variations here.

Here's a cool discussion-game that can lead to deep math: Which One Doesn't Belong? More ideas on my Pinterest page: Playful Math for Preschool & Early Elementary

Saw this on twitter recently. It might be fun to have kids make their own version with the math they've learned. The basic divisions of number, algebra, geometry, and data seem to be a common way to divide and define topics/standards. With my math club kids, I've used divisions like number, shape, puzzle, pattern. Whatever you use, the categories tend to overlap more than this diagram shows...

This I completely agree with, though my recommendation to take a break from the topic still stands. Encouraging students to estimate before calculating is one of those never-ending tasks, worthwhile but very discouraging to the teacher. They are kids, even the teenagers who want to be treated like adults. They have a short-term worldview, because their whole life is short-term so far. They see no value in taking the time to make an estimate, but think of it as busywork. The stuff that I meant doesn't matter was the multitude of algorithmic steps to get a hand calculation of several digits (whether integer or decimal), where a mistake or lapse of attention to detail in one intermediate step can make the whole answer wrong. To require much of this sort of thing is to guarantee that most students will be convinced that they are no good at math, and that it doesn't matter because math is picayune and ridiculous. To be able to estimate the answer to 15.432 x 13.759 (or 15,432 x 13,759) is valuable. To calculate the exact value by hand, in either case, is not important.

Fourth grade math is stinking hard work! The difficulty level ramps up in fourth grade, by a significant factor. it's not surprising for a student to hit a wall at this level. For instance, I once analyzed a "simple" fraction calculation for a blog post. I counted *seven* steps the student has to remember and perform in order to solve that single problem -- and several of those steps had sub-steps of their own. No wonder kids get confused! So a new math program might help, if it speaks to your daughter's learning style better than Singapore Math does. But don't be surprised if she still struggles. That's pretty much the nature of fourth grade math. Whatever program you end up using, it may help to work it buddy style. That made a huge difference in my daughter's attitude toward math at this age.

Here's a radical bit of truth: It doesn't matter! Really, it doesn't. Nobody in their right mind does multidigit decimal calculations by hand after 6th grade. She will NEVER have to use this, and she will ALWAYS have a calculator, now that they build them into phones. She knows how to multiply. She knows a lot about decimals. She can even do quite a bit of decimal calculation, according to what you've described. Even some decimal multiplication. The things she is getting confused on are NOT important things. My recommendation would be to drop the topic entirely. Do something completely different. If you want a suggestion, I'd recommend the free Exploding Dots course by James Tanton --- work it together, at her pace, as slow as it takes. Don't feel like you have to finish the course, but just keep going as long as it's fun. The course touches on decimals a bit and even algebra topics, too, so it's not like you'd be skipping out on math. It can be refreshing to try something totally new. And then, if you absolutely have to do the multidigit decimal multiplication worksheets, come back to them in a few months, after her brain cells have had time to settle and it doesn't feel so frustrating.

Here are a few tidbits to tempt your son who thinks that "fractions aren't math": A mathematician teaches fractions: sure, it starts easy, but it's fun and it goes deep. A mathematician teaches other math topics -- not directly related to fractions, but who can resist math that explodes? A fraction game from my math blog.

I agree that we are talking past each other. If what has come across in my posts is the idea that "writing numbers on paper is wrong," then I am clearly not communicating what I intended. That's frustrating, because from reading your posts I suspect we would agree on most aspects of teaching. As for my "program" for teaching math, um, I have a blog. And I've expanded some of the more popular blog posts into an inexpensive ebook. Does that make it a program? But you are right, my feelings on this topic are coming out overly strong -- not so much in reaction to this thread (which I will stop hijacking), but probably as a boil-over from other things I've been reading online. My sincere apologies!

Perhaps a better question is, how old do you think children should be when they learn to line up numbers on a page because we said so. If the child is lining up the numbers because it helps him express the thoughts that are in his mind, then he can do it whenever he wants. But if he is lining up numbers because an adult has told him it's the right thing to do, that is harmful. And it really doesn't matter how many times you try to demonstrate with blocks what is happening. It doesn't even matter if the child can repeat back to you a description that makes it sound like he understands place value and what the algorithm is doing. Children are very good at figuring out what makes an adult happy, but when you push a little harder, you find that what sounded like understanding was just a Clever Hans or Benny's Rules effect. Seriously, watch the video at the second link above. It's amazing. As Ben Blum-Smith said, "Never underestimate your ability to fool yourself into believing your students understand something when really what they are doing is watching you." Yes, children can be trained to do things that go against their intuition. They can be trained to line up numbers on a page to make Mom and Dad happy. They can be trained to remember and follow certain steps with those numbers. They can be trained to say certain words about those numbers, like "tens" and "ones." They can be trained to perform well on standardized tests -- or at least, many of them can. But is this a worthwhile use of a 7-year-old boy's time? Only if it's HIS idea in the first place. Otherwise, there are plenty of deeper and more interesting things to play around with in math! The boy in the original post showed excellent understanding of mental addition and subtraction until the nitpicky steps of the algorithm messed him up. If the parent focuses on the details of the algorithm now, she risks damaging his relationship with math as a whole. But if she allows him to use whatever intuitive method he wants on the numbers, and together they go on to explore new topics -- and there are some seriously interesting things to study in the rest of Singapore level 2 -- then she can easily come back to try the algorithm again next year. By then, he just may have figured it out on his own, or he will at least have some additional maturity to help his attention span. P.S.: An algorithm is just a set of procedures to get from point A to point B. Like a recipe. You have these ingredients (numbers), you follow these steps, you automatically get the result. No understanding required. A computer can do it.

You misunderstand my point. I'm not criticizing Singapore math for the way it teaches this, only for the timing. EVERY textbook that teaches the algorithm tries to explain it and tie it back to fundamental concepts. And almost every teacher tries to emphasize the meaning of the place value columns and build conceptual understanding. A few teachers resort to meaningless mnemonics like "More on the floor? Go next door!" to teach borrowing, but by far most people really are trying to communicate to kids WHY the algorithm works. The problem is in the algorithm itself. Standard algorithms unteach place value. To use any algorithm efficiently, you have to follow the steps without stopping to think about them. If you try to think, you will probably forget what step you are on and make mistakes. Therefore, every time the student does a calculation, he or she is training the mind to skip out on math. And because the algorithm goes against how most children intuitively think about numbers, he or she may also be training the mind to think, "Math is not about making sense. It is about following steps. And if I mess up the steps, I must not be good at math."

He just turned 7? I would drop it entirely and focus on mental math and on other topics that encourage him to think about what the numbers mean and how they behave. The point of the pencil-and-paper borrowing/carrying/renaming/whatever technique (called an algorithm) for calculating answers is that it can be done automatically, without any understanding, just by following programmed steps. It can be done by computers, and even by dollar-store calculators. In earlier days, it could be done by low-paid scribes in an accounting house (think "Christmas Carol"). All you have to do is turn off the mind and crank through the motions. This is what Alfred North Whitehead meant when he 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." BUT what we really want from our children at this age is that they THINK about the numbers, that they build up a foundational understanding of number properties and behavior. The algorithm works against this sort of understanding. We say numbers from left to right, big place value to smaller, and children naturally work with them that way. But the algorithm forces them to go against intuition and work from right to left. We want children to think about the meaning of numbers and their relationships to each other. But the algorithm treats each individual digit as a stand-alone entity, independent of its place in the number. That is, you use the same methods whether you are adding/subtracting a number in the millions or millionths. This is one of the few places where I disagree with Singapore math (and most other curricula---and most of the people who have posted answers here, too). I think that teaching the algorithm at this age is developmentally unsound. I don't think it does quite as much damage in Asia, where most of the teachers understand math and where the style of teaching emphasizes discussing alternative methods and using mental math. In American classrooms, however, I think this topic is one major contributor to the math anxiety epidemic.

There are so many wonderful algebra resources online that it's hard to choose among them! Here is a great list to explore: Such a Thing as Free The list was compiled by one of my favorite math bloggers from suggestions by high school and college math people around the internet. Rich and rigorous, yet plenty of variety.

Sounds like a poorly-posed problem to me, but I found several of those when we were going through Singapore math. It seemed like it was part of the "game" of school math in their eyes, that if the lines LOOKED straight/perpendicular and if you needed that assumption to make the problem soluble, then you were allowed to assume it. I would much rather have all facts clearly stated and never, ever trust the diagram. Just have your student clearly state whatever assumption he or she is forced to make in order to solve the problem. That way, you clarify in their minds that there is an assumption being made, so at least their thinking is clear, even if the book is not.

This is an important and very valid complaint! Please don't let math turn into some sort of arcane ritual where the authority figure knows all the answers and the student's job is to jump through the hoops. If you like MEP overall, you don't necessarily have to change the program, but just tweak it. For instance, when the program offers a challenge puzzle like that one, think about what it is trying to do --- in this case, to make the student work backward (algebraically) using the information she knows to discover the mystery number she doesn't know. So how can you get to the same goal, but make the problem fit her better? Start with a simpler puzzle: "I have a mystery number. If I had 3 more, that would be 10. Can you tell what my number is?"And then --- and this is the most important point --- let HER make up a mystery number puzzle for you. Take turns making up math together, so neither of you gets to have all the secrets. This one, empowering change could transform her attitude toward math forever. I wrote a blog post about this "taking turns" approach to math, if you're interested: Tell Me a (Math) Story

Well, they DO call the book "challenging." :) This is a fairly common pattern for a Singapore math problem. One person has more of something than another person, then one of them gives some of the somethings to the other one. The key is to realize that the relationship between their amounts changes by double whatever was given, as Maize explained above. Bar diagrams (like Cosmos drew) make it easier to see how the numbers are related.

First, refresh your own understanding of multiplication. There's SO much more to it than most of us think! Check out this collection of examples: 12 models of multiplication Next, try to make or find physical examples of multiplication, of as many models as you can. Take time at this stage --- multiplication is a BIG idea, and learning to recognize it in all its guises is worth whatever time it takes. One thing that helps in recognizing multiplication situations is that there will almost always be a "this per that" type of relationship. Fingers per hand, cookies per package, buttons per snowman, panes per window, chairs per row, legs per spider, wheels per car, shoes per foot (one shoe per foot, right? times two feet = two shoes per person, times three people = six shoes per family...), window per door, pennies per nickel, etc. When you are ready to move on to more abstract models, here are a couple of fun ones: How to Make Multiplication Tables with Perler Beads Multiplication Towers For more tips on teaching multiplication, you might enjoy this post on my blog: PUFM 1.5 Multiplication, Part 2

Be careful with this rule! It works in the examples you give, but what if only part of your number repeats? Or what if the repeat doesn't start at the decimal point? Are you going to make your student memorize new, different rules for every situation? What a load of arbitrary stuff to clutter up the memory!

I always have trouble remembering rules. In this case, you really don't need the rule -- just a bit of common sense, and a determination to make the problem look simpler. Whatever part bothers you, focus on that part, and find some way to simplify it. This is how I think it through: There is some fraction that equals 0.63636363... I don't know what that fraction is, so I'll just give it a name. Things are always easier to talk about if they have names. Call it F: F = 0.6363636363... But I don't know how to work with all that decimal junk. This would be a whole lot simpler to work with if I had a nice number like I'm used to, something I could round off and hold in my mind. Let's multiply by enough to get a nice number on the "good" side of the decimal point. For one repeating digit, I could multiply by 10, but to move two digits out of darkness into the light, I need 100. 100 × F = 63.63636363... That's better, but I still have a bunch of decimal junk. It would be simpler to throw that all away, but wouldn't that be cheating? At least it's still the SAME decimal junk. In fact, it exactly matches the decimal junk that I already gave a name to. That's cool! So I can call this junk by the same name: 100 × F = 63 + F Hey, now, that looks ever so much better! If a hundred of something (our mystery fraction F) equals a number plus a single something, then the number (63) must be worth ninety-nine of the somethings. 99 × F = 63 And if 99 of something is 63, then if I cut 63 into 99 pieces, I will find out what my original something was...F = 63 ÷ 99 = 63/99 = 21/33 = 7/11

Try supplementing MM with some (free) enriching activities from Moebius Noodles. I bet your daughter would enjoy several of the ideas in the Multiplication archives, and the Inspired by Calculus series would be a wonderful spin-off.

At this age, the most important rule for teaching math is the old doctor's adage: "First, do no harm." If she finds the curricula interesting and enjoys playing with the ideas, that's great. But if math begins to feel like a chore, then stop. There are plenty of interesting ideas to play with, and you gain nothing but stress by rushing ahead in math. One great resource for playful activities that touch on deep math concepts: Moebius Noodles, blog and book (pdf available free or pay-what-you-want) And my favorite how-to-teach-math resource for parents of young children: Talking Math with Your Kids, blog and book (and twitter @Trianglemancsd)

We always skipped around in Math Mammoth, using post-it bookmarks to keep track of our places. I like to have variety in our lessons, so we would work in several places in the book at once, doing a bit in each section each day. That way I could mix the more fun topics in to balance the important-but-boring, repetitive parts.

[i put this on the high school forum yesterday, but since my dd was actually in 8th grade when she wrote the book, I'm crossposting here as well. We're excited: her book has made it onto the Amazon Bestseller List for teen & young adult fantasy books. :) ] Do you have a budding writer? Here's some inspiration: Homeschool teen Teresa Gaskins' fantasy adventure novel is free for Kindle now through Monday. Banished: The Riddled Stone, Book One She wrote this book when she was 13yo. Now she's 15, and just finishing up Book Two of the series, which is scheduled for publication in spring 2015. Teresa also writes a blog (occasionally), and your kids may enjoy reading her older stories there. She started by writing much shorter tales and gradually worked her way up to full novels: Kitten’s Purring — Stories for a Cozy Lap