I like Elementary Mathematics for Teachers by Parker and Baldridge. The PUFM lessons on my blog are based on that book.

Both of the problems you mention are comparison problems, which are particularly troublesome for young students because there is no movement inherent in the story. Nothing is being "put together" or "taken away" to guide the student's intuition on whether to add or subtract. In such a situation, you may need to rephrase the question slightly to help your student visualize the situation. Instead of asking, "How many fewer stamps does John have than Bob?" you might ask, "How many stamps would we have to give to John so that he would have the same amount as Bob?" for beginners, such a little change in the wording can make a huge difference in understanding, because it lets the student visualize an action: giving stamps to John.

Do you blog about your homeschooling adventure? We would love to have you submit a blog post about learning or teaching math to the upcoming math blog carnival. Details here: Do You Blog About Math?

Do you blog about your homeschooling adventure? We would love to have you submit a blog post about learning or teaching math to the upcoming math blog carnival. Details here: Do You Blog About Math?

A child who has decided she won't/can't do something in math is like a toddler who has decided she doesn't want to walk and just goes limp. She's not going anywhere, and you can't make her do it. With the toddler, you have two choices: leave her to pout, or pick her up and carry her. Similarly, with the math student, you have two options: In math, the "leave her" option means taking a break and doing something completely different. Instead of working on multiplication, play with story problems or logic puzzles or symmetry or geometry or read a math book from the library. There are plenty of other things to do in math, and in a few weeks you can come back to multiplication refreshed, without the emotional struggle. Your other option isn't so much to "carry her" (as with the toddler) but to "entice her." You need to find something that SHE will think is interesting about multiplication, so that she wants to do it. You've heard the saying, "You can lead a horse to water, but you can't make him drink." But enticement is like salting the horse's oats so he wants to drink --- or in your daughter's case, she wants to think about multiplication. Now, obviously, she does not find the idea of memorizing multiplication tables very enticing. But multiplication is a big idea with lots of applications and plenty of cool patterns, and I'm sure you can find something to interest her enough to get over her "won't do it" hump. Here are some blog posts to get you started: PUFM 1.5 Multiplication, Part 1 Several different ways to think about multiplication, and a rectangle game from Education Unboxed. PUFM 1.5 Multiplication, Part 2 More ways to think about multiplication, and a card game to help learn the most important multiplication models. Math Monday: What’s cooler than a number line? A number CIRCLE! Math Monday:: More Number Circles! Math Monday:: The Patterns of Numbers. Three posts on visual ways to play with numbers, creating cool designs that bring out patterns you probably never noticed before. How to Conquer the Times Table A conversational, cooperative approach to learning the math facts, with an emphasis on pre-algebra understanding and with minimal memorization.

Everyday Number Stories is an old, out-of-copyright book in pdf format, so it could be printed rather than on a tablet. It's a predecessor of the Verbal Math Lessons series that someone else mentioned --- basically, it's a conversational prompt. The idea is to get you and your son talking about math, and eventually making up little problems for each other. At this age, I think anything that encourages conversation about math is great. Young children learn best through interaction with other minds, even more than through manipulatives (although those are also useful) and certainly more than through workbooks. Even if you get something like the magnetic rods, use them to spark conversation about sizes, patterns, which are greater or less, how many ways they can be put together, etc. Ask (and encourage your son to ask) questions: What do you think about? What do you notice? What do you wonder?

For multiplication, you might try the methods in my Times Table Series (blog posts), which are heavily slanted toward developing pre-algbra understanding.

Well, dd has read all the Danica McKellar books. The last organized math program she did was MM4, plus assorted worksheets from 5 (can't remember how much of that we got through). And about half of Zaccaro's Becoming a Problem Solving Genius, and some of his Real World Algebra. And some of the counting lessons from Competition Math for Middle School, because I love that sort of problem. Can you tell we're not so good at sticking with things long term? Neither of us much likes repetition... Edited to add: Her main preparation is that she's always been expected to think her way through a math problem and to explain her reasoning. I never let her get by with just memorizing procedures. And she's stubborn with a strong perfectionist streak, so she hates to let a tough problem beat her.

How about Everyday Number Stories on a tablet? And then make up stories like them for each other?

Hooray that she could do it, and hooray that she could explain her thinking! Good job to both of you :hurray:

Yes, on Smashwords you can download the Kindle book, epub and pdf (all three, if you want!) and go back to get updates at any time. The latest version is always available, which is nice since I'm still making changes (until whenever the paperback is finished). The downside is: the Smashwords files are created by an automatic program, which they call the "Meatgrinder", and the result can be a bit funky. It was designed to work with simple books like novels, and it didn't handle very well the endnotes, cross-references, and multiple layers of sub-headings that make up a nonfiction book. I managed to edit and resubmit the epub file, so I think it's working now (please let me know if you find errors!), but it's the only file that Smashwords will let me tweak by hand. If you read on a Kindle, the hand-formatted file at Amazon.com is much more pleasant to read. But unfortunately, Amazon does not automatically give customers a new version when I update the file. For instance, I just uploaded some major changes (including this blog post and an big expansion of the high school math chapter) last night, and I sent an email to Amazon's customer support begging them to release the new file to old customers. We'll see what happens... Of course, new customers get the most recent file, and I'm also posting all major additions to my blog (the stuff on high school will go up sometime next week, when I get it transferred), so anyone who liked the book well enough to subscribe won't miss anything. Still I wish Amazon would allow old customers to update if they wanted.

If you want something you can print out and give to her, there are some nice problems at the AGMath site, first drafts of the Competition Math book. (No answers, unless you buy the book, but she should be able to check herself by doing the problems two different ways, which is always a good habit.) Counting is a good place to start. MathCounts is also a good source of problems. Some are routine, but some are very interesting. Check out their handbook and past competitions. And the problems of the week, too.

My daughter is 14 and in 8th, doing AoPS pre-algebra, and I'm in no hurry to rush her on. IMO, there's no reason to worry about a normal student doing algebra in 9th grade. For many people, that extra bit of maturity is very helpful! As for what algebra book to use, can you get sample lessons from the ones you are considering? Then your son could try them out and see which ones he likes. If you can find one that he enjoys (or at least doesn't hate), that makes it much easier to stick with it. As for being able to help him, you might refresh your own math skills by reading the Danica McKellar books (you can probably get them through your library). She has creative ways to explain math concepts, and they really clicked with my daughter.

I have instructions to several math games on my website, if you're interested: 20 Best Math Games and Puzzles

Math is the only thing I keep track of by time. Since we use "real" books for everything else, we just go by chapters. As for the extra stuff and playing online, I either count that as part of the "school" math time or count it as part of play time, depending on the day and the child and my mood and whatever. We're pretty laid back around here!

This is important. Keep it in mind for when you get to fractions, because it will make it much easier to recognize when a fraction is in simplest form or whether the numerator and denominator have factors in common. Also, when you have to rewrite numbers as a product of prime factors (pre-algebra), being able to recognize quickly what a number is divisible by can help. And then, proving WHY these rules work is a wonderful exercise for pre-algebra or algebra students to demonstrate their understanding and mastery of the concept of place value and of how numbers work.

Yes, if you don't have Open Office installed, you will have to download it first. Or you might be able to open the doc file with another word processor. What do you usually use to write documents with? Most word processors should be able to open a doc file. Here is the link for Open Office: http://www.openoffice.org/

Oh, wait: Is your computer trying to open the file with a trial copy of Microsoft Word? Many new computers come with a trial version installed, which you can use maybe five times before you have to buy the thing. Try right-clicking your file and then choosing the "Open With" option from the drop-down menu. Tell the computer to always open that sort of file with Open Office, and that should solve the problem.

Did you download the KISS file to your computer first? Open Office should be able to read a doc file without giving you goofy messages, but if you're trying to read it through your browser, that might be the problem. (Just guessing!)

From the little that she said in her books about math, I'd say that Mason especially appreciated two things: (1) In many math problems (though not in all!), our children come up against a firm rule or law, something that is solidly right when any other answer is wrong. She felt that this was a valuable and humbling experience. (2) Math gives children the chance to grapple directly with ideas, to learn how to justify their reasoning. These two ideas are related, since it is the justifications (or proofs) that convince us an answer is right or wrong. How do we know that we got a sum correct? We can take the numbers apart and add them another way, to see if we get the same answer. Or we can subtract one of the numbers from the sum and see if we get the other number. Or... well, how would YOU prove it? From the very beginning, children should be doing this sort of informal proof, explaining how they figured things out. Don't wait until high school geometry to let your children wrestle with ideas! This is why stories and manipulatives are so important when working with elementary children. Do not rush to abstract math notation, because children cannot reason with it. They need the physical presence of manipulatives or the mental images of a story to give them something "real" to reason with, so they can grapple with ideas and make justifications. Not until MUCH later will they be able to reason using only abstractions. This is true for teenagers and adults as well. As W. W. Sawyer wrote in his wonderful little book, Mathematician’s Delight: Jimmie has collected several of Mason's comments on math here: Charlotte Mason on Math But I think some of the earlier comments are right, that math was not one of Mason's primary interests. She didn't think or write as deeply about it as she did about other subjects. And because of that, it's easy for us to read our own interpretations into what she wrote and to come to different conclusions. (I certainly have had significant disagreements with the AO ladies about how to interpret Mason's comments and about how to apply ideas such as narration to math.)

Oh, that reminds me of another good resource for moms who are trying to relearn math: the Danica McKellar books. Here's a quote from Math Doesn't Suck: Working on math sharpens your brain, actually making you smarter in all areas. Intelligence is real, it’s lasting, and no one can take it away from you. Ever. And take it from me, nothing can take the place of the confidence that comes from developing your intelligence — not beauty, or fame, or anything else “superficial.†Math isn’t easy for anyone. It takes time and persistence to understand this stuff, so don’t give up on yourself just because you might feel frustrated. Everyone feels like that sometimes — everyone. It’s what you do about those feelings that makes you who you are. It’s in those moments when you want to give up but you keep going anyway that you separate yourself from the crowd and build the skills of patience and fortitude that will allow you to excel throughout your entire life — no matter what you choose as a career. And here is something from Kiss My Math: Whenever you don’t understand how to do a math problem, this is actually a good thing, because now you have an opportunity to exercise the part of your brain that makes you stronger, more capable, and successful in life: the part that does not give up. When you’re struggling with something but you believe in yourself and you keep trying until you succeed, you not only become stronger, but also much more powerful. Doing math has a funny way of expanding our brains, making us better problem solvers, and strengthening our mental fortitude and stamina. As hard as math can seem sometimes, you’re actually benefiting from it in ways you might not realize. Seek out the things you don’t understand, and seize opportunities to learn how to think in entirely new ways. Believe me, math will keep giving you these opportunities, so take them!

It sounds to me like you (and probably the private school before you) are pushing him into abstract math work before he is really ready for it. 9yo is still very young for such stress! Instead of doing so much workbook-style math, I would suggest: (1) Go to Moebius Noodles and look at their projects. Find something your son would enjoy, and explore it together. The Moebius Noodle activities focus on advanced topics (like symmetry, functions, infinity) at a playful level. (2) Go to the library and check out all the Murderous Maths books you can find, or the Time-Life "I Love Math" series, or other math books that look interesting to you. Ask a librarian to help you find something fun. (3) If you feel that your son needs some workbooky math, try this: Fold a sheet of plain paper in half both ways, making four parts. (Or used colorful markers to divide a white board. My kids love to work on a white board.) Write one math problem in each part. Choose them from Math Mammoth or any of your other math books, and make sure each one is different --- one addition, one fractions, one multiplication, etc. You can make up a whole week's worth of these at once, with a nicely balanced mix of problems for each day. And your son shouldn't find that so overwhelming, since it's just four problems, but he'll be practicing and reviewing his number skills. (4) Let him play with geometry! That is certainly math, isn't it? Ask him questions about the things he builds, what he is thinking, what he notices, what he wonders about. Such questions are the heart and soul of mathematics.

You might try doing everything orally, rather than having them do it "homework style." I find oral work goes much faster and is more enjoyable, and it also gives you the best idea of exactly what they understand (that is, whether their mistakes are from misunderstanding concepts or just from carelessness). When you work orally, you can also judge whether you can skim ahead or whether today's lesson is an area that needs more work. I always tried to limit our work to 10 minutes or less per grade level. More time than that seemed to just cause burnout. Mental work is tiring! Even now, with my youngest in 8th grade, she can't go much more than 45 minutes without getting grumpy and frustrated. If you have to do more than that, try splitting it into two sessions with a nice break (and maybe a snack) in between. Don't worry too much about "catching up" -- if your students are learning and making progress, then they're doing fine! Grade levels are artificial in the max, and it's a shame that we parents put so much stress on them. You will find that there will be some topics that go quickly for your children, and some that go slower. For my kids, the B books of Singapore math always seemed to go much faster than the A books, because A has more of the just-plain-arithmetic stuff that they and I find boring. For your kids, the pattern may be something different. But take the pace that seems to fit their learning level, make sure their understanding is firm, and don't worry about being "behind" an arbitrary number.

When you read it, stop at the beginning of each of the four question chapters and decide (or even write down) how you would answer that question. That primes your mind, so you will be able to think more deeply about the answers given by the American and Chinese teachers Ma interviewed. I have a blog post series that is going (very slowly!) through this textbook and relating it to how we teach homeschool math. You might find it helpful: Homeschooling Math with Profound Understanding (PUFM) Series

My son (raised on Singapore Math) did this, too. He was so good at it that he never really saw the point of learning the standard "borrowing" method. Basically, what they are doing is taking advantage of our base-ten place value system, that numbers are easier to work with when you get them to a multiple of 10. Therefore, you can take away 6, which gets you down to 30, a nice multiple of ten. And then you have three more to take away. An even more advanced way to think about this is reflected in your son's "crossing out the six's" explanation. Subtraction finds the difference between two numbers, how far apart they are. Imagine the numbers on a number line, and subtraction will tell you the number of spaces in between them. Now, if you move both numbers UP the number line by the same amount (add the same number to both) or DOWN the number line by the same amount (subtract the same number from both), can you see that the distance between them would stay exactly the same? Therefore, your son can "cross out 6" from each number without changing the difference, the answer he is looking for: 36 - 9 is exactly the same as 30 - 3.