your original solution works fine when the numbers of each type of thing are close together. the other solution (assume all the things are one type, and then adjust as needed to make the total come out right) works more generally -- even if there were 9 bikes and only 2 trikes, it would find the answer for you.

conceptual math is when the student can explain why she's doing whatever she's doing. not just because it's the rule, but why it works. i'd say this was definitely conceptual math for you. if your daughter understood why it worked to add and subtract that way -- recognized the connection between what she was supposed to do in the problem and the trick you gave her to do it -- then it was conceptual math for her. if she just followed your rule, then not.

In reading about research on how children think about mathematics, I was surprised to learn that "how many more/less" is the absolute most difficult type of question to ask a young student, because the question doesn't prompt any activity in the child's imagination. Simply rewording the problem can make a world of difference in the child's understanding: "How many should we give Jack so they will have the same number of things?"

Even "equal groups" is being a little bit too simplistic, since most things in life are continuous rather than discrete. When teaching elementary students, we do tend to think in terms of discrete items and groups, but in real life "this per that" units are everywhere. For instance, we can travel 100 miles at 65 miles per hour (or more?), and get 19.48 miles per gallon of gas, which costs us $3.49 per gallon and rising. Being able to work with "this per that" units will be especially important in high school science. Better that our children get used to noticing and working with them while the problems are still relatively simple.

Actually, you could subtract the four. Imagine handing out the cookies one at a time. You hand out 4 cookies, one to each student, and you have 4 left. Subtract those (hand them out), and your cookies are gone. You subtracted twice, and each child got 2 cookies.

Try reading through my elementary problem-solving series of blog posts. The grade-level posts explore a variety of bar model problems of gradually-increasing difficulty.

The problem with slogans in teaching: Whether we are the teacher or the student, once we accept a slogan as dogma, we stop thinking. "____________ is simply ____________ ." Fill in the blanks however you want, and you have a BAD statement for a teacher to make. "Division is simply repeated subtraction?" As to the original question: I agree with the other commenters that "repeated subtraction" is a poor way to introduce division. On the other hand, it IS a great mental math technique to have in your toolbox for solving certain math problems. And the slogan does encapsulate ONE way of looking at division. Let's consider the types of story problem (or real life) situations our student might meet which require division. First, we might have some amount of stuff that must be shared evenly among a certain number of whatevers, and we need to find out how much stuff each whatever will receive. Second, we may have some amount of stuff that must be measured out in chunks of a certain size, and we need to find out how many chunks we can make. The latter situation looks much like subtracting the size of the chunk over and over until we run out of stuff. For instance, we might need 3/4 yard of fabric to make a certain type of pillow cover -- so how many pillows could we cover with 6 yards of fabric? Also, as another commenter has pointed out, this understanding of division is at the heart of the standard long-division process. Conclusion: Don't teach with the slogan. But do, as a teacher, think about what might have inspired the slogan and how it might help you develop a deeper, more flexible understanding of division. "Multiplication is simply repeated addition?" This slogan has the SAME problem as the statement about division (it encapsulates ONE way of looking at one very limited application of multiplication), but because the multiplication slogan is so familiar to us, we teachers don't recognize the problem. We have a familiar, comfy slogan, and we don't think deeply enough to realize the problem this can cause for our students. If we train our students to think "multiplication is repeated addition", then we have no cause to complain when those same students can't solve story problems or when they get confused trying to remember the fraction rules. Consider: (2/3) x (5/6) = (2 x 5) / (3 x 6) but (2/3) + (5/6) is NOT = (2 + 5) / (3 + 6) Why not? Isn't multiplication just a special type of addition? So WHY are the rules so different? The Fibonacci Series is created by repeated addition of the two previous numbers. Is that multiplication? We can form the square numbers by adding up the odd numbers: 3^2=1+3+5, and 4^2=1+3+5+7, and 5^2=1+3+5+7+9. That's definitely repeated addition, and squaring a number is a sort of multiplication... The problem with the definition "multiplication is repeated addition" is that it leave unstated the MOST IMPORTANT difference between the two operations. That's why so many students are reduced to staring blankly at a story problem, asking, "Do I add or multiply?" We haven't given them any way to recognize the difference. For more examples of how not understanding the difference between addition and multiplication makes learning fraction rules difficult: Quiz: Those Frustrating Fractions For a more thorough exploration of the "repeated addition" debate: If It Ain’t Repeated Addition, What is It? What’s Wrong with “Repeated Addition”? Then how SHOULD we teach multiplication? If we accept this argument, if we agree to no longer define basic multiplication as "repeated addition", then what? How does that affect the way we teach? Mainly, we need to change our focus from how to why. We can teach multiplication in much the same way that we do now, using manipulatives arranged in groups or rows, pictures of multiplication situations, and rectangular arrays of dots or blocks. But instead of drawing our student’s attention to the process of adding up the answer, we want to focus on the fact that the items are arranged in equal sized groups. In other words, we teach our students to recognize the multiplicand: Teach children the useful word “per” and how to recognize a “this per that” unit. Have them label the quantities in their workbook: 3 cookies per student, 5 flowers per vase, 1 eye per alien, or whatever. If your story problem has a "this per that" quantity, then it must be a multiplication or division problem. You may be able to solve it with an addition or subtraction approach (especially if the numbers are small), but the heart of the problem is multiplicative.

I have a few games for multiplication (or for mixed operations including multiplication) on my blog: Game: Times Tac Toe The Game That Is Worth 1,000 Worksheets Contig Game: Master Your Math Facts Game: Target Number (or 24) 20+ Things to Do with a Hundred Chart

While it is easy to do simple algebra without fractions (my children start algebra in kindergarten), it is impossible to do well in an algebra 1 course without a good understanding of fractions. Is the course your son is doing just at home, or is he signed up for an online or classroom course? If at all possible, I would suggest stopping the algebra for awhile to go back and firm up the fraction concepts. Khan Academy might be a good option, and it's free. Then when his understanding of fractions is solid, come back to the algebra.

Is there a charge for the conversion? And does it work for purchased, copyrighted pdfs?

I do something similar to this, except I stretch it out much longer than a week and use it to teach several basic pre-algebra ideas: Times Table Series

Here is one of the recent "massive" KISS Grammar posts: http://www.welltrainedmind.com/forums/showthread.php?p=3326468#poststop It should give you a good idea on how to get started.

Here is an article that might help you, no matter which program you decide to use: Relational and Instrumental Understanding "Instrumental" understanding means just learning and following the rules of math. "Relational" understanding means seeing the underlying patterns and reasons behind the rules. Most of us had a school education that focused on instrumental understanding, but what a student like your daughter really needs is relational understanding -- and knowing the difference will help you learn math along with her! Example: Area of a triangle. Instrumental understanding means memorize the formula (area = 1/2 * base * height) and know how to apply it. Relational understanding means knowing why the formula is true and how it grows out of the more basic concept of finding the area of a rectangle, as well as knowing how to apply the formula.

Short lessons, real books, no busywork -- to me, that sums up KISS Grammar. We do a couple sentences at a time (more in the younger years, when the sentences were shorter), a few times a week, sitting together on the couch.

Miquon does not need a supplement, but it is different enough from "normal" school math to make some people nervous. For the rest of us, who know it's a fine program on its own but supplement anyway, I guess we're just math program junkies. :D If you were to do Miquon alone, be aware that the workbooks are only part of the program. Most of the program is just playtime -- free play and directed exploration both. Lots of ideas at the beginning of each section in the Lab Notes.

No matter what curriculum, I have the best success with couch-schooling math using the buddy system, sitting side by side and taking turns doing the problems out loud. We also like to use a whiteboard and colorful markers, and when we can, we just work orally without writing anything down. And games are almost always more popular than worksheets.

I would just play with language in 1-2nd grade and not worry about studying grammar at all, except for whatever comes up naturally. Have lots of conversations, learn fun words, let the little ones narrate while you take dictation, encourage them to write captions for their drawings, etc. That's one reason we have taken so long to work through the levels -- that, and natural laziness. My daughter and I have been doing KISS for at least 4 years, and we're only now starting KISS Level 3, which is "officially" the 3rd year of study. We're learning, but slowly, and the pace doesn't worry me. We still have plenty of school years to go...

I said "Yes" because a well-written blog post follows style guidelines (frequent breaks for headings or bullet lists, lots of white space, short paragraphs, generous linking, etc.) that do not work at all for other genres of writing (such as narration or essays). Bloggers who approach blogging like essay writing end up with horribly unreadable posts.

My times table series of blog posts may work for your daughter. My approach minimizes memorization and stresses building a foundation of prealgebra-style understanding.

You're welcome! I'm glad you and your son are enjoying KISS so far. Perfectionism is tough. My daughter tends that way, and if I try to bite off more than she can chew (as happened today), she ends up face down on the couch, refusing to deal with the world. We had an extra-long holiday break, and for our first day back I should have just tried two sentences (one for each of us to analyze). But they were short, and I decided to go for four of them. She ran into the phrase "to harness", which sounds like an infinitive (as in "to harness the horse") -- but in this particular sentence it happened to be a prepositional phrase ("breaking a horse to harness"). Facesplat!

I wrote a blog post about conversion factors: How Old Are You, in Nanoseconds?

Moebius Noodles! This was designed as a 6-week enrichment course to expose children to big math ideas. Scroll down to the bottom (the earliest posts) and work your way up, exploring a few activities each week.

Warning: Geometry is the hardest math course to grade. Answers that look completely different from the book's version may still be correct -- especially when you get to doing proofs. Also, when I taught a geometry course that had tests, I always offered students a chance to re-test (with slightly different questions) until they scored above 80%. Some students need the extra "kick in the pants" of getting a lower grade at first to help them recognize how much they didn't understand.

We always do KISS as a "cuddle on the couch" activity, with the printout on a clipboard or notebook shared between us, taking turns with the marking as described in my blog post Buddy Math. We started with the 2nd grade book and did a 5-10 minute lesson once or twice a week -- sometimes more, when my daughter asked for extra. Now, in 7th grade, she still insists that we snuggle for our grammar lessons. If you love diagramming, you could always add it to the simpler levels of KISS, but in the upper levels the sentences are MUCH too complicated to diagram them. The KISS mark-up that the students learn forces them to pay attention to the logical structure of the sentence -- isn't that what diagramming is all about, too? I think the KISS mark-up is clearer and easier to learn than diagramming, so if the logic of language is what you want your child to master, I wouldn't worry too much about not having the diagrams.

:iagree: The equal sign gives you a good way to get your brain around predicate nouns. If the sentence you want to analyze says (in some way, not always with a "to be" verb) that A=B, then B is a predicate noun (PN). Another possible complication: If the word that answers the question "subject verb what?" is an adjective that describes the subject, you have a predicate adjective (PA) -- for instance, if your first sample sentence had been "The quetzal is beautiful."