There are three key elements to memorizing anything, and these three elements must be in place for memorization to occur.

The first key element for memorization is repetition - just repeating the information over and over and over again, whether saying it, writing it, hearing it, singing it, or staring at it - simply repeating it again and again and again.
The second key element for memorization is repetition.
The third key element for memorization is repetition.

In summary, the three elements necessary for memorization to occur are: repetition, repetition, and repetition.

Most adults had the repetition via flash cards and drill sheets. There IS a place for these items, I agree - so I am not going to delve into the various ways of using flash cards, except for one recommendation listed below. For my time, money effort, and variety, I have found it better to supplement the regular use of drill sheets and flash cards with other activities, such as the following:

1. Music/singing: I recommend skip count song tapes. Skip counting is not multiplication per se, but it does form the foundation for number sense and for +, -, x, and . My website (http://www.AlgebraForKids.com/audio.aspx) shows two skip count tapes that we carry, and there are other places where such tapes can be found. I recommend skip count songs before multiplication songs because skip counting is wider and broader than mere multiplication as such. Skip count songs are great for kids from age 2 to about age 8 or so, so your 9-year-old may be a bit too old for this item.

2. Games: any game where a child has to count, multiply, or do repeated addition (multiplication) is great - such as Yahtzee, Monopoly, Cribbage, Rummy, etc. While some families prefer not to play cards, with my children, card games were both fun and educational in that the games allowed them situations in which they had to use/practice their basic math skills in a safe, enjoyable setting that was full of contextual associations. It is exactly this type of repeated counting, adding, subtracting, and multiplying - found in card and dice games - that lends itself so readily to practice-practice-practice necessary for the eventual mastery. In other words, put kids in situations often enough where they will be confronted with the *reality* (not the advice) that memorizing arithmetic facts is simply faster than figuring each fact out from scratch every time.

3. Oral recitation: after narrowing and identifying the particular facts that my sons did not know, I found (as usual) that the unknown/weak facts usually fell into families - my older son years ago was weak on the "times 8" facts. So during a period of about 2-3 weeks, every time something he wanted to do arose - like having dessert, going to baseball practice, going biking, etc. - I said, "Before you do this, you have to recite the 8s table 3 times completely." This meant he had to recite out loud, "One times eight - eight; two times eight - sixteen; three times eight - twenty-four; four times eight - um, um, um, thirty-two; five times eight - forty; ..." etc., etc., etc. If he protested about doing this, I would just smile and good-naturedly say, "Your ice cream is melting..." or "You are going to be late for practice..." So he worked his way through this, building up the mental concentration necessary to master multiplication, and noticing connections along the way. The really odd thing is that I only remember doing this for maybe 1 or 2 weeks, after which he seemed to have it mastered.

4. Flash cards: when flash cards are used, be sure at some point to have your child start separating the flash card facts into two groups or piles: the mastered pile, and the not-yet-mastered pile. Then your child should focus on the cards in the not-yet-mastered pile, while occasionally reviewing the cards in the mastered pile. The mastered pile should grow continuously larger over time, while the not- yet-mastered pile should grow smaller. If you and your child do not identify and isolate the facts that are not yet mastered, then the use of flash cards is very inefficient and almost pointless.

The main thing to keep in mind is to not always make the practice of basic arithmetic facts be the tedious, serious, pressure-filled associations of drill sheets, timed tests, and such. While there is a place for these types of pressure-related activities, they simply cannot be ALL the time. Look for your own ways to practice math, especially with games.

You might be surprised at my answer here: visually examining a basic facts table can be part of the repetition-repetition-repetition that is necessary for memorization to occur! Should he have access table ALL the time? I don't think so - but neither do I think you should always deprive him of access to the table.

Play some math games in which computation is required, such as Yahtzee, Cribbage, or Rummy. Let your child have the basic facts math table out. He will start to see that the other players are doing the same computation without the table - because they've memorized the basic facts - and it's faster.

Get a copy of my booklet "Math Games to Supplement Any Math Curriculum" (at the Math Products/Books link at www.AlgebraForKids.com). It has a chapter on competitive math computation games you can play with your child. When playing these games with your child, let him have access to the table as he plays against an opponent (you, or a sibling, or a friend) who has memorized the basic facts. He will quickly see for himself that memorizing the facts is a lot quicker than checking a table!

If your child is 7 years old or younger, get a copy of the CD or cassette of the Skip Count songs that I recommend on my website. Music is a powerful tool. See answer 1.A in the above section that discusses the power of skip count songs.

In fact, see ALL of my answer to Question 1 above.

With boys as young as yours are, I would (as you said) "keep on with the games and concept development using manipulatives." In other words, let the necessary practice, practice, practice be done in the form of games rather than drill - at least at this young age. Sure, you can (and should) check them with drill sheets every now and then - just keep in mind that what you are after is fluency in basic fact recall, and that desired fluency can, in my book, be achieved better from games and music than from the strict use (and overuse) of drill sheets.

Obviously, you should use your selected curriculum properly and regularly, with this one caution: some curriculum have what I consider an overemphasis upon drill, drill, drill that is boring (only worksheets and flash cards). While I am ***completely*in*favor*** of immediate recall of basic facts being mastered at an early age, I am also aware of the benefit of music and games in facilitating this very goal. Even some of the "basic fact mastery" materials advertised in PHS magazine are programs that use games or game-like conditions to practice, practice, practice those basic facts to the point of mastery. I am very aware that many kids love flash cards - but I would also contend that what they like about them is when they are used in game-like conditions - even if the game is a solo game of competing against oneself or practicing by oneself by trying to improve one's own best time or best score.

More precisely, what I mean is this: that desired fluency (in basic fact recall) can, in my book, be achieved better from a combination of the regular curriculum work, drill/practice, AND games, AND music, rather than from the supplementing of curriculum work solely and only with drill sheets and flash cards.

I also highly recommend skip counting at the age of your boys. Skip counting is counting in multiples of an amount - e.g., 7, 14, 21, 28, 35, 42, etc., or 4, 8, 12, 16, 20, 24, 28, etc. The best way to acquire skip counting mastery is with skip counting songs. There are several sources for skip counting musical cassette tapes - check local stores or my own website. Skip counting forms a skeletal foundation for addition, multiplication, division, factors, and fractions and is enormously helpful in building up number sense for kids.

Another variation on games: for any game in which counting by ones is done, try introducing the idea of counting by tens instead. For example, in Yahtzee, instead of the "three of a kind" set of 3 fives being worth 15 points, make each five face be worth 50 points instead. So "three of a kind" with 3 fives would be worth 3 x 50 or 150 points. This variation builds familiarity with place value, multiples of 10, and multiplying by 10. Similar variations could be done with hundreds instead of tens.

To answer your question of "how much time for math each day?" Your teacher's advice is sound - and so is yours - think of this as providing a math-rich environment for your boys - music; games of computation; games of geometry and shape; games of logic and thinking (chess, etc.). Check the "Math Games" link on my website (www.AlgebraForKids.com) for the BEST math games out there.

Besides the alternatives with the reference to manipulatives (albeit the proper use of manipulatives) - another type of alternative to paper-pencil math is math games. Check my webpage (www.AlgebraForKids.com) for several of the best math-type games, and check local game stores for traditional games such as Yahtzee, Cribbage, Monopoly, and 'Smath in which sheer counting and the basic arithmetic operations are used repeatedly. You can also order my booklet "Math Games to Supplement Any Curriculum" from my webpage for a modest price. Yes, I definitely would combine these with paper-and-pencil math.

Think of yourself not strictly and only as a teacher per se but also as your child's supervisor-administrator. Depending on laws in your state, there are varying degrees of latitude you have between the two positions of being your child's instructor (the one who does the actual teaching) versus also being your child's supervisor (the one responsible for ensuring that your child is being taught well). Find out what latitude you have where you live.

Keep in mind what will probably happen with your younger children as they progress, mature, and learn over the years, especially (hopefully) with math: beginning especially in upper elementary grades, they not only learn math, they learn HOW TO learn math - from *reading*the*texts (what a concept). Starting in the 4th-5th-6th grade area, students will for the most part learn how get meaning from the written math text and its examples - IF you insist that they do this *especially* during the years of elementary arithmetic when YOU CAN answer their questions. I've learned with both my own two sons in homeschool as well as my students at public school to engage in something along the following, when they come to me with a question:

1. Did you actually, factually read the text?
2. Show me what you read.
3. Don't read it to me, but do tell me what it means.
4. Tell me as precisely as you can what you don't understand.

If they have not read the text, they have to go read it alone, then come back to me, at which point I repeat the same above sequence all over again.

The reason I do this is because I want them to learn how to learn and to learn how to be more self-reliant by learning how to get meaning from written explanations (instead of being reliant on the teacher too much of the time)..

Anyway, after a number of weeks or months or years of doing this, they will be well on their way to not only learning math but also learning how to learn. At this point, if you still find yourself beyond your competence, you can still provide some guidance and resources:

1. You can engage a private tutor to work with your child when they get into an area beyond your competence. This does not necessitate that the tutor is so much teaching your child as much as the tutor is there to competently answer questions from your child that you cannot answer, especially if/since your child has already for some time been learning how to learn.

2. Check your vicinity for homeschool co-ops, homeschool academies, etc., that might provide a tutor, a tutoring opportunity, or perhaps a class for your child.

Re your comment about having your sons doing calculus by age twelve: I am all for enrichment, and I'm all for providing genuine "whiz kids" a math level that provides them success and challenges up to their abilities, and I am all for getting number sense down, and I am aware that many curricula spend too much time on arithmetic.

Now having said all that! - I've also seen that *unless* a child is a genuine whiz kid (which yours well may be), I want to be cautious about accelerating kids *too* fast mathematically. The reason for my caution here is that even for kids who master algebra early on (say, grades 6-7), there are still development/maturational factors to consider. Developmentally, there is still a maturity that kids come into about age 14 or so that allows them to have a perspective, a range of insight, a depth of understanding, a "with-it-ness" that only arrives at this age and not before. I saw this with my own older son: he mastered algebra as a 2-year course in grades 6-7, then mastered geometry in 8th grade, both times getting grades of solid A to A-, with a depth of conceptual understanding and excellent number sense and problem solving skills.

But it wasn't until 9th grade that I saw for the first time a capability in him to reflect-generalize-connect different aspects of what he was studying - "Oh, Dad, what we're doing here is just like what we did over there..." I consider this a maturational development, not an academic-intellectual issue. I've even heard that there is research indicating that kids can master algebra better when they're older - although I hasten to add that that does *not* mean we should bore them to tears with 7-8 years of paper-pencil arithmetic - *and* I hasten to add that this doesn't mean kids should not see algebra before a first-time formal Algebra I course in 9th grade. In fact, we have taught substantive algebra to kids as young as 2nd-3rd grades in our Summer Algebra Institute for Kids since 1995, and I also taught algebra to FIRST graders in a year long program in 1993.

So my older son certainly wasn't harmed by having algebra relatively early (grades 6-7) - but on the other hand, my son had a father who was a math teacher, too. I just wonder about kids who get pushed into algebra at an earlier age than they perhaps ought, if it wouldn't have been better for them to have been a bit older. Again, none of what I say here applies to the genuine whiz kids who are so obviously bright that it would be a disservice if they *didn't* take algebra earlier on.

Bottom-line: if your son is a genius, go for it. If he's "merely" just very bright, then consider other factors such as maturational development - and consider providing some serious and substantive enrichment along with age-appropriate math.

We ran into this problem, too, with both our sons. The first thing that comes to mind is along the lines of avoiding the connection of anything negative (like punishment) to math studies, including the seemingly "harmless" practice of giving additional math problems for mistakes.

The most effective response was to take a casual, matter-of-fact, but firm approach that any problem or exercise they got wrong, they simply had to do again correctly. Just having to correct their errors is "penalty" enough - although it is not necessarily a penalty as much as it is a consequence of poorly done work. This approach with math errors was similar to the same approach we still have with chores and duties around the house - be careful to do the task right the first time, because if you don't, you going to have to go back and do it over again - whether the task is mowing the lawn, doing the dishes, or cleaning the shower.

So this casual, matter-of-fact, but firm strategy fit well with other stuff we were already doing. It seemed that the key thing was conveying clearly to our boys that them doing a task over again was not a punishment but rather a simple reality - do it right, or go back and do it again. This also worked much better when we didn't threaten, argue, or get angry. BTW, the best advice I've ever heard regarding issues of discipline and training is "Don't threaten, don't argue, don't plead - just inform clearly once, then enforce." So it was a family policy (which we could have communicated more clearly) of "Do it right the first time or do it over" - whether chores or school work or thank-you notes. I should say, too, that we blew this any number of times, even with the math.

Another aspect of this that I want to elaborate on is this: that the need for the necessary repetition-repetition-repetition does not need to be only in the written mode. One of the most effective ways to help a student reach that necessary degree of repetition-repetition-repetition is through the use of games - games, for example, in which the skills of addition or multiplication are practiced repeatedly. Such games include board games (depending on age) such as Yahtzee, or Chutes & Ladders, and almost any board game involving play money and/or dice such as Monopoly. There are also many card games in which simple addition is used repeatedly, such as Rummy, Cribbage, and variations of the standard card game of War in which cards are added, subtracted, or multiplied (instead of just the higher card winning in traditional War). These types of regular activities can be deliberately and overtly used as part of one's homeschool repertoire of basic math skill practice-practice-practice. There is more addition - and more meaningful addition - occurring in some of these games than page after page of addition exercises. You can also check my website (the MathProducts/Games link at www.AlgebraForKids.com) for some commercial games that we carry.

Finally, try some of the various "arithme-trick" books in book stores, which show effective use of shortcuts for many different types of arithmetic operations. Shortcuts are effective ways for children to see some of the flat-out cool ways to do arithmetic operations more efficiently and develop that mental fluency we want our kids to have with number operations. Think about it: page after page of arithmetic textbook exercises would bore most of us and make us more likely to commit silly mistakes along the way.

The identifiers I use as to readiness for algebra are if the student has mastered:

1. basic fact operations;
2. place value work (dividing and multiplying by 10; 100; 1,000; etc.);
3. fractions, decimals, and percents;
4. operations with negative numbers;
5. operations with exponents;

- then the student is ready for Algebra I.

If the student is weak on these, then I usually suggest pre-algebra.

Additionally, several issues come to mind here: basic fact mastery; learning styles; boring vs. challenging.

Basic fact mastery: has she mastered basic facts or is she still stumbling over them? If mastery has not occurred, do what you can to see that this happens. There are three keys to memorization: repetition, repetition, and repetition. This repetition can take numerous forms besides traditional flash cards and drill sheets: oral recitation and games come to mind. Oral recitation is simply the out-loud repeating of the basic fact or facts in question, and it can occur repeatedly throughout the day. Games can also help by providing the necessary repetition, repetition, repetition - games like Yahtzee, Cribbage, etc., where basic facts are needed repeatedly and so practice occurs.

Learning styles: for some kids, Saxon is great because of the review, review, review (which is the main reason I like it). For some kids, Saxon is too dry and boring. However, I usually steer away from the Saxon 87 and try to get kids into the Algebra 1/2 and/or Algebra 1 sooner, as long as they are ready for it. The Saxon 87 can be good for kids who still have not mastered arithmetic procedures; on the other hand, I think most kids don't need another year of arithmetic, arithmetic, arithmetic.

Boring vs. challenging: consider whether your daughter's need for the review- review-review of Saxon might outweigh the possible boredom of 87's arithmetic- arithmetic-arithmetic. She may need the challenge of Algebra 1/2 or Algebra 1 or some other algebra program.

Another option that I have arranged with several students I have worked with (including my own two sons): knowing that these students have mastered basic facts and understand place value, fractions, decimals, percents, exponents, and operations with negative numbers, my thinking was that on the one hand I want students to move out of arithmetic as soon as possible while also knowing on the other hand that developmentally many kids in grades 6-7-8 (even bright kids) are not quite ready for a full-blown algebra course. So the balance I have struck is something along the lines of a two-year Algebra 1 course: algebra lessons on M, W, F; arithmetic enrichment on T (like the book Rapid_Math_Tricks, which is available at my website); and other enrichment on Th (such as classic geometric compass-and-straightedge constructions or math games).

What I like to see before kids begin algebra is that they have mastered and understand basic facts, place value, fractions, decimals, percents, exponents, and operations with negative numbers. A child who has done this is probably ready for a traditional Algebra 1 course.

Those are the topics I like to see already mastered before beginning algebra. However, there are also other considerations to make, such as maturity and motivation. There are some younger kids who have mastered all those specified topics who are also so fascinated with - and motivated by -math that it would be a crime not to have them go on to algebra, no matter what their age. There are also some kids who are intellectually ready for algebra but not ready maturity-wise in terms of the necessary rigor and focus. In that case, I recommend doing some algebra but at a slower pace, while supplementing the algebra work with math enrichment topics and activities (which are mentioned below).

I can mention some guidelines for what to look for in math texts:

1. For this level of mathematics texts, you want to choose texts that are essentially self-teachable from the student's point of view. IMHO, most algebra texts almost *require* adult intervention-explanation, which is something to be avoided. However, if you are looking for some possibilities here that do include explanations not limited to written text-based instructions, then some possibilities are:
   1. algebra via video taped instruction: they're better than they might initially sound, especially since she (and even you) can always push rewind for parts that weren't initially clear.

   2. algebra via satellite/cable feed: this is live instruction, with varying degrees of interactive capability; there may be such "distance-learning" options available locally for you, and there are some homeschool companies that specialize in this as well;

   3. seeking a local tutor-instructor for your child to check with on a regular basis.

2. In terms of a self-teaching, minimal-adult-intervention text, you want to look for texts that do not require and/or recommend and/or carry lots of teacher supplemental materials like "Reteaching Sheets" or transparency masters, etc. IMHO, especially for the homeschool situation at this level of math where the teacher-parent may not be that well-versed in math and the student should by this age be well on the way to being self-taught, if such supplementary materials are recommended and are effective, then they should have been included in the regular text in the first place. Keep in mind: most such recommended teacher supplementary materials are for large classroom use where it *can* perhaps help to have something to give some of the 25-35 students who want another way of looking at a topic.

3. Look for texts that have substantial amounts of built-in review of previous topics, so that your child remembers in April and May what they learned in September and October. Again, IMHO, almost all traditional math texts (and also the newer so-called integrated math texts) have nowhere near enough review of skills, procedures, and concepts previously studied. Dropping previous topics altogether while tackling new topics, with no regular review or use of all topics covered so far, is what a colleague of mine calls "planned forgetfulness" on the part of the text.

4. Note on Algebra 1 topics: check how much time is spent on factoring quadratic expressions and on solving quadratic equations by factoring. This would be exercises like "Factor 2x^2 + 7x + 6" which factors into (2x + 3)(x + 2), or "Solve 2x^2 + 7x + 6 = 0" whose solution is found by solving (2x + 3)(x + 2) = 0. SOME exposure to factoring is necessary, especially for recognizing and factoring each of the following: differences of squares; sums and differences of cubes; perfect square trinomials; and a few other special cases. The hidden secret of much algebra instruction is that very, very few quadratic equations encountered in the real world of math, science, medicine, business, etc., can be factored at all and are usually solved with other methods, such as by using the quadratic formula, or by graphing, or by tables of values.

   Back to my point: while some special cases of factoring are very helpful and should be taught, and some exposure to factoring is good, too many traditional algebra texts spend far too much time on this procedure. [There are other reasons for teaching factoring at an earlier age that are related to topics like place value or evaluation, but that is not relevant to your question at this point.] Several algebra texts that I have seen spend two entire chapters on factoring quadratic expressions and solving quadratics by factoring.

5. Another issue is learning style. For some kids, for example, a typical math text with color photos and color graphics on every page is hard for them to work with, because it's visually overwhelming and they have trouble concentrating. For them, a no-frills black-and-white text like Saxon works better. For other kids, a no-frills black-and-white text is dull and boring, so they would do better with books that have lots of color graphics and color photos. While there is a lot of psychobabble "out there" in the education world, the issue of learning styles is, IMHO, a valid and sound consideration.

6. Almost finally, sort through this forum and examine previous posts related to your question.

7. Finally, research and ask around! This can be simply asking your circle of homeschool friends what's worked well for them, or doing a websearch and visiting sites of various publishers.

Yes, teach her math at a 9th grade level and the reading at the 7th grade level.

Not only is this a typical development for someone at your daughter's age, it is a common development for virtually all kids - that they are more developed in certain academic-mental-topical areas than they are in others. The main reason why your situation seems "odd" is that we are so accustomed to having all learning filtered through the template of the one-size-fits-all approach of large-group, same-age classroom settings. Given that most large-classroom education on a topic for a particular grade level is aimed at the middle 80% of kids, and that many kids retain 80% or less of what they learn, after a few years you are looking at topic selection or depth being sometimes based on 80% of 80% of 80% of 80% of what the middle 80% of kids retain or what a teacher-text-author-committee thinks they are ready for.

Think about your own education: remember that classmate who was really, really good at history but not so good at math? Or the one who was really knowledgeable about science but didn't like literature that much? Or the tech- expert who could fix almost anything but hated to read? This happens all the time. It is normal. This is actually part of what true diversity is. You have the opportunity to make it happen for your daughter that she can accelerate when needed and adjust in other ways as well.

Several thoughts come to mind:

1. Get a skip count tape or CD (at the AudioTapes & CD's link at my website www.AlgebraForKids.com), which are catchy tunes that teach him how to count in multiples of numbers (6, 12, 18, 24, 30, etc.). Skip counting forms the foundation for multiplication, division, fractions, factoring, and common multiples, along with enhancing addition and subtraction skills, too. There are two skip count tapes at my own website (www.AlgebraForKids.com), although you can probably find them elsewhere, too. BTW, do NOT get multiplication tapes yet - skip counting is more powerful and more versatile.

2. Give serious consideration to homeschooling him. Your situation is exactly the type of setting in which a child like yours can blossom and mature and in which you can choose a curriculum to fit him (rather than making him fit the system).

3. Get some math enrichment resources to provide a math-rich environment in the home for him. Get some of the excellent math games at my website (the MathGames link at www.AlgebraForKids.com) which can truly enrich your child's learning. There are also some excellent math books available at large bookstores like Barnes & Noble, and I've also written a booklet "Math Games to Supplement Any Math Curriculum" which is also available at the "Books" link at my website.

4. Give serious consideration to making these math enrichment activities (in #3 above) being a deliberate, overt part of your planned curriculum for his day. The way to think about this is that you are going to provide a math-rich environment in which this child's unique talents can blossom with all the stimulus and sunshine you'll be providing.

5. Finally, if you are not a math type yourself and you see an early end to your own competence in terms of mathematical guidance - sensing, for example, that he's going to be beyond you by 4th or 5th grade - then start praying now for a mentor and for opportunities for him down the road. I myself had the privilege of working with such a young man several years ago (something I normally don't do) - he was exceptionally good at math, and he loved math. Things just converged for it to make sense for me to tutor him for about 2 years, both with a text and with some enrichment activities, too. Keep in mind the vital importance of not only fostering his obvious talent but also training him up in the wisdom, so that he is as much "normal" in the usual positive, character trait sense of that word - polite, thoughtful, self-controlled, etc. In working with the boy I tutored several years ago, both the parents and I felt strongly about keeping an eye on his character training, too, and so I also kept a close eye on things like honesty, responsibility, respect, initiative, etc.

Math is for both skill use and for learning how to think. Beyond memorizing basic facts and learning how to solve equations and work with the common topics of fractions, decimals, and percents, the study of math should also teach a person how to think, how to learn, how to reason, how to memorize, how to recall important information, and so forth.

Despite your own poor and unfortunate experiences with math yourself, part of your job as a homeschooling mother is, defensively, to do what you can to ensure that your son doesn't have similar experience, and pro-actively, to provide a math-rich environment for him. A couple ways to provide a math-rich environment come to mind:

1. use skip count songs and any other music - including music lessons if possible;

2. use games; see my comments elsewhere on this forum about the use of games. I've also written a book about using math games to supplement any math curriculum.

3. teach him how to use math around the house - baking, simple carpentry, working with nails, screws, bolts, nuts, wrenches, lumber, and drill bits that have measured sizes in fractions and mixed numbers that provide a highly meaningful and motivational context in which math is used and discussed and seen. You say that you cannot do simple fractions - but can you bake a batch of chocolate chip cookies from the recipe on the bag of chocolate chips? This requires the use of fractions, plus you get to eat the results. Baking and following basic recipes for brownies, bars, cookies, cakes, and so forth are a great way to work with fractions.

Regarding your husband's remark that "he never used geometry or algebra in his adult life" - well, I can raise the ante here quite a bit - my sister is a medical doctor who had to take 2 years of college calculus along with physics, chemistry, biology, etc., in order to even be *admitted* to medical school. Recently I asked her a series of questions about her math background. I highly suspected what her answers were going to be, too - and I was right:

-Q: As a doctor, have you used calculus, like integrals and derivatives? [A: "Never."]

-Q: Have you used pre-calculus topics like sequences and limits? [A: "No."]

-Q: Have you used trigonometry, like sine and cosine? [A: "No."]

-Q: How about geometry and proofs? [A: "Mostly no, although some use of process-of-elimination-type thinking."]

-Q: How about algebra, like the Quadratic Formula and solving for a variable? [A: "No."]

-Okay, Doctor, what's the most math you use in your life as a doctor? [A: "Two things come to mind. First, the arithmetic I need to balance my checkbook. Second, the proportional thinking skills I need to mentally calculate that if I can give 200 mg of a drug to a 140 pound patient, then how many milligrams can I give to a 180 pound patient?"]

-Okay, Doctor, then all those years of math were just a waste of time, right? [And bless her heart, here's my dear sister's answer - A: "No, they taught me how to think, how to reason, how to learn, how to memorize things both long-term and short-term, how to recall information immediately, how to notice the details if necessary, and how to keep the big picture in mind at the same time." ]

Although your husband may not realize it, he almost certainly has used what he's learned in math - both for skill use and for having learned how to think. Even if he's never used the skills or procedures of algebra or geometry, he still has probably benefitted from having studied higher math like algebra, geometry, and trigonometry. Studying those topics teach a person how to think in general, how to think abstractly, how to reason, how to memorize, how to recall important information, how to sort and prioritize, and so forth. I'm certain he's used those types of skills. Those skills are skills that the study of algebra and geometry helps to develop and sharpen. For another example of the value of studying a math topic that is seldom itself later used, see the following question on Roman numerals.

To answer your question directly - most adults use math in at least some of the following ways:

1. The mental arithmetic of addition and subtraction to balance checkbooks.
2. The mental math of finding unit costs - 12 buns of Brand A cost $1.60 but 8 buns of Brand B cost $1.20. Which package is less expensive per-bun cost?
3. Calculating square footage for carpet, floor tiling, sod, grass seed, etc.: How many square feet of carpet-tiles-sod-seeding do I need? This requires a knowledge of formulas for finding areas of rectangles, triangles, and circles, at least.
4. Calculating perimeter for trim board, kickboard, fencing, etc.: How many linear feet of board-fencing-edging-tape do I need? This requires an understanding of the difference between area and perimeter.
5. Not only calculating area and perimeter (in C and D above) - but *knowing* which one to calculate and how to calculate it as well.
6. The mental math of ratios and proportions:
   1. The trip-ometer on my dash board shows I drove 157 miles since I last filled up the gas tank, and now I just filled up my tank with 6.8 gallons gallons of gas. How many miles per gallon am I getting in my car?
   2. The bucket of paint says that 1 gallon should cover about 250 square feet of wall surface - so how many gallons do I need to paint the living room and dining room?
   3. The recipe says that one full batch using 3/4 pound of hamburger should serve 4 adults. I've got 14 hungry teenage boys coming over for dinner - how many batches of the recipe should I make, and how many pounds of hamburger should I buy?
7. In Job A, I can earn $5.75 per hour but can work only 20 hours per week. In Job B, I can earn $5.50 per hour and can work 25 hours per week. In which job will I earn more money?

While any such list like this will be incomplete, this list should prompt readers to realize various other ways that math is used by adults in the real world.

A reason for learning Roman numerals in general is that one learns how to analyze, memorize, code, and decode information both to and from one system to another. That is a valuable process to learn, and it generalizes, IMHO, to a greater ability to do similar analysis, memorization, coding, and decoding with other system translations as well. This ability transfers to other situations where we work with putting information into (or getting information out of) systems. This could be any kind of system - from symbol representation in algebra, geometry, trigonometry, and calculus; to pictorial representation of information in geometric shapes; to translating concepts and phrases into proper code in computer programming language; to learning other languages and translating back and forth from one language to another; to identifying structures of grammar (such as prefixes, suffixes, prepositions, sentence structure); etc. The ability to analyze, recall, decode, and encode information is one broad topic that tests like the SAT are getting at. Such thinking is relevant to virtually any career, from real estate to business to science to linguistics to computers.

One specific (but often unmentioned) value of working with Roman numerals is that they are a contrast with our place value system - and working with that contrast helps highlight the distinctives of base ten place value system. For example, in our base ten place value number 777, the left-most 7 represents 7 hundreds (or 700), while the middle 7 represents 7 tens (or 70), and the right- most 7 represents 7 ones (just 7). So "the left-most 7" stands for an amount greater than the right-most 7!

Now, in the Roman numeral XXX, how much does the middle X represent? It represents 10, period - just like each of the three X's do. Roman numerals have no place value, and if taught properly, this distinction can, by contrast, help one appreciate the distinctiveness and value of a place value system, in which each place has a different value.

In other words, sometimes we gain a deeper understanding of what Topic A is by studying something that Topic A is not.

See my webpage link of MathProducts at www.AlgebraForKids.com for the BEST math games out there.

Math games provide fun, context, motivation, and variety for practicing important math skills.

1. Fun: math games are just fun. Kids like these games and usually will ask to play them!

2. Context: the skills learned from the basic fact practice involved in Cribbage or Monopoly are associated strongly with the smells, sounds, sights, and emotions of a fun family night around the kitchen table or family room. "Dad, remember the time I scored so high in Cribbage?" - "Mom, remember when I beat you for the first time at SET?"

3. Motivation: kids want to WIN! Math games can draw upon that motivation and position that drive in the direction of mathematical skill mastery.

4. Variety: math games can provide another format for practicing math skills. Instead of flash cards ALL the time - or drill sheets ALL the time - get out some computational math games and make them be a deliberate, overt, planned part of your child's mathematical experience.

You do so by allowing the child to see for herself that the use of fingers is slower than memorizing the facts. This means being able to see how others - siblings, peers, parents, relatives - can calculate so quickly without fingers because they've memorized basic facts.

This means... use math games! Play Addition War (as described in my booklet "Math Games to Supplement Any Math Curriculum" which is available at my website www.AlgebraForKids.com), so she will see that others get their answers faster without fingers because of memorization. Play Math Dice (also at my website), or Cribbage, or Yahtzee. Put her casually into situations where she will simply experience for herself the reality that fingers actually slow us down. The experience will speak louder than any nagging from Mom or Dad.

Finally, be patient - finger-counting is a developmental phase that kids go through. They need to confirm to themselves that 5 + 7 really is 12 every time. Be patient. She'll move on - if she's put into game situations where she loses consistently if she keeps using her fingers and if she sees other players NOT using fingers who consistently win.