Mental Arithmetic?

PASSWORD>aa4um5Lp2CxdUI don’t really have any experience with this, but I wonder if your son has developed his own algortithms for doing computations in his head, but has not figured out how the standard algorithms work. For example, I always add from left to right when I am doing computations in my head, but I do understand why you move from right to left when using the standard pencil and paper approach, and why carrying works.There’s nothing wrong with using your own methods to make computation easier, but there is a lot of value in understanding the standard algorithms and being able to use them as needed.Jean: This may be a stupid question but why would my son excel in all
: aspects of math when he computes in his head, but when faced with
: algorithims he is not consistent?: His quantitative reasoning score on the Stanford Binet was 130,
: Abstract/Visual Reasoning was 122, on the WISC 111 he scored a 120
: on the arithmetic section, Object assembly 130 and Block Design
: 140.: He is in 3rd grade and has been dx with dyslexia, ADHD and language
: disabilities. Oh! His visual perceptual abilities were all in the
: gifted range. Maybe I am just expecting too much?: Please let me know if anyone has an idea what could be going on.: Thanks Terri

: Hi,: This may be a stupid question but why would my son excel in all
: aspects of math when he computes in his head, but when faced with
: algorithims he is not consistent?: His quantitative reasoning score on the Stanford Binet was 130,
: Abstract/Visual Reasoning was 122, on the WISC 111 he scored a 120
: on the arithmetic section, Object assembly 130 and Block Design
: 140.: He is in 3rd grade and has been dx with dyslexia, ADHD and language
: disabilities. Oh! His visual perceptual abilities were all in the
: gifted range. Maybe I am just expecting too much?Most mental arithmetic problems are — pretty much have to be — simple direct one-step computations. The algorithms exist for times when the problem gets beyond that. Your son has some kind of personal visualization and/or counting system that works well for him under limited conditions. This is a good thing, not a bad one! But he needs to extend that system and continue developing, not stall out here.I personally do a lot of arithmetic in my head, up to and including square roots. I use non-standard algorithms for many of these, a lot of successive approximations. BUT — and this is a huge qualification — I know how and why my algorithms work. I can demonstrate and prove each one to you concretely. They aren’t a hodgepodge of quick tricks picked up at random.My experience with “I do it all right in my head, I just can’t put it down on paper” students (whom I meet frequently, one or two at least in every class) is that they slam into a brick wall when you go beyond the introductory stuff and start doing real math. Unfortunately, with the help of untrained teachers and a spiral curriculum, some people stay on the introduction until senior high school or even college. The later you hit the brick wall, the harder you hit, and the more bad habits need to be untaught.Your son needs those algorithms. The first and most basic, adding with carrying, will definitely be needed when balancing a checkbook or calculating a total purchase. Even those of us who are good at mental arithmetic rarely add four or more numbers of five digits each mentally. (dollars in the hundreds plus cents = five digits; a common occurrence)Same for subtracting with borrowing. Multiplication and division will be needed for everything from doing your taxes to calculating the cost of re-tiling your kitchen and bathroom. And so on. Yes, you can punch calculator buttons — and accept bizarrely wrong answers if you have not developed a feel for numbers.Work with him on the “why” of the algorithms. The structure of the base-10 system is a thing of beauty. Use abacus and/or base-10 blocks to help him get a literal grip on what he is doing. This should be done in school, and generally is not, so most students get an aversion to math because they are taught bizasrre and complicated algorithms as magic incantations rather than as logical and sensible shorthand for counting and measuring.If you keep working on the logic behind the rules, you as well as your son will discover some fascinating and interesting inner workings of math that you never thought of before.

Interesting! Have you ever read Math Magic by Scott Flansburg? He developed a whole system for doing mental math left to right. It’s a fascinating read.If so, what did you think?I don’t really have any experience with this, but I wonder if your
: son has developed his own algortithms for doing computations in
: his head, but has not figured out how the standard algorithms
: work. For example, I always add from left to right when I am doing
: computations in my head, but I do understand why you move from
: right to left when using the standard pencil and paper approach,
: and why carrying works.: There’s nothing wrong with using your own methods to make computation
: easier, but there is a lot of value in understanding the standard
: algorithms and being able to use them as needed.: Jean

Victoria,Can you suggest your own additional ideas or books, websites, or other sources of information that might be helpful when working with a kid who has it all in his head and can’t get it on paper. My 10 year old who is dysgraphic and was diagnosed as GT/LD three years ago was recently retested using the Woodcock Johnson battery. His score on the mathematics aptitude cluster was SS 152 (influenced by a score of 184 on the analysis-synthesis subtest and similarly high scores on the other subtests dealing with fluid reasoning. His SS on math reasoning was 132, broad math was 115, and math skills was 96. On the subtests, calculation was 94, applied problems was 132, and quantitative concepts was 100. He is a kid who uses lots of algorithms of his own design to deal with weak calculating skills. He finds it almost impossible to explain these algorithms or to put anything down on paper. He is working with a wonderful tutor, who uses lots of hands-on materials, including base ten blocks and other math manipulatives so that he will understand the reasons why, for example we “borrow” or “carry” numbers, but we are looking hard for any thing else that will help him get things from his head to the paper. Thanks!Andrea

PASSWORD>aa4um5Lp2CxdUI have heard about his book, but I haven’t read it- maybe I’ll look for it at the library. I noticed a couple of weeks ago that my son was using the same method for mental addition, although I have made no effort to teach him this approach.Jean.: Interesting! Have you ever read Math Magic by Scott Flansburg? He
: developed a whole system for doing mental math left to right. It’s
: a fascinating read.: If so, what did you think?

: Victoria,: Can you suggest your own additional ideas or books, websites, or
: other sources of information that might be helpful when working
: with a kid who has it all in his head and can’t get it on paper.
: My 10 year old who is dysgraphic and was diagnosed as GT/LD three
: years ago was recently retested using the Woodcock Johnson
: battery. His score on the mathematics aptitude cluster was SS 152
: (influenced by a score of 184 on the analysis-synthesis subtest
: and similarly high scores on the other subtests dealing with fluid
: reasoning. His SS on math reasoning was 132, broad math was 115,
: and math skills was 96. On the subtests, calculation was 94,
: applied problems was 132, and quantitative concepts was 100. He is
: a kid who uses lots of algorithms of his own design to deal with
: weak calculating skills. He finds it almost impossible to explain
: these algorithms or to put anything down on paper. He is working
: with a wonderful tutor, who uses lots of hands-on materials,
: including base ten blocks and other math manipulatives so that he
: will understand the reasons why, for example we “borrow”
: or “carry” numbers, but we are looking hard for any
: thing else that will help him get things from his head to the
: paper. Thanks!: AndreaSounds like you have a real winner in tutoring! Certainly stick with what’s working. Written materials are a real problem. I keep looking for new stuff out there, and I keep getting sick to my stomach at the amount of wasted effort going into making things as difficult, confusing, and wasteful as possible. For younger kids, I pick and choose among workbooks. None I really love, but Scholar’s Choice Check and Double-Check math is OK. (Their phonics is outstanding) For older students, I keep looking for newer things but keep falling back out of necessity on my old pre-1955 texts, which I keep photocopying at need. You are welcome to Grade 5 and up photocopies at cost if your tutor would like to try them.It often helps to back up and work the written at a lower level than the oral until the hands catch up with the mind. He might do well to go back and review some Grade 3 or even Grade 2 written work and then build up to where he is in mental arithmetic. Let him know that this is review, just to go back and look things over, and he won’t be stuck in baby stuff forever but will work forward quickly. This can sometimes bring together the missing connections.It’s not so much which book or workbook, but what you do with it. The math stays the same, and the standard algorithms are still the ones you are expected to know. Your wonderful tutor could try working with a decent or fair Grade 3 book with the manipulatives right there on top of the page until the connections are made. Ignore the book’s plan and order and instructions and just use it as a starting point to teach the logic and give you written practice.

Many people on this website speak highly of Saxon math. There is a website, I believe just saxonmath.com I haven’t seen or worked with this personally, but the description of the program given by real parents sounds very good — progressive, concrete, and detailed. If you do decide to use something like this, again you would want to back way down for review (and do explain clearly and repeatedly to the child that this is review, and he isn’t stuck in baby work forever). I would go as low as Grade 2 or if necessary right back to Grade 1, with the assumption that you can review three or four grade levels in one year — you don’t need to do every single problem when reviewing and catching up; you can either do every second question, or “test out” of the sections he has already mastered using chapter reviews as a test. As I suggested above, you want your good manipulatives that he does succeed with right there on the page as you do the work.Another program that I used successfully for primary students was Alpha-Omega. This is a Christian publisher that produces sets of workbooks for individual study. I don’t recommend their other curricula, but the math was quite good when I used it some years ago (always necessary to check if revisions and updates have kept the good in a program). There are ten or twelve workbooks for each grade level, complete with a mastery test in each. These workbooks are very very very detailed and go over every topic several times, with two or three unit self-tests in each book and then the chapter test to be done at the end of the book. Students who are able to move faster may find this tedious, but for kids who need lots of support, it’s there. Again, going back for review and rebuilding foundations and with manipulatives there on the desk, this could work well.

Title of this note is a joke — I have dealt for fifty years with what I now discover is NLD. I have a very, very poor sense of direction, and have to think consciously which hand is right or left. On the other hand, I do three-dimensional calculus and never had any trouble at all with those problems where you have to rotate things mentally — never could make the image sit still anyhow …On a more serious note: It actually makes much more logical sense to start at the (let’s see now, which is it?) left end of a number, the largest digit, when doing arithmetic. If you have five digits of money, hundreds of dollars and some cents, you want the answer accurate in the hundreds of dollars and a small error in the cents is trivial. So when doing mental arithmetic, the majority of use our common sense and life experience and work large to small. We deal with the largest and most significant values first, then make corrections as needed as we work down.So why do the standard written algorithms (except division) work exactly the reverse of our common sense, going right-to-left and small to large? Simply because we get tired of crossing out and changing numbers. Try some time $23.25 - $17.59 both small-large (standard) and large-small (mental).Mentally we work something like this: 17.59 is just a little less than 18. I’ll subtract 18, which is a bit too much, and then add back the too much (aha! another negative minus negative problem. But I digress). 23 dollars - 18 dollars is 5 dollars. I also had 25 cents to start, so I have $5.25. But I took off 18 - 17.59 too much, let’s see, 59 is 41 cents less than the next dollar, so I took off 41 cents too much. So I take the $5.25 I got by taking out $18, and add back the extra .41, and the answer is $23.25 -$17. 59 = $5.66 This answer is correct, and you can do this kind of estimate-and-correct mental math in fractions of a second, and it all makes sense — but try writing that calculation down on paper, and try making up a consistent set of rules that you can use to teach it as an algorithm. This is just too complex, too many steps, too much experience and number sense involved, to teach to a beginner.Now, paper and pencil: what happens if I try to do $23.25 - $17.59 large to small? In the tens, 2 - 1 = 1. OK, now in the ones — oops, 3 -7 won’t work, I need to borrow. Back to the tens, scratch it out, make the 2 into 1 ten and 10 ones. Start over. In the tens, 1 - 1 - 0. In the ones, 13 - 7 = 6. OK, now down to the tenths or dimes. Darnit! 2-5 won’t work. Back up to the tens, scratch it out again … You see why we don’t work this way on paper.Standard algorithm, small to large: $23.25 - $17.59 First hundredths or pennies, 5-9 won’t work, borrow from the tenths or dimes, one dime makes ten pennies, and 15 - 9 =6. Now the tenths or dimes, 1 left after borrowing, 1 - 5 won’t work, borrow from the one dollars, one dollar makes ten dimes, and 11-5 = 6. Now the dollars, 2 left after borrowing, 2 - 7 won’t work, borrow from the ten-dollars, one ten-dollar bill makes ten ones, and 12 - 7 = 5. Last of all the ten-dollars, one left after borrowing, and 1 - 1 = 0. Answer is 6 pennies, 6 dimes, 5 dollars, and zero ten-dollars, or $5.66You see how neat and accurate and systematic the standard algorithm is compared to the others. The price we pay for the neatness and system is that we have to work backwards.Mathematically, all three systems above (and several others) are perfectly correct! The question is NOT mathematical correctness, but understandability and practicality. The standard algorithm is the simplest paper method that humans have yet devised (try roman numeral arithmetic some day for amusement. No translating out of roman system allowed.), it’s neat and quick, it can be explained and taught by regular rules, and it is well-known so other people can work with it too.It’s important for students to learn the standard algorithms for exactly that reason — communication. Math isn’t much use if it isn’t attached to a real-world meaning. And especially in teaching, but also in real-world work (estimating the cost of that tile job) you will be wanting to communicate with others about what the numbers mean. If people only see a random and apparently disconnected bunch of numbers, they won’t trust that estimate, whether in school or in Home Depot.: I don’t really have any experience with this, but I wonder if your
: son has developed his own algortithms for doing computations in
: his head, but has not figured out how the standard algorithms
: work. For example, I always add from left to right when I am doing
: computations in my head, but I do understand why you move from
: right to left when using the standard pencil and paper approach,
: and why carrying works.: There’s nothing wrong with using your own methods to make computation
: easier, but there is a lot of value in understanding the standard
: algorithms and being able to use them as needed.: Jean

PASSWORD>aa4um5Lp2CxdUYep- that’s pretty much the way I do mental arithmetic- a nightmarish tangle of computation if you actually have to explain it step by step. I like your point about the standard algorithms being the best way to communicate mathematically. I also think that working with the standard algorithms (if you actually understand them) helps develop the number sense that makes mental computation easy and reliable- at least that’s what I think I see happening with my sons.As far as algorithms are concerned, have you ever used the “subtractive method” for introducing long division, progressing on to the standard algorithm?Jean

: Many people on this website speak highly of Saxon math. There is a
: website, I believe just saxonmath.com I haven’t seen or worked
: with this personally, but the description of the program given by
: real parents sounds very good — progressive, concrete, and
: detailed. If you do decide to use something like this, again you
: would want to back way down for review (and do explain clearly and
: repeatedly to the child that this is review, and he isn’t stuck in
: baby work forever). I would go as low as Grade 2 or if necessary
: right back to Grade 1, with the assumption that you can review
: three or four grade levels in one year — you don’t need to do
: every single problem when reviewing and catching up; you can
: either do every second question, or “test out” of the
: sections he has already mastered using chapter reviews as a test.
: As I suggested above, you want your good manipulatives that he
: does succeed with right there on the page as you do the work.: Another program that I used successfully for primary students was
: Alpha-Omega. This is a Christian publisher that produces sets of
: workbooks for individual study. I don’t recommend their other
: curricula, but the math was quite good when I used it some years
: ago (always necessary to check if revisions and updates have kept
: the good in a program). There are ten or twelve workbooks for each
: grade level, complete with a mastery test in each. These workbooks
: are very very very detailed and go over every topic several times,
: with two or three unit self-tests in each book and then the
: chapter test to be done at the end of the book. Students who are
: able to move faster may find this tedious, but for kids who need
: lots of support, it’s there. Again, going back for review and
: rebuilding foundations and with manipulatives there on the desk,
: this could work well.

: Yep- that’s pretty much the way I do mental arithmetic- a nightmarish
: tangle of computation if you actually have to explain it step by
: step. I like your point about the standard algorithms being the
: best way to communicate mathematically. I also think that working
: with the standard algorithms (if you actually understand them)
: helps develop the number sense that makes mental computation easy
: and reliable- at least that’s what I think I see happening with my
: sons.: As far as algorithms are concerned, have you ever used the
: “subtractive method” for introducing long division,
: progressing on to the standard algorithm?: JeanYes, the more you work with numbers, and the more organized your view of numbers, the better your number sense gets; and then you CAN use shortcuts and mental tricks successfully. I find too many people trying to teach kids to skip backwards on one foot (forget running) before they can walk, and that is why I am so adamantly against quick tricks as a teaching method.For division, I like to keep in mind reverse multiplication, so I use estimate-and-check as a way to lead into it. The standard algorithm is just an organization of estimate-and-check anyhow.Subtractive division wouldn’t be so bad with really little problems, but a nightmare with larger numbers.Funny story about standard algorithm and division: I was taking a math education class with a nice young instructor who was spouting word-for-word the theories she was learning in her grad-school studies. OK but lacking depth of understanding. Anyway, she was French, but perfectly fluent in English. Things went along reasonably until one day she was telling us how absolutely horrible the English system of long division was and how vastly superior the European system is. The European system involves writing down a very long list of trial estimates and is even more tedious than ours, by the way. Also it leaves the answer hanging off on the side apparently unrelated to the problem. So we were all skeptical. She said she would prove her point by doing a problem on the bosrd. She started writing it out, and the whole class — mostly elementary teachers and only two of us math majors — started yelling “No, no! You have to put the 3 over the 5!” She stopped, absolutely flummoxed, and argued with us about it. Turns out that, after lecturing us for weeks about the base-10 system etc., that she did not even know that the English algorithm for division is strongly based on place value. A moral tale to remind us to know what it is we are criticizing …

PASSWORD>aaM9oFgb1kg3YThis is in response to your trouble with telling your left hand from your right. I’ve always thought it just had to be a memory thing. Guess what? There is a trick. I learned it from one of my second grade LD kids. You won’t believe how easy it is. Hold your hands in front of you with your palms facing down or away from you. Hold your thumb and your pointer fingers in the shape of an L. Your thumb being the bottom line and your pointer finger being the vertical line. The hand that has the L facing the correct way is the left hand. The other is the right. Get it? Left starts with L. Try it. It really works.

: This is in response to your trouble with telling your left hand from
: your right. I’ve always thought it just had to be a memory thing.
: Guess what? There is a trick. I learned it from one of my second
: grade LD kids. You won’t believe how easy it is. Hold your hands
: in front of you with your palms facing down or away from you. Hold
: your thumb and your pointer fingers in the shape of an L. Your
: thumb being the bottom line and your pointer finger being the
: vertical line. The hand that has the L facing the correct way is
: the left hand. The other is the right. Get it? Left starts with L.
: Try it. It really works.Thanks for the thought.Well, actually I do fairly OK if I stop and think for a second about which is right and which is left. I do it by internal body sense — this is my right hand, I know if I check inside my head, so that direction must be right.The trouble with us NLD types is that the “easy” “quick” solutions are, for us, harder than the original problem. Hold out my hands — which way again? Why down instead of up? What fingers do I look at? What am I looking for again. Darnit, which way does an L point anyway?? Give me a little time, and I can get a whole classroom confused.I lost it in physics 3 class; had the right-hand rule almost memorized, and then they came along with the left-hand rule. That was it for physics.

: Most mental arithmetic problems are — pretty much have to be —
: simple direct one-step computations. The algorithms exist for
: times when the problem gets beyond that. Your son has some kind of
: personal visualization and/or counting system that works well for
: him under limited conditions. This is a good thing, not a bad one!
: But he needs to extend that system and continue developing, not
: stall out here.: I personally do a lot of arithmetic in my head, up to and including
: square roots. I use non-standard algorithms for many of these, a
: lot of successive approximations. BUT — and this is a huge
: qualification — I know how and why my algorithms work. I can
: demonstrate and prove each one to you concretely. They aren’t a
: hodgepodge of quick tricks picked up at random.: My experience with “I do it all right in my head, I just can’t
: put it down on paper” students (whom I meet frequently, one
: or two at least in every class) is that they slam into a brick
: wall when you go beyond the introductory stuff and start doing
: real math. Unfortunately, with the help of untrained teachers and
: a spiral curriculum, some people stay on the introduction until
: senior high school or even college. The later you hit the brick
: wall, the harder you hit, and the more bad habits need to be
: untaught.: Your son needs those algorithms. The first and most basic, adding
: with carrying, will definitely be needed when balancing a
: checkbook or calculating a total purchase. Even those of us who
: are good at mental arithmetic rarely add four or more numbers of
: five digits each mentally. (dollars in the hundreds plus cents =
: five digits; a common occurrence)Same for subtracting with
: borrowing. Multiplication and division will be needed for
: everything from doing your taxes to calculating the cost of
: re-tiling your kitchen and bathroom. And so on. Yes, you can punch
: calculator buttons — and accept bizarrely wrong answers if you
: have not developed a feel for numbers.: Work with him on the “why” of the algorithms. The structure
: of the base-10 system is a thing of beauty. Use abacus and/or
: base-10 blocks to help him get a literal grip on what he is doing.
: This should be done in school, and generally is not, so most
: students get an aversion to math because they are taught bizasrre
: and complicated algorithms as magic incantations rather than as
: logical and sensible shorthand for counting and measuring.: If you keep working on the logic behind the rules, you as well as
: your son will discover some fascinating and interesting inner
: workings of math that you never thought of before.Victoria,Sorry - I haven’t been here in awhile. Are you saying that the spiraling program contibutes in some way to the difficulties of using algorithms? If so, his teacher also mentioned the same idea to me. He is using the Everyday Math program in school which you probably already know is very language based(he also has language disabilities)- his teacher mentioned that we just practice more using the algorithms at home because conceptually he understands.I also know that he has trouble paying attention to details and if he proceeds too quickly without *thinking* he will get one of those wrong/right answers (right answer, wrong procedure). He was tested by a neuropyschologist last year and in the report it was stated that higher level math skills were in the superior range or higher and that at times he does not access his reasoning skills to perform simple tasks. It was also stated that reasoning should be used for insruction with him. What does she mean by this - do you have any clue? Any feedback would be great. Thankyou!Terri

: Victoria,: Sorry - I haven’t been here in awhile. Are you saying that the
: spiraling program contibutes in some way to the difficulties of
: using algorithms? If so, his teacher also mentioned the same idea
: to me. He is using the Everyday Math program in school which you
: probably already know is very language based(he also has language
: disabilities)- his teacher mentioned that we just practice more
: using the algorithms at home because conceptually he understands.: I also know that he has trouble paying attention to details and if he
: proceeds too quickly without *thinking* he will get one of those
: wrong/right answers (right answer, wrong procedure). He was tested
: by a neuropyschologist last year and in the report it was stated
: that higher level math skills were in the superior range or higher
: and that at times he does not access his reasoning skills to
: perform simple tasks. It was also stated that reasoning should be
: used for insruction with him. What does she mean by this - do you
: have any clue? Any feedback would be great. Thankyou!: Terri(a) Yes, the spiralling more than contributes but is a main cause for difficulties in using algorithms, problem-solving, fractions, and anything else that requires time and application and real work to learn.In a spiral curriculum, you have only two weeks to teach an entire topic, and then you have to rush on to another topic and another and another, or else as a teacher you’re in *&%^$ with the school board and well-meaning but misinformed parents for not “covering the material”. I keep trying to explain to parents and college students that it doesn’t matter that **I** can cover the material that fast — of course I can. What matters, and what has gotten lost, is whether my students retain anything useful out of this 100-mile-per-hour rush.Given only two weeks, maybe four at most, to cover an entire mathematical topic, say fractions, or say long division, I don’t have time to present any real-life problem that will motivate the students to understand it, I don’t have time to work in detail through the reasoning of the problem, I don’t have time to have the students try to work the problem logically step-by-step with me; all I can do is to present a quick formula to memorize or else, drill and kill, and test fast before they pitch this out of mind to clear the decks of the memory for the next onslaught of meaningless trash.Also in a spiral curriculum, being so rushed, I am sorely tempted to throw out stuff that gives trouble — things that students have trouble memorizing and regurgitating in two weeks, things that bring up long and uncomfortable questions, things that are hard and lead to crying over homework and complaints from parents to my boss; so I ignore or don’t grade or leave to the last week of the summer party all the “hard stuff”, usualy including fractions and problem-solving and frequently long division. (With the grade pressure and inflation of the last few decades, it has become standard practice for teachers to be “nice” and say they will omit the lowest grade on each report card — so any academically challenging topics get trashed — you think the kids don’t know this means they can skip fractions this term, division next, and problems next? The kids actively plan which subjects not to do.)(b) I’m not personally familiar with “Everyday math.” It hasn’t had good reviews on this board, and that’s all I know.(c) Practical advice. Gee, I’ve been snowing this board under with practical advice. Somebody else today posted a nice system for modelling long division using money — check for a new post under “Division”. Can you look at my old posts?Also, I copy some of what I type — if you can tell me what topics you’re looking for, instead of reposting the whole thing I can see if I have a copy and email it to you.General advice: be concrete and applied. Mention real-world places where you use arithmetic, exactly the things he is learning to do. For example, adding with carrying — show him a checkbook or a bank statement. And so on.Use counters, abacus or base-tens blocks and/or money (ten multiples only) for models.Access your son’s good logical and reasoning ability by giving him the models and letting him try to work out a solution his own way; only help when he gets bogged down. Let him take the time now to develop foundations for later.*After* he has a grip on how the concrete real world works, show him how the standard algorithm is a model of what he’s been doing with pennies and dimes and dollars. (as in trading ten pennies to make a dime; that’s all you do in carrying. Dime traded for ten pennies — that’s what you do in borrowing)Ask again for more ideas. Got lots, but tired now.

: (a) Yes, the spiralling more than contributes but is a main cause for
: difficulties in using algorithms, problem-solving, fractions, and
: anything else that requires time and application and real work to
: learn.: In a spiral curriculum, you have only two weeks to teach an entire
: topic, and then you have to rush on to another topic and another
: and another, or else as a teacher you’re in *&%^$ with the
: school board and well-meaning but misinformed parents for not
: “covering the material”. I keep trying to explain to
: parents and college students that it doesn’t matter that **I** can
: cover the material that fast — of course I can. What matters, and
: what has gotten lost, is whether my students retain anything
: useful out of this 100-mile-per-hour rush.: Given only two weeks, maybe four at most, to cover an entire
: mathematical topic, say fractions, or say long division, I don’t
: have time to present any real-life problem that will motivate the
: students to understand it, I don’t have time to work in detail
: through the reasoning of the problem, I don’t have time to have
: the students try to work the problem logically step-by-step with
: me; all I can do is to present a quick formula to memorize or
: else, drill and kill, and test fast before they pitch this out of
: mind to clear the decks of the memory for the next onslaught of
: meaningless trash.: Also in a spiral curriculum, being so rushed, I am sorely tempted to
: throw out stuff that gives trouble — things that students have
: trouble memorizing and regurgitating in two weeks, things that
: bring up long and uncomfortable questions, things that are hard
: and lead to crying over homework and complaints from parents to my
: boss; so I ignore or don’t grade or leave to the last week of the
: summer party all the “hard stuff”, usualy including
: fractions and problem-solving and frequently long division. (With
: the grade pressure and inflation of the last few decades, it has
: become standard practice for teachers to be “nice” and
: say they will omit the lowest grade on each report card — so any
: academically challenging topics get trashed — you think the kids
: don’t know this means they can skip fractions this term, division
: next, and problems next? The kids actively plan which subjects not
: to do.): (b) I’m not personally familiar with “Everyday math.” It
: hasn’t had good reviews on this board, and that’s all I know.: (c) Practical advice. Gee, I’ve been snowing this board under with
: practical advice. Somebody else today posted a nice system for
: modelling long division using money — check for a new post under
: “Division”. Can you look at my old posts?: Also, I copy some of what I type — if you can tell me what topics
: you’re looking for, instead of reposting the whole thing I can see
: if I have a copy and email it to you.: General advice: be concrete and applied. Mention real-world places
: where you use arithmetic, exactly the things he is learning to do.
: For example, adding with carrying — show him a checkbook or a
: bank statement. And so on.: Use counters, abacus or base-tens blocks and/or money (ten multiples
: only) for models.: Access your son’s good logical and reasoning ability by giving him
: the models and letting him try to work out a solution his own way;
: only help when he gets bogged down. Let him take the time now to
: develop foundations for later.: *After* he has a grip on how the concrete real world works, show him
: how the standard algorithm is a model of what he’s been doing with
: pennies and dimes and dollars. (as in trading ten pennies to make
: a dime; that’s all you do in carrying. Dime traded for ten pennies
: — that’s what you do in borrowing): Ask again for more ideas. Got lots, but tired now.Victoria,Thanks so much for all of your input and advice. I will review the other posts that you have replied to. I’ll ask you if I have further questions and appreciate all you have done so far.Thanks - Terri