The Quantum Pontiff

Do We Teach Addition Backwards?

Addition, for me, is intimately connected up with my concept of a number. When I think of numbers in my head, I often think of the number in connection with its constituent parts, and when I divide these parts up into equal pieces I “get” multiplication. However, on top of this bare bones thinking, I also conceptualize numbers strongly by their size, thinking about the number first as the most significant digit in the number and proceeding down to the less significant digits. Which makes me wonder, do we teach addition backwards?

The standard grade school algorithm we are taught for adding numbers starts with the least significant digit and then proceeds to higher significance digits. Thus, for example, to add 123 and 237 we first add 3 and 7, obtaining a 0 and a carry of 1. Then we add 2 and 3 and that carry of 1 to get a 6 with no carry. And then we add 1 and 2 to obtain 3. So our total is 360. But when I was learning to add numbers I didn’t work this way. I worked the other direction. I started with 1 and 2 and add them to get 3. Then I would take 2 and 3 and add it to get 5. Finally I would add 3 and 7 to get a 0, but then I needed to add that carry of 1 back to the 5 and obtain 6. Of course my answer was the same, 360, but I did the addition starting with the numbers that matter the most in the magnitude of the number (1 and 2).

Which makes me wonder if teaching addition starting with the least significant digits first and then moving to the most significant digits doesn’t cause kids to never really grasp how a number represents magnitude or size? By performing the largest magnitude terms first, it seems that you’d get a better idea about the size of the resulting sum, which you then increasingly refine as you go to lower significant digits.

Of course I think teaching both ways of performing addition (as well as any other variation) is probably better than teaching just one way of performing addition. But it would be interesting to see if kids who are taught to add “backwards” have a better grasp about orders of magnitude than those who are taught to add “forward.”

Comments

  1. #1 Coturnix
    December 7, 2007

    Left to right is ho I add in my head. On paper, I go from right to left. Good question – I will think about it…

  2. #2 Eric Lund
    December 7, 2007

    I’d say it’s a question of whether you are emphasizing magnitudes, or the mechanics of the process. If getting the magnitude right is the important thing, then it is correct, as you say, to start from the most significant digit and go to the least. But if you need to deal with the mechanics (say, because you are trying to design a circuit that adds two numbers), then you need to start with the least significant bit (LSB) and work your way up. The reason for the difference is that in the traditional (LSB first) method, the results of subsequent steps have no effect on earlier steps, while in your example the carry from adding the units digits modifies the result of the previous step, adding the tens digits.

    I can see why it makes sense to teach small children (first and second graders) the LSB method, as it is usually done. It’s important at that age to understand the mechanics, and to see how it generalizes to adding an arbitrarily large number of arbitrarily large numbers. But it does cause problems later on when they have to deal with adding numbers that are known to different levels of precision.

  3. #3 RNB
    December 7, 2007

    I agree with the article and using Eric’s acronyms – although a computer program may be simpler as LSB to MSB, for an adult thinking about genuine quantities in the real world it makes more sense to think from MSB to LSB…

    …but I think the more practical issues with adding/comparing numbers occur when they are used to represent date and time. The sensible way to talk about this is with decreasing precision – we don’t show time as secs-hrs-mins so why do we continue with mm-yy-dd or equally convoluted ordering – we should always place date before time and stick with MSB-to-LSB convention.

  4. #4 Wim van Dam
    December 7, 2007

    As a Dutchman it is very important to me that people give me back the right amount of change and not a cent less. Learning how to add and subtract the standard way has served me well.

  5. #5 agnostic
    December 7, 2007

    You’ve never taught math to the average student, I take it! You’ve never spent much time around people with average, or only somewhat above-average IQs. I.e., those who won’t go to college, or only to a very low-level college.

    A lot of “innovative” — destructive — math teaching takes the approach you do: take a math nerd approach and hope that everyone is a budding math nerd. Unfortunately, this screws everyone up who won’t eventually get a near-perfect SAT math score.

    The great thing about the standard algorithm: it works, is very simple to understand, there is no ambiguity or juggling or moving back-and-forth, and it can be committed to memory by rote practice rather than having an on-the-fly aspect to it. It doesn’t get at the concept of importance of orders of magnitude, but it does get at the concept of smaller things adding up to bigger things.

    In the end, though, teaching anything conceptual in math is always the last thing to do, and even then only with very bright students. The sub-brights will not understand what you’re saying at all, and in fact most bright students who aren’t math-inclined will be bored to death (even if they get it).

  6. #6 Davis
    December 7, 2007

    I’ll have to express some assent with agnostic’s comment. Students at the lower levels have a tendency to learn only the mechanics. My experience is that this group has no real awareness of the conceptual backing to a method — even if they’ve been taught the concept, only the memory of the mechanics remains long-term.

  7. #7 JYB
    December 7, 2007

    I generally teach it that way too. The same with multiplication. Left to right so they can know if they’re in the correct order of magnitude.

  8. #8 Dave Bacon
    December 7, 2007

    …I guess if a consistent algorithm is all you need, then teaching someone to use a calculator or computer is more important than teaching addition :)

  9. #9 Markk
    December 7, 2007

    “…I guess if a consistent algorithm is all you need, then teaching someone to use a calculator or computer is more important than teaching addition :)”

    Actually even here – no – there are many real world places where you need to do math in you head if you are to function well. You can get by without, but it bites, it is a disadvantage. I have a niece who always used a calculator and many times lost money, games, etc. She started playing dominoes a lot and actually got her addition in her head better and started to realize all the times she “lost out” before. It was very interesting for the rest of us to have her tell this.

  10. #10 -Q
    December 7, 2007

    I think I’m going to touch on what almost everybody has said.

    I think the LSB first method has the advantage of being simpler for many to learn, and gives a firm grasp on the mechanics. Plus it just seems to save another line of paper when showing all the work, which younger ones would be happier about.

    They should learn both ways. I think learning happens either way. What’s important, is that they actually keep using it. There are too many that just use a calculator for very basic operations, which is really a shame.

  11. #11 Dave Bacon
    December 7, 2007

    I think I’m going to start using more smiley faces to indicate when I’m joking!

  12. #12 Robin Blume-Kohout
    December 7, 2007

    The comment by agnostic really hit a nerve for me — in much the same way that a dentist will sometimes literally hit a nerve while mucking around in my mouth, causing me to twitch both in shock and in pain.

    I found myself reacting viscerally to “teaching anything conceptual in math is always the last thing to do“. This is radically opposed to my strong beliefs about teaching. But… (I thought) maybe agnostic has a point here — am I luxuriating in a viewpoint only available to the privileged intellectual elite, who don’t have to teach kids with IQ=100 in the trenches?

    I guess I don’t know. It’s plausible that teaching math concepts-first to average students would fail… but I cling to the conviction that if it does, it’s not because of their average intelligence, but because they’ve tragically been taught (so far) to be intimidated by concepts. Taught to fear intellectual failure, rather than encouraged to be curious.

    It’s a fact that, in this system, bright kids are the ones that are rewarded for learning and curiosity… and I desperately hope that it’s possible to sustain the curiousity of averagely smart kids. And I still believe that if kids are curious and eager to learn, then concept-first is a good way to teach — regardless of IQ.

    That makes me wonder, though — if you had to choose between teaching kids to successfully but mindlessly do addition, and teaching them to understand the concept of magnitudes without the ability to do it bit-by-bit… which would you confer?

  13. #13 The Ridger
    December 7, 2007

    I have to say – not brag – that I’ve got a pretty decent IQ. Also, that I grasp the adding 6XX plus 3XX will work out to between 9XX and 11XX. But I really don’t see the utility of adding the 6 and 3 first, especially since you would so often have to backtrack and change the answer.

    Now this may simply be because I don’t get math, and maybe I would have had I been taught it differently. But 646 + 375 “6 and 5 is 11 carry 1 and 4 and 7 is 12 carry 1 and 6 and 3 is 10 so 1021″ is much more satisfying than “6 and 3 is 9 and 4 and 7 is 11 so change the 9 to 10 and then 6 and 5 is 11 so 1 and change the 1 to 2 so 1021″. And I’m not sure I could do it in my head without a lot of practice … though of course it took practice to learn to do the other in my head.

    OYMMV…

  14. #14 Emily
    December 7, 2007

    Although it was a couple years removed from when I actually learned addition (1st grade? K?), I remember being in 5th grade and learning to divide and add not just forwards or backwards, but by starting in the middle, etc. Maybe that was just a privilege of the amazing (but low-income) honors program at my elementary school, but I never really thought about it. However, as a consequence, whenever I do any sort of simple math, I don’t even think about the order. I can just look at it and conceptualize which is the fastest way.

    I’m not going to pretend I’m that smart, though. I had to be showed how to do that.

  15. #15 Peter Morgan
    December 7, 2007

    I believe I do a check for whether there are likely to be any carries first. If it looks as if there are not going to be any carries, I add MSB first, if it looks as if there are going to be carries, I add LSB first, taking more care. I think I assess at a glance what method I will use on a given addition.
    I’ve been watching my 8-year-old daughter being taught to group numbers in a column of numbers that is being added. They have been taught to look for convenient ways to add columns so that the carries are less trouble — if there’s a 13 and a 7, make them into a 20, etc. There is also a strong emphasis on estimation at her grade level; adding 13500+12870, is the result near 20000 or near 30000? That indirectly addresses Dave’s worries about the understanding of quantity.
    As something of an aside, we did some additions and subtractions of hours and minutes a few days ago, for which the carrying of 60 minutes from the hours column reduced my daughter to tears. Actually, it was not being able to do it instantly in front of her mathematician father that led to the tears. Being English and old enough to have had to learn how to add columns of Pounds Shillings and Pence (20s and 12s to carry, for the uninitiated) when I was her age, I probably have less sympathy than I should.

  16. #16 Andrew Brummer
    December 8, 2007

    I think agnostics comments highlight a weak point in the standard approach to teaching arithmetic. There are teaching systems that start out teaching the concept of number through concrete physical objects, and young students learn the concepts through manipulating and comparing quantity, length, area, and volume physically before they are every introduced to the concept of numeral.

    Only after the student has demonstrated mastery of manipulating the physical objects are they introduced to the more abstract symbolic manipulation. While that sounds similar to what agnostic said, when you have 163 and 456 composed of 5 hundred squares, 11 ten rods, and 9 units carrying is just reducing the representation to the simplest number of elements and can be thought of either way.

    I really think the “average” students that don’t get it are those that haven’t had the opportunity to manipulate enough objects to develop a real concrete sense of number.

  17. #17 cm
    December 8, 2007

    Due to this post, I just invented a new system for adding that feels much better for me. It’s a little like your system, but it eases memory demand. In your example of 123 + 237, I’d (now) do it like this:

    Step one: just add the digits individually, whatever order you want. Don’t worry about carrying.

    1 2 3
    2 3 7
    ———
    3 5 10

    Step two: Use these rules;

    1) scan for two digit numbers; if found in this new row, remove the 1 and increase the number to its left by 1.
    2) if that number is a 9-18, 9 becomes 0, all others just lose their 1, and the number to its left increases by 1. if it is also 9-18, apply this rule again.
    3) All other numbers just drop down.

    3 5 10 (the two digit number loses its 1 and
    3 6 0 turns the 5 into a 6, then all drop down.

    Here’s another:

    4 8 3
    7 1 9
    ———–
    11 9 12
    ———–
    12 0 2

    and this works quite well for rather large numbers:

    Step one:
    7 5 3 8 3 9 2 1 9
    9 3 5 2 7 5 2 3 4
    ————————————
    16 8 8 10 10 14 4 4 13
    ———————————— Apply the rules…
    16 8 9 1 1 4 4 5 3

    Sure, this is similar to the normal method, but somehow I just found it easier because first, you never have to carry any 1 and thus never add three numbers sometimes and 2 other times. I think carrying the one breaks the mental rhythm when you have either a mental “placeholder” or you have jotted a small 1 above the printed numbers.

    For me it is remarkably faster and far less effortful! It’s almost weird. There are of course other trick methods, such as one Harry Lorayne provides in a book.

  18. #18 cm
    December 8, 2007

    RE: my post above…The HTML formatting sort of ate the columns on my examples, but if you try it on paper it’ll make sense. The rules are far less cumbersome to keep in mind than they may read, especially if you get in the habit of recognizing two digit numbers and then “zapping” the number to its left.

  19. #19 Jonathan Vos Post
    December 8, 2007

    I first taught impovershed teenagers in summer school, some of whom did not “get” addition, the four basic operations: Ambition, Distraction, Uglification, and Derision. This is from mathematician Charles Dodgson,” I told them. Does anyone know him by his pseudonym, Lewis Carroll?”

    Oh, yes,” said one young lady. “He was the freak who molested little girls.”

    That sent the math lesson off in unexpected directions.

    The Jesuit Baltasar Gracian y Morales wrote:

    Keep the extent of your abilities unknown. The wise person does not allow his knowledge and abilities to be sounded to the bottom, if he desires to be honored by all. He allows you to know him but not to comprehend him. No one must know the extent of a wise person’s abilities, lest he be disappointed. No one should ever have an opportunity to fathom him entirely. For guesses and doubts about the extent of his talents arouse more veneration than accurate knowledge of them, be they ever so great.

  20. #20 aram
    December 8, 2007

    I think it’s more important to be able to approximately add than to be able to exactly add.

    As for conceptual math, there’s a book called The Teaching Gap which talks about how in the US children are math skills, while in Japan they’re taught math concepts. This is now part of the culture of teaching (just like using a blackboard at the university level is), so that kids in the US would become confused and dismayed if they started being asked conceptual instead of technical questions. This doesn’t mean that only bright kids are capable of understanding, though, but that a cultural change will be necessary in order for math to be taught in a different way.

    As evidence for this, try asking a kid a somewhat conceptual math question (like “how do you think the car can figure out how many miles we have left to drive?”) and listen to the “I don’t know.” that you hear. Usually it has a sound of finality to it, instead of a “I don’t know – maybe I can figure it out.” sound. This is evidence, I think, that kids are used to being asked questions that they immediately know how to mechanically solve.

  21. #21 Paul
    December 9, 2007

    The MSB order isn’t terrible for adding 2 numbers but it is increasingly awkward when there are many more numbers to add as occurs in the multiplication tableau for large numbers.

    Subtraction is more interesting and there might be something to be said about the MSB-based approach. Even with LSB-based subtraction did you learn borrowing as reducing the top digit or increasing the bottom digit? I learned the former but my parents learned the latter which yields a more online algorithm – no side-trips far left to borrow. (Check out “New Math” by Tom Lehrer.)

  22. #22 Jonathan Vos Post
    December 9, 2007

    “Imagine the following situation. For twelve years you are forced to acquire skills in using some tools that become more and more intricate, complicated and finally utterly unwieldy — like some gardening tools — trowels, shovels, rakes, hoes, all the way to combines. You practice using them in a gym, with a patch of soil about 1 foot by 1 foot in size and about 1 inch deep, and you are NEVER EVER allowed to get to a real garden and use your tools and skills there. Would you love and cherish your tools? Would you strive to learn and perfect your skills? Preposterous as it sounds, this is exactly what our K-12 math education currently is. Kids are forced to learn algorithms and techniques without ever being allowed to apply them to a situation for which these algorithms and techniques were invented. We believe that this is a main cause of the present crisis in math education, and the only way out of it is by opening the doors of a real and beautiful garden and letting our students do their best in cultivating it. This is what problem solving is about…”

    Tatiana Shubin, “The Teacher’s Circle: an AIM initiative”, AIMath. The Newsletter of the American Institute of Mathematics, Autumn 2007, p.8. Avaiable as PDF online.

  23. #23 Sandra Porter
    December 10, 2007

    Welcome to ScienceBlogs!

    It’s time for a meme!

  24. #24 Kaleberg
    December 11, 2007

    Liping Ma wrote an interesting book on the problem of teaching basic arithmetic and contrasted the American and Chinese methods of teaching. (She got into math teaching thanks to Chairman Mao and his Great Cultural Revolution). The Chinese tend to get better results for two reasons. (1) They use the abacus which provides a mechanical, tactile means of learning the algorithms. (2) Chinese math teachers understand the math better, so they can diagnose student problems better and answer questions meaningfully. The abacus algorithms work from LSB to MSB, though I suppose you can work in any direction if you don’t mind slapping around a few extra beads.

  25. #25 Jesoph
    December 12, 2007

    I add faster when I do it L-R. As a child I hated math, probably because I found R-L awkward and slow, but was discouraged by teachers from doing it that way. It just felt more natural to start at the left and add carries later. I am also Left Handed, and wonder how many Lefties(Right Brainers) do it this way.

  26. #26 Kevin
    December 14, 2007

    I’ve been teaching my second grade son:

    A farmer has 1,000 bales of hay and need to sell some to buy grain for the 5 months of winter. He has 5 horses and 10 cows. A horse eats 3 bales a week and a cow eats 2. A bale of hay sells for $2 and a bag of grain costs $60.

    How many bags of grain can the farmer buy?

    Bonus: The clerk at the feed store said grain was 10% off today. How much does the farmer save?

  27. #27 Jonathan Vos Post
    December 14, 2007

    “How many bags of grain can the farmer buy?”

    Depends. How much can he get if he sells some or all of the 5 horses and 10 cows? You didn’t say that he couldn’t. Boundary conditions…

  28. #28 Kevin
    December 15, 2007

    No market for horses or cows.

    have to just shoot them I guess so you can buy lots of grain. or say who cares if they need hay! let them eat grain!

    but no, you can’t sell or kill the animals. There. you are boundaried. and you have to save enough bales to feed them the hay. anything else?

  29. #29 Jonathan Vos Post
    December 15, 2007

    What do horses or cows do with a mixture of hay and grain? With a pile of hay next to a pile of grain?

    I grew up in New York City and in greater Los Angeles. I have little first hand data on horses or cows or farms. To me, this is a hard problem, precisely because of the hidden assumptions that I may be making, and I hypothesize that you may be making.

    Arithmetic is easy. The real world is complicated. Teaching is complicated because of how one endeavors to connect Arithmetic as the teacher knows it to the real world as the student knows it. Your mileage may vary.

  30. #30 KeEvin
    December 15, 2007

    Heh heh. Yeah I grew up in NYC too but I used to go to Vermont where people had horses.

    The grain is measured out in buckets I think like 1/2 bucket per horse per day. The hay is put into the hay crib for them ot eat and onto the floor for them to stand in. It makes cleaning up the poop easier.

    G = bags of grain = S (bales available for sale) * $2 / $60

    S = 1,000 – B

    B = number of bales needed = H #eaten by horses + C #eaten by cows

    H = 1 horse * 5 horses * 3 bales / week * 4 week/month * 5 mths = 300

    C = 1 cow * 10 cows * 2 * 4 * 5 = 400

    B = 300 + 400 = 700
    S = 1,000 – 700 = 300
    G = 300 * $2 / $60 = 10 bags

    Bonus = $60.

    the 10% bous is a clue to the answer. The discount should never be something like $45.75.

  31. #31 Kevin
    December 15, 2007

    Here is another one.

    The class is going on a trip. There are 400 students and ten teachers. Each bus has two rows of ten seats that seat two on each side of the aisle. Each bus needs two school monitors. Parents will fill in if you don’t have enough teachers.

    How many parents need to go on the class trip?

    Bonus: If all the other buses are full, how many people are on the last bus?

  32. #32 Jonathan Vos Post
    December 15, 2007

    Are horses or cows allowed to be school monitors? If they are, and are not yet parents, then no human parents need to go on the class trip.

    I can see that my city-boy childhood makes me ill-equipped for this higher math.

    I’m not sure if horses can be Math teachers, but one does have an Erdos number, as a published coauthor of a Math paper.

  33. #33 Kevin
    December 15, 2007

    I don’t think its your childhood at fault.

    Must be an organic metabolite imbalance….

  34. #34 Jonathan Vos Post
    December 15, 2007

    “The class is going on a trip.” Assuming that they have a good metabolite balance, they usually reason this way…

    “There are 400 students and ten teachers.”

    Thus we have 400 + 10 people in the class, where people = {students} U {teachers}.

    We make the additional assumption that no person is both a student and a teacher, even though, in the real world, I am currently a teacher, but also a student, while I attend (in night classes) a teacher’s college to get additional teaching credentials. That is, {students} and {teachers} are disjoint sets.

    “Each bus has two rows of ten seats that seat two on each side of the aisle.”

    2 rows/bus x 10 seats/row = 20 row-seats/row-bus. Cancel the rows, to get 20 seats/bus.

    “Each bus needs two school monitors.”

    We make the additional assumption that no school monitor is a student.

    We make the additional assumption that students, teachers, and school monitors all need to be sitting one per seat.

    We make the additional assumption that the bus driver has his/her own seat in each bus.

    “Parents will fill in if you don’t have enough teachers.”

    We make the additional assumption that parents can fill in for teachers only in the role of school monitor.

    “How many parents need to go on the class trip?”

    Okay, now. We have 410 {x: x is a student or a teacher} and an unknown number M of school monitors.

    With 20 seats/bus for the 410 teacher-or-students, and 2 school monitors on each bus, we have the following table:

    Bus # Who is on the bus besides driver
    —– ——————————–
    1 18 teacher-or-students + 2 monitors
    2 18 teacher-or-students + 2 monitors
    3 18 teacher-or-students + 2 monitors
    4 18 teacher-or-students + 2 monitors
    5 18 teacher-or-students + 2 monitors
    6 18 teacher-or-students + 2 monitors
    7 18 teacher-or-students + 2 monitors
    8 18 teacher-or-students + 2 monitors
    9 18 teacher-or-students + 2 monitors
    10 18 teacher-or-students + 2 monitors
    11 18 teacher-or-students + 2 monitors
    12 18 teacher-or-students + 2 monitors
    13 18 teacher-or-students + 2 monitors
    14 18 teacher-or-students + 2 monitors
    15 18 teacher-or-students + 2 monitors
    16 18 teacher-or-students + 2 monitors
    17 18 teacher-or-students + 2 monitors
    18 18 teacher-or-students + 2 monitors
    19 18 teacher-or-students + 2 monitors
    20 18 teacher-or-students + 2 monitors
    21 18 teacher-or-students + 2 monitors
    22 18 teacher-or-students + 2 monitors
    ——————————————
    1-22 398 teacher-or-students + 44 monitors

    The last bus follows:
    23 410-398 = 12 teacher-or-students,
    + 2 monitors
    ——————————————
    total 410 teacher-or-students + 46 monitors

    Now, 10 of those 46 monitors are teachers, and the other 46-10 = 36 are teachers.

    “Bonus: If all the other buses are full, how many people are on the last bus?”

    We have 14 non-busdriver people on the last = 23rd bus.

    Different assumptions can give somewhat different answers. Those students who naively divided and implicitly assumed fractional busses get only partial credit.

    This answer deserves partial credit, but is not complete. Different students may have different but acceptable answers. Those who show no equations, but draw the right picture, get at least half credit in my class.

    One of the purposes of the class trip is to observe horses, cows, and mathematicians in their native habitats.

  35. #35 Jonathan Vos Post
    December 15, 2007

    (scrawled in margin):

    I mean 10 of those 46 monitors are teachers, and the other 46-10 = 36 are parents.

  36. #36 Kevin
    December 15, 2007

    “Each bus has two rows of ten seats that seat two on each side of the aisle.”

    2 seats per side. 2 sides per row. ten rows

    40 seats.

    if you only had students, it would take 10 buses. so you need 11.

    that’s 22 monitors

    # parents = 12

    bonus answer = 23

  37. #37 Kevin
    December 15, 2007

    ok here is an old one that I did not make up.

    A man is going to make a square of beads ten on each side. He has $3,000 dollars. Gold beads cost $50 each and silver ones cost $10.

    what is the maximum number of gold beads the man can buy to make the square?

  38. #38 Jonathan Vos Post
    December 15, 2007

    “Each bus has two rows of ten seats that seat two on each side of the aisle.”

    The ambiguity: “that seat two.” That seat two what?

    I explained what comes from the assumption “one person per seat.”

    You are assuming “two people per seat” but not say so explicitly.

    So you get “2 seats per side. 2 sides per row. ten rows. 40 seats.”

    I gave you an example of the type that I routinely get from students who demand that they be given credit for working consistently from their stated assumptions. You didn’t eliminate ambiguities in your testing instrument. So how do you handle students who give examples such as mine?

  39. #39 Jonathan Vos Post
    December 15, 2007

    0000000000
    0…………….0
    0…………….0
    0…………….0
    0…………….0
    0…………….0
    0…………….0
    0…………….0
    0…………….0
    0000000000

    Each of the “0″ is a silver bead that costs $10. There are 40 beads making the edges of the hollow square. So that costs $400.

    Of his $3,000 budget the man spends $400 as shown for silver beads, gives that to his client saying that he met the requirements. From the remaining $3,000-$400 = $2,600 budget he buys 52 gold beads at $50 each, which he keeps as profit. If audited, he shows the invoice from the jeweler which shows $3,000 for beads” and survives the audit. It is clear that he spent the $3,000 and gave the client something meeting specifications.

    At least that’s how I learned how to do things for NASA and the Pentagon, often with gold-plated components.

  40. #40 Kevin
    December 16, 2007

    “So how do you handle students who give examples such as mine?” examples? what? I asked for an answer. Your answer was wrong. you fail.

    number of beads = 10 * 10 = 100

    G = gold beads and S = silver beads, G + S = 100

    G * $50 + S * $10 = $3,000.

    (S-100) * $50 + 10S = $3,000

    etc

  41. #41 Jonathan Vos Post
    December 16, 2007

    Kevin, you’re missing my point again.

    “a square of beads” is ambiguous. It could mean either of two things.

    The term “square” can be used to mean either a square number (“x^2 is the square of x”) or a geometric figure consisting of a convex quadrilateral with sides of equal length that are positioned at right angles to each other as illustrated above. In other words, a square is a regular polygon with four sides.

    I gave a consistent interpretation stemming from the convex quadrilateral with four right angles. You had the meaning 10^2, the area of my illustration:

    0000000000
    0000000000
    0000000000
    0000000000
    0000000000
    0000000000
    0000000000
    0000000000
    0000000000
    0000000000

    To be an effective Math teacher, one has to be extraordinarily clear. One must eliminate such ambiguities in lecture, homework assignments, and exams.

    One must be open to valid interpretations by students that can result from your ambiguities, or those in the textbook.

    With all due respect, the problems that you gave would have been trivial to me back when I mastered Calculus at age 13, received a perfect score on the Math Level II AP exam, and was accepted to Caltech on full scholarship at the age of 16. They would likewise have been trivial to my son, the best of all my students so far, who started fullt-time at university at age 13, and grdauated by 18 with a double B.S. in Mathematics and Computer Science.

    I’m not sure you understand how trivial your problems are, or how poorly you presented them. It was not hard to find loopholes in your ill-posed problems.

    After you’ve had a few thousand students, as I have, and half a dozen semesters of teaching at college and university level, and written a few dozen refereed publications, where referees point out ambiguities in presentation that force rewrite, then you might come to realize why I have to fail you in this public test of your teacher training.

    Your failure comes from your naive attitude that “I asked for an answer. Your answer was wrong. you fail.”

    Some problems have more than one correct answer. Your problems had multiple answers. You were unable to understand that in advance, or after alternative solutions were shown to you. You were unable to apply an effective rubric to evaluate a student’s answers, and determine whether their problem-solving methodology, algorithmic capabilty, and numerical or geometric insight was appropriate.

    Don’t give up, though. You are not stupid. You are merely ignorant of modern educational theory and practice. Ignorance can be corrected through proper instruction.

    Mit der Dummheit kaempfen Goetter selbst vergebens. (Against stupidity the gods themselves struggle in vain.)
    [Friedrich von Schiller]

  42. #42 Kevin
    December 16, 2007

    blah blah blah….

    from someone who can’t multiply 2x2x10

    oh well…

  43. #43 Jonathan Vos Post
    December 16, 2007

    Kevin: with all due respect, I believe that insufficiently experienced people, limited in their advanced education, lacking in the flexibility to figure out the mental and emotional functions of others, rigid in their thinking, imprecise in their statements, limited to convergent thinking, dismissive of divergent thinking, who believe that every problem (however sloppily phrased) has one and only one right answer — are themselves a big part of the failure of American primary and secondary education, be they teachers, staff, or administrators.

    Does anyone here, including Dave Bacon, significantly agree or disagree?

  44. #44 Jonathan Vos Post
    December 17, 2007

    For a deeper look on how Mathematics (including addition R-to-L or L-to-R) and how it relates to the physical world, see the PDF by Minhyong Kim entitled “Mathematical Vistas” or see a hotlink to it embedded in a related discussion “Is Mathematics Special?” at the n-Category Cafe’
    http://golem.ph.utexas.edu/category/2007/10/this_weeks_finds_in_mathematic_18.html#c013828

  45. #45 Andrew
    December 20, 2007

    Okay, so I’m like a week late, but uhm.

    Kevin: “Bonus: The clerk at the feed store said grain was 10% off today. How much does the farmer save?”

    10%, obviously. ;) (Parents, don’t let your children grow up to be economists)

  46. #46 Jonathan Vos Post
    December 20, 2007

    “10% off today”

    My wife, who generally looks as if she know’s what’s going on, possibly as a side-effect of being a Physics professor — was asked a question at a fabric store.

    ‘”This fabric costs $2 per yard, but is 50% off. How much is it?”

    My wife tried several times to explain that 50% off means half price, and that half of $2 per yard is $1 per yard.

    The customer did not get it.

    I’ll also point out that one result of the failed American public school system is the several times that a cashier has been unable to give me change, not knowing how to count, and/or how to subtract, and/or how to add. The excuses given to me include:

    “I can’t figure out the change because the cash register is down.”

  47. #47 mrG
    December 27, 2007

    Samuel Weiser, I think, in the early 1980′s or perhaps late 70′s, published a book called “Vedic Mathematics” which I no longer own, but I do recall that it outlined methods for addition and subtraction precisely as you and the commenters state here, and also did similar exercises for multiplication and division. The author claimed this was the standard practice for teaching math in India and wondered if it might have something to do with the disproportionate number of math savants we see who are Hindi.

    Whether that is true or not, I don’t know, but I did create an early C-language library for arbitrary-precision math based on the methods in this book and my library was used for several years in the Linux community as a high-speed alternative to the very slow and limited fixed-point methods. Whether my library was faster than any other integer-based method, I don’t know, all I can say is that it worked for what I needed for my early PC-based 3D graphics apps.

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