# New Math, New New Math....WTF???? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [August 31, 2010, 9:00pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182 "2010-08-31T21:00:24Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 5, 2010, 7:23pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/41 "2010-09-05T19:23:01Z")
As a professional mathematician, I say… who gives a shit whether your kids do or don’t remember whatever standard traditional algorithm for how to do long division with pen and paper, or decimal expansions of square roots, or what have you? We live in a world with calculators and computers. **LHoD** ’s approach emphasizing understanding, while also presenting the algorithm in this context without making its unthinking memorization and application the goal, sounds just fine to me. Of course, I’m not a professional teacher of young children (but he is). ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [September 6, 2010, 2:41am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/42 "2010-09-06T02:41:20Z")
> [@needscoffee](#): > > Math is one subject where oftentimes the full understanding sinks in after the algorithms are mastered, through the process of doing the algorithm. Oh, really? That’s not remotely my experience. And I’ve tried it out: I’ve asked adults to explain to me why the traditional algorithm for 2-digit multiplication works, and very, very few adults can do it, even if they can manage to do the algorithm correctly (in college, I designed a 2-week unit that would build students’ understanding of this algorithm, precisely because at the time I couldn’t possibly explain why it worked). Is your understanding of this fourth-grade algorithm good enough that you can give an explanation without having to think it through very carefully first? So that’s why, if I taught fourth-grade, I’d use an array model to explore multiplication, dividing each side of the array into tens and ones and showing the four sections thereby created, giving kids a visual image of how 2-digit multiplication works. As **Indistinguishable** says, nobody does calculations on paper anymore: that’s what your phone is for. What you need to know is how to set up the equation, and in order to do that accurately, you need to understand the underlying math. But even then, I _do_ teach algorithms. (I should’ve said before now that I don’t like absolutists on either side of the argument, I think they’re dumb). I just teach them as the end of the process, and I don’t let students use an algorithm they can’t explain. And I make it very clear to students that there are multiple ways to solve a problem. It’s funny to me that you dismiss the sentence you bolded: what I was showing you there is that I teach kids to go _beyond_ the algorithm. If neither of us has a calculator on us, and if you’re not Rain Man, I can probably do most problems using flexible strategies than you can using the traditional algorithm, because I’ve practiced finding and using shortcuts to solve problems, and I can do it because I can hold multiple numbers in my head at once, and because I know several go-to strategies. I’m no math genius; it’s just a skill like solving crossword puzzles. If your kids need math tutors, there are several possible reasons why, but it just strikes me as intellectual laziness to blame it on the pedagogy itself. ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [September 6, 2010, 6:01am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/43 "2010-09-06T06:01:26Z")
> [@Left\_Hand\_of\_Dorkness](#): > > > [@needscoffee](#): > > > > Math is one subject where oftentimes the full understanding sinks in after the algorithms are mastered, through the process of doing the algorithm. > > Oh, really? That’s not remotely my experience. And I’ve tried it out: I’ve asked adults to explain to me why the traditional algorithm for 2-digit multiplication works, and very, very few adults can do it, even if they can manage to do the algorithm correctly (in college, I designed a 2-week unit that would build students’ understanding of this algorithm, precisely because at the time I couldn’t possibly explain why it worked). Is your understanding of this fourth-grade algorithm good enough that you can give an explanation without having to think it through very carefully first? Yes. > [@](#): > > So that’s why, if I taught fourth-grade, I’d use an array model to explore multiplication, dividing each side of the array into tens and ones and showing the four sections thereby created, giving kids a visual image of how 2-digit multiplication works. > > As **Indistinguishable** says, nobody does calculations on paper anymore: that’s what your phone is for. What you need to know is how to set up the equation, and in order to do that accurately, you need to understand the underlying math. And if you forget your phone? Or your high school math teacher doesn’t let you use your calculator? > [@](#): > > But even then, I _do_ teach algorithms. (I should’ve said before now that I don’t like absolutists on either side of the argument, I think they’re dumb). I just teach them as the end of the process, and I don’t let students use an algorithm they can’t explain. And I make it very clear to students that there are multiple ways to solve a problem. > > It’s funny to me that you dismiss the sentence you bolded: what I was showing you there is that I teach kids to go _beyond_ the algorithm. If neither of us has a calculator on us, and if you’re not Rain Man, I can probably do most problems using flexible strategies than you can using the traditional algorithm, because I’ve practiced finding and using shortcuts to solve problems, and I can do it because I can hold multiple numbers in my head at once, and because I know several go-to strategies. I’m no math genius; it’s just a skill like solving crossword puzzles. > > If your kids need math tutors, there are several possible reasons why, but it just strikes me as intellectual laziness to blame it on the pedagogy itself. Yes, clearly the reason that the US is trailing behind the rest of the world in math education is because of my intellectual laziness. In the state of Washington, parents and higher educators have begun a group/website called [WheresTheMath.org](http://WheresTheMath.org) to fight the horrible math education in the state. Maybe it’s better where you teach. The biggest problem seems to be the switch over to discovery-based math. This is from University of Washington’s professor of atmospheric science Cliff Mass, [from Where’sTheMath.org](http://www.wheresthemath.com/Pages/Where%27s%20The%20Math.aspx): > [@](#): > > \*\*How Good are UW Students in Math? by Cliff Mass > > As many of you know, I have a strong interest in K-12 math education, motivated by the declining math skills of entering UW freshmen and the poor math educations given to my own children. Last quarter I taught Atmospheric Sciences 101, a large lecture class with a mix of students, and gave them a math diagnostic test as I have done in the past. > > The results were stunning, in a very depressing way. This was an easy test, including elementary and middle school math problems. And these are students attending a science class at the State’s flagship university–these should be the creme of the crop of our high school graduates with high GPAs. And yet most of them can’t do essential basic math–operations needed for even the most essential problem solving. > > Consider these embarrassing statistics from the exam: The overall grade was 58% > > 43% did not know the formula for the area of a circle > 86% could not do a simple algebra problem (problem 4b) > 75% could not do a simple scientific notation problem (1e) > 52% could not deal with a negative exponent (2 to the -2) > 43% could not do simple long division problem with no remainder! > 47% did not know what a cosine was. > > I could go on, but you get the message. If many of our state’s best students are mathematically illiterate, as shown by this exam, can you imagine what is happening to the others–those going to community college or no college at all? > \*\* There’s more on his excellent blog: [Cliff Mass Weather Blog: How Good Are UW Students in Math?](http://cliffmass.blogspot.com/2010/01/how-good-are-uw-students-in-math.html) , including: > [@](#): > > Math remediation rates have soared at community colleges to the 50% level. The math tutoring industry has exploded (over 300% in the past decade and a half). and from the math, science and engineering faculty: > [@](#): > > **Public Statement by University of Washington Faculty on Math Preparation of Incoming Students.** > We the undersigned faculty in math, science and engineering at the University of Washington have become increasingly concerned about the declining level of math competency of students entering the university. Many students arrive with poor mastery of essential mathematical skills, such as algebra, manipulation of fractions, trigonometry, and basic mathematical operations. Increasing numbers of students are forced to take math remediation courses after admission to the UW. Over the past decade many of us have lowered the mathematical levels of our courses as math skills have declined. We believe that it is essential that steps be taken to ensure that Washington State students are provided with world-class mathematics standards, curricula, and instruction. > > This statement has been signed by nearly 300 faculty members at the UW. [Here’s another thread](http://boards.straightdope.com/sdmb/showthread.php?t=539249&highlight=math) a little while back where this was discussed as well, starting with post #18. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 6, 2010, 6:05am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/44 "2010-09-06T06:05:50Z")
> [@](#): > > My kids were barely taught adding columns of numbers, never taught how to multiply two 2-digit numbers on paper, and never taught how to do long division, because all that was ever emphasized was the blocks and the strips. My district’s parents are frustrated and the elementary school learning specialist is frustrated and the high school math teachers are dumbfounded at how little math the students are able to actually do; the local colleges have to teach remedial math, and one of the biggest growth industries here is math tutoring. A large part of the problem here is the assumption that “knowing math” means “knowing how to do long division”, or the like. There’s no connection whatsoever between them. Now, understanding why the long division algorithm works, that’s math. The student who has never seen the long division algorithm, but who, on seeing it, can figure out how it works, knows far more math than the student who can turn the crank to do long division, but has no clue why. Most students won’t have any math classes at all until at least high school. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 6, 2010, 6:43am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/45 "2010-09-06T06:43:59Z")
> [@needscoffee](#): > > And if you forget your phone? Then you get one from someone else. The “What if you’re stranded in the woods without access to technology and need to carry out lengthy arithmetic calculations but never managed to learn the traditional algorithms? What then, smart guy?” challenge isn’t very realistic. I’m sure you survive just fine resorting to the convenience of calculators for logarithms, sines, nth roots, and all the other such, and if you were honest with yourself, you’d recognize that everyone gets along just as fine doing the same for long division, nontrivial multiplications, and even additions of any burdensome length. The only point there is to teaching kids arithmetic algorithms in the modern age is in order to be able to explain how those algorithms work, as a launching point for discussions of more general mathematics. But if one’s goal is merely to have people possess the means to obtain the answers to numerical queries, well, that’s why we invented the calculator in the first fucking place. Show kids how to use one, and the ubiquity of their availability (standalone pocket calculators, pre-bundled OS calculators (Windows, Mac, Linux, what have you), Google, iPods, cellphones, …), and they’ll be fine. Really. Truly. Restaurants don’t care whether you use your phone or your pencil to determine the tip. > [@](#): > > Or your high school math teacher doesn’t let you use your calculator? Any skill can artificially be made to appear useful by contriving to test for it specifically. But why shouldn’t your high school math teacher let you use a calculator for tedious arithmetic? A student who has never learnt to use a calculator is far more ignorant than one whose only sin is recognizing the convenience it offers. ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [September 6, 2010, 6:53am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/46 "2010-09-06T06:53:15Z")
> [@Chronos](#): > > A large part of the problem here is the assumption that “knowing math” means “knowing how to do long division”, or the like. There’s no connection whatsoever between them. Now, understanding why the long division algorithm works, that’s math. The student who has never seen the long division algorithm, but who, on seeing it, can figure out how it works, knows far more math than the student who can turn the crank to do long division, but has no clue why. It’s pretty hard to demonstrate understanding of the long division algorithm without being able to carry it out. The student who turns the crank and doesn’t understand why is far more likely to be holding a calculator than doing it on paper. (And again, good luck if you forget your calculator.) Obviously, though, lack of long division isn’t all of what’s holding students back in math; it’s just one small element. > [@](#): > > Most students won’t have any math classes at all until at least high school. ??? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 6, 2010, 6:55am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/47 "2010-09-06T06:55:40Z")
> [@needscoffee](#): > > ??? Learning how to mechanically follow instructions is of course a basic life skill and has, in many contexts, gone by the name “math”, but it has very little to do with the work of mathematicians. **Chronos** ’s point is that classes along these lines, which are for many students the only classes they take by the name “math” until high school, are no more teaching math than classes focused on the rules of staff notation without any actual singing or playing of instruments would be teaching music. ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [September 6, 2010, 7:01am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/48 "2010-09-06T07:01:43Z")
> [@Indistinguishable](#): > > > [@needscoffee](#): > > > > And if you forget your phone? > > Then you get one from someone else. The “What if you’re stranded in the woods without access to technology and need to carry out lengthy arithmetic calculations but never managed to learn the traditional algorithms? What then, smart guy?” challenge isn’t very realistic. I’m sure you survive just fine resorting to the convenience of calculators for logarithms, sines, nth roots, and all the other such, and if you were honest with yourself, you’d recognize that everyone gets along just as fine doing the same for long division, nontrivial multiplications, and even additions of any burdensome length. The only point there is to teaching kids arithmetic algorithms in the modern age is in order to be able to explain how those algorithms work, as a launching point for discussions of more general mathematics. But if one’s goal is merely to have people possess the means to obtain the answers to numerical queries, well, that’s why we invented the calculator in the first fucking place. Show kids how to use one, and the ubiquity of their availability (standalone pocket calculators, pre-bundled OS calculators (Windows, Mac, Linux, what have you), Google, iPods, cellphones, …), and they’ll be fine. Really. Truly. Restaurants don’t care whether you use your phone or your pencil to determine the tip. The point it that you have to make sure they can understand it without the calculator first. > [@](#): > > Or your high school math teacher doesn’t let you use your calculator? > [@](#): > > Any skill can artificially be made to appear useful by contriving to test for it specifically. But why shouldn’t your high school math teacher let you use a calculator for tedious arithmetic? A student who has never learnt to use a calculator is far more ignorant than one whose only sin is recognizing the convenience it offers. I’ll have my daughter tell her algebra teacher this when she doesn’t allow her to use her calculator on the test. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 6, 2010, 7:24am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/49 "2010-09-06T07:24:35Z")
> [@needscoffee](#): > > The point it that you have to make sure they can understand it without the calculator first. I’m all for understanding. So is **LHoD**. And **Chronos**. Everyone here is pro-understanding. But let me ask you a question… do you understand the method your calculator uses to perform division? Do you even know what it is? It’s almost certainly not the long division you were taught in school, for example. Is that a problem? Does it matter? If a child with no particular interest in numerical analysis or algorithm design should happen to grow up to know what division is (“A divided by B is the number which you multiply by B to get A”), how to reason about its properties, and how to use a calculator to carry out decimal expansion of its results, but not be practiced or proficient at carrying out such decimal expansion by hand, what harm will they suffer? Now to be able to see that this operation on integers generalizes to that of splitting polynomials into a quotient and a remainder modulo a given divisor, to realize how to mechanically produce canonical representations of such, and to appreciate that such an algorithm is the same thing as “long division”, well, that’s a fine level of understanding to reach (albeit it’s _still_ the case that the second step of those three is far outweighed in significance by the others). But I reckon less than a hundredth of the people who are taught long division ever realized or were shown such a way of looking at it, and for those among that majority who can still remember the algorithm anyway, there’s not terribly much difference between the lives they’ll have with their unused and unnecessary ability to reckon with pen, paper, sliderule, and abacus, and the lives of their more forgetful peers… ---
**Author:** ![needscoffee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/needscoffee/32/1076_2.png) [@needscoffee](https://boards.straightdope.com/u/needscoffee)\ **Post date:** [September 6, 2010, 7:55am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/50 "2010-09-06T07:55:02Z")
I have no idea why you are nitpicking long division. The topic here is whether “new new math” is serving our students well. The evidence indicates that it’s turning out a generation of math illiterates. Apparently, demonstrating math concepts to kids and then throwing them a calculator isn’t working. In my district, they aren’t even being made to memorize math tables because they can just use a calculator if they need to know what 4x6 is. Again, [here is the blog by the UW professor](http://cliffmass.blogspot.com/2010/01/how-good-are-uw-students-in-math.html) where he describes the sad state of the math education of his incoming students. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 6, 2010, 8:04am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/51 "2010-09-06T08:04:26Z")
The way I see it, calculators are fantastic tools for learning, not obstacles to it. What a wonderful era we live in, to have them available to us. As an anecdote from a slightly older level, my TI-89 essentially _was_ my calculus teacher, and its manual my textbook, and I seem to have turned out alright. Having the ability to ask it questions of my choosing didn’t spoil my mind and stunt my mathematical growth; it opened doors while I was still young and curious enough to care to explore them. Calculators are the infinitely generous sherpas of an entire second universe, and by lessening the burden of making one’s way about this unfamiliar territory, they provide the sufficient leisure to appreciate its sights. We should be embracing these marvelous tools, not masochistically shunning them with the equivalent of grunts about hard work and character and paradoxical hills. That anyone, decades into the aftermath of the computer revolution, can still feel differently is frankly astonishing to me. And if the cost should be that young children fail to memorize the product of 4 and 6 prior to such experience as would naturally set it in? Very well; I haven’t memorized the product of 44 and 66. If a student doesn’t realize how to turn 4 \* 6 into 6 + 6 + 6 + 6, then I’ll be worried. ---
**Author:** ![Tabby\_Cat](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Tabby\_Cat](https://boards.straightdope.com/u/Tabby_Cat)\ **Post date:** [September 6, 2010, 9:24am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/52 "2010-09-06T09:24:17Z")
Perhaps the problem is that some children simply don’t understand math, and simply showing them cubes and strips doesn’t teach them anything at all, except how to memorise cubes and strips instead of an algorithm. And the algorithm is much more useful in calculating your change than cubes and strips. I wish more people understood how to turn 4 \* 6 to 6 + 6 + 6 + 6. But most people won’t ever need to. They can still get the answer, but with a complete lack of understanding about the fundamental nature of multiplication. I think the answer is “that’s okay, for most people. After all, that’s why we have mathmaticians”. Personally, as a product of the Singapore system, I say drill the algorithm and the formulae. Those interested in math will get into it at some higher level. I actually learned my math by programming. Before I ever did algebra, I was typing x = x + 6 in my BASIC compiler. When I finally got to algebra, it took me a while before I adjusted to having x = 6 and learning that x = x + 6 was a nonsense statement in math. ---
**Author:** ![Mk\_VII](https://avatars.discourse-cdn.com/v4/letter/m/5f9b8f/32.png) [@Mk\_VII](https://boards.straightdope.com/u/Mk_VII)\ **Post date:** [September 6, 2010, 11:06am UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/53 "2010-09-06T11:06:59Z")
Someone who can’t do everyday mathematical calculations with just a pencil and paper isn’t being educated properly. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 6, 2010, 3:15pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/54 "2010-09-06T15:15:29Z")
> [@Tabby\_Cat](#): > > I wish more people understood how to turn 4 \* 6 to 6 + 6 + 6 + 6. But most people won’t ever need to. They can still get the answer, but with a complete lack of understanding about the fundamental nature of multiplication. I think the answer is “that’s okay, for most people. After all, that’s why we have mathmaticians”. What is the value of knowing by memory that 4 \* 6 = 24 without even the most rudimentary understanding of the meaning of this statement? How can this possibly be of any use to anyone? That having been said, I think, whatever the numerous failings of our standard mathematics education, that most people _do_ understand the equivalence between 4 \* 6 and 6 + 6 + 6 + 6, but perhaps I am naive. (On another note, your having learnt math by programming is, I think, a fine example of what I am talking about as far as the pedagogical value of calculators/computers in mathematics.) ---
**Author:** ![davidm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/davidm/32/225_2.png) [@davidm](https://boards.straightdope.com/u/davidm)\ **Post date:** [September 6, 2010, 3:29pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/55 "2010-09-06T15:29:23Z")
> [@Left\_Hand\_of\_Dorkness](#): > > My current curriculum uses some pieces from new math. My basic take on it is that there are two competing schools of thought–traditionalists and constructivists–and the folks who ascribe to both theories are ideological idiots. You gotta combine both approaches, just like you gotta combine phonics teaching with reading whole books. > > So I teach (following the fairly good curriculum) the addition facts, such that by the end of the year my second-graders should know everything from 0+0 to 10+10 instantly. But I don’t teach them all at once. For example, tomorrow we’ll start learning the facts with a sum of 10, and in doing so we’ll examine the commutative property. We’ll use 2x5 tables in this discussion, make it clear that the table has 10 cells, and put dots in some cells. How many cells have dots? (This also helps kids decompose numbers, seeing, for example, that 7=5+2, a really useful skill for flexible addition strategies). How many cells don’t have dots? If 7 cells have dots and 3 don’t, what do you need to add to 7 to get 10? > > That kind of thing. > > Next we’ll work on facts that add 1 to a number, then facts that add 2 to a number, then ones that are doubles, then ones that add 10 to a number, then ones that are doubles-plus-one, then ones that add 9 to a number–and when we’ve done all those clusters, we’ll have something like 16 facts left in 8 pairs, instead of having 121 facts to learn. Throughout the process we’ll use games and manipulatives (I have a fun little magic trick I use to help kids visualize doubling a number). > > One of the big controversies in math education is whether kids can use algorithms. Quick, y’all: add these numbers > > 347 > +538 > > Most of y’all are adding the 7 and 8, carrying the 1, adding 1+4+3, adding 3+5, and arranging the digits from right to left, yes? And that works fine. Plenty of folks when challenged on this can’t explain why it works, however. And plenty of kids who learn this method screw it up: they’ll write 8715 as the answer, or 876, or 985. And they’ll have a reason for each of these specific answers related to an imperfect understanding of the algorithm. (Bonus points: figure out what misunderstanding leads to each of these answers). > > The constructivists don’t want me to teach the algorithm. The traditionalists want me only to teach the algorithm. I think they’re both kind of idiots. I model the equation for the kids, and I pound place value into their heads over the entire year, and I force them to build the equation using base-10 blocks (a 1 block is a tiny cube, a 10-strip is 10 1-blocks put together, a 100-flat is 10 10-strips put together, and a 1000 cube is 10 100-flats put together). I give them my patented “Build a Borg” game to play. I compare place value to pennies, dimes, and dollar bills, and remind them how you could make equal trades between these money amounts. > > And when they understand addition, including regrouping (or borrowing, or trading, or carrying, or whatever you want to call it) from the ground up–when they can explain how the system works–then I’ll show them the algorithm. I don’t emphasize it, and indeed I explain that it’s often not the fastest way to solve a problem: you can solve 495+342 much faster if you do 342+500-5, for example. But I let 'em use it if they can explain why it works. > > You gotta combine approaches. If I’m doing it in my head I tend to do it the opposite of how you describe it (starting with the 7 and 8, carrying the 1, etc). I go left to right, starting at the most significant, (the 3 and 5 in this case), then moving on to the next, etc., adding any carries to the previous result (next highest decimal position) retroactively as needed. For whatever reason this works much better for me than going right to left, where I have difficulty keeping the intermediate results in my head for some reason. I don’t think I was ever taught this way, it just naturally came to me. I doubt that it would have if I hadn’t had an understanding of decimal places, etc. I’ve never heard of the “blocks and strips” stuff. Is that considered “new math”? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 6, 2010, 4:01pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/56 "2010-09-06T16:01:32Z")
Put it this way: Do you know how to use the algorithm for finding logarithms by hand? I’m betting the answer is no. Does it bother you that you don’t, or do you just get a calculator every time you need to know a logarithm? Is the log algorithm really any less important than the long-division algorithm? Personally, I don’t know the algorithm for finding logs, either, but I understand logs well enough that, if I were ever stranded on a desert island without a calculator and needed to do logarithms, I could work out a method for doing them. Quoth **Indistinguishable** : > [@](#): > > As an anecdote from a slightly older level, my TI-89 essentially was my calculus teacher, and its manual my textbook, and I seem to have turned out alright. Reminds me of the time in one of my high school math classes, when I asked my teacher “What’s the significance of the number 2.71828? My calculator appears to have a button for calculating logs to that base.”. ---
**Author:** ![Gagundathar](https://avatars.discourse-cdn.com/v4/letter/g/a183cd/32.png) [@Gagundathar](https://boards.straightdope.com/u/Gagundathar)\ **Post date:** [September 6, 2010, 4:54pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/57 "2010-09-06T16:54:47Z")
> [@Left\_Hand\_of\_Dorkness](#): > > … > Today I showed the kids some 10-frames–2x5 grids with dots in some of the spaces. The idea was for them to ascertain the number of dots very quickly. When I showed 7 dots, someone saw 4 dots above 3 dots: awesome. Someone saw a column of 5 (which they automatically knew, since it filled one complete column of the grid) plus 2. Keen. > > Then one girl explained her method: she saw a column of 5, plus a column of 5 missing three, resulting in 5 + (5-3)=7. Kind of similar to what you did, JWK. This jumped out at me while I read this thread. I believe it has been fairly well established that humans group objects in no more than 5 units when they are visually breaking down a large set of objects. That means that if a human sees 8 units, they see it as 5 + 3 (or 4 + 4 in some cases). This is done on a primitive level and may explain why some numerical methodologies are easier to grasp for a larger percentage of humans. I really should get a cite for this… Ah… there we go: Corbetta, M., Shulman, G.L., Miezin, F.M., & Petersen, S.E. (1995). “Superior parietal cortex activation during spatial attention shifts and visual feature conjunction”. Science 270 (5237): 802–805. doi:10.1126/science.270.5237.802. PMID 7481770 And, I was wrong. Apparently the operation is called ‘subitizing’ and applies to natural groupings of 1 to 4 units. Not 5 as I said above. Hey! I learned something new today! ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [September 6, 2010, 4:56pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/58 "2010-09-06T16:56:01Z")
> [@davidm](#): > > If I’m doing it in my head I tend to do it the opposite of how you describe it (starting with the 7 and 8, carrying the 1, etc). > > I go left to right, starting at the most significant, (the 3 and 5 in this case), then moving on to the next, etc., adding any carries to the previous result (next highest decimal position) retroactively as needed. For whatever reason this works much better for me than going right to left, where I have difficulty keeping the intermediate results in my head for some reason. > > I don’t think I was ever taught this way, it just naturally came to me. I doubt that it would have if I hadn’t had an understanding of decimal places, etc. > > I’ve never heard of the “blocks and strips” stuff. Is that considered “new math”? I’m addressing current pedagogical theory (and also my own approach, which is somewhere between the extremes, as I think is the approach of most actual teachers, good or bad). Under current theory, your approach is absolutely fine: we encourage kids to come up with a method, or better yet methods, that make sense to them and that they can prove is true. The use of blocks and strips and such are just good tools for showing kids why it works. Very few kids understand why you’d want to carry a 1, but if you show them that what you’ve really done is put 7 and 8 blocks together to get 15, and that you can trade 10 of those blocks in for a 10-strip, which you then group with the other 10-strips, they get it. I get alternately amused and annoyed by people whose experience of education is limited to their own time in grade school who think they know better than teachers what’s developmentally appropriate for most kids. Sure, if you’ve really put the time into studying child psychology and how mathematical schemata are modified, then by all means your opinion is valuable. But if all you’ve done is read the blog of a cranky atmospheric science professor who gives freshmen an inauthentic exam, well, that’s not exactly an informed opinion. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [September 6, 2010, 5:17pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/59 "2010-09-06T17:17:18Z")
> [@Gagundathar](#): > > This jumped out at me while I read this thread. > I believe it has been fairly well established that humans group objects in no more than 5 units when they are visually breaking down a large set of objects. > That means that if a human sees 8 units, they see it as 5 + 3 (or 4 + 4 in some cases). This is done on a primitive level and may explain why some numerical methodologies are easier to grasp for a larger percentage of humans. Heh–when I was showing these 10-frames to the kids, I actually told them about this research. Not because I expected them to remember it, but because it’s awesome, and I figure sprinkling addition facts with awesome science will make it more memorable. (However, I thought the cutoff was 6: I’ll correct my misunderstanding for the students on Tuesday. I also figure it’s a good lesson for them that when you make a mistake, you fix the mistake enthusiastically and without shame). ---
**Author:** ![Gagundathar](https://avatars.discourse-cdn.com/v4/letter/g/a183cd/32.png) [@Gagundathar](https://boards.straightdope.com/u/Gagundathar)\ **Post date:** [September 6, 2010, 5:26pm UTC](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182/60 "2010-09-06T17:26:33Z")
> [@Left\_Hand\_of\_Dorkness](#): > > > [@Gagundathar](#): > > > > This jumped out at me while I read this thread. > > I believe it has been fairly well established that humans group objects in no more than 5 units when they are visually breaking down a large set of objects. > > That means that if a human sees 8 units, they see it as 5 + 3 (or 4 + 4 in some cases). This is done on a primitive level and may explain why some numerical methodologies are easier to grasp for a larger percentage of humans. > > Heh–when I was showing these 10-frames to the kids, I actually told them about this research. Not because I expected them to remember it, but because it’s awesome, and I figure sprinkling addition facts with awesome science will make it more memorable. (However, I thought the cutoff was 6: I’ll correct my misunderstanding for the students on Tuesday. I also figure it’s a good lesson for them that when you make a mistake, you fix the mistake enthusiastically and without shame). I wonder if the cutoff point is different for different individuals. I ‘remembered’ 5 as the cutoff because I _think_ I group things in fives. Of course, the study was done using PET scans, not subjective assumptions, but I wonder if there was any variation found in a population. You sound like a truly terrific teacher! Kudos. [Previous page](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182.md?page=2) [Next page](https://boards.straightdope.com/t/new-math-new-new-math-wtf/552182.md?page=4)