Intuitional Math

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Will Truman

Will Truman is the Editor-in-Chief of Ordinary Times. He is also on Twitter.

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  1. Avatar zic
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    says:

    There is something missing here — age of the child, and the goal of what’s being taught. We want to teach 2nd graders how to add and subtract; and back in the day, that meant memorization.

    But there are many, many ways to skin a cat, and memorization is not necessarily the best. Learning that you can reason out a way to solve the problem, not just that you can apply pre-approved solutions, is pretty valuable.

    I was really lucky to have a 2nd grade teachers who believed in giving children the ability to find solutions; it gave my kids a ‘can do’ attitude toward math (and a whole lot of other reasoning skills.) The algorithmic math happened still, it has to to build additional layers of skills; kids don’t need to reinvent the wheel with every problem. But having them work through the confusion to understanding has immense value; and it’s very easy for us to forget that it’s the moment of confusion that proceeds deep learning.

    I suspect FP might be so engaged in the basics that he takes them for granted, and really doesn’t have much understanding of how to build understanding. This is pretty common; understanding something and being able to use it competently is not the same skill as understanding something and breaking it down for novices.Report

    • Avatar Kim in reply to zic
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      says:

      It’s fine to teach most 2nd graders rote memorization.

      When the kid’s in the 8th grade, and still not able to add and subtract, it’s time to throw shit at the wall and see what sticks. “Gee, you got close. Well, you won’t be a cashier, but you at least won’t be losing as much money.”

      My perspective is very much influenced by knowing someone with dyscalcula.Report

  2. Avatar Jim Heffman
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    says:

    He’s got a bachelor’s degree? That’s cute.

    I’ve got a master’s degree in aerospace engineering and A: I understood the “number-line” thing just fine, and B: I understand why they’re doing it that way.

    The intent is to get students to internalize the idea that a number can be broken down into a group of constituents which all sum up to get the final answer. This is important for later concepts like algebra; it’s easier to get to “x-squared-minus-y-squared is also x-plus-y times x-minus-y” if you’ve learned that “three-hundred-fifty-two is also three-times-a-hundred plus five-times-ten plus two-times-one”.

    *****

    The problem, I think, is that the teachers responsible for this stuff don’t understand it all that well either, and so they can’t explain it to anyone. So they’re still teaching an algorithm, just a different one than the old “stack up the numbers and add up the columns” way.Report

    • Avatar Jonathan McLeod in reply to Jim Heffman
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      says:

      I’m with Jim… though I only completed first-year math courses, so I can make no claim to expertise here.

      It seems most of the reaction to “new math” (at least that I witness) is that parents and other people just don’t get it. Many of these people (again, in personal experience) just absolutely hate math and never did well in it.

      And, yet, they want to rigidly adhere to the learning methods they so hated and never responded too. Perhaps there are other lessons we need to learn.Report

      • Avatar Will Truman in reply to Jonathan McLeod
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        says:

        Well, all I can say that is that my experience coincides almost perfectly with the criticisms here. I don’t mind teaching alternative methodology in combination with reasoning it out, but by the sixth grade even a lot of the smarter kids seemed to have bypassed algorithms altogether. It’s not that they weren’t taught it. Just that they never learned to use it and couldn’t use it to (for instance) check their work against the method they were using (or trying to use).Report

    • Avatar Patrick in reply to Jim Heffman
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      says:

      The intent is to get students to internalize the idea that a number can be broken down into a group of constituents which all sum up to get the final answer.

      Yes. It’s a first introduction to basic algebraic operations, too.

      The problem, I think, is that the teachers responsible for this stuff don’t understand it all that well either, and so they can’t explain it to anyone.

      Yes, as well. Although the whole facebook meme rage I’ve been seeing is also kind of misplaced because it focuses on “here’s a problem I can do easily the old! way!” rather than the greater context in which the lesson is being taught, it’s not unlikely that a bunch of the teacher don’t understand the concept either.

      The number line bit is confusing for people who aren’t used to it, but conceptually the approach is actually a better way to leverage the tools of algebra to do math in the real world.

      Long form addition and subtraction, with carrying, is a terrible way to do math, it’s prone to errors and it’s basically impossible to do in your head.

      Leveraging the algebraic properties of addition and subtraction can let you do a lot of short-cutting in your head.

      The number line thing is actually analogous to how most people mentally make change, if you think about it that way. And you make change every day.

      $4.34 means 4 ones, a quarter, a nickle, and four pennies. Operationally, you got there by jumping on a number line, by the currency values.

      Here’s another example of using basic algebraic operations to turn something that you’d do the long way on paper into something you can do in your head:

      117 x 23
      (120 – 3) X (20 + 3)
      (120 x 20) + (120 x 3) + (-3 x 20) + (-3 x 3)
      2400 + 360 – 60 -9
      2691

      Also relevant: https://www.youtube.com/watch?v=UIKGV2cTgqAReport

  3. Avatar Katie S
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    says:

    My son’s 4th grade teacher was teaching him math but she told him something wrong and he wasn’t getting the right answer. He was frustrated. I sent her a polite note asking if what he had learned was correct and she was happy to correct herself. I had to figure out what they were doing so I could explain it to him better. My only problem with it was that he already knew how to do it the way I knew how so this alternate way was a little confusing for him. I could see how it might help students that were having trouble with the traditional way. Math seems to be one of those topics where one bad teacher can easily ruin it for you. I like math and one professor in college nearly ruined it for me. After everyone flunked the first test his remark was, ‘you people should know this’. Fortunately you can drop classes in college.
    I think some people stay in a kid’s sort of thinking that how they learned something is the only best way to do it. I think it’s fun to learn new ways once you understand how your way works.Report

  4. Avatar Kolohe
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    says:

    I’ve always liked this essay (pdf) particularly this quote:

    Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions.

    because I myself had a real hard time with any 3000 level math course that wasn’t part of the science/engineering core.

    Yet, I still find the algorithmic method of teaching preferable. (I’m reminding of that Dead Poet’s Society quote “when they find out they’re not artists, they’ll hate you for it, John”).

    We don’t object to rote learning and repetition in learning athletic skills, for instance. (lay up drills, batting practice, hitting the sled a gazillion times in late summer two-a-days, etc). Just the opposite – we count on people ‘learning the fundamentals’ and then growing from that base to be more improvisational and inspired. And we look for people with ‘natural talent’ to take these leaps of inspiration somewhat on their own.Report

    • Avatar Jim Heffman in reply to Kolohe
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      says:

      “We don’t object to rote learning and repetition in learning athletic skills…”

      Well, that’s because even the most-talented baseball player who gets drafted up to the Major League still has to contact a ball with a bat at some point. It’s not like he takes his glove off and starts playing chess instead, which is about what happens when you go from arithmetic to algebra.Report

    • Avatar Kim in reply to Kolohe
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      says:

      K,
      But what if the rote memorization is hitting severe mathematical deficiencies?
      I do know a guy who does math using 2’s complement, as that’s easier in his head than counting 17 pebbles (you’d lose track too, I suspect).Report

  5. Avatar Michael Cain
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    says:

    I have a couple of math degrees, so I’m sure that my perspective is badly out of line with most of the population. And I have, on a few occasions, been suckered into teaching beginning calculus (why doesn’t anyone ever ask me to teach graph theory?). Given that…

    There’s a progression of things that have to be learned. It’s one thing to say that you don’t need to understand the distributive property in order to do simple multiplication with actual numbers. But you need to understand it in order to do the symbolic manipulation you need for algebra. And you need that algebraic manipulation in order to handle calculus (the typical beginning calculus student’s skills at manipulating algebraic formulas improve immensely in first-semester calculus). And you need calculus for, well… a whole lot of things.

    If the goal is simply to get the right answer, Mathematica is to calculus as calculators are to simple arithmetic. I will never be as good at symbolic integration as Mathematica is (nor can I use the same techniques Mathematica does under the hood, unless I want to spend a week on each problem). But integral calculus and differential equations are the way that I can express a whole lot of things about reality (or approximations thereto). I need to understand it well enough that I can write down the proper integral, even if I’m not going to solve it by hand.

    OTOH, one of my ongoing complaints is that there are a whole lot of fields within mathematics that are useful ways of thinking about problems (eg, graph theory) that most students are never exposed to.Report

    • Avatar Michael Cain in reply to Michael Cain
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      says:

      …why doesn’t anyone ever ask me to teach graph theory?

      And yes, I already know the answer to this one. Graph theory is the kind of class that you get to teach, so the PhDs keep it for themselves. Beginning calculus is the kind of class that you have to teach, so push that off onto the peons.Report

      • Avatar Kim in reply to Michael Cain
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        says:

        Calculus is so much more fun to teach than business calculus.
        *used to tutor math at a community college*Report

      • Avatar Michael Cain in reply to Michael Cain
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        says:

        And business calculus is more fun that remedial algebra. At our local community college, the instructors on staff fight over the business calculus sections and remedial algebra falls to the adjunct faculty, many of them retired engineers looking to make a few bucks. Adjuncts don’t last very long when all they get to teach is the remedial classes.Report

      • Avatar zic in reply to Michael Cain
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        says:

        So how come we’re not getting hot graphs that illuminate the world @michael-cain?

        I’d love to see stuff like how ACA subsidies decline as income increases (with the poverty line marked) until there are no subsidies. I’ve been looking for one, haven’t found it yet. Might be possible to set up API call to the kaiser calculator to generate the data.Report

      • Avatar Michael Cain in reply to Michael Cain
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        says:

        So how come we’re not getting hot graphs…

        I’m thinking (hoping?) that there are implied sarcasm tags here. Yes?Report

      • Avatar zic in reply to Michael Cain
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        says:

        No sarcasm tags; serious request for skill sharing.Report

      • Avatar Mad Rocket Scientist in reply to Michael Cain
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        says:

        @zic

        Err, that isn’t the graph theory he’s talking about (hence the comment about sarcasm tags).Report

      • Avatar zic in reply to Michael Cain
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        says:

        I thought I was missing something (perhaps my specific request did), but I don’t think I am; we’re talking methods of showing relationships between different bits of information, no?

        I would still request to see resulting work; this is a seriously interesting time to do this kind of stuff considering the vast wealth of data available on the internet, often available in real time from things like twitter streams and user logs.

        My sweetie actually does work very similar to this, but uses the results to control other processes — mostly sound/video generation — as art. An example I’ve been begging him to do is an animated map of bird migration using Audubon sighting data and (of course) pre-recorded calls to indicate amplitude of sightings. Where the birds are.Report

      • Avatar Michael Cain in reply to Michael Cain
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        says:

        @zic
        Graph theory is the study of a particular type of discrete structure. The concept of this type of graph is almost trivially easy to grasp. The use of that term goes back to the 1870s; the first formal use of the structure is usually credited to Euler in 1736. Because the structure is simple, it can provide a nice introduction to formal proofs about the properties of graphs. At the same time, it’s also easy to state hypotheses that are very difficult to prove. The four-color theorem is still a source of controversy because the currently-accepted proofs require computer software, raising the problem of “Is the software implementation actually correct?” A lot of real-life problems can be cast in graph-theoretic form. Solving those can range from easy to very difficult; it’s a good field for introducing people to formal studies of algorithms.

        What you’re describing sounds more like what I would call “data visualization.”Report

      • Avatar zic in reply to Michael Cain
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        says:

        Data visualization; sonification, too; depending on the output media chosen. But it’s often involves signal processing, mapping input form one set of variables to another scale of some sort, as well.

        Conceptually, I have a difficult time separating the output forms; all are methods of showing change in two (or more) sets of variables in relation to one another, aren’t they?

        And yes, a very good method of teaching logic.Report

      • Avatar Patrick in reply to Michael Cain
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        says:

        Graph Theory: Topology for The Simple-Minded.

        #mathnerdslamReport

    • Avatar Kim in reply to Michael Cain
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      says:

      Michael,
      Yeah, graph theory, probability, those seem like really good things to start teaching kids early. (below 7th grade) — not purely those things, but life doesn’t need to be purely memorization. That’ll just make everyone hate math.Report

  6. Avatar Saul DeGraw
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    says:

    The culture war fight over the Common Core is interesting especially because liberals are hardly united on the common core. There are plenty of liberals like me who are suspicious of Michelle Rhee, charter schools, and aspects of the common core. This is a real split in the Democratic faction.

    I generally agree with Bouie when he wrote that conservatives are against the Common Core because a chunk of the Democratic Party supports it.

    http://www.slate.com/articles/news_and_politics/politics/2014/04/conservative_tribalism_conservatives_hate_anything_barack_obama_and_liberals.htmlReport

    • Avatar Will Truman in reply to Saul DeGraw
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      says:

      As near as I can tell, the partisan dynamics of common core are mostly a product of who has the megaphones. There seems to be a fair amount of conservative support for and opposition to it, as well as liberal support for and opposition to it.

      Personally, I view the above as a distinct issue from Common Core (which, honestly, I have no idea how I feel about). The discussion pre-dates Common Core.

      The biggest tie-in, and the concern that I have, is that Common Core means rolling out methodology nationally before finding out too late that a mistake was made. Conceptually, anyway, I like Common Core as a notion of benchmarks – things to be learned – and am somewhat less comfortable with it enforcing teaching methodology (to the extent that it does that).Report

      • Avatar Saul DeGraw in reply to Will Truman
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        says:

        Good points.Report

      • Avatar J@m3z Aitch in reply to Will Truman
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        says:

        Ditto Will’s last paragraph.

        There’s an old habit in education of latching on to new ideas and promoting them as huge improvements, the one best way, without adequate testing of them. When education professionals say there’s a “lot of literature” on something it usually turns out that there are a lot of education journal articles advocating something, or relating personal anecdotes of using a method, but no data, controlled experiments, or otherwise careful assessments.

        With the particular case here, I’m not qualified to speak (I’m a mathematically minded guy who was ruined by a couple of lousy teachers in 7th through 9th grades). But I do know that back in the early ’70s there was big hoopla over “the new math.” Maybe the common core way really is better, but in general the excited promulgation of new educational fads is old hat. Even if this way is the best way to do it, a new way will be proposed in another generation.Report

    • Avatar Ken S in reply to Saul DeGraw
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      says:

      The backlash I have read against Common Core doesn’t concern its content at all. Rather, it’s the boilerplate bogeyman “Uncle Sam wants to dictate a national curriculum.” Question: Of the countries whipping us at teaching their children math and science, how many have a centrally-determined curriculum? I don’t know, but my guess is “all of them.” (I would love to hear from someone who does know.)
      As usual, Mark Twain said it best. “First God made an idiot. That was for practice. Then he made a school board.”Report

  7. Avatar Kazzy
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    says:

    It’s important to note that many of the “horror stories” being put out there about what the Common Core is or isn’t doing are complete or near-complete fabrications. I won’t pretend to know enough about CC to weigh in on its various pros and cons, but I do know that there is a lot of misinformation being spread about it — some of it rather deliberately.Report

    • Avatar Jonathan McLeod in reply to Kazzy
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      says:

      No. Way.

      I refuse to believe falsehoods are peddled on the internet.Report

    • Avatar alkali in reply to Kazzy
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      says:

      You can read or download the actual Common Core standards at corestandards.org.Report

      • Avatar Alan Scott in reply to alkali
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        says:

        Or, if you want a shortcut, you can get the guy who’s training to be a math teacher to post the relevant ones in the thread. Here’s a selection of 2nd grade math standards related to the subtraction methods used in the assignment:

        2.NBT.5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
        2.NBT.7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
        2.NBT.9. Explain why addition and subtraction strategies work, using place value and the properties of operations.

        Students still need to learn the typical method, but not until 4th grade:

        4.NBT.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.Report

      • Avatar Kazzy in reply to alkali
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        says:

        It is important to note that the Common Core is not a curriculum, but a set of standards. That many districts are turning it into a curriculum is something else entirely.Report

  8. Avatar Mad Rocket Scientist
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    says:

    @michael-cain & @jim-heffman

    Agreed

    I struggled at math all through high school, mainly because none of my teachers really understood the math they were teaching, so they never understood why I couldn’t get it. It wasn’t until College Algebra that someone figured out what key concepts I had missed so many years ago & spent a couple of days getting that drilled into my head. The understanding that followed was nothing short of phenomenal.

    I can see the value in letting younger kids explore solution paths, but the school should be making sure the parents understand what is being taught & why. If my kid suddenly came home from school with homework like that, I wouldn’t be sending snarky notes along to the teacher, I’d be on the phone, or in their office after school seeking an explanation.Report

    • I always wondered how much trouble I got my kids in in elementary school. When they learned multiplication, one of the things they got from me during homework at the dining room table was the importance of the distributive property (Me, over and over again: “You have to really understand this. It’ll be important in a few years.”). The other thing they got for why we do multiplication in columns was in math, notation is important (Me again: “Over hundreds of years, this is the way we’ve figured out to keep all of the steps correct and make the fewest mistakes.”). One evening when my daughter had started long division and was carefully drawing the division bracket, she looked up and asked, “This is that notation thing, isn’t it?”

      Didn’t manage to turn her into a mathematician. But there’s a new granddaughter now, and maybe I’ll have better luck with her.Report

      • Avatar Mad Rocket Scientist in reply to Michael Cain
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        says:

        I suspect I’ll do something similar with Bug. I don’t want to necessarily turn him into a mathematician or engineer/scientist (although I think I’d be giddy if he did), but I do want to make sure he doesn’t suffer the debilitating panic & anxiety I had as a kid with regard to math.Report

  9. Avatar Patrick
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    says:

    As I said on Facebook when this thing came around:

    I maintain (perhaps optimistically) that anybody can be taught math, if you have a good enough math teacher.

    I maintain (perhaps pessimistically) that there aren’t very many good math teachers.Report

    • Avatar Mad Rocket Scientist in reply to Patrick
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      says:

      @patrick

      Too true.

      I often wonder how many teachers, public school or private, are teaching subjects they are not well versed in. I mean, not every mathematician is a good math teacher, but I think it’s OK to ask that a math teacher have a background in mathematics beyond some basic requirement.Report

      • Avatar Patrick in reply to Mad Rocket Scientist
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        says:

        You can make the same point about a lot of subjects, really.Report

      • Part of the issue is the lack of specialization in k-5, which I’m not sure how to address.Report

      • Avatar Kazzy in reply to Mad Rocket Scientist
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        says:

        Generally, as students get older, there is more emphasis put on subject matter than on teacher training. As Will notes, the lack of specialization in the elementary grades coupled with the fact that social-emotional curriculum and other components of teaching are more important in those grades is a huge contributing factor to those teachers being less well-versed in certain subject matters. The primary means of addressing this — higher standards for teachers — is harder to implement without drastically increasing pay. If you are going to make it a basic requirement that a teacher has multiple degrees in both pedagogy and multiple content areas, you’re going to have to have a starting salary higher than $35K.Report

      • Avatar Mad Rocket Scientist in reply to Mad Rocket Scientist
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        Perhaps this is asking too much, or maybe I’m not well versed on teacher education, but is there no room in the college course work for aspiring teachers to say you should have a subject matter focus beyond the core.

        Say, someone who wants to be a math teacher actually takes classes in advanced math, or maybe takes classes specifically addressing math education?Report

      • Well, a lot of schools would need to reconfigure how they do their instruction. In Arapaho, for example, the teachers taught all of the subjects all through the sixth grade. Back home, they didn’t and from the third to the fifth grade there was a degree of specialization. So you could possibly do things the latter way and have it work. My impression is that that’s not the norm, though, and there may be a reason for that?Report

      • The math dept at my undergraduate alma mater does work with math instruction as well as theoretical math. IIRC from the math dept newsletter I still get, the School of Education has a special endorsement in math education, the classes for which are jointly developed and taught by people from both faculty groups.Report

      • Avatar Patrick in reply to Mad Rocket Scientist
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        says:

        Actually, a good chunk of kids in the pre-12 grades will “get it” if they have just a halfway decent teacher, and another good chunk of kids in the pre-12 grades will “get it” if they have reasonable active and engaged parents and a middlin’ teacher.

        It’s the rest of the kids you worry about, and for that you really need someone who is good at both math and teaching. In the pre-12 grade level, that’s hard to get… they either need to be pretty natural teachers, or you’re talking multiple degrees and extrie bonus cash.

        A floating expert is not a terrible potential solution for some school districts.Report

      • Avatar Kazzy in reply to Mad Rocket Scientist
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        says:

        @mad-rocket-scientist

        My undergrad required education majors to have an “interdisciplinary major”… basically a few credits short of a full second major. In the secondary ed program, this meant that you worked on your subject matter in the School of Arts and Sciences. For early childhood and elementary, you could go that route, or you could take another ed major. I took “Math Education”, which was a pretty thorough program around not just math content, but theory and pedagogy. A lot of students did ‘Human Development’; I don’t know what that means. And a bunch did some form of SpEd because it significantly boosted their job prospects.

        The thing is… in early childhood/elementary, you are teaching all (or most) of the subjects: definitely math, writing, and reading; probably social studies; and sometimes science. You might have a math coach or reading specialist who supports you, but you’re still it. You’d be hard pressed to really master all of those — on top of all the other ed stuff and the student teaching — in a four-year-degree. And it is hard to ask for entry-level people to have multiple degrees if you are only going to pay them 35K.

        That is why the move has been towards standardizing curriculum: try to make it idiot proof.Report

      • Avatar Mad Rocket Scientist in reply to Mad Rocket Scientist
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        @kazzy

        Well you know when they say about making something idiot-proof. All is well and good, until along come better idiots.

        Is $35K a starting salary, or average? If so, for a profession that requires that much education, that is offensive. I remember when I was graduating with my BS, I had an interview & an offer for a job at a nearby company. It sounded interesting, it involved a lot of travel (about 75%), and was highly technical, but it did not require a degree in engineering, and they only wanted to pay me $37K. I declined the offer & wondered privately what they were smoking.

        I am really starting to wonder where all my tax hikes for better teacher pay are going?Report

      • Avatar Kazzy in reply to Mad Rocket Scientist
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        says:

        @mad-rocket-scientist

        I was kind of guestimating with the $35K, but I was actually pretty close: https://www.nea.org/home/2012-2013-average-starting-teacher-salary.html

        Part of the problem is that a lot of teacher education programs are piss poor. I’ve taken certification tests (practice and real) in multiple states. For early childhood and elementary, they ask for a mastery of the subject matter itself. Basically, if you can do 3rd grade math, you are presumed able to teach 3rd grade math. The problem is — when I took the tests for real — I spoke with other test takers who were relieved to have finally passed after three prior attempts. HOLY SHIT!!!Report

      • Avatar Mad Rocket Scientist in reply to Mad Rocket Scientist
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        says:

        @kazzy

        I spoke with other test takers who were relieved to have finally passed after three prior attempts.

        Is it Halloween? Are you trying to scare me!?Report

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