Simple math made complicated — for a reason

The Common Core makes simple math more complicated in order to teach understanding, writes Libby Nelson on Vox.

In the past, “students had this sense that math was some kind of magical black box,” says Dan Meyer, a former high school math teacher studying math education at Stanford University. “That wasn’t good enough.”

Students will learn different ways to multiply, divide, add, and subtract so they can see why the standard method works, writes Nelson. “They can play with them in fun, flexible ways,” says Meyer, who blogs at Dy/Dan.

Using a number line for subtraction lets students visualize the “distance” between two numbers. A father’s complaint about a confusing number line problem went viral on the Internet. Nelson provides a clearer version. 

Students put the two numbers at opposite ends of the number line.

Screen_shot_2014-04-17_at_5

It’s 4 steps from 316 to 320, 100 steps from 320 to 420, 7 steps from 420 to 427.

Screen_shot_2014-04-17_at_5

Then they add the steps together: 4 + 100 + 7 = a distance of 111. LearnZillion, a company that creates lesson plans for teaching to the Common Core standards, has a 5-minute video explaining this technique.

“Students should be able to understand any of these approaches,” said Morgan Polikoff, an assistant professor of education at the University of Southern California who is studying how the Common Core is implemented in the classroom. “It doesn’t mandate that they necessarily do one or the other.”

“A key question is whether elementary school teachers can learn to teach the conceptual side of math effectively,” writes Nelson.

If not, number lines and area models will just become another recipe, steps to memorize in order to get an answer, Polikoff says.

This is a real risk: Many elementary teachers are strong on reading and weak in math (and science). Perhaps we need math/science specialists in elementary school who understand their subject deeply and can teach kids to understand too.

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Comments

  1. Common Core does not require such silliness. The failed “reform math” of the 90s is making a comeback with people using Common Core as the excuse for implementing it.

    Common Core is bad enough without adding false accusations against it.

    • common Core does require (thus) “using concrete models or drawings and strategies” for addition and subtraction in, e.g. Grade 2.

      Whether that qualifies as “such silliness” is an open question, of course – but I don’t think it’s silliness. I think it’s a decent way to teach mental math.
      No specific strategy or model or drawing is required, but something vaguely analogous to this example is required.

      This is not problematic.

    • Teaching the underlying principles in an introductory class definitely courts silliness.

      No responsible person would dream of teaching beginning cooking students organic chemistry so they’ll understand the underlying principles of frying an egg. The result – a properly fried egg – is, at that level of skill and understanding, more relevant to the student then are the underlying principles.

      Under the best of circumstances you’d end up with someone who knows why an egg fries but not how to fry an egg and the rather more common outcome would be someone who knows neither or at least doesn’t learn the skill in the class.

      I think that qualifies as silliness.

  2. Ann in L.A. says:

    This is essentially the same, albeit a harder way, to how to make change. Seems to me the same thing can be more easily taught with pennies, nickles, dimes, dollars, $5′s, $10′s, $100′s (or facsimiles.)

  3. Miller Smith says:

    Teach math neophytes an algorithm to quickly find the answers to common arithmetic functions so they get the right answers and THEN, later, teach them the number theory for the functions. This way they can quickly calculate the answer and then delve into the theory with full knowledge of what the process should produce.

    • Ann in L.A. says:

      I ended up auditing my high school calculus class (spent the time sitting at a desk studying for the AP bio exam.) As a result, I knew the shortcut way of doing derivatives before I ever took calc for credit. I’ve always thought that was the better way to teach it. Show how to do it the quick way, show some of the things you can do with it, and only then show why it works.

      • What is the shortcut way of doing derivatives?

        • Ann in L.A. says:

          Just: pull the exponent down and multiply it to the term, then subtract 1 from the exponent.

          • Roger Sweeny says:

            That’s only true for “power functions,” functions that have one variable raised to a power. There are lots of other functions that are used in intermediate math, e.g. trig functions and exponential functions (a number raised to a variable power).

  4. What obfuscation to claim that they’re just teaching these other methods so that kids understand why the traditional algorithms work. To do that, they’d actually have to teach the traditional algorithms. Here is EM’s official page showing how to do “a variety of algorithms”: http://everydaymath.uchicago.edu/teaching-topics/computation/ Notice some missing?

    • Michael E. Lopez says:

      So CT, I looked at those powerpoints. In fact, they DO use the Traditional Addition algorithm.

      They use it (although I think they don’t realize they’re using it) in Lattice Multiplication to add up the diagonals, presumably because Lattice Multiplication is already so damn complicated that anything else would make it unbearable.

      • Yes, I was referring to the multiplication and division traditional algorithms. They supposedly have a link to something on long division, but I can’t access anything when I click on it. If they’re not even going to teach long division, it seems to me that they are basically saying that these kids will never go on to factor polynomials. No STEM-hyping parties will overcome that deficit.

  5. Jerry Doctor says:

    As Charlie Brown’s sister Sally complained decades ago, “All I want to know is how much is two and two!”

  6. We wonder why so many kids do poorly in math in the U.S., this is a classic example of EPIC FAIL, IMO.

    Sigh

  7. Roger Sweeny says:

    Is it better to teach mechanics first and then go for understanding or to teach “why it works” and then go for the right answer? NO ONE ___ING KNOWS!

    To get a drug approved, you have to do randomized controlled trials: a group of people with some condition is randomly divided; one division is given the new drug, and the other is given an existing drug or a placebo. The groups are tested for how successful each drug was and the results are compared. The results for the new drug have to be better for it to be marketed.

    All a new curriculum has to do is get the support of enough political actors with enough power. Then not only is it approved, everyone in that jurisdiction has to buy it.

    • This is only the case in a country where the citizens have allowed the federal government to control education. No federal DoE, no Common Core.

      • Roger Sweeny says:

        There might be no Common Core without the federal DoE. But there are unproven ed fads all the time that have nothing to do with the feds–because we set very high standards and nobody knows how to achieve them.

        • Mark Roulo says:

          “But there are unproven ed fads all the time that have nothing to do with the feds–because we set very high standards and nobody knows how to achieve them.”
           
          The business world has fads without the feds implementing them or setting “high standards.” There are a lot of people who are out of work if things aren’t changing often …

    • Ann in L.A. says:

      With the example above, it’s almost moot. I would guess many teachers, if not most (or all but a few,) use the number line extensively when teaching addition and subtraction. I was teaching a 6th grader last night, and it is indispensable when you start dealing with negative numbers.

      But, at the end of the day, you need a quick, robust algorithm to actually get the job done.

      • The number line is not used here. It is pictorial, not good for the inclusive classroom that has many students who are in the concrete stage.

        The preAlgebra with negative numbers is done with examples such as elevators and houses, or counters and jars. Inclusivity demands concrete or pictorial that is closer to concrete rather than abstract.

        At the end of the day, they’ll be on a calculator unless they are being tutored privately.

      • palisadesk says:

        Yes, we use the number line extensively, from first grade on (and perhaps in K, but I can’t say).We have number lines on the children’s desks, and whiteboards with number lines for children t use; the “concrete” issue is a non-issue, even though we have full inclusion, because there are concrete number lines too, such as tracks for unifix cubes and counting frames for color-coded materials like Cuisinere and Math-U-See.

        The primary grade teachers make regular use of the number line for addition and subtraction particularly, but it is also used for fractions. A number line is the best way I know to demonstrate how multiplication and division of fractions work — it’s useful to remember “invert and multiply,” but using the number line makes it clear why that works.

        The major problem I see is that there is insufficient instructional time to get most (let alone all) students to mastery on either skills or concepts (and these are synergistic and work together). In order to address the problem we would have to give a serious look to how we organize elementary instruction, as well as what methodologies to use. Irrespective of ability, which is an important variable, processing speed and time to mastery varies widely and we don’t provide enough time for those who need more practice to mastery. When they don’t get it, they are building on an increasingly shaky foundation each year.

        Instructional failure is part of the problem ,but a more fundamental one is that we hold time constant for all, when THAT is what we should be “differentiating.”

        • All through my math education (’50′s, 60′s) we were taught the most efficient algorithms, and tested on them. We were also taught, frequently, alternative ways of handling the basic operations, such as the ones illustrated above and others. These helped cement understanding of how the basic functions work, and they were interesting and for some kids provided their first understanding of why they work. BUT, we were not tested on the ability to use these alternative methods, and there was a clear sense that the most efficient method (standard method then) was the one to commit to memory. Since you can’t expect children to commit multiple methods to memory, this seems to me like the best option. (We did use number lines too, though not on each desk).

  8. I have been substitute teaching for the past 2 months. It has been eye opening. The methods that they are teaching children are not aiding in understanding. It’s so convoluted, there’s no way they’ll be Math literate. I have to be honest with the students and tell them that the way they are learning Math was very different than how I was taught. So unfortunately, I couldn’t help them at all. They do not know what is going on.

    • Your confusion is understandable.

      Those methods you find convoluted aren’t meant to aid in teaching kids math but to put a smile on the face of a superintendent or school board members.

      Using methods of recent vintage is an assurance that old, unfashionable methods aren’t being used and modern, and thus superior, methods are being used.

      The fact that the methods are convoluted is immaterial since the people creating these new methods won’t be held responsible for their ineffectuality nor will those who choose to employ those methods.

      The only people hurt are the kids and they won’t have a clue how badly they’ve been hurt for quite a few years so, no worries.

  9. In my past (I”m older than the person quoted), students did not think math was a black box. They liked math for its rules. They could get a problem right. They could determine an answer..and it would be the same answer the second time through. It was a refreshing break from English class and its subjective grading.

  10. Jerry Doctor says:

    Something happened today that made me think of this thread. I took my son-in-law to lunch. The bill was $19.15. I looked in my wallet – four ones, two five and a hundred dollar bill. The thought that immediately flashed into my head was: “85 cents to make $20 and four twenties.”

    Wow – exactly the way these people want me to do the problem! Except, I didn’t think about how I was going to calculate my change. The fully formed thought was instantly there. Why? Because I went to grade school in the 50′s. Drill, drill, drill. As a result I instantly recognize mathematical relationships. Not because I was taught to look for them as a way to do subtraction but because the basics were so thoroughly pounded into my little brain that as an adult it just comes naturally.

    I often think about all the workshops I had to attend over the years where they talked about Piaget and marvel that apparently no one listened to what he had to say.

    • Jerry,

      I learned math in the 70′s the same way, endless drills on addition/subtraction/multiplication/division, so you didn’t have to really think about the answer, it more or less popped into your head.

      I can usually figure out the change due at the register faster than the cashier can do it with the computer, but that’s due to a solid working knowledge of math basics.

      • Jerry Doctor says:

        Bill,

        A friend of mine did all the programing for a theater chain’s computerized ticket and concessions sales. He included the procedure to calculate change. When the managers saw it they immediately demanded that it be removed.

        The managers explained to him that when they hired kids they first tested their ability to make change. If they couldn’t do it immediately without assistance, they were relegated to cleaning and other low level tasks, never sales. They explained to my friend that when there is a line of people at the concession stand and only a few minutes until the movie starts there isn’t time to have the computer calculate the change.

        • Jerry,

          Makes sense, i’ve been in restaurants where the POS broke down and no one could compute sales tax. I’ve gotten a lot of free food for doing what amounts to basic math (percentages) from restaurants.

          Making change and having clerks count it back to the customer (which is the way I was taught) is the best way to ensure mistakes don’t happen. I get handed money back all the time, and I don’t take it for granted the cashier/clerk got the amount correct :)

    • Michael E. Lopez says:

      There is a trend — one I think not particularly helpful — of taking the sort of thing that a practiced, advanced veteran of a discipline does and making it a training tool.

      The thought process that you just described, Jerry, is one example.

      Another are the endless “scaffolding” exercises that many teachers make their students go through these days. They are explicitly practicing the sort of reading techniques that advanced readers do without thinking about it. But they are doing it before the students have mastered the fundamentals.

      It’s as if I opened up Week 1 of a fencing school with broken-time compound feints. Yes, *I* can do those. But I’ve been fencing for over a decade. When you start off, you have to do 200+ hours of footwork drills so that the movements come naturally. Then you can start to play with them, to understand them at a more advanced level.

      If you ignore fundamentals, if you ignore the basic rules of how to train someone in something, then you’re really just engaging in cargo-cult thinking: advanced readers/math students do X, so if our students do X, they’ll be advanced readers/math students!

      • Jerry Doctor says:

        Michael,

        “advanced readers/math students do X, so if our students do X, they’ll be advanced readers/math students!”

        That is exactly the nonsense that prompted my move from high school teaching to college, even though it meant a cut in pay. Our administration noticed that many of the best students used day planners. They raised the funds and bought day planners for everyone. When (surprise!) grades didn’t improve the teachers were directed to check at the end of class and make certain students were recording assignments in their planners. Naturally the grades still didn’t improve but we did create a whole new type of discipline problem.

        The fact we not only allow fools like this into education but actually make them leaders is probably the main reason the general public will never consider us to be “professionals.’

        • This nonsense has been going on since the 80s (and likely before), WRT Latin, debate, 8th-grade algebra I, AP courses, modern foreign languages and higher-level math/sci courses (alg II to calc, physics, chem). Kids who took such courses did significantly better (GPA, SAT, HS grad, college completion etc) than kids who didn’t take them —- so taking those courses will make students succeed!!!!!! Of course not – because, at that time, only the most capable, well-prepared and motivated kids took those courses – which acted as a proxy variable for identification of those students. Dumping unqualified kids into courses for which they have no preparation – like Prince George’s County, MD, mandating an AP course prior to graduation (even though most kids are colossally unprepared, as in reading at 5th grade level, according to one teacher) – is somewhere between idiocy and willful blindness.

        • Roger Sweeny says:

          I think the problem is not that we “allow fools like this into education” but that when it comes to education almost all of us are “fools.”

          Your administration noticed something that was true–that students with day planners were generally better students. No doubt they thought of reasons why using a day planner would make someone a better student: you don’t have to try to remember what should be done when, or search for that information; it’s all in one place. You will then be better able to do what has to be done, and to do it in an efficient and productive manner.

          That inevitably leads to the question of why other students don’t have and use day planners. An answer that jumps into the mind of anyone in the education business is “lack of education.” And, depending on your politics, “they can’t afford one” (if you’re on the left) or “their significant adults don’t know enough/care enough to get them one” (if you’re on the right).

          So, the obvious solution presents itself: educate the students on the value of day planners and, since it would be unfair to force the kids’ families to buy one, provide each student with one for free.

          But then it turns out that lots of the students don’t use them. Well, education is more than listening. It’s also practicing. So make sure the students practice. Check to make sure that students record assignments in their planners. After all, the whole idea of school is “We know better than you. You are legally required to be here for twelve years so we can make you a better person.”

          To have stopped anywhere along this progression would have required thinking unpleasant thoughts. Maybe school isn’t that powerful; maybe it has only limited power to change people for the better. Maybe there are major differences between young people as to what they care about, how organized they are, what their aptitudes for schoolwork are–and there isn’t a lot schools can do about that.

          Almost nobody wants to go there. Or even come close.

          • Michael E. Lopez says:

            Roger-

            NANANANANANANANANANANANANANANANANA!

            I’m not listening!

            NANANANANANANANANANANANANANANANANA!

            I’m not listening!

  11. This … is a simple way to handle math.

    It’s just not vertical-stack subtraction.

    “Different from how you [and I] learned it in grade school” does not make it complicated. Just different.

  12. Jerry Doctor says:

    Likewise, different doesn’t make it better.