Count the Tropes, Dept.

Yet another in an unending series of articles about math education and what parents should (and should not) be doing to help.  This particular one is from Chicago Parent and contains the usual tropes/mischaracterization about how math used to be taught and why the new ways are so much better.  We start with the usual ever-popular one:

STOP TEACHING THE TRICKS: A large amount of research has gone into the progression of teaching students mathematical concepts. The shift has moved away from teaching students to blindly follow rules and toward making sure they understand the larger mathematical ideas and reasoning behind the processes.

I have written extensively about how math was taught in the past, citing and providing examples from textbooks from various eras. It might be interesting to look at some of the books used in previous eras that have been described as teaching students to “blindly follow rules”.  Many, if not most, of the math books from the 30’s through part of the 60’s were written by the math reformers of those times.  It makes the most sense to start with the series I had in elementary school: Arithmetic We Need.  The reason is because not only is it from the 50’s, but also one of the authors was William A. Brownell, considered a leader of the math reform movement from the 30’s through the early 60’s. Today’s reformers also hold Brownell in high regard, including the prolific education critic Alfie Kohn, who talks about him in his book The Schools Our Children Deserve .

In arguing why traditional math is ineffective, Kohn states “students may memorize the fact that 0.4 = 4/10, or successfully follow a recipe to solve for x, but the traditional approach leaves them clueless about the significance of what they’re doing.  Without any feel for the bigger picture, they tend to plug in numbers mechanically as they follow the technique they’ve learned.”  He then turns to Brownell to bolster his argument that students under traditional math were not successful in quantitative thinking: “[For that] one needs a fund of meanings, not a myriad of ‘automatic responses’. . . . Drill does not develop meanings.  Repetition does not lead to understandings.”

Brownell, however, requires students to do the practice and exercises held in disdain by those who believe traditionally taught math did not work.  The series contained many exercises and drills including mental math exercises. Such drills might appear to run counter to Brownell’s arguments for math being more than computation and “meaningless drills,” but their inclusion ensured that mastery of math facts and basic procedures was not lost. Also, the books contained many word problems that demonstrated how the various math concepts and procedures are used to solve a variety of problem types. Other books from previous eras were also similarly written—most authors were the math reformers of their day—and provide many counter-examples to the mischaracterization that traditional math consisted only of disconnected ideas, rote memorization, and no understanding.

But let’s move on. It gets worse:

…By teaching the trick before a child has this foundation, you may be inadvertently doing more harm than good. Students become reliant on tricks and fail to master the conceptual understanding needed to use the tricks appropriately.  Remember, kids will be more successful in the future as a problem-solver than as a memorizer.

What this article and many look-alikes caricature as “tricks” are actually mathematically sound algorithms. The idea that teaching standard algorithms “too early” eclipses the underlying conceptual understanding most likely stems from Constance Kamii’s infamous study “The Harmful Effect of Teaching Algorithms to Young Children” which was published by the National Council of Teachers of Mathematics (NCTM) in an annual review.  It has become the rallying cry that has garnered more believers than the idea that the substance called “laetrile”, extracted from apricot pits, is a cure for cancer.

With respect to the math books of earlier eras, they started with teaching of the standard algorithm first.  Alternatives to the standards using drawings or other techniques were given afterwards to provide further information on how and why the algorithm worked.  This is opposite of how reformers are advising it be done now.  What happens, typically, is the first way a child is taught to do something becomes their anchor, with everything else being supplemental.  By teaching the supplements first, there is a mix-up of main course versus side dish, with many students unable to tell the difference.  The popular theory is that students now have a choice and can pick the method that works for them.  I have tutored students showing profound confusion, asking me what method they should use for particular problems, feeling that various problems demand different versions of the same algorithm.

Note also the ever-popular warning against memorization: Memorizers are not problem solvers apparently. Well, sure, there may have been teachers who taught math poorly and had students memorize day in and day out with no conceptual context. To listen to the people who write these articles, it seems that the nation was plagued with such teaching, as if poor teaching was/is an inherent quality of traditionally taught math.  I would argue otherwise and go so far as to say that many of the “understanding-based”, student-centered, collaborative techniques that dominate many of today’s classrooms are inherently ineffective and damaging.

Memorization is the seat of knowledge. Eventually students just have to know certain facts and procedures and do them automatically. The idea that memorizing eclipses the understanding of what, say, multiplication is presumes that students are taught the times tables with no connection to what multiplication is, and what types of problems are solved using it.

Other than that, the article is pretty good, I suppose.

 

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8 thoughts on “Count the Tropes, Dept.

  1. “What happens, typically, is the first way a child is taught to do something becomes their anchor, with everything else being supplemental. By teaching the supplements first, there is a mix-up of main course versus side dish, with many students unable to tell the difference.”

    Bingo!

    “Memorization is the seat of knowledge. Eventually students just have to know certain facts and procedures and do them automatically.”

    I’m sure the author of the article hasn’t memorized how to get home. She looks up the directions on her phone… Wait, she hasn’t memorized how to use a phone, she reads a manual on how to use her phone…
    The author’s screed against memorization is sophomoric stupidity.

    Please post math books with solution manuals that parents can use to teach their kids. This will help parents that are struggling with this nonsense.

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    • JUMP Math books for home are excellent resources; Kumon also has some. You can probably get them through Amazon, although many stores might have them now for back to school. Ironic, isn’t it? Selling books for parents for when the kids return to school? That says a lot about how our school system has deteriorated.

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      • Hi Tara, thanks for the response.
        I teach my kids using Saxon math.
        Yes, I agree that parents having find books to teach their kids math does say a lot about the present day school system.
        When my kids started school they were “taught” Everyday math. I say “taught” because the “teaching” involved the teacher giving my kids problems with no instruction on how to solve the problem. I remember when my older daughter was in second grade she came home with a problem like 2.3 + 4.01. I had to explain the decimal system to her, I had to explain how you line up the decimals when adding, I had to explain… After about 3 or 4 more problems in which I had to explain everything and after getting told by the teacher that I didn’t “understand” math very well, I decided it was time to teach my kids myself.
        This was about 7 or 8 years ago. When I started teaching my kids I searched the public libraries for books to use. I must have looked in a dozen libraries for books to use, but I couldn’t find anything worthwhile. I also ordered some books off the web thinking that they were in line with what I was looking for, but was sorely disappointed. I can’t tell you how long I looked and how many textbooks I looked at.
        Finally, I was introduced to a high school math teacher that thought like I do (and Barry, I think) and the math teacher introduced me to Saxon math. I’ve been teaching my kids Saxon math ever since.
        I greatly enjoy Barry’s posts. One thing I think is missing is a list of recommended text books along with the answer keys. I think there is a disconnect between Barry and some parents, like me, that have given up on present day math teaching. I know the math teaching these days is horrible, but I need books that I can use to teach my kids. I was lucky to find books with answer keys. Barry should have a list of recommend books with answer keys. That would really help people like me. I find Barry’s observations spot-on and his writing is very good and humorous, but parents like me need to be told where we can find good math books so we can teach our kids.

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      • I thought I had provided you such a list, and apologize if I did not. I’m aware of some books from personal experience and some from what I’ve heard from others so I’ll provide you info on what I know. In the meantime, since you are actively homeschooling and in contact with others doing the same, perhaps you can do a blog or an op-ed with a list of recommended books. You’re certainly welcome to do a guest blog here so let me know. So here are my recommendations:

        I used Singapore Math Primary Math series (US edition) with my daughter, though I find that the bar modeling problem gets a bit excessive in 6th grade, and I tended to avoid the more complicated problems. Still, it’s a useful way to do percentage and proportion problems and it can easily be used as an entre to the algebraic approach in 7th and 8th grades. You can find Singapore books at http://www.singaporemath.com For the primary math series, (non Common Core) go here: http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39.htm Haven’t tried the Common Core editions so can’t comment on those.

        Saxon Math is good but I prefer the older editions by Hake to the newer ones. I started using Saxon 7/6 for 6th grade. The thing about Saxon is that it’s a total commitment because of its spiraling technique. They do spiraling in a good way, not like Everyday Math’s spiraling. But if you’re looking for help in a specific area like fractions (which was what I was looking for with my daughter), doing surgical strikes is a bit difficult with Saxon. You have to start from the beginning and stay the course. I’ve heard from homeschool parents that it’s an excellent series to use.

        Sadlier Oxford publishes textbooks that have been used in Catholic schools for years and they are good for the K-8 grades. These can be found on Amazon. I have heard good things about the Homeschooling series “Math U See” but have not tried it myself, so can’t comment on it from any personal experience. The website is here: https://www.christianbook.com/page/homeschool/math/math-u-see?event=Homeschool|1000117

        Algebra books: You can get books by Dolciani; I prefer the ones written in the 60’s and 70’s. If you got beyond the 80’s her name is on the book, but she had died by that time. They are rewritten and not as good. I currently use the 1962 “Modern Algebra” by Dolciani, Berman and Freilich, which is available on Amazon fom $25 up though prices vary with supply and demand. I used it in my 8th grade algebra class. You do have to supply some explanation because her written explanations are somewhat formal at times. (The books were written in the 60’s New Math era so contain a bit more set theory than is really necessary. But not too bad, and one can easily skip over some of the overly formal sections.) The word problems are excellent and are distributed throughout the book; they use whatever skill/procedure the chapter they appear in is using. Thus, word problems in the chapter on fractions use fractional equations, etc. Excellently laid out and presented. Some of the used books contain only odd numbered answers, so keep your eye peeled for teacher’s editions.

        Algebra by Paul Foerster. This is also an excellent series. I recommend the ones written in the 80’s, early 90’s. He does a good job in scaffolding word problems.

        As far as answer keys, ordering from Amazon is hit or miss. Some of the books have them; others do not, so you have to pick and choose. Generally the books I find useful are the older ones, so they will be used, and the number with answer keys are limited.

        I agree with Tara about JUMP Math and am going to be using it for my 7th grade math class this year.

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  2. This is another rote answer from those claiming to know what non-rote learning is in a subject area they know little about. OK. I’ll bite. Where are the success examples after 20+ years of their reform math nonsense? Why are all successful high school graduates in math created from traditional (AP/IB) math classes and not from curricula using their silly and simplistic ideas? Why do all successful math graduates now have to get help at home or with tutors to survive or recover from K-6 math?

    My traditional math experience (50’s and 60’s) ended memorization with the times table. However, drill did cause us to use understanding “tricks” like remembering 6*7 from the more easily remembered (at least for me) 7*7 – 7. As for “rote” long division, like 23 divided into 479, how many 23s are in 47? It was never rote and now pedagogues are just using these ideas to cover for low expectations. They want to eliminate testing and assurance of skills with some sort of natural IQ learning, while ignoring all of that assurance and drill that goes on at home. Just ask us parents. A survey of the parents of the best students will give you the answer.

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  3. re: this is a response to Marc. I’m sorry you feel disconnected from Barry’s endeavours. I find that besides working as a middle school math teacher, writing books, maintaining a blog, and keeping a visible online presence, might ensure that Barry’s doing the most that he can for his readership, and for regular parents out in the blogosphere. I’m a parent to 2 kids; I get where the frustration is when it comes to determining how to help our kids. But I don’t think blaming others is the right path here. There are multiple resources, and information readily available to help in this situation; all one needs to do is put in a bit of time researching the issue.

    Saxon Math is great. If kids are a bit more advanced, try Singapore Math. Try any and all; see what works. Goto http://www.wisemath.org for more resources. Sign their petition while you’re there. INSIST that your children learn properly by speaking to your school principal and teacher. Go to a PTA meeting and tell them that as well. And also make a presentation at your School Board to show how frustrated and upset you are about your children not learning proper arithmetic in the classroom.

    Do all those things, like all of us out here advocating for better math instruction, have already done. Lobby your State and Local Representative. Start a writing campaign…the list is endless. But to sit back and suggest that one individual isn’t doing enough for the parent community…how is that helpful?

    I encourage you to start a blog. Write an op-ed. A magazine article. I look forward to reading your articles. Until such time, say THANK YOU to people like Barry. He’s the one who’s trying really hard for more than just his own kid…let’s all take a lesson from that.

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    • Hi Tara,
      I’ve signed the petition.

      As far as speaking to teachers, I tried speaking to my kid’s teacher, but was blown off.

      I didn’t mean to blame Barry for anything. The reason I’m encouraging Barry to publish a list of books is that Barry has a unique position in that he has written books about math education (he has made his position clear), he has a math degree (not a math education degree), and he has worked in the private sector. If I had found him before I’d started my kids on Saxon math I would have followed his recommendations. I think that if he had a web page of recommended books including version numbers and links to where people could buy the books including solution manuals, that would be very helpful and carry great weight. My recommendation of a book wouldn’t help anything.

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  4. For my “math brain” son, I found that I had to help him directly in K-8, but not once in our high school that uses a traditional AP Calculus track and textbooks. The kids I tutor need help because of the poor, low expectation Everyday Math (EM) in K-6, not because of the traditional math in high school. EM specifically tells teachers to keep moving and to “trust the spiral.” This fits in with full inclusion, not the higher expectations required to keep all STEM career doors open. These low expectations, including on homework, failed to teach them how all math from 7th grade onwards generates success – with mastery of individual homework problem sets.

    When I tutor high school kids, I find they “understand” the material when I break it down and tie it to what they previously learned, but then I tell them that they have to go back and do every homework problem completely (!) – to prove to themselves that they can translate that understanding into results. Words are weak understanding and doing lots of problem variations is strong understanding. Pedagogues may complain about drill and repetition, but they need to study all traditional math textbook problem sets to see the variations and increasing difficulty of the problems. They are fundamentally wrong in their views of drill and kill. What my tutees never learned in K-6 was that full understanding comes from doing ALL of the nightly problem sets. If you open up EM to any page, you will see reasonable math problems. What you never see, however, are high expectations and thorough coverage of material that requires any sort of sufficient individual homework and mastery. My son had 4/5 problems to do at night and that’s all they ever expected. They trust the spiral (circle), nature, and IQ and then blame the kids (and parents, poverty, peers) for lack of success and point to the kids who are successful. However, they never ask us parents of their best students what we had to do at home. Never, but it’s a very easy thing to do.

    The fundamental incompetent flaw of K-6 math is that they try to break the connection between understanding and mastery of complete homework sets. They see it as repetition (drill) and not variation even though it’s right there in plain sight in all proper traditional math textbooks. That’s what creates students ready for STEM careers – IF they can somehow manage to survive silly, low expectation K-6 math.

    What parents have to do in K-6 is not really onerous. I used Singapore math, but not rigorously. I had mixed feelings about some of their things like bar models. I mostly used worksheets and taught him what I knew was important in terms of mastering the basics. K-6 schools now officially care only about CCSS proficiency, but even the highest level of CCSS success leads only to no remediation for college algebra. That low slope starts in Kindergarten. If a student is capable of more (due to IQ or effort) they are on their own. No amount of silly low expectation engagement will get the job done naturally.

    The big K-6 hurdle comes in the second half of 6th grade when most schools decide on the 7th grade math track for your child. Parents need to know how that works and make sure that their kids get to a proper Pre-Algebra class in 7th grade followed by a proper Algebra I class in 8th grade. If your school does not have those classes, then you have a much bigger problem. As an overall K-6 parent job (done over time), preparing for Pre-Algebra in 7th grade is not difficult. If you have to cover for bad middle school math to get your child ready for Geometry as a freshman in high school, then you might need to pay for tutoring or be good at algebra. We parents were able to drive out CMP in our middle school to bring in proper Glencoe algebra textbooks because we could show that they offered no proper curriculum path to Geometry as a freshman. It’s one thing to blab on about fuzzy math and understanding, but another thing to explain away a proper curriculum path to high school. Our middle schools were forced to deal with their connections to both Geometry and a second year language class as freshmen in high school. Once my son was in the proper AP Calculus track in high school, I never had to do a thing.

    As for college prep, it depends on what you want to major in, but most full STEM departments like to see success (grade of 80+) in calculus. Success depends on having few skill gaps and your ability to do full nightly homework sets. Beyond that level, then you have to aim for AP Calculus BC, SAT II Math success, and involvement with the yearly AMC test, because colleges want to see those scores.

    Now that my son is in college and one of his majors is abstract math, what serves him best is his ability to grind out the nightly problem sets. When we were at MIT for a tour (he didn’t go there), they made a big point about “hacking” and creative sorts of learning outside of class, but what it’s all based on are their huge numbers of P-Sets in traditional math classes. That’s what all college STEM programs have – huge numbers of individual p-sets. MIT might promote their engaging “Splash” events for high school students, but what will get them into MIT are their grades on traditional classes and the AMC.

    K-6 educational pedagogues are all “Splash” and no P-Sets. They just don’t “understand” what’s going on. They just have rote knowledge and live in their own echo box.

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