One of the many popular tropes of the internet is referring to someone who offers criticism of a piece as a “troll”. Not that there aren’t bellicose and baiting-type comments, but the word is often applied to any type of criticism as in : “If you don’t agree with this, you’re a troll.”

So I thought I’d provide some comments I received in private regarding the discussion following Keith Devlin’s piece in the Huffington Post, as well as what I wrote about Devlin’s piece in the post below this one. Chances are pretty good that these commenters would be viewed as trolls by many, so if what they have to say is disturbing, I won’t hold it against you if you want to call them that. If it helps you to ignore what they have to say, go for it! They won’t mind.

**Troll No. 1:**

With the consultants I’ve met, who always push this stuff and insist that kids aren’t fluent unless they can explain everything to you, it seems that they themselves just figured out that there are reasons behind procedures in math as adults. Then they’re angry that their teachers (supposedly) didn’t explain all these things to them. They’re certain that they would have liked math more and done better if only their teachers would have focused on understanding. So, their mission is to make sure that all kids are forced to explain their thinking at every step. Pure torture, really. Funny thing is, that the understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.

**Troll No. 2**

**Troll No. 3**

Does what they do work? Can they even define what success means? I’ve seen no explanation. They don’t show how any sort of discovery for one problem translates into other areas. Discovery in basic fractions will help very little with factoring quadratics, let alone rational expressions. It also takes a lot of class time and few topics can be covered. What “deep” concepts helps students solve all of the variations of Distance/Rate/Time problems? Draw a picture? Think backwards? How about studying the governing equation, seeing how the variables relate, and doing a whole lot of individual homework problems of all sorts of variations?

In preparing for the AMC math competition, you don’t just study concepts or some general top-down process. You prepare by going to their web site and doing ALL of their past problems. There are tricks and understanding subtleties that will never be discovered with some general process. By working on all of these problems, you start at a much closer solution point when you encounter a new variations. This is how success for any highly-valued math test is created. You don’t have to have any conceptual knowledge if you can define the variables and equations. Just let the math give you the understanding.

The people pushing these new ideas need to show how a full curriculum based on their ideas works, not just some sort of small delta success for kids already damaged by fuzzy K-8 math curricula. They define their own version of success without regard to the needs of where these kids are going after high school. A local highly-regarded vo-tech school in our area requires the AccuPlacer test. Do the techniques of these reformers help those students do better on that test than a traditional process – done well?

I can’t imagine people calling any of the above three comments “troll” comments (although I totally agree with you about people calling others “trolls” if their opinion is different). These are all great and thoughtful comments.

I suggest providing a link in your post above to Devlin’s piece in the Huffington Post (I haven’t seen it).

Regarding the above comments, there is something not mentioned in any of them, which is what I see personally. I think there are a LOT of parents who don’t like their children having homework, or having to SUPERVISE the children getting their homework done. I think THIS is the real reason why math workbooks (at elementary level) have cut the nightly homework problems to half or one-third of what was normal a decade ago. In the first decade of the 2000’s and before, a normal page of elementary school problems for one night’s lesson was about 30 problems, twenty-eight of which would be calculations (subtraction, addition, or multiplication, for example), and two of which would be story problems at the bottom of the page. The new workbooks contain ONLY 10-15 problems for the lesson; each lesson contains about three concepts, and 3-4 problems are devoted to each concept. In my experience, this is JUST NOT ENOUGH practice to really master these concepts.

The people who are focusing on “understanding” are of several types. There are some good things about understanding. But what SOME of them are attempting to do is to GET RID OF REPETITIVE PRACTICE because THEY DIDN’T LIKE IT. Well, none of us liked it. But it was how we MASTERED problems.

The other problem is that most teachers seem to spend a lot of class time devoted just to children working problems on their own and then they grade the problems, and children look at their grade only, but don’t learn from their homework because they don’t really know what they did wrong, even if the teacher has marked it.

What I always did in my classes and still do in my tutoring is spend MOST of my class time (half to two-thirds) GOING OVER homework problems in great detail, showing students HOW and WHERE they went wrong.

I used to give my third-graders an A+ for having all their homework DONE and SHOWING ALL THEIR WORK, EVEN IF IT WAS WRONG. Why? Because then they showed up at class READY TO LEARN. We went over answers and children marked answers (in INK, with a BIT A+ already at the top of their page, so that they didn’t erase work they did to change it). If their work wasn’t FINISHED, or if they didn’t SHOW their work, they got a BIG RED F at the top of the page EVEN IF ALL THEIR ANSWERS WERE CORRECT. (Any time this happened, they had to get the page signed by their parents.) They ALL did their homework EVERY NIGHT and WELL after the first few weeks of this!!!

We couldn’t work every problem, but we worked two in each row, chosen by a show of hands of the two problems most missed by lots of people. Then we’d go through all the WRONG answers, and I’d be able to show them on the board HOW they got that wrong answer. For example, one of the main errors in grade three subtraction of three-digit numbers is to subtract the ones place, mistakenly add the tens place, and subtract the hundred’s place. When students see THEMSELVES making the SAME mistake like that in several problems, they really learn quickly what to watch out for and leave school energized. It also makes it WORTH IT for them to DO their homework, and they are not just looking for the GRADE (which they already HAVE), but are instead really looking to learn something and become internally motivated. Teaching this way, most of my students got A’s and B’s on their tests (on which their report cards were really based).

This idea for teaching wasn’t my own–I went back to America after teaching a few months in my first Grade 3 class and had a chance to observe for a week in the elementary school I’d attended as a child, and this was the system they were using there. I adopted it coming back here and it worked SO WELL.

I think it’s REALLY IMPORTANT that students get SUFFICIENT PRACTICE in concepts and ADEQUATE FEEDBACK on the work which they ARE doing. In MANY math classrooms BOTH of these are now missing.

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The link to Devlin’s article is now in place; thanks.

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This statement really caught my attention in Devlin’s article, because I think he is WRONG about it.

“The shift began with the introduction of the electronic calculator in the 1960s, which rendered obsolete the need for humans to master the ancient art of mental arithmetical calculation.”

Children DO need to learn how to calculate. Most children are CONCRETE thinkers; only a few can think abstractly in elementary school, and about a third can in middle school. Many high-school students, and MANY ADULTS STILL cannot not think very well abstractly. It’s a brain maturation issue, and different brains mature at different rates.

In my many years of experience as an elementary teacher, the “number sense” which is lacking is best obtained by BOTH concrete mastery and practice IN ADDITION to discussion. But it is NOT obtained by SUBSTITUTING DISCUSSION and handing calculators to children before middle school!

Two-thirds of students in elementary school have problems with NOT KNOWING IF AN ANSWER MAKES SENSE. In story problems, they may calculate an answer, but no one has taught them to ADD THE ADDITIONAL STEP BEFORE DOING THE CALCULATION simply to ask themselves IF THEIR RESULT SHOULD BE A LARGER OR SMALLER NUMBER THAN THEY ARE STARTING WITH. The first way to check if an answer makes sense is to compare the result of the calculation with that ORIGINAL PREDICTION that it should be larger or smaller. If that doesn’t match, then something is wrong, something doesn’t make sense.

For example, many students would not recognize that something was wrong with the following.

John has 90 apples that need to go in equal amounts to three different parties. How many should each party get? A student without number sense might say, 90 x 3 = 270. They don’t recognize that you wouldn’t be able to give 270 apples to each of three different people when you are only starting with 90. This if because they have NO CLEAR PICTURE IN THEIR HEAD of what is going on in this problem, or what someone is trying to do.

The other problem that is hurting number sense is that instead of learning about a particular type of problem and practicing that until THAT TYPE IS RECOGNIZED AND MASTERED, instead, DIFFERENT TYPES of problems are MIXED UP and the student ends up mastering NONE of them (which is what happened to me when I was young, with story problems).

Many teachers in elementary school SKIP the story problems all together and don’t do them at all, probably because they don’t know how to explain them effectively themselves.

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I certainly wouldn’t call these comments ‘troll’ comments either. They seem to be teachers with some good insight into the issues. Unfortunately, they have to remain anonymous because holding such opinions in a professional forum, will cause them much grief. I fully agree with their statements and sentiments and as a retired teacher am now free to use my name. It is teachers, who work with students, day in and day out, who are best equipped to discern what children need. Experienced teachers, whose professional opinions cannot even be spoken, are symptomatic of much graver problems.

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Thanks for your comment; very good point.

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Reblogged this on The Echo Chamber.

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