We’ve always been at war with Eurasia, Dept.

What with Robert Pondiscio’s welcome and well-written article extolling the benefits of Direct Instruction (Zig Engelmann’s method for instruction) and thereby praising direct instruction in general, there are indications that others may be following suit.  I just read a blog piece by a math teacher who has reached the eye-opening conclusion that conceptual understanding doesn’t always have to precede procedural fluency. In fact, procedures may not be the bogeyman that math reformers have been saying they are for the past hundred years or so.

And it isn’t as if math teachers have routinely refused to teach the conceptual understanding. It’s just that if you’ve spent any time at all in a classroom, you will have noticed that your students glom on to the procedures.  And unless the conceptual understanding piece was part and parcel of the procedure (as is the case with adding and subtracting with regrouping) few if any remember the underlying concepts.  This has led to math texts now “drilling understanding” by making students do the conceptual understanding piece as if it were the algorithm itself; i.e., 3 by 5 rectangles and shading the appropriate parts to represent 2/3 x 4/5 as a means to “understand” what fractional multiplication is.

The belief still persists that in order for students to understand, math must be made relevant.  It just can’t be that if students know the procedures and can do them, and use them to solve problems, they really do not care if the problems are relevant or not.  And so we have statements like this which appeared in a recent Education Week testimonial/polemic that passes as evidence-based, research-based, relevance-based, brain-based and any other kind of “base” you can think of:

Math lessons, on the other hand, have historically focused less on real-life connections. Like many students, I excelled in math by memorizing rules and tricks. In college, I trained to teach social studies, but became a math teacher by accident because I had earned enough math credits to qualify for a math teaching certification.

Never mind that the author of the article may have benefitted from what she calls “rules and tricks”.

In any event, it appears that there may be more teachers who had insisted we are at war with Eastasia now coming out from the woodwork to say that we’ve always been at war with Eurasia, though it comes out more like: “Hey, procedures aren’t that bad, and most kids don’t really get the understanding til later.”

I will leave you with the words of a math teacher I know from New Zealand who puts it this way:

A few years back I started explicitly telling my students “I don’t care if you understand it, provided you can do it” when they complained that they “didn’t understand”. I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.
Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.
Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding because they never have.  To the fury of many reformers, most kids really don’t want to understand very much.

 

7 thoughts on “We’ve always been at war with Eurasia, Dept.

  1. I just read a blog piece by a math teacher who has reached the eye-opening conclusion that conceptual understanding doesn’t always have to precede procedural fluency.

    Funny: I just wrote a blog piece about the same thing!

    https://problemproblems.wordpress.com/2018/02/13/learning-is-weird/

    The way I think about procedures/concepts reminds me of that old nugget of writing advice, “no ideas but in things.” Concepts and procedures should be hard to disentangle in the classroom, I think.

    Also, check out this research article that the researcher shared with me on Twitter, it’s interesting stuff:

    Click to access ATME_Rittle-JohnsonandAlibali_1999.pdf

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  2. Houston, we have a problem. DI = Direct Instruction or Differentiated Instruction?

    It’s the first, because when my son was growing up in a differentiated instruction full inclusion classroom, we quickly learned that their DI was really “differentiated learning (DL). They even called it that. “Trust the spiral” and process. Reaching your highest level of educational development and opportunity was thought to be natural. I can’t tell you how many times we heard that “Kids will learn when they are ready.” This is another way of blaming the student, peers, parents, society and poverty. This philosophy leads them to what I call their two+ generational solution. They are thrilled if little Urban Suzie is the first in her family to get to the community college. Apparently, she will then have the knowledge to do for her kids what we parents had to do with our kids at home or with tutors to achieve a one-generation education. El Sistema clearly shows that the solution can be one-generational if you push and focus on mastery of content and skills from the earliest grades. Almost all of the All-State musicians in the US have taken years of private music lessons. It wasn’t PBL band that made the difference. Band is neither necessary or sufficient. Now, math requires home or tutor private lessons for mastery of skills. I HAD to do that for my “math brain” son.

    I call this a systemic academic turf problem. If you are a K-6 educator and go to a cocktail party, do you want to talk about how well you teach content and skills and ensure mastery with homework and tests? What’s worse, not only do they claim hegemony over the teaching process, but they claim ownership of what constitutes understanding in math. You don’t have this problem with high school AP calculus or AP English teachers. They teach directly and they are certified in their subject area. That’s their turf and they understand what’s needed. That’s also why all high schools and colleges are, and have always been (for most cases) bastions of direct instruction.

    Hello?!? Constructivists first have to explain why DI still dominates high schools and colleges and why it all changes for K-6. In our schools, parental (and real life) push back caused the elimination of CMP math in middle school to be replaced by direct instruction using proper Glencoe textbooks. They are losing the war and the boundary between reality and fantasy land is now between 6th and 7th grades.

    Also, my view is that this is not a battle of skills versus understanding or whether understanding can be built on top of mastery at a later date. There are many levels of understanding, and there is a LOT of low level understanding that comes with mastery of even the most basic skills. Nobody EVER masters the basic algorithms with no or rote understanding. People just don’t see the understanding because they now take it for granted.

    Later, in algebra, understanding is not possible without mastery of skills. Words don’t show understanding. Homework and tests where you have to do all variations of problems show understanding. Invert and multiply is never rote – there are very simple explanations. The proof of understanding is not whether you can explain that in words or even in a proof, but whether you can show how it applies to all variations of rational terms. Simplify (a/b + c)/(1/bc). In my algebra class, everyone had to do only one step at a time and had to put the rule or identity next to each line to justify the change. This was for direct instruction.

    All constructivists directly teach or they will get very little done in the class. We never hear about how they do that and how they assign and provide feedback on homework and tests. We never hear about grading and how they push (or not) and how they determine whether kids move on to the next grade. In many K-6 schools, they almost never hold anyone back. Let’s not look at that systemic problem, but just focus on their claim of better understanding for all. More with less, naturally. It’s not quite believable..

    Just asking. For the last 20 years.

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  3. It took me years and years, but I finally realized that the reason I had trouble learning something was that I did not have the right teacher or textbook. I also realized that really learning something required incremental mastery of the material, as in textbooks with unit chapters and homework variations. I wonder why constructivists never refer to homework and p-sets as hands-on learning? Do they really think that practice is only about speed? We all love engagement, but engagement really has to support mastery. It doesn’t automatically drive mastery after the fact. Mastery generates engagement, not the other way around.

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