Here's a more personal follow-up to the topic of educational malpractice. A couple of months ago, I wrote about working with a group of French African immigrant children whose school-based math education consists of Everyday Math. I noted how they weren't able to subtract 91 from 1000 because they didn't know how to borrow (regroup) across multiple digits.

As I noted then,

You can’t blame the mathematical deficiencies of these 4th and 5th graders on their parents: both the private school and the after school program select for parents who care about education. You can’t blame it on the kids: my kids, who clearly wanted to learn, had been admitted [to our after school program] in part based on their behavior.You also can't blame it on language problems; these kids are fluent in English. In fact, there's really only one thing outside the Everyday Math curriculum that one can possibly point a finger to, and that is that these immigrant parents (many of them don't speak English) don't realize what many native-born parents already know: namely, that they can't count on the schools to fully educate their children.

So these kids are a case study in what happens when you leave math instruction entirely up to Everyday Math practitioners. And the answers to this question are slowly coming in.

For several of the 5th graders I work with, it turns out that not only do they not know how to borrow across multiple digits; they also don't know their basic addition and subtraction "facts." In other words, they don't automatically know that, say 5 plus 7 is 12, or that 15 - 8 is 7; instead they count on their fingers.

This got me thinking about addition and subtraction "facts." Back in my day, there was no issue of kids learning these facts as such. Yes, we memorized our multiplication tables. But we never set about deliberately memorizing that 5 plus 7 is 12. Why? Because the frequency of the much-maligned "rote" calculations we did ensured that we, in today's lingo,

*constructed*this knowledge on our own.

Back in my day, a typical third grade arithmetic sheet looked something like this:

And a typical fourth grade arithmetic sheet looked something like this:

But in Reform Math programs like Everyday Math, such pages filled with calculations are only occasional, and each problem involves a much shorter series of calculations. Here's a set from 4th grade Everyday Math:

Each multi-digit addition problem amounts to a series of simple addition problems. For example, adding two two-digit numbers involves adding at least two pairs of numbers; three if one is regrouping. Adding three three-digit numbers can involve 8 iterations of simple addition. Some of the problems in the second traditional math sheet involve as many as 17 iterations of simple addition.

In the traditional 4th grade math scenario, we may have had 25 problems per day like those in the first two sheets above, 5 days a week. With Everyday Math, you might get, at best, 25 problems like those in the second two sheets above

*per week*.

Putting it all together, the resulting difference in the amount of practice with basic addition "facts" is quite large. 5 (days) times 25 (problems) times (say, as an average of iterations of simple addition) 10 for the traditional math curriculum versus 1 (day) times 25 (problems) times (average iterations) 3 for the Everyday Math curriculum. Assuming I'm not screwing up

*my*arithmetic, that's 1,250 vs. 75 basic addition calculations per week. No wonder so many of those who are educated exclusively through Everyday Math don't know their "addition facts" by grade 5!

Ah, but surely their "conceptual understanding" is deeper. Note the calls for "ballpark estimate" at the bottom of each Everyday Math problem, where traditional math simply has you calculate. Stay tuned: in my next post on this topic, I'll discuss the state of conceptual understanding in my Everyday Math mal-educated 5th graders.

## 14 comments:

In my state, Virginia, there's a huge fight going on to remove calculators from the state end of course exams. The teachers and state Dept of Ed are fighting the move, I believe because of what you demonstrated here. If you can give kids calculators and they're allowed to use them on exams, then you don't have to teach arithmetic beyond the "understanding" achieved with programs like EDM and Math Investigations. If you take the calculators away, then you have to teach arithmetic.

I was in a local bakery when the power went out and, even with a calculator, none of the four counter employees (20s-30s)could calculate sales tax. They clearly had no conceptual understanding whatsoever. It took me 10 minutes to show them how to total purchases and multiply the total by 1.06 to get the amount due. They had no idea how to set up and enter data or that 6% could be represented as .06. Their reaction? "Wow, you must be a math teacher!" (no, just someone who learned 6th-grade arithmetic)So much for "higher order skills" and "critical thinking"; a large, repeated dose of "drill and kill" would have been useful for them. Yes, the district uses Everyday Math.

They clearly had no conceptual understanding whatsoever.Interesting, considering that EM is supposed to provide that. An example of how "procedural understanding" is closely tied to "conceptual understanding".

They also had no number sense. I stayed a bit to help them with other customers, and they made repeated data-entry mistakes, then failed to realize the resulting number could not possibly be correct; complicated stuff like the (supposed) amount due was double or triple what it should have been, because they had entered a decimal point wrong. $10 purchase with 6% sales tax adding up to $16. Beyond pathetic. I'm temped to take that anecdote, and a few others, to the School Board meeting.

To be fair, if they were good students, maybe they wouldn't be working retail food service.

To be fair, if they were good students, maybe they wouldn't be working retail food service.To be fair, there was a time when people working retail food service could do those calculations.

I am a regular customer and the staff has always seemed to be of normal IQ and are always very pleasant and helpful. However, if kids learn that they will be passed along regardless of what they learn, many (if not most), won't put in too much effort. As a summer hire, I worked a cash register in the 60s and had to do all the calculations by hand. Mistakes meant my pay was docked. None of the people working retail in my childhood small town had more than a HS education and probably some of the older generation had less. The were all, however, literate and numerate. I never saw situations like the one I described and I never saw signs like Happy Holiday's", "Celebrate and Festive With Us" and "Sign-up Now For Classes" (at a local CC, no less). Maybe kids aren't putting in much effort at school, but schools aren't demanding much and aren't requiring mastery before advancement. Also, their curriculum and instructional methods are weak or seriously flawed.

I am doing some math tutoring at a school that uses Everyday Math. The number sense of the lowest two thirds of the class is mind-boggling. Fourth graders are counting on their fingers for almost all addition and subtraction facts. Some of them start at the number 1- they don't even count on. It's so depressing.

My son's school uses Everyday Math. I am so thankful he didn't start there until fourth grade, so he knew all four operations when he enrolled. The teachers were kind enough to let him use traditional algorithms.

We are in a district using Everyday Math, but with New Common Core implementation they are switching to a new program called Stepping Stones. Anyone heard of it? Looks like it will be light years ahead of Everyday Math.

My daughter spent K-2 learning Singapore Math. We moved to a district utilizing Investigations, and she's rapidly losing interest. Here is an example from her mid-3rd grade "homework" (and only 3 problems at that):

1. In Kelley's picture there are 6 shirts. Each shirt has 6 buttons. How many buttons are there altogether?

For each problem, they are supposed to write a multiplication equation, solve the problem, and show the solution. She had to show 4 (4!!) methods - a picture (seriously??), repeated addition, skip counting, and "grouping." Isn't this a fact that she should just "know" at this point? 6x6=36? She has to spend (waste) her time drawing little cubes with a "6" in each cube, write out the skip counting, etc....where is just the basic math fact mastery? This stuff is just unbelievable.

It is NOT necessary to "borrow (regroup) across multiple digits" to subtract 91 from 1000. It is NEVER necessary. The author should learn about the so-called "Austrian method" of subtraction, see http://en.wikipedia.org/wiki/Subtraction (and google it).

"It is NOT necessary to "borrow (regroup) across multiple digits" to subtract 91 from 1000. It is NEVER necessary."

That's right. It is never necessary. One can also use blocks, cuisenaire rods, tally marks, or one's fingers. But for efficiency and mastery, some sort of consistent algorithm is necessary. The Austrian method is one candidate. Regrouping is another. As far as elementary school math is concerned, picking one of the various general algoriths and sticking with it until students achieve mastery is preferable to teaching multiple methods all at once. Or to teaching no general algorithms to any level of mastery.

The Austrian Method of subtraction uses regrouping. It just uses different regrouping. One could say it loans instead of borrowing. Or that you carry under rather than carrying over. But it's incorrect to say that it doesn't use any regrouping.

I googled the Austrian method. I don't see that it is essentially any different from "the" standard algorithm for subtraction. It seems just to use a different notation. By the way, for the example of 1000 - 91, for that case I would hope kids would just take away 100 then add 9 back in. It is good to recognize when mental math works and when you should plug and chug. It is not so easy to learn to make this decision. Sometimes my 10-year old wants to take a real long time trying mental strategies when he would have been done long ago if he had stacked 'em up. But I think there is real value in having kids learn to say: is this an easy case or a general case? Sometimes the standard algorithm is the easy way, sometimes it isn't. Learning to tell one case from another leads to greater fluency.

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