While I was cleaning up my office yesterday morning, I uncovered (literally) the Teacher’s Reference Manual for Everyday Mathematics ®, Grade 1-3, 2007 (the edition currently used in our schools). Over the last few months I have marked various pages for ease of reference to material which I find particularly interesting, where you can define interesting any way you want. I flipped to the page (page 96) which discusses Algorithms and Procedures. As I read, I “reacquainted” myself with all of the concern and angst associated with this failed program.
The authors include the following paragraph at the start of the section on Computational Algorithms (section 11.1.1, page 97): “Several teachers have asked the Everyday Mathematics authors about the role of computational algorithms in elementary school mathematics. Should traditional paper-and-pencil algorithms be taught? Should children be expected to use these algorithms to solve complex computational problems? Should calculators be used in the classroom? If so, in which circumstances and under what conditions?”
The authors then spend the next one and one-half pages discussing what, to me, are obvious answers (yes, yes, no, with no answer needed for the fourth question).
Their answer, however, included two stories, “the need to rethink the school mathematics curriculum in light of the widespread availability of calculators and computers outside of school,” a discussion of estimation and approximation, a discussion of “buggy algorithms”, and a rationale for introducing different algorithms. When they finally got to the end of the section, which I took to be their position, which they call a “moderate” position, they conclude that :
“(D)uring the early phases of learning an operation, Everyday Mathematics encourages children to invent their own procedures. Children are asked to solve arithmetic problems from first principles about situations in which operations are used, before they develop or learn systematic procedures for solving such problems. This helps them to understand the operations better and also gives them valuable experience solving nonroutine problems.
Later, when children thoroughly understand the concept of the operation, several alternative algorithms are introduced.”
As we have come to expect with Everyday Math, there are plenty of cautions and caveats which make it appear that they have considered all of the angles. For example: “Children certainly still need:
· To know the meanings and uses of all the arithmetic operations in order to function in the practical world and to succeed in mathematics in high school and beyond;
· To know the basic addition and multiplication facts automatically, especially to solve mental-arithmetic problems in our technological society [I guess subtraction and division facts don’t count anymore];
· To understand and be able to apply paper-and-pencil algorithms for addition, subtraction, multiplication, and division of whole numbers, decimals, fractions, especially in an environment of standardized testing [heaven forbid that you would actually have to know this outside of a test].”
If you are looking to find the core of what is wrong with Everyday Math, look no further. There are so many issues with this philosophy (a philosophy reflected throughout the program), it is hard to know where to start. Let me begin with:
1. Inventing incorrect algorithms, which stick with the child
2. Inventing algorithms which do not apply to the generalized case.
3. Putting the burden on teachers to correct all of the errors, when a teacher may not have the background to realize all of the flaws (see number two).
4. The introduction of calculators.
5. The introduction of estimation and approximation, with a justification that they need to be able to judge the output of calculators and computer spreadsheets.
6. The concept of solving from first principles for a grade school aged student.
Regarding the first three, several teachers with whom I spoke said that this was not a problem for them, since they don’t teach it that way to begin with. My guess is that these are the more astute teachers, who recognize the deficiencies of the Everyday Math program. This does raises two questions, however. First, why not use a curriculum which allows proper teaching of these critical elements of arithmetic? Second, if the teacher is not astute enough to recognize the deficiencies, are they really equipped to fix the errors?
And speaking of fixing incorrect invented algorithms, the authors of Everyday Math wrote: “Research carried out in the past 30 years by Kurt Van Lehn and others has shown that many children develop “buggy” algorithms that resemble standard procedures but do not work properly. In subtraction, for example, some children always subtract the smaller digit from the larger digit. Van Lehn has shown that bugs such as this develop because children are trying to carry out procedures they don’t understand and can’t remember well enough to reproduce accurately. Procedures that are well understood, on the other hand, are more easily recalled, are more easily “repaired” when they are not recalled accurately, and are more easily modified to fit new situations.”
So EDM says invent first, and solve problems from first principles, “before they develop or learn systematic procedures…. This helps them to understand the operations better…” A little later they say, “prematurely teaching traditional paper-and-paper algorithms can foster persistent errors and “buggy” algorithms…” But the research they quote says by inventing first, they introduce bugs because they don’t understand. This is a perfect example of the authors of Everyday Math getting, as an old boss of mine use to say, “tangled in their own underwear.”
Correct me if I’m wrong, but isn’t it the teacher’s role to teach a topic so that the child understands? But that is not how Everyday Math works. The student needs to learn for himself.
Interestingly, the authors also highlight research done by Van Lehn, who “said this about using the traditional subtraction algorithm in some of his research: ‘(O)rdinary multidigit subtraction….is a virtually meaningless procedure [for] most elementary school children… When compared to procedures they use to operate vending machines or play games, subtraction is as dry, formal and as disconnected from everyday interests as the nonsense syllables used in early psychological investigations were different from real words. This isolation is the bane of teachers…’ ”
So the researcher doesn’t like the traditional subtraction algorithm, because it is meaningless and dry (i.e., from the Everyday Math school of fun math, no doubt). The authors of Everyday Math make the researcher sound as confused as they are. And you can provide your own joke about what vending machines have to do with this topic.
But it gets more confusing. The Everyday Math authors go on to say “(T)he program aims to make children active participants in the development of algorithms. Such participation requires a good background in the following three areas:
· The decimal system for number writing In particular, children need to understand place value.
· Basic facts To be successful at carrying out multistep computational procedures, children need to know basic facts automatically.
· The meaning of operations and the relationship among operations To solve 37 – 25, for example, a student might reason What number must I add to 25 to get 37?”
Since Everyday Math fails miserably at getting anywhere near mastery of basic facts, the whole premise of this invented algorithm route falls apart. And doesn’t having “a good background” mean that you understand? And if you understand what addition and subtraction mean, should you really have difficulty with learning a method (the efficient traditional algorithms, developed over centuries by intelligent people) for adding or subtracting multi-digit numbers, instead of spending (wasting?) time inventing a method that may not be correct, and may not generalize?But back to our list. Calculators! The authors of Everyday Math say “In the modern world, most adults reach for a calculator when faced with any moderately complicated arithmetic computation. This behavior is sensible and should be an option for children, too.” (Wait for it) They then backpedal and say: “Nevertheless, children benefit in the following ways from developing their own noncalculator procedures:
· Children are more motivated when they do not have to memorize traditional paper-and-pencil algorithms without understanding why they work [stop and think about this last sentence and see if it means anything to you]…(C)hildren generally understand their own methods, as obscure as they may be to others.”
Notice, “their own” method, not traditional methods. So calculators are okay, but children benefit from using invented algorithms.
So we have calculators and computers, and we use apps and spreadsheets. Then, in order to check that these tools are giving us the right answer, we need to be able to estimate and approximate. However, if we just spent the time wasted learning estimation and approximation (which should be learned later, but not in grades 1-4) actually practicing our number facts and algorithms, we not only would learn the facts and procedures to mastery, but we would be developing number sense. This would make learning estimation and approximation very easy, at the appropriate time. And it should be noted that Everyday Math fails as a curriculum for estimation and approximation. How do I know? Check the “estimating solutions to problems” strand results on the grades 3-5 CMT’s. The only scores that come close to being as low as this strand are the “approximating measures” and “mathematical applications” strands. The mathematical applications strand is education-speak for solving word problems. Isn’t it strange how the lowest scores are in the areas that Everyday Math considers important?
I must admit to a small moment of panic when I saw the words “first principles.” I flashed back to several engineering and graduate school classes, where we had exam questions asking us to begin with first principles and prove this, or solve for that, or show how to do something. Example: Imagine you are stranded on a deserted, sandy island. Describe step-by-step what you would do to manufacture a certain silicon-based polymer (polydimethysiloxane). Be sure to include all chemical reactions necessary, and indicate where you would source the materials required for both the synthesis of the polymer and the production machinery. You have three blue books (exam notebooks) and three hours. Go.
Now I recognize that this may not be exactly the type of learning experience the authors of Everyday Math were getting at when they used the phrase in the manual (or maybe they had a similar experience in education school, and are now getting back at the world), but solving arithmetic problems and inventing algorithms from true first principles for first through third graders is an interesting, albeit comical, notion. This is like asking a student to act like an expert, without the appropriate knowledge and experience.
In summary, I am not sure if the authors ever clearly answered the questions asked, and if they did, I am sure the qualifications and provisions they attached to those answers made any conclusion meaningless.
Oh, and my office is still not clean. Another thing to blame on Everyday Math???