The following report on Everyday Math
was submitted to the California State Board of Education.

EVALUATION OF SUBMITTED CHANGES
FOR EVERYDAY MATHEMATICS

July 5, 1999

by

David Klein
Professor of Mathematics
California State University, Northridge

I was asked by the California State Board of Education to review materials submitted by Everyday Learning Corporation which are intended to address shortcomings of its K-6 basic adoption submission, Everyday Mathematics. I was also asked to consider the program as a whole.

I am in possession of the Grade 3, and Grade 5 Everyday Mathematics curriculum materials, as well as the revised California Operations Handbook, Third to Sixth Grade Everyday Mathematics lesson pages with corresponding California Standard and references to the Operation Handbook, the original Operation Handbook submitted to California, a set of miscellaneous changes made as noted in the CRP/IMAP reports on this curriculum, the original CRP#2 and IMAP#4 reports.

As of this date, in spite of requests to Everyday Learning Corporation made by Cathy Barkett to send them to me, I have not received the curriculum materials for the remaining grades, beyond the Teacher's reference Manual for Grades 4-6 and the Teacher's reference Manual for Grades K-3 which came with the Grades 3 and 5 curriculum materials.

My recommendation to the State Board of Education is that the K-6 Everyday Mathematics submission be rejected.

My recommendation is based on the following:

I elaborate on each of these points below.

1) Standard Algorithms of Arithmetic

Arithmetic and Number Sense is the centerpiece of any viable K-6 curriculum. No other strand of elementary school mathematics is as important. California State law recognizes the importance of basic computational skills. The Emergency Regulations Implementation of Education Code Section 60200.1(a)(5) as added by Assembly Bill 2519 (Chapter 481, statues of 1998) includes the following statement:

"a basic program or partial program for reading or mathematics shall be based upon the fundamental skills required for these subjects, including, but not limited to . . . basic computational skills."

In response to criticisms from the IMAP and CRP reports that arithmetic algorithms are poorly covered by this curriculum, Everyday Mathematics submitted its revised California Operations Handbook. But with or without this handbook, students are not required by this curriculum to learn the standard arithmetic algorithms.

The Everyday Mathematics revised California Operations Handbook is a putative collection of what Everyday Math refers to as algorithms for addition, subtraction, multiplication, and division. Some of these lessons are misplaced. For example, the first algorithm for fraction addition on page 23 includes an "algorithm" for adding fractions with a common denominator. The "algorithm" is to add the numerators. This is not really an algorithm. It is part of the definition of fraction addition. In the case of multiplication of fractions, the definition of fraction multiplication is listed as an "algorithm." Another "algorithm" is to draw a picture of a rectangle and partition it into sub rectangles to see how to multiply fractions. This should not be portrayed as an algorithm. It is a pedagogical device to motivate the definition of fraction multiplication. Similar problems exist with the treatment of fractions throughout this handbook.

Far more serious a shortcoming is that all of the algorithms of arithmetic are treated on an equal footing. The standard algorithm for multiplying two numbers has no more status or prominence than an Ancient Egyptian algorithm on page 54. This is a serious disservice to students and is contrary to the California Mathematics Standards. The Grades 4 and 5 California Mathematics Standards address proficiency in the standard arithmetic algorithms as follows:

In the revised California Operations Handbook submitted by Everyday Learning Corporation, additional practice sets on basic arithmetic computation have been added to the submission to compensate for the near absence of such exercises in the original submission. But the added problems in these practice sets fail to rise to the coverage required by the California Mathematics Standards. For example, in the case of Grade 5, a total of 15 division problems have been added as part of new practice sets and NONE OF THESE EXERCISES INVOLVE DECIMAL NUMBERS.

That means that the Grade 5 Number Sense standards listed above are not satisfied.

The problems added for Grade 6 in the practice sets include only 19 division problems, only two of which involve decimals. In all cases, students are instructed to choose their own algorithm to carry out the computations.

STUDENTS ARE NEVER REQUIRED TO USE THE STANDARD LONG DIVISION ALGORITHM in this curriculum. This in itself is a fatal omission of this curriculum.

2) Material Contrary to the California Mathematics Standards

The materials, even with the submitted revisions, continue to promote the inappropriate use of calculators and to disparage standard algorithms of arithmetic. The Kindergarten Everyday Mathematics Program Guide & Activity Masters says on page 5, "Calculators are an integral part of Kindergarten Everyday Mathematics," and indeed, they are not exaggerating. On the same page, the manual says:

"Calculators can also be useful to Kindergarten children. They allow children to display and read numbers long before they can write them. There are nice ways to make calculators count, by 1s, 2s, 10s, or other numbers, both forward and back."

Beginning on page 99 of the Grades K-3 Everyday Mathematics Teacher's Reference Manual, it says:

"Calculators also provide a new medium for instruction. Their speed makes possible some activities that simply could not be done on a chalkboard or with paper and pencil. Beginning in Kindergarten Everyday Mathematics, for example, children learn algorithms (key sequences) for calculator counting."

The Reference Manual continues with detailed instructions to teachers on how to make calculators count 1's and other numbers, forward and backward for their Kindergarten students.

In the 5th Grade Resource Book, "Study Link: Parent Letter" (for use with Study Link 53) one finds the following passage in a letter to parents of students:

"In Fourth Grade Everyday Mathematics, students were introduced to a nonstandard division algorithm. This algorithm was emphasized because it is easy to learn, builds upon a student's multiplication and number knowledge, and can be quickly employed by students who struggle with traditional computation. Equipped with this algorithm or a calculator, every student should be able to solve even the most complicated division problems."

Contrary to the Grade 5 Number Sense standard 2.2 listed above, this passage says that the standard long division algorithm is not necessary to learn in part because of the availability of calculators. It is perhaps worth noting here the mathematical significance of the standard long division algorithm. It is a foundation for mathematical ideas which appear in later grades in the California Mathematics Standards. For example, an understanding of the relationship of repeating decimals to rational numbers relies crucially on facility with the standard long division algorithm. This is a 7th grade standard in California. In algebra, long division of polynomials is essentially the same algorithm and without the practice in elementary school, students will be at a disadvantage in algebra classes. At the university level, division of power series which arises in calculus, differential equations courses, and physics courses relies on facility with long division. The omission or abridgment of this important algorithm is short-sighted.

Other passages contrary to the California Mathematics Standards abound in this curriculum. For example, in the Third Grade Resource Book, the Parent Letter to be used before or with Lesson 85 says:

"Everyday Mathematics gives children a starting point to model how algorithms work. The traditional algorithm is one that most adults grew up with. Please keep in mind that children will have a choice in selecting algorithms that work best for them."

Later in the Third Grade Resource Book, the Parent Letter for Lesson 107 states:

"Lastly, children will review multiplication using partial products and lattice algorithms for whole numbers multiplied by 1-digit numbers. Such a review will then be extended to 2-digit numbers multiplied by 2-digit numbers, again looking at the two algorithms. Following are examples of how children might solve such problems, using one or both of these algorithms."

Students are not required to learn the standard algorithm for multiplication. They are free to choose their own method and the partial product method and lattice method are favored in this curriculum.

The Parent Letter for Lesson 22 in the Third Grade Resource Book warns parents that "Drill tends to become tedious and gradually loses its effectiveness." There is precious little drill in this curriculum. This is no accident and it is freely admitted by the publisher, as for example on page 22 of Everyday Mathematics Creating Home & School Partnerships: A Guide for Administrators and Teachers, where it says:

"Everyday Mathematics believes in children sharing their own invented algorithms rather than teachers continuing to teach the standard paper-and-pencil algorithms as such."

"Furthermore, research shows that teaching the standard algorithms fails with large numbers of children."

"However, in spite of the demonstrated lack of success in teaching paper-and-pencil computational algorithms, Everyday mathematics still believes that children should be exposed to such algorithms. If taught properly and without demands for 'mastery' by all children by some particular time, paper-and-pencil algorithms help reinforce children's understanding..."

Much of this booklet is devoted to defending Everyday Mathematics from the criticisms of parents. For example, on pages 8 and 9 of Everyday Mathematics Creating Home & School Partnerships: A Guide for Administrators and Teachers it says:

"Parents are also sometimes critical of a program that is unfamiliar in both its appearance (no textbooks) and its terminology."

"We do not use traditional drills and countless pages of repeated practice. Rather, we encourage practice through games."

"By supporting one another, administrators, teachers, and parents can work together to make sure that the children of today grow up embracing the NCTM standards and become confident problem-solvers who value and enjoy mathematics."

The Everyday Mathematics curriculum makes clear its hostility to proficiency in arithmetic through the standard algorithms, its opposition to drill and practice, and its support for reliance on calculators in arithmetic. The introduction to the California Mathematics Standards circumscribes the use of calculators as follows:

"Students require a strong foundation in basic skills. Technology does not replace the need for all students to learn and master basic mathematics skills. All students must be able to add, subtract, multiply, and divide easily without the use of calculators or other electronic tools. In addition, all students need direct work and practice with the concepts and skills underlying the rigorous content described in the Mathematics Content Standards for California Public Schools so that they develop an understanding of quantitative concepts and relationships. The students' use of technology must build on these skills and understandings; it is not a substitute for them."

The inconsistency of the Everyday Mathematics curriculum with this statement in the introduction to the California Mathematics Standards is a serious shortcoming.

3) The criticisms of the IMAP #4 and CRP #2 reports were insufficiently addressed

The submitted revisions simply do not address most of the criticisms of the IMAP/CRP reports. For example, one of the CRP reports indicates that Grade 3 Everyday Mathematics does not meet the Number Sense Standard 3.2. Specifically, it says, "DOES NOT MEET STANDARD, and this is a very important standard!" Everyday Mathematics has shortcomings in all the strands, as well as some strengths, but in the context of the overwhelming flaws described in this report, there is little reason to consider other shortcomings at this time.

4) No Textbooks for Students

The failure of Everyday Mathematics to provide textbooks for students is a serious shortcoming of this submission. The Emergency Regulations Implementation of Education Code Section 60200.1(a)(5) as added by Assembly Bill 2519 (Chapter 481, statues of 1998) include the following:

(2) A basic program shall include, but is not limited to the following elements:
(E) Program organization that ... (ii) enables the materials to be used for both classroom instruction and individual independent study.

An identical statement appears in the same document for partial programs. The absence of textbooks in this curriculum is by itself significant enough to warrant possible rejection on that basis alone.

It is also worth noting that some of the activities for students in Everyday Math have little or no merit. For example, the Grade 5 Journal 1 has a worksheet on page 34 entitled "Time to Reflect" which has the following fill-in-the-blank problems*:

* Much more space is allowed for the student responses after the word "because" in the submitted material than is indicated here.

A Grade 3 activity called Project 3 on page 276 Teacher's Manual & Lesson Guide volume A also contains a time consuming lesson plan of limited value based on an equally vacuous project in Lesson 109. The following quote from the overview of the project describes it adequately:

"Children construct multiple-thickness pattern blocks by taping two or more pattern blocks of the same size and shape together. They perform an experiment to determine the least number of pattern blocks of each shape that need to be taped together so that the block is more likely to land on an edge than on a base when tossed in the air. This experiment is an extension of the pattern-block experiment performed in Lesson 109."

Conclusion

I strongly recommend against the adoption of Everyday Mathematics either as a basic or partial program for any grade ranging from K to 6. Everyday Mathematics dramatically fails to meet the California Mathematics Standards in the important strand Number Sense. The high degree of integration of calculators in the curriculum--even as devices to teach Kindergarten children how to count--defies common sense and could cause significant educational harm to children. The failure to provide textbooks to students is contrary to the California Education Code relevant to this adoption process.

End of report

A committee of the American Mathematical Society, formed for the purpose of representing the views of the AMS to the National Council of Teachers of Mathematics published a report which stressed the mathematical significance of the long division algorithm, as well as addressing other mathematical issues. An excerpt from this report published in the February 1998 issue of the Notices of the American Mathematical Society is illuminating:

Standard algorithms may be viewed analogously to spelling: to some degree they constitute a convention, and it is not essential that students operate with them from day one or even in their private thinking; but eventually, as a matter of mutual communication and understanding, it is highly desirable that everyone (that is, nearly everyone--we recognize that there are always exceptional cases) learn a standard way of doing the four basic arithmetic operations. (The standard algorithms need not be absolutely unique, just as there are variant spellings between, say, the U.S. and England, but too much variation leads to difficulties.) We do not think it is wise for students to be left with untested private algorithms for arithmetic operations--such algorithms may only be valid for some subclass of problems. The virtue of standard algorithms--that they are guaranteed to work for all problems of the type they deal with--deserves emphasis.

We would like to emphasize that the standard algorithms of arithmetic are more than just "ways to get the answer"Pthat is, they have theoretical as well as practical significance. For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not by accident, but by virtue of the construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials. The division algorithm is also significant for later understanding of real numbers. For all its virtues, decimal notation suffers a significant drawback over, say, standard notation for fractions: decimal numbers (meaning decimal fractions with finitely many terms) do not allow division. This can be remedied at the cost of using infinite decimal expansions, but this is a big leap, and the general infinite decimal is not rational. To understand that rational numbers correspond to repeating decimals essentially means understanding the structure of division of decimals as embodied in the division algorithm. We do not see that naive use of calculators can be of much help here: the length of repeat of a decimal will typically be comparable to the size of the denominator, so that 7/23 or 5/29 will not reveal any repeating behavior on standard calculators.