Weaknesses of Everyday Mathematics K-3
by David Klein
November 13, 2000
The K-3 Everyday Mathematics series has major shortcomings relative to the California Mathematics Framework and should not be adopted for classroom use in California. Among these shortcomings are:
1) The failure of Everyday Mathematics to adequately satisfy 12 of the 17 major grade level standards, as identified in the California Mathematics Framework, for grade 3.
2) The failure to follow the requirement of the California Mathematics Framework which states, "The foundation for the mastery of later standards should be built at each grade level." In particular the program fails to develop the standard algorithms of arithmetic to support California's requirements for student proficiency in later grades. It also fails to require memorization of basic addition and multiplication number facts at the grade levels specified in the California Mathematics standards.
3) The excessive and detrimental use of calculators in each of the grades K, 1, 2, and 3 curricula in contradiction to the California Mathematics Framework.
1) The Failure Of Everyday Mathematics To Satisfy Major Standards In Grade 3
According to the CRP report (Content Review Panel #1) for Everyday Mathematics, the grade 3 submission only partially satisfies, only weakly satisfies, or completely fails to satisfy the following principal standards, as identified in the California Mathematics Framework for grade 3:
Number Sense 1.3, 1.5, 2.1, 2.2, 2.3, 3.2
Algebra and Functions 1.1, 2.1
Geometry and Measurement 1.2, 2.1
Statistics Data Analysis and Probability 1.2, 1.3
The grade 3 Everyday Mathematics submission fails to meet, or only partially meets, other grade 3 standards as well, and these shortcomings are identified in the CRP report. However, it is especially significant that the series has shortcomings in 12 of the 17 major standards for this grade level.
2) The Failure of Everyday Mathematics to Develop Number Sense Standards and the Standard Algorithms of Arithmetic
In direct contradiction to the California Framework, Everyday Mathematics advises against requiring memorization of addition facts in first grade. The Everyday Mathematics first grade Teacher's Lesson Guide on page 308 says:
"First graders are not expected to memorize all of the addition facts by the end of the year. But the class should make significant progress in memorizing +0, +1, +2, and doubles (for example 4 + 4) facts by the end of the year."
This directly contradicts the following principal first grade California mathematics standard:
NS 2.1 Know the addition facts (sums to 20) and subtraction facts and commit them to memory.
The Everyday Mathematics second grade Teacher's Lesson Guide on page 810 says:
By the end of second grade you should expect near mastery for multiplication facts with 2, 5, and 10.
This contradicts the following principal second grade California mathematics standard:
NS 3.3 Know the multiplication tables of 2's, 5's, and 10's (to "times 10") and commit them to memory.
There is a similar failure of Everyday Mathematics at the third grade level. According to the CRP report for third grade, "there is insufficient practice to bring to automaticity" the memorization of the multiplication tables in grade 3 in contradiction to the principal third grade standard:
NS 2.2 Memorize to automaticity the multiplication table for the numbers between 1 and 10.
The Everyday Mathematics K-3 series de-emphasizes standard algorithms and this is contrary to the California Mathematics Framework. The standard algorithms for arithmetic are inadequately covered. The consequences of this shortcoming to later grades is explained below. Nonstandard algorithms are favored by Everyday Mathematics at the expense of the standard algorithms. This is not merely an oversight or unintentional failure to provide practice and exposure to the standard algorithms. The booklet, Grades K-6, Everyday Mathematics: Creating Home & School Partnerships, A Guide for Administrators and Teachers makes clear that the lack of support for the standard procedures of arithmetic is intentional. On page 22 it says:
With calculators available, should children be exposed to paper-and-pencil algorithms at all? Everyday Mathematics answer is "Yes." But many advocates of change in school mathematics believe otherwise—that children need not be exposed to computational algorithms done on paper. 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 properly taught and without demands for "mastery" by all children by some particular time, paper-and-pencil algorithms help reinforce children's understanding of our number system and of the operations themselves. Exploring different algorithms also helps build estimation skills and "number sense."
The California Mathematics Framework on page 230 of chapter 10 states:
The foundation for the mastery of later standards should be built at each grade level.
By failing to develop the standard algorithms of arithmetic at the K-3 level, the Everyday Mathematics series, if adopted would undermine California's expectations for students to later satisfy the following grade 4 and 5 mathematics standards:
Grade 4 Number Sense 3.1 Demonstrate an understanding of, and the ability to use, standard algorithms for addition and subtraction of multidigit numbers.
Grade 4 Number Sense 3.2 Demonstrate understanding of, and ability to use, standard algorithms for multiplying a multi-digit number by a two digit number and long division for dividing a multidigit number by a one digit number; use relationships between them to simplify computations and to check results.
Grade 4 Number Sense 3.3 Solve problems involving multiplication of multi-digit numbers by two-digit numbers.
Grade 4 Number Sense 3.4 Solve problems involving division of multidigit numbers by one-digit numbers.
Grade 5 Number Sense 2.0 Students perform calculations and solve problems involving addition, subtraction and simple multiplication and division of fractions and decimals.
Grade 5 Number Sense 2.1 Add, subtract, multiply and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results.
Grade 5 Number Sense 2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multiple digit divisors.
One particularly negative consequence of the failure of Everyday Mathematics to develop mastery of the standard algorithms for subtraction and multiplication of whole numbers is that students would be unable to master the important long division algorithm in later grades. In order to use the long division algorithm efficiently, students must first master the standard algorithms for subtraction, and for multiplication at least for the case when one of the factors is a single digit whole number. The long division algorithm is required to convert fractions to decimals. Moreover, it is essential for students to understand the long division algorithm in order later to meet the seventh grade Number Sense standards 1.4, and 1.5. These standards lead students to an understanding of rational and irrational numbers in terms of decimal notation, and the long division algorithm is essential to this purpose. Higher grade level standards at the high school level also require knowledge of the standard long division algorithm, and the basis for this later understanding is not provided in the Everyday Mathematics K-3 submission.
Even within the framework of the K-3 grade level mathematics standards, there are significant shortcomings of Everyday Mathematics to comply with the California Framework. For example, in Everyday Mathematics Home Links (for grade 2) on page 97 is a worksheet for addition of two and three digit numbers. The standard algorithm for addition is not included on the "Addition Strategies" for this lesson given on the preceding page. On page 97 students are also asked to give a "ballpark estimate" for each problem and from the Teacher's Lesson Guide for this lesson (Lesson 4.8). The idea is to first round the numbers to the nearest 10 or 100, and then add. However, ROUNDING IS NOT EXPLICITLY TAUGHT AND DOES NOT EVEN APPEAR IN THE INDEX OF THE GRADE 2 TEACHER'S LESSON GUIDE. This runs counter to the California Mathematics Framework which says on page 129, "Rounding is a critical prerequisite for working estimation problems." The failure to deal explicitly with rounding occurs in grade 3 as well. This shortcoming is surprising considering the great emphasis on estimation of this series, even at the expense of exact calculation.
Everyday Mathematics is not free of mathematical errors. For example, in Everyday Mathematics Home Links (for grade 2) the exercise on page 105 is called, "What's My Attribute Rule" and is part of Lesson 5.1. This whole approach is ill-conceived and the worksheet is especially bad. A much better way to deal with this topic is to present a rule and ask students to identify objects which satisfy the rule, not the other way around, because there can be many rules for a given collection of objects. For this worksheet, students are shown geometric shapes and asked to identify a rule consistent with the shapes identified by dots in their interiors. Evidently the rule is supposed to be that the shapes have four sides and that is the answer given on page 104, but that answer is incorrect since one of the excluded figures also has four sides (but is not convex). A collection of shapes not satisfying the rule is also given as a hint, but that collection makes the exercise even more incomprehensible.
3) The Excessive and Detrimental Use of Calculators
According to the California Mathematics Framework (Chapter 9, pages 223, 224):
The Mathematics Content Standards for California Public Schools was prepared with the belief that there is a body of knowledge—independent of technology—that every student in Kindergarten through grade twelve ought to know and know well...Teachers should realize that understanding basic concepts requires that the students be fluent in the basic computational and procedural skills and that this kind of fluency requires the practice of these skills over an extended period of time...The extensive reliance on calculators runs counter to the goal of having students practice using these procedures. More to the point, IT IS IMPERATIVE THAT STUDENTS IN THE EARLY GRADES BE GIVEN EVERY OPPORTUNITY TO DEVELOP A FACILITY WITH BASIC ARITHMETIC SKILLS WITHOUT RELIANCE ON CALCULATORS
[my capitalization and bold font]
The introduction to the California Mathematics Standards, even independently of the rest of the California Mathematics Framework, also warns against 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.
In contradiction to the California Mathematics Framework, Everyday Math makes excessive use of calculators starting in Kindergarten in ways that are completely inappropriate and in some cases manifestly destructive. Specific examples are given below. Calculators are an integral part of the Everyday Mathematics K-3 series. This is made clear by the Everyday Mathematics booklets themselves. In Grades K-6, Everyday Mathematics: Creating Home & School Partnerships, A Guide for Administrators and Teachers on page 26, the following statement appears:
Evidence is growing that students' intelligent use of calculators enhances understanding and mastery of arithmetic and helps develop good number sense. Moreover, teacher experience and considerable research show that most children develop good judgement about when to use and when not to use calculators. Students learn how to decide when it is appropriate to solve an arithmetic problem by estimating or mentally calculating, by using paper and pencil, or by using a calculator.
Calculators are useful teaching tools. They make it possible for young children to display and read numbers before they are skilled at writing numbers. Calculators can be used to count by any number, forward and backward. They also allow children to solve interesting, everyday problems.
Please encourage children to use their calculators whenever they encounter interesting numbers or problems that may be easier to handle with calculators outside of the mathematics period. Do not worry that the children will become dependent on calculators and will be unable to solve problems with paper and pencil or in their heads.
The inconsistency of the Everyday Mathematics curriculum with the California Standards and Framework on the use of calculators in Kindergarten, first grade, second grade, and third grade is so serious a shortcoming that even by itself, it should prevent the adoption of this series. The CRP reports on Everyday Mathematics raised concerns about the use of calculators. The grade 3 report included the warning, "There are a number of pages with exercises, but they are difficult to find, and there is a strong tendency for the problems to be very easy if the students are expected to do them without the aid of a calculator. "
Examples of Calculator Use in Everyday Mathematics K-3
Below is a list of examples of the use of calculators in Everyday Mathematics K-3 and some comments about those lessons. This list is far from exhaustive and does not necessarily contain the most egregious misuses of calculators. However, these examples demonstrate the inconsistency of Everyday Mathematics with the California Mathematics Framework.
Teacher's Guide to Activities page 128
The lesson, "Counting Shortcut," is an extension of the previous calculator lesson which directed children to count on calculators by repeatedly pressing the sequence of buttons: "+" "1" "=". This lesson on page 128 shows Kindergarten children a shortcut to counting on calculators. Instead of pressing the three buttons "+" "1" "=" over and over again, students can just press the "=" or "=/R" button again and again. This causes the calculators to iterate the first sequence "+" "1" "=". The window display on the calculator then shows 1, 2, 3, ...
It is a bad idea to use calculators in Kindergarten classes for any purpose, but a particularly negative feature of this lesson, and others like it that follow, is its potential to cause lasting confusion over the meaning of the equal sign, "=". The equal sign should not be used to represent an instruction to "do something yet again;" it has a very specific meaning in mathematics, and this lesson works against that meaning. The subsequent lesson on page 130, "Counting on with Calculators" again reinforces this bad habit, as do the lessons, "Counting Backward with Calculators" on page 188, and "Skip Count with Calculators" on page 206.
The lesson, "Place Value on Calculators" on page 286 attempts to explain place value using calculators. The lesson fails to do anything substantial, and only acclimates Kindergartners to calculators for no good reason.
Student Math Journal.
On pg. 82, problem #4 asks students to measure their calculators with a ruler. There is nothing wrong with this exercise, except that it focuses undue attention to calculators in first grade.
On pg 84, problem #4 says, "Use a calculator. Count up by 7's.
0, 7, 14, ____, ____, [etc. 8 blanks in all]."
On pg 104, problem #1 says, "Count up by 10's. You may use a calculator.
77, 87, ____, ____, [etc. 9 blanks in all]."
On pg 136, problem #45 says, "Count up by 10's. You may use your calculator.
155, 165, ____, ____, [etc. 7 blanks in all]."
On pg 155, problem #2 says, "With a calculator, I can count to _____ in 15 seconds. (Press 1 + = = ...)"
On pg 157, the exercise is entitled "Class Results of Calculator Counts." Students are instructed to tabulate data for how high classmates can count in 15 seconds, using a calculator."
On page 180, students are asked to complete a table in a lesson called, "The Broken Calculator." They are told, "A key on your calculator is broken. Can you still use your calculator? Show how. Make up your own last line." The lesson is ambiguous and does not adequately support the concept of place value.
On pg 197, problem #1 says, "Use your calculator. Count up by 50's.
550, 600, 650, ____, ____, [etc. 8 blanks in all]."
On pg 214, problem #2 says, "Count up by 25's on your calculator.
200, 225, 250, ____, ____, [etc. 8 blanks in all]."
On pg 245, problem #4 says, "Count down by 1's on your calculator.
3, 2, 1, ____, ____, [etc. 6 blanks in all].
The smallest number is ______"
Student Math Journal.
On page 14, there is a lesson entitled, "Counting with a Calculator." The lesson gives practice on how to use a calculator to count up or down by a specified number.
On page 169, problem 5 says, "Use your calculator. Start at 92. Count by 5's.
92, 97, ____, ____, [etc. 7 blanks in all]."
On page 175, there is a worksheet entitled, "The Wubbles." Problem 1 has the instructions, "On each line, write the number of Wubbles after doubling. Use your calculator to help you." Students are to find the number of "Wubbles" for 8 consecutive days if on the first day there is one "Wubble" and they double each day. Problem 2 on the same page the instructions say, "On each line, write the number of Wubbles after halving. Use your calculator to help you. Remember that '1/2 of' means 'divide by 2.'" This worksheet goes with lesson 7.5 in the Teacher's Guide which includes a story about "Wubbles."
On page 186, problem 5 says, "Use your calculator. Enter 42. Change to 70. Write what you did." There is a similar problem on page 187 as well as a different calculator problem on that page.
On page 244, there is a lesson entitled, "Calculator Dollars and Cents" The lesson gives children practice pressing the correct buttons to enter in dollar amounts.
On page 264, problem 4 says, "13 calculators. 2 children for each calculator. How many children?" This problem may or may not involve the use of calculators, but it illustrates the obsession of this series with calculators in the early grades.
Everyday Mathematics Home Links
The exercise sheet, "Calculators and Money" on page 227 has two problems for students. The first problem requires students to type in money amounts into their calculators. That is the full extent of problem 1.
Problem 2 on this same page says, "Ask an adult to think about a time that he or she remembers from when he or she was a child. Ask the adult to compare how much the item cost then and now. Make a record below of what you find out."
The instructions to parents urges them to compare prices directly as a difference in price, and also as a ratio, e.g. "A bicycle costs about 4 times as much now as it did then." What are first grade children to make of these ratio comparisons when the concept of multiplication is not well developed and is not even a first grade standard? Arithmetic needs to be developed in a coherent systematic fashion.
Student Math Journal
On pages 216, 220, 223 and 228, exercises appear called, "Using the Partial-Products Algorithm." Students are to multiply a single digit number times a two digit number, compare answers with a partner, and then use a calculator to resolve disagreements.
On page 238, problem 2 says, "Solve. Use your calculator. Pretend the division key is broken." Students are then asked to divide 153 by 8 and 285 by 3 (using repeated subtraction on their calculators).
On page 239, "2-Digit Multiplication" is a worksheet with the instructions, "Multiply using the partial-products algorithm. Compare your answers with your partner's answers. Use a calculator if you disagree. If you did a problem wrong, work it again.
On page 244, problem 4 says, "Solve. Use your calculator. Pretend the division key is broken." The two problems given are 144 divided by 9 and 465 divided by 3.
On pg. 269 there is an exercise called, "Calculator Memory" The point of this exercise is for students to guess the final display on a calculator following a specific sequence of button presses, ending with a memory key. There seems to be little point to this exercise, other than acclimating students to calculators.
"Cracker Fractions" on page 245 is a lesson designed to teach the fractions 1/2 and optionally, 1/4. This lesson is dubious because it does not emphasize that both "halves" need to be the same size. One of the pitfalls in teaching fractions is that children can come away thinking that when you break something into two pieces, each one is a half. Care should be taken to emphasize that the two pieces must be the same size.