Spiraling Through UCSMP Everyday Mathematics

By Bas Braams, New York University, March 2003

Related pages: The Many Ways of Arithmetic in UCSMP Everyday Mathematics # Reviews of UCSMP Everyday Mathematics # New York City HOLD # Mathematically Correct.


The Everyday Mathematics (Everyday Math, aka Chicago Math) curriculum is among the best known and most widely used of the "reform" curricula in the philosophy of the 1989 NCTM Standards for school mathematics. Everyday Mathematics was developed by the University of Chicago School Mathematics Project (UCSMP) and published by Everyday Learning Corporation, a part of SRA McGraw-Hill.

Everyday Mathematics is a "spiral" curriculum. As expressed in a research note on the Everyday Math Web site [Distributed Practice: The Research Base]:

From the beginning, accordingly, Everyday Mathematics was designed to take advantage of the spacing effect. An explicit attempt was made to explore multiple exposures to important concepts and skills, spread over two or more years. As the First Grade Everyday Mathematics teacher manual states, "If we can, as a matter of principle and practice, avoid anxiety about children 'getting' something the first time around, then children will be more relaxed and pick up part or all of what they need. They may not initially remember it, but with appropriate reminders, they will very likely recall, recognize, and get a better grip on the skill or concept when it comes around again in a new format or application - as it will!"

The notion that important skills need regular reinforcement and that practice in such skills should reappear multiple times throughout the course of study will not be controversial and is not original. The key issue of spiraling and distributed practice by which the authors of Everyday Mathematics distinguish their program is then that mastery and fluency in basic skills is only aimed for long after concepts are first introduced. Many teachers and reviewers of the program believe that Everyday Mathematics has taken the spiral nature to an extreme where it makes teaching and learning very difficult.

This Web page, intended as a contribution to a collection of Reviews of Everyday Mathematics, displays some examples of the spiral nature of the Everyday Math curriculum. The description is based on the the 3rd through 6th grade teacher's lesson guides, student reference books, and student math journals of Everyday Mathematics, 2nd edition (SRA/McGraw-Hill, 2002).

At this time I only cover a part of the spiral through whole number multiplication and division. A description of the EM approach to the arithmetic of fractions, decimals, and percents is left for another day.

Grade Four Whole Number Multiplication and Division

In grade three Everyday Mathematics students have explored multiplication of a multi-digit number by a single digit number and have explored simple division problems in an intuitive setting such as sharing some objects. Systematic treatment of multi-digit multiplication and of division starts in grade four.

Classical Approach

Before going through the Everyday Mathematics treatment of multi-digit multiplication and elementary division it may be useful to review the ingredients of a systematic traditional approach to the multiplication procedure.

Necessary prerequisites for obtaining mastery of the traditional algorithm for multi-digit multiplication include, certainly, the concept of place value, easy facility with whole number addition problems, and mastery of the tables of multiplication through 10*10. The student should also understand and be at ease with the operations on powers of ten in multiplication problems; i.e., the student should know and understand how to multiply two numbers each of which is a single digit number times a power of ten. (In Everyday Mathematics such operations are called extended multiplication facts.)

With that in place, the student learns to multiply a multi-digit number by a single digit number. At first the student learns this by a strategy of explicitly decomposing the multi-digit number in the individual digit place values. (To compute 6*327 the student will write down 6*300=1800, 6*20=120, 6*7=42, and add it up.) A "left to right" strategy is certainly appropriate at first, also with a view towards fluent mental arithmetic. The student will understand that the decomposition can be done left to right as well as right to left, and at some point the student is taught the efficient traditional right to left method of paper and pencil multiplication, first with carries written down explicitly and then with carries done mentally. The student receives extensive practice in such single-digit times multi-digit multiplication problems, both for mental and for paper and pencil arithmetic. Along the way the student also becomes comfortable with the related extended single digit times multi-digit problems, in which one of the factors is a product of a single digit number and a power of ten.

When the student has a certain fluency in multiplying a multi-digit number by a single digit number, as well as in the extended variant in which the numbers may appear multiplied by some power of ten, then paper and pencil multi-digit by multi-digit multiplication follows in a straightforward manner. The student learns and understands to take one of the numbers (preferably the shorter one) and decompose it according to digit place value, do the products, and add it up. After some practice with this procedure the student learns how to write it in the traditional efficient way. Assuming that the addition is done fluently, this is just the traditional method of multiplication.

Grade Four Everyday Mathematics Approach

There are twelve Units in Grade Four Everyday Mathematics, with on average ten lessons per unit. Some review of whole number addition and subtraction is found in Unit 2, and the multiplication tables to 10*10 are reviewed as a component of Unit 3. This material is, of course, not new for the student. New material on multiplication is found in Unit 5, and on division in Unit 6.

Lesson 5.1 concerns the extended multiplication facts, which have not been treated in a systematic manner earlier in the program. Students compute 6*40 by repeated addition, by use of base-ten blocks, and by moving about the factor of ten, and observe that the result, 240, bears an apparent relation to the result of the multiplication fact 6*4 = 24. They reinforce their facility through a game of "beat the calculator".

Lesson 5.2 contains some practice in identifying place value (an old topic), but the main focus is a game of "multiplication wrestling". Pairs of students work together. Using a die or a set of cards they draw random pairs of two digit numbers and then form and add the partial products. E.g., if they draw 75 and 84 they compute and add 70*80=5600, 70*4=280, 5*80=400, 5*4=20. At this stage it is presented as a game, and is not explicitly connected to the multiplication problem 75*84.

Lesson 5.3 reviews rounding and estimating sums. Lesson 5.4 is about estimating products, using numbers that come from a Food survey. The students also estimate averages.

The student math journal exercises for lessons 5.1-5.4 include multiplication exercises, but also exercises in "sharing" (intuitive division problems), identifying place value in large numbers and in decimal numbers, measurement and the rounding of measurements, order relations among large numbers and decimal numbers, addition problems revolving around a driving trip, some logic problems, and problems involving averages.

The core treatment of multi-digit whole number multiplication is provided in lessons 5.5 through 5.7.

Lesson 5.5 concerns the partial products algorithm for single-digit multipliers. In the same lesson students also practice estimating the length of objects. The partial products algorithm is preceded by some "easy" single-digit times two-digit problems that students may solve in any way that they know. Then the teacher does a more difficult problem such as 6*869 on the blackboard using the partial products algorithm. Students practice a few problems of this kind. (The student math journal contains four such problems for this lesson, interspersed among measurement problems, estimation problems, and division problems.)

Lesson 5.6 proceeds to the partial products algorithm for two-digit multipliers. Each multiplication problem is coupled with an estimation problem. The math journal for this lesson has five such problems, and then again a place value, a measurement, and a division exercise.

Lesson 5.7 features the lattice method of multiplication of one- and two-digit numbers. The lattice method is shown in class, and there are nine related exercises in the student journal for this lesson, together with further practice in place value and in estimation.

Unit 5 now starts to leave multiplication behind. The core topic in lesson 5.8 is large numbers, in the millions and billions; in lesson 5.9 the exponential notation using powers of ten is introduced for large numbers; and the core topic in lesson 5.10 is rounding of large numbers. The student math journal pages for these lessons have a few more multiplication exercises, along with some large number place value and rounding exercises and further exercises on measurement, sharing, and estimation. Lesson 5.11 is a "World Tour" lesson, drawing various numerical exercises from a hypothetical world tour that recurs throughout the Everyday Mathematics year, and lesson 5.12 is for review and assessment.

Unit 6 starts with a first systematic treatment of division, but then quickly moves on to geography and geometry. My description will be brief.

Lessons 6.1 and 6.2 concern the partial quotients division algorithm. In these lessons the divisor is a single-digit or small two-digit number. Lesson 6.3 concerns the translation of number stories to division problems, and in lesson 6.4 the various meanings of the remainder are discussed. Then division leaves the focus area. Lesson 6.5 concerns coordinates on maps and estimating distances on maps. In lesson 6.6 the topic is rotations and angles. Lessons 6.7 and 6.8 focus on measuring angles with use of a protractor. In 6.9 and 6.10 the topic is coordinates of a global map: latitude and longitude. Lesson 6.11 is for review of unit 6.

Remarks on EM Multiplication and Division

We reviewed the Everyday Mathematics grade four treatment of multiplication and division as an example of the implementation of the program's philosophy with respect to spiraling and distributed practice. I note the rapid progression from single-digit times multi-digit multiplication using partial products (lesson 5.5) to two-digit times two-digit multiplication (lesson 5.6) without expectation of student fluency for the easier problem class; the immediate introduction of the alternative lattice method of multiplication (lesson 5.7); and the very brief treatment of division.

The Everyday Mathematics philosophical statement quoted earlier describes the rapid spiraling as a way to avoid student anxiety, in effect because it does not matter if students don't understand things the first time around. It strikes me as a very strange philosophy, and seeing it in practice does not make it any more attractive or convincing.

(Return to Reviews of Everyday Mathematics or to the author's Links, Articles, Essays, and Opinions on K-12 Education or to the New York City HOLD Web site or to BJB Essays and Opinions.)

The opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by New York University.