A Review of the Voyages K-5 Math Program

by Bill Quirk (E-Mail: wgquirk@wgquirk.com)

Voyages Attempts to Offer Both Traditional and Reform K-5 Math

Initially developed by the Hillsborough County Florida School District, the Voyages K-5 math program is currently marketed by Metropolitan Teaching and Learning. Voyages attempts to support both the traditional and reform philosophies of K-5 math education. Anchors textbooks support the traditional development of K-5 math, and Excursions textbooks support the constructivist "hands-on" reform development of K-5 math. Why attempt to serve two diametrically opposed philosophies? It appears that Voyages developers understand the importance of traditional K-5 math content, but likely felt they had little choice but to attempt to follow 15 years of reform recommendations from the National Council of Teachers of Mathematics (NCTM).

But the NCTM is now promoting traditional K-5 math! With their September 2006 publication of Curriculum Focal Points, the NCTM has significantly backed off from prior reform recommendations and now encourages elementary school teachers to focus on arithmetic, geometry, and measurement. Voyages covers all of the NCTM's K-5 math focal point topics. Teachers using Voyages should now be free to emphasize these topics and not be required to waste valuable time on non-essential topics and time-consuming "hands-on" activities. It's difficult enough to effectively cover the essential topics in the available time.

Coverage of K-5 Math Content

Mastery of standard pencil-and-paper arithmetic should be the primary focus of elementary math. Without such mastery, students have no hope of mastering algebra, and algebra is the gateway to higher mathematics. Voyages offers above average coverage of standard arithmetic content, but improvements are needed:

Automatic recall of single digit number facts is the key necessary condition for later mastery of the standard computational algorithms. Students who don't know single digit number facts will get bogged down when they encounter the multidigit computational algorithms. By ingenious design, these algorithms reduce any computation to a series of single digit facts. Unfortunately, Voyages currently covers single digit facts in a minimal way, with no clear indication that students eventually need to remember all these facts. Although tables of single digit facts are covered in Voyages, they are not presented as something to commit to memory, but offered only as lookup aids, available to help solve computational problems.
Mastery of fractions is the number one predictor of later success in algebra. Voyages coverage of fractions could be improved by offering more difficult practice exercises. Current examples and exercises are limited to small denominators.
Although calculators are not emphasized in Voyages, there are "real world" suggestions that students should think of using a calculator for more difficult computations. In the real world this is good advice, but this is not good advice for the elementary math classroom. It suggests that students don't really need to master pencil-and-paper arithmetic.

Presentation of K-5 Math Content

Voyages Uses the Spiral Method

Traditionally, elementary math educators have used the mastery (or building block) method. They attempted to get it right the first time. They emphasized practice and other reinforcement activities. They recognized the critical importance of knowledge retention. They recognized that it's usually not possible for students to move to a higher level, if they don't remember the background knowledge they should have mastered at a lower level. A student must remember how to add and multiply, before attempting to learn long division.

The Voyages program employs the reform spiral method, where a topic is regularly revisited, perhaps once or twice per grade over a period of two or more years. Teachers and parents are told not to worry about mastery now, because the topic will be "spiraled back to" at a later time. Noteworthy aspects of the spiral method include:

Keeping to the schedule takes priority over mastery of content. Mastery will somehow occur later.
Regular exposure is valued over adequate "time on task."
Quick hit topic visits don't allow enough time for practice and other reinforcement activities. Concentrated practice is necessary for knowledge retention. Forty exercises at one point in time is far more effective than 40 exercises spread over 3 years.
The spiral method is time consuming. Each revisit requires topic review. It all adds up to more total time on topic, relative to the mastery (building block) method.
Mathematics is a vertically-structured knowledge domain. Legitimate higher-level math topics can't be properly presented and understood without background knowledge acquired at lower levels. But advocates of spiral theory believe that any subject can be presented to child of any age in some intellectually respectable form. This belief justifies the premature introduction of K-5 math topics.

Premature introduction is frequently a misleading introduction. The topic must be dumbed-down and redefined to make it accessible to those who don't have the necessary background knowledge.
Premature coverage may inadvertently assume knowledge that has not yet been introduced. .

Voyages goes beyond premature introduction of K-5 topics. Voyages introduces topics that are beyond the K-5 level. This usually does more harm than good. Consider the following examples:

Probability: Voyages promotes probability as a major topic for all 5 grades, but students never get beyond time-consuming "spinner" experiments to estimate the probability of a simple event. This is perfectly understandable, because nothing more is possible without prior mastery of operations with fractions. Better to wait until at least the 6th grade. Then all the probability that is "covered" in Voyages can be properly covered in less than one hour. Better to wait rather than wasting valuable time and giving a misleading impression about probability.
Statistics: Voyages promotes statistics as a major topic for grades 3, 4, and 5. But only measures of central tendency (mean, median, and mode) can be covered at this level, and only in the trivial context of small finite sets. The concept of arithmetical average (mean) is enough for K-5 math. The term "statistics" shouldn't even appear at this level.
Functions: This is one of the most important concepts in mathematics, but the abstract concept of function shouldn't be introduced prior to some experience graphing linear functions. But the topic of linear functions isn't covered in Voyages. Nevertheless, Voyages attempts to introduce the abstract concept of function in the second grade. The topic of functions is revisited 4 times in grades 3, 4, and 5.

Grade 4 students are given a definition for three terms: "In an ordered pair of numbers such as (347, 305), the first number is called the input. The second number is called the output. If a set of ordered pairs has a rule for how to find the output from the input, it is called a function." All subsequent function-related Voyages activities revolve around finding the "rule" for various finite sets of ordered pairs.
Most abstractly, a function is a set of ordered pairs, but there's no requirement of a "rule" relating the "input" to the "output." The only requirement is that each distinct "input" value can be associated with at most one "output." value. Thus, for example, a set of ordered pairs is not a function if it contains the pair (2, 4) and the pair (2, 5). The value 2 can't be associated with two different values, in this case both 4 and 5.
Voyages students go away thinking that the topic of functions is primarily about finding the "rule" for a finite set of ordered pairs. This is totally misleading and does more harm than good. The long-term mathematical study of functions has nothing to do with properties of finite sets of ordered pairs.
Before getting to the abstract concept of function, it's better to wait until students have experience with graphing linear functions, such as y = 3x + 2. The infinite solution set consisting of all ordered pairs of the form (x, 3x + 2) gives a simple, but mathematically important illustration of the concept of function. Experience graphing such infinite sets provides the orienting context knowledge that prepares the student to appreciate the abstract concept of function.

Voyages Uses an Adult Presentation Style

Voyages employs a Schaum outline method of content presentation, as if written for an adult who simply needs to be reminded of what they once knew. This is somewhat understandable if a topic is being revisited, but this style is employed the first time a topic is introduced. Consider the following example:

The following "definitions" (bold in the original) are compactly given, back-to-back, at the beginning of page 193 in grade 3 Anchors:

A line is a straight path that goes on and on in both directions.
A line segment is a part of the line with two endpoints.
A ray starts at a point and goes on with no end in one direction.
An angle is formed when two rays or line segments meet at the same endpoint.

Just 8 pages latter the following definitions are compactly given at the beginning of page 201.

A right angle is an angle that forms a square corner.
An angle that shows a smaller turn than a right angle is called an acute angle.
An angle that shows a greater turn than a right angle is called an obtuse angle.

Comment: Only angle and right angle are needed for K-5 math. Both can be introduced in the 3rd grade. The other "definitions" aren't needed at this level. Stick to the necessary terms. Avoid information overload.
Comment: This attempt at formal definitions is not appropriate for K-5 math. At this level there's a strong tendency to define a new term in a sentence that uses one or more undefined terms. Examples of undefined terms above include path, direction, end, square corner, and turn.

Voyages Developers Need to Communicate More Clearly

There are problems with the use of standard mathematical terms

Example: Some standard terms for fractions are given in the Teacher Edition for Anchors, but not the Student Edition. Fifth grade teachers are given the compact term "common denominator," but students are only given the wordy "you first have to rename the fractions so they have like denominators."

There are problems with language clarity.

Example: In grade 4 Anchors students are given this definition: "To multiply a large number by a small number, you can separate the larger number into two smaller parts that are easier to multiply. Then multiply each part by the smaller number, and add the two products. This is called the distributive property."
Comment: It's not necessary to explicitly define the distributive property at the 4th grade level. It's probably best to convey the idea though examples and rectangular arrays. But if a definition of the distributive property is given, it should minimally convey the idea of a number property that shows how multiplication and addition interact. The distributive property should not be defined as a process for how to "multiply a large number by a small number." The use of "small' and "large" is further misleading. It may make sense to first write the "small" number as a sum of two numbers. Example: 50 x 22 = 50 x (20 + 2) = (50 x 20) + (50 x 2).

There are problems with language consistency.

Here examples from three lessons related to the addition of fractions:

Grade 3 Anchors: "Before you add fractions with unlike denominators, you must rename one fraction so they are like fractions."
Grade 4 Anchors: "When you add fractions, they must have the same denominator."
Grade 5 Anchors: "You can add fractions by adding or subtracting the parts, as long as they have the same denominator."

Comment 1: An adult who understands how to add fractions will possess the background knowledge needed to make sense of these statements. But think of the typical student in grades 3, 4, and 5. Will a third grader be perplexed when faced with 1/3 + 1/4, because both fraction must be renamed (converted to a common denominator)? Will some spiraling-back 4th graders forget whatever they learned in the 3rd grade and conclude that it's not possible to compute 1/3 + 1/4, because the fractions don't have the same denominator? Will some spiraling-back 5th graders think that to add the "parts" means to compute 1/2 + 1/3 as (1 + 1)/(2 + 3) = 2/5 ?
Comment 2: If any variation of the spiral method continues to be used, it's particularly important to first decide on the best way to use language to communicate an idea and then consistently use the same language each time a topic is revisited.
Comment 3: The minimal necessary vocabulary should be used. Better to rely on pictures and examples.

It's About Time: Voyages Developers Need to Eliminate "Hands-on" Activities

There's a place for concrete learning methods. Singapore math uses the concrete, pictorial, abstract approach, usually moving quickly from concrete to abstract. The key defect with constructivist philosophy is the fundamental belief in heavy, ongoing use of "hands-on" methods at all grade level. Recent research indicates this is a mistake, and the NCTM has now backed off from their strong endorsement of these methods.

Voyages packages "hands-on" activities in the Explorations textbooks. Here's a grade 5 example:

Lesson Name: Discovering Pi
Lesson Materials: Ten circles cut out of construction paper, string, 8 cans of soup or vegetables, can of tennis balls, 8 CDs, 8 paper places, centimeter rulers, and calculators
Lesson Overview: "Using a picture of a circle, students learn about circumference as it relates to perimeter. The terms chord, diameter, and radius are also introduced and modeled. Students find the radius, diameter, and circumference of different-size circles using a piece of string and a centimeter ruler. Pi is introduced and students discover a formula to find circumference. Finally students use the formula to find the circumference of real-world objects."
Comment: A very time consuming project for a topic that shouldn't be introduced in K-5 math.

Voyages Developers Need to Correct Errors

Example: Voyages does a good job providing word problems, but there are errors in the solutions provided in Teacher Editions. The following example is found on page 6 of the grade 5 Anchors Teacher Edition:

Problem: Terence filled his car's gas tank and drove 0.6 of the distance to Wendover. He has 1/4 tank of gas left. Should he buy more gas before continuing to Wendover?
Teacher Edition Solution: I wrote 1/4 as a decimal, 0.25. To find the distance Terence still has to drive, I subtracted 1 - 0.6 = 0.4. Since 0.4 is greater than 0.25, he needs more gas.
Comment: More gas is needed, but the logic is bogus because it doesn't use the key fact (boundary condition) that Terence used 75% of the tank to travel 60% of the distance. This fact tells us the rate that gas is being used and it easily follows that Terence needs 50% of a full tank (or an additional 1/4 tank) to travel the remaining 40% of the distance.

If Terence stated with only 1/2 tank and had 1/4 tank left after traveling 0.6 of the distance, then the details of the Teacher Edition solution would be the same, with "0.4 is greater than 0.25," but Terence wouldn't need more gas because he would have only used 1/4 tank to travel 60% of the distance and he would have 1/4 tank available for the remaining 40% of the distance. Only 1/6 tank is actually needed.