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: 
  1. 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. 
  2. 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.
  3. 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:
  1. Keeping to the schedule takes priority over mastery of content.  Mastery will somehow occur later.
  2. Regular exposure is valued over adequate "time on task."
  3. 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.
  4. 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.
  5. 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.
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:

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:

Voyages Developers Need to Communicate More Clearly

There are problems with the use of standard mathematical terms
There are problems with language clarity
There are problems with language consistency.

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:

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:
Copyright 2006 William G. Quirk, Ph.D.