- 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.

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. .

- 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.

- 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.

- 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.

- 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.

- 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."

- 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).

- 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.

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.

- 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.

- 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.