Non-standard techniques, inadequate practice and low expectations lead to poor conceptual understanding of mathematicsIn Fall 2006, the National Council of Teachers of Mathematics (NCTM) released a Curriculum Focal Points document. Pearson Scott Foresman, publishers of the elementary mathematics textbooks Investigations in Number, Data, and Space 2nd edition subsequently released a document claiming Investigations is aligned with the NCTM Focal Points.
First, a word about the Investigations 2nd Edition materials. Each teacher receives a box measuring 12" x 14" x 14" containing nine Teacher's Unit Guides, a three-ring "Resources Binder" containing student worksheets for photocopying, and an "Implementation Guide." Students receive a workbook "Student Activity Book" and the "Student Math Handbook." The nine units have catchy names that obfuscate the mathematical content they purport to contain. For example, the unit on addition and subtraction is called "Thousands of Miles, Thousands of Seats;" the unit on multiplication and division is called "How Many People? How Many Teams?"
Focus on Fractions Since reliable research indicates that algebra is key to post-secondary academic success (Answers in the Toolbox, 1999) and fractions are key to algebra (National Math Advisory Panel report, 2007), I will focus on the NCTM Focal Points that directly relate to fractions. The Investigations unit on fractions is called "What's That Portion?"
The first NCTM focal point regarding fractions is:
"Understand fractions and fraction models to represent addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators."
Contrast the language of the NCTM document with the Grade 5 mathematics benchmarks published by Achieve, Inc., a national consortium of state governors and business leaders:
"Understand how to add and subtract fractions.
Specifically: Add fractions with unequal denominators by rewriting them as equivalent fractions with equal denominators.
Understand and use the general formula
a/b + c/d = (ad + bc) / bd
Note: There is no need to find a least common denominator. The easiest common denominator of ab and cd is most often bd."
NCTM focal points do not provide examples of the type of problems students would be expected to solve. But disregarding the problems with the NCTM Focal Points, Investigations fails to meet even these nebulous standards. The publisher claims that Investigations meets this focal point with several activities in Unit 4: What’s that Portion? specifically:
pp. 95-96 “Introducing Adding Fractions”
pp. 96-98 “Clock Fractions”
pp. 98-100 “Adding Clock Fractions”
In these activities, students learn to mentally visualize “familiar” fractions on a clock face to add them. If sufficient drill is provided in class, this technique might work for problems with special denominators easily represented on a clock face (fractions with denominators that are factors of 60, although this particular fact is never pointed out in the Investigations lessons), including:
12 (5 min), 10 (6 min), 6 (10 min), 5 (12 min), 4 (15 min), 3 (20 min), and 2 (30 min)
A typical student reaction is presented in the teacher's guide: “I’m starting to remember halves, thirds and fourths. Like I know 1/2 is 6, 1/3 is 4, ..." However, this technique would be useless for adding other simple fractions with denominators that are not factors of 60 like 7, 8, or 9 for example.
The concept of common denominators is specifically delayed for “later grades.” On page 100 of Unit 4: What's that Portion? a note to the teacher states: "One useful strategy students encounter in later grades for adding and subtracting fractions is finding equivalent fractions with a common denominator.”
The "clock face" activity and fraction tracks are not just examples of activities used to teach adding and subtracting fractions; these are the only procedures presented for adding and subtracting fractions in 5th grade Investigations materials.
No universally applicable, accurate and efficient procedure for adding and subtracting fractions is ever presented in the Investigations lessons. Instead a hodge-podge of short-cut methods and estimating techniques displace fast accurate calculations.
Decimals
NCTM Focal Points also state:
"Apply understanding of decimal models, place value, and properties to add and subtract decimals."
Investigations claims to meet this standard with several activities in Unit 6 entitled "Jeweler's Gold" (p. 93-97). During the “Jeweler’s Gold” activity in one full hour of class time, students solve exactly 1 problem:
.3 + 1.14 + .085
The majority of class time is spent making a poster to explain their solutions. If students finish quickly, another problem is suggested for “an additional challenge”:
2.05 + 0.76 + 1.3
The next section covers "Strategies for Adding Decimals" p. 98-104. During one more hour of class time, students order the decimals .625, .025, .3, .8, .75 and then add together the three largest:
.8 + .75 + .625
using a “hundredths grid” if necessary. Nowhere do the Investigations materials discuss alignment of the decimal point for stack-and-carry addition.
A "Teacher Note" on page 133 of Unit 6 states: "Often students are taught 'rules' or 'tricks' to add decimals, including lining up the decimal point." The professional development notes go on to likewise dismiss the value of filling in blank spaces with zeros. Students are never let in on these "tricks."
Failing to provide adequate practice with calculations is as bad as failing to cover essential concepts; lack of repetition masquerades as “deep thinking.” Standard procedures for adding and subtracting fractions and decimals
NCTM Focal Points also states that students should:
"Develop fluency with standard procedures for adding and subtracting fractions and decimals."
Investigations claims to meet standard 3 with:
• Unit 4: What’s that Portion?
pp. 102-103 “Roll around the Clock” game
pp. 105-106 “Writing Fractions Problems”
p. 108 “Subtracting Fractions”
pp. 122-123 Activity 1
pp. 129-131 Discussion 1
pp. 132-134 Math Workshop 2
• Unit 6: Decimals on Grids and Number Lines
pp. 105-106 Activity 3
pp. 108-109 Activity 1
Investigations never presents "standard" methods for adding and subtracting fractions; therefore, it is baseless to claim that Investigations develops "fluency" with such methods.
Summarizing Investigations 5th grade lessons on subtracting fractions, we have:
- “Subtraction Fraction Equations” chart (A list of subtraction facts like: 8/10-3/10=4/8 presumably to be memorized since no explanation of how they are calculated is given.)
- Using a clock face (discussed above)
- Playing the "Fraction Track" game (Students move between zero and one by fractional amounts; they use premarked number lines with denominators up to tenths but have no way of determining if one fraction is greater or less than another except by using the track as a measuring stick.)
These are not standard procedures nor are they universally applicable. Without the concept of common denominators, students can not even compare simple fractions. For example, on page 51 of the "Student Math Handbook" (textbook), the fractions 7/12 and 4/10 are placed in order on a number line by noting that 7/12 is a "little more than 1/2" and 4/10 is a "little less than 1/2" so therefore 7/12 must be greater than 4/10. However, could students order 8/9 and 9/11 using any method presented in the Investigations materials? No; the "fraction tracks" provided don't have denominators greater than 10, and the denominator 11 is not easily represented on a clock face. This is just one example of how problems that are not cherry-picked to be solvable by Investigations' methods are ignored by the curriculum, yet this is supposedly the way to make children think critically about math.
Investigations vs. Singapore Math
Investigations students will be drawing pictures of clocks in 5th grade while Singapore students will be solving problems like these presented on the Singapore Math web site 5th grade placement tests:
•Find 32% of $96.
•Express 8 5/8 as a decimal correct to 2 decimal places.
•Express each as a percentage:
(a) 0.47
(b) 6
(c) 215 out of 500
Investigations' non-standard techniques, inadequate practice and low expectations lead to students having a poor conceptual understanding of mathematics and poor problem-solving skills as reflected in dropping test scores in 4th and 7th grade at Columbia Public Schools.
The average ACT score is highest for students in the honors algebra pathway. Note that the average ACT score of students in the regular (non-honors) algebra pathway is higher than the average ACT of honors integrated math (IM) students, with regular (non-honors) IM students having the lowest scores. But the year-to-year increase in ACT scores is similar for the algebra, honors algebra and honors integrated pathways. The increase for the regular IM pathway is less from year 2 to 3, but similar for year 3 to 4.
Looking at how much ACT scores change year-to-year in the course sequence suggests where students might have a difficult time moving up to higher level math. For example, regular IM students that try to take either of the AP Calculus courses may struggle since the jump in ACT scores is quite big. (As a matter of fact, very few regular IM students enroll in AP math courses.) Honors algebra pathway students may not be adequately challenged by AP calculus AB (one semester of college calculus in a year-long high school course) since the average ACT score actually drops from precalc to AP calc AB; they should likely be encouraged to enroll in AP calculus BC (one year of college calculus in a year-long high school course).
