Math Connects



Math Connects
Macmillan McGraw Hill Math Connects home page

McGraw-Hill's Math Connects Receives Top Scores in Washington State Superintendent's Report
Wed Sep 24, 2008 Reuters Publisher Press Release


Independent Study of Washington State K-8 Curriculum Review Final Report
October 27, 2008 Strategic Teaching

Review of Mathematical Soundness Background Notes for WA State Curriculum Study
W. Stephen Wilson, Ph.D.
Elementary
This review looks at the mathematical development of whole number multiplication, area of a triangle, and adding and subtracting fractions in the elementary K-5 programs from TERC Investigations, Math Expressions, Bridges in Mathematics, and Math Connects. This review found Math Expressions and Math Connects to be mathematically acceptable. TERC Investigations and Bridges in Mathematics were found to be not mathematically acceptable.
Middle School
This review looks at the mathematical development of multiplication and division for fractions and proportions in the middle school 6-8 programs for Math Connects, Prentice Hall, Holt, and Math Thematics. This review found Math Connects ranking first and Prentice Hall and Holt tied at second with Math Thematics at the bottom.

Review of how Math Connects presents fractions.

Characteristics of the Math Connects program

  • Some content lacking
  • Arithmetic of fractions well covered
  • Needs some supplementation
  • Scaffolding, examples worked, check for understanding, guided practice, individual practice, some spiral review
  • Depends on direct instruction

·


A Comparison of Math Connects, Math Expressions, and enVisionMath

Overview

I set out on this effort to compare what I could find in the online sample materials available that develop the skills required to divide fractions, and compare these findings between the three curricula. The main obstacle I encountered is the fact that the three publishers don’t make the same material available. The manner in which the materials available differs substantially as well, further confounding direct comparison. Given the differences, it was not practical to view the three at once, as I might have done if I had three actual textbooks. So I reviewed each text on its own first, created the three documents that present my findings in each text series, and in this document will present what I can as a comparison.

Review of how Math Connects presents fractions.

Review of how Math Expressions presents fractions.

Review of how enVisionMath presents fractions.


Summary

Math Expressions is the weakest of the three curricula here. It discounts fundamental principles to teachers as being “complicated and error prone” and relies instead on semantic arguments to convince students that certain procedures are valid. The concept of the “reciprocal”, essential to fraction division, is absent, and a contrived and confusing argument takes its place. Differences between the format and availability of online material for the other two texts make it difficult to find stand-out distinctions between the other two. Both appear to handle basic principles behind all four fraction operations with comparable clarity and completeness, with each of the two having highs and lows that leave them pretty much even. The grade six Math Connects text, where multiplicaiton and division of fractions was not available for me to review.


Online Materials

Math Expressions made the teacher’s manual and student “Activity Books” available for viewing online, but not the student “hardbound” book. Apparently “Activity Books” and “Homework and Remembering” books are available as softbound books to students, and the teacher’s manuals make reference to these. It isn’t clear if the “hardbound” version is just a combined and bound version of the other two, or if it contains more material. If the former is the case then the students don’t get an adequate reference of definitions and examples to take home. The teacher’s manual appears to direct classroom activities, and from what I could see was the sole source of reference material, as it is structured in large part as a script for teachers to follow.

EnVision Math Online materials were easy to use and contained both teacher and student texts. The teacher’s text is essentially an expanded and annotated version of the student text, so it is easy to see and follow what the student has in front of them and what the teacher has in the way of hints and emphasis. The student book is complete in that it contains the definitions, examples, problems, a glossary, index, etc.

Math Connects Online materials were comprised of the student book only, so I wasn’t able to see anything that the teacher had to back it up. That said, the student books appear to be a complete and conventional text with reference material, examples, etc.,


Discussion


The three texts cover the four operations either in fifth grade or straddled over fifth and sixth. All three address addition and subtraction of fractions in grade 5. Math Expressions covers multiplication and division in grade 5 as well, where Math Connects appears to hold both until grade 6. I didn’t have the G6 MC material available for this review. Math Envisions covers addition, subtraction, and multiplication in Grade 5, leaving division for Grade 6.

Fraction division is based on an understanding of what fractions are, and how operations work on them. So I couldn’t start with any of these books at the point where division occurs. I had to go back to where fractions were introduced and developed. What are they? What does the numerator represent? The denominator? All three of these texts address these basic questions in essentially the same manner. Pictures of wholes are presented, divided up into equal parts, etc. All four operations on simple fractions can be represented pictorially with pictures, and all three texts do this with comparable results. Where the three differ is in how long is spent with pictures and when more abstract yet efficient methods are justified and employed. Math Expressions seems to have students use pictures the longest, where EnVision Math weans students away sooner. Math Connects lands somewhere in the middle. Although coverage of multiplication and division is held off until sixth grade, somewhat more emphasis is placed on laying a foundation for it in fifth.

Facility with factors is fundamental to skill with fractions. Math Expressions’ standout weakness, compared to the other two, is its lackluster treatment of factors. Although all necessary content is present in the teacher’s manual, it is presented in an “oh, by the way” sort of fashion, as if a teacher should use their own judgment on whether or not they bother to present the material. The greater emphasis is on tying the meaning of verbal phrases that imply multiplication or division to operations that can be proven to work, rather than teaching the foundational mathematical principles that guarantee that they will. As a result, operations are taught as “Do this. See? It gives you the answer you’d expect, so you can believe it” as opposed to “Here are some named properties, and examples of why they are always true with the integers we’re familiar with. Now, let’s apply them to fractions and see what we get.”

The concepts of “Common Denominator” and “Least Common Denominator” are not developed with any coherence in the student material. They appear in the teacher’s manual, but transmission to the students is apparently left to happen in during the scripted classroom discussion. From page 504 of the teacher's book:

Teaching Note

Math Background The GCF of unlike denominators can be used as a greatest common divisor (GCD) to simplify fractions, and the LCM of unlike denominators can be used as a least common denominator (LCD) to add or subtract the fractions. However, it may be easier for students to simplify fractions and find common denominators using other methods.

Also, the teacher’s manual discourages systematic prime factorization as being too “complicated and error prone”. This discounts not only the value of the exercise but the intelligence of the students.

Equally damning is the omission of any coverage of the concepts of multiplicative inverse or identity; missing the fork in the road that shows how since division is the inverse of multiplication, division by a fraction is equivalent to multiplication by the reciprocal of that fraction. The text uses an odd and confusing (to me, at least) method in place of this, which will not serve a student well when they get to algebra. The “parent’s letter” presents a summary informing them that their children will learn to “invert and multiply” yet does not go into detail as to the odd manner in which this happens. The word “reciprocal” appears in the teacher’s manual, but not in the student book’s glossary.

EnVision Math develops the number properties behind fraction operations reasonably well. Since the teacher’s manual follows the student book directly, it is clearer to see how this is likely to happen in a typical classroom. Factor-based reasoning behind simplifying fractions (such as finding the GCF, why “canceling works” etc.) are all developed. Grade 6, section 5.1 has an excellent run-down on factoring and divisibility. Prime factorization is covered in 5.2. The idea behind the Greatest Common Factor is presented in 5.3. Sections 5.2 and 5.3, however, are under an “Enrichment” heading. Finally, however, students are reminded that multiplication and division are inverse operations, and that division is the same as multiplication by its reciprocal. This is the key concept behind dividing fractions that will prepare students for similar operations in algebra.

Math Connects also seems to do a decent job with the presentation of factors. The material is presented in dribs and drabs, though, which may be a good thing, or bad. The pace, as it appears in grade five, may explain why both multiplication and division of fractions are pushed out to grade six. At any rate, prime factorization and full development of the concepts of LCM and GCF are present in the student’s book.

As is difficult to avoid, multiplications of fractions appears in the development of how to simplify fractions, without introducing it as such. In my experience teaching remedial algebra in a community college, it is best to present multiplication of fractions before addition, and this is how it is done in many texts. But that’s neither here nor there.

I can only imagine how Math Connects presents multiplication and division, since I don't have the sixth grade text to look at, but the groundwork is well established in fifth grade.

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