Review of Connected Mathematics Project, 2nd (and 1st) Edition vs Saxon
Mathematics student / teacher / parent textbook for
Middle / Junior High school By Arthur HuRev 3/12/2008
The reviewer is a MIT graduate in computer science with 3 sons between elementary school and high school who have taken TERC, CMP, Everyday Math and Core Plus and has a collection of many math textbooks K-12 from 1960s to 1990s.
A - excellent book, great explanations and samples
B - complete content, good explanation
C - complete but not great
D - incomplete but has some content
F - no standard methods taught
== Average ==
== 6th graders play with blocks in the name of Geometry ==
== A unit on circle area without PI R SQUARED. ==
== DECIMAL MATH IN BITS AND PIECES ==
== Algebra in 7th grade? ==
About the books
Some reform books such as Mathland were so badly received they are no longer in print. Mathland's massive failure was largely responsible for the backlash in California with revised standards that no longer permit books such as this or CMP. CMP and TERC have survived for revised second editions, also after enduring scathing reviews and complaints from parents and mathematics professionals. A recurring theme I have in reviewing such books is that kids work twice as long, twice as hard, to learn one-tenth as much useful mathematics compared to the disparaged traditional books. No parents have risen up in anger against school boards to demand the adoption of reform textbooks, or to discard traditional books. Investigations take much longer to cover than the traditional method of show the method, explain why it works, and practice. Many reviews of teachers who were sold on the program still mention that the pace is not possible to cover most or even half of the material in one school year. One entire booklet is devoted to leading up to comparison of fractions, yet still omits the key concept of using common denominators, instead introducing two "toy" methods, fraction strips and "benchmarks" which are of no use in college or real life.
Expensive Shelf Full Of Booklets
CMP is formatted as a series of paperback books, each on one topic, with an edition for the teacher, which evidently contains most of the actual instruction (if it can be called that, since lectures aren't allowed, and "investigations" don't actually allow the teacher to give away any actual methods that weren't "constructed" by the students). A complete set for grade level takes up about half a bookshelf, and is probably much too expensive and impractical for any homeschooler to use.
The Parents' "Edition"
The parent's edition consists of letter to send to the parent, with the front side essentially telling the parent that the math may be different and shocking, but there's no need to panic. Reviews of the first edition indicate that the original letters discouraged or forbade parents to show children the standard methods they were taught. The back side actually lists the important concepts that students are supposed to learn such as using common denominators and pi r squared, but these "conclusions" and even their terminology are curiously absent from the student books. The student books give the framework for the investigations and problems, but don't actually contain any explanations for any of the methods that students are supposed to construct. For example, the unit on area of a circle does not mention the formula "pi r squared", and the unit on adding fractions makes no mention of "common denominators" in the text, glossary or index, even though the term is used in the teacher edition, on their website, and in the parent letter. The teachers editions also lack any formal explanation of any of the methods. There is often a one paragraph explanation of a method, but this is given only as a "possible answer to (do something such as add fractions)" from the student not as something to actually present in class by the teacher.
Nonstandard derivations / explanations
If the formulas are often standard, the methods used to derive a formula are sometimes not. The area of a circle consists of cutting out 3 quarters of a circle, and filling in the fourth piece with scraps from outsides of the 3 pieces, with no connection to why PI is involved other than "When is the last time you saw a number a little bigger than 3?" To conclude on the basis of a few measurements that PI is involved is no more than a wild guess, but standard methods show how and why 2 pi r is part of the answer. I have an extensive collection a half-dozen math books for the 6th grade from 1965 to 1995, and every single one has "pi r squared". According to previous reviews, the 1st edition did not even teach standard mathematical methods. The 2nd edition appears to have investigations which usually do lead to standard methods, they're just left out of the student text to insure they can't "peak" at the final method before the investigation.
Nonstandard methods and constructivism
The structure of CMP appears to be due to two reform beliefs. The first is that mathematics must be reformed so completely that standard methods should be discarded in favor of new or nonstandard methods, but this appears to be abandoned in the first edition. The second that knowledge must be "constructed", not just presented or memorized.
No explanation of methods allowed
CMP is the most extreme example of actually prohibiting access to every basic method from the student text, rendering it useless as a reference, a problem also mentioned with TERC Investigations first edition, which provides no student textbook at all, and also contained no standard elementary arithmetic. The 2nd edition has a textbook which is useful as a reference, and claims to teach standard methods. When you look up "adding fractions, algorithm to compute" you get a page that says "you will now write your own algorithm to add fractions" with a small list of examples that it should work with. There are no examples showing how a problem was actually solved using any particular method. In the wrap up, you are asked to write up "your method" to add fractions or whatever again. The teacher's manual usually also lacks any formal presentation of how to actually do anything, but often has a short 1 paragraph explanation under "a possible answer to (how to add fractions) ", along with a statement such as "you can multiply both sides by 10 to convert both sides to whole numbers" in the parent's letter.
Lowest Common Denominator Not Allowed
Saxon does not use the term "lowest common denominator", but "common denominator" is used in the text and index, and what they actually use is in fact how to get the LCD. Some methods such as comparing fractions using common denominators, or lowest common denominator are missing entirely. Both are neccesary for high school and college mathematics, and taught to practically the entirety of the current parent and professional math generation with more than a 6th grade education. While Elizabeth Phillips, the lead on CMP answered my e-mails, she confirmed that lowest common denominator is not included because their "research" determined it was not important. Interestingly, the term "common denominator" is used in the book on adding decimal fractions, where it is normally not used by the standard method. So "standard" terms appear to be used in non-standard, but not the standard methods.
Researched, Approved My Eye.
The cover emphasizes that CMP is backed by research and piloted (more like guinea pigs) by thousands of students and hundreds of teachers. Pages of every booklet contain the names and pictures of the authors (including Phillips), and dozens of teachers and professors at universities who have allegedly reviewed the series, defects and all, and given their approval. Most of the authors have made careers on making mathematics as completely different as possible from proven, traditional methods. The logo of the National Science Foundation is prominently featured. However, this should be viewed as a warning label as the NSF has funded a dozen or so projects, all severely criticized by professional engineers and mathematicans as being severely deficient if not "the worst textbook I have ever seen" as stated by many reviewers.
Saxon Contains Methods, No fluff or research.
By contrast, my edition of Saxon consists of one book, so you can use it to look up skills from the entire year. In fact, previous skills are reviewed throughout, telling where to look up the skills in case you forgot. The student edition contains explanations, formulas, and examples, and even standard derivations for every important skill. There is no NSF logo on the cover, no statement that is piloted and based on research, and no pages of people who will testify that this textbook is not a pile of junk. What this means for parents who looks at these books is that if you have a textbook that requires letters to the parents, research, and lots of people who will testify that the book is not junk... it's probably junk.
In Saxon, all important concepts are in the index. There are no ommissions, deliberate or otherwise of information that you would find in every other traditional math book. You don't need a teacher edition because the student edition isn't phobic about actually telling students how to do something, and doesn't hide anything where they can't find it. There is no need to supplement or need to look up basic methods for reference because the student book has everything you need for a reference. It's also evident why this series is more popular for homeschoolers as you can outfit your home classroom for the price of one textbook less than $100 instead of what I would guess would be thousands of dollars in materials and reacher re-education neccesary for CMP since it is so radically different from how we've taught math before.
== Average ==1st edition: No standard method
The first edition spends an entire booklet leading up from median to finding the average by moving stacks of blocks around until you get an equal height. However, it does not cuminate in the standard definition in a college statistics book or dictionary. That's adding all the items, and dividing by how many you have. That's a huge omission of the most important thing anybody needs to know about computing the average.
The 2nd edition does have this formula, but still wastes the rest of the book on useless time wasting investigations. A traditional K6 sequence only includes average, not median or mode which were traditionally covered in college stats. Median and mode cannot be computed with a simple four function calculator, the mean can. One particular exercise required scaling and charting dozens of data points which might not even fit on a sheet of notebook paper before a median could be found. As an adult who knew exactly how to do this task, it would have taken half an hour to complete, it could easily take all night for a student who didn’t have any help. I got an email from a student who also complained about how long it took
Saxon simply gives you the standard formula for average, and examples. There are no problem sets from hell which take all night and two pages of drawing x's, everything is short and simple.
Grade: F for 1st for omitting standard method, D for 2nd taking so much time on topics of little use and just one line for the standard method as a "by the way, there's this other way".
== Prime numbers ==
Pick and write about a favorite number
One of the first exercises in prime numbers asks students to pick a number “they are especially interested in”, and write everything they know about it. This was mentioned in the nationally published Christian Science Monitor as a problem that parents though was particularly silly and unmathematical. This is very similar to the widely lampooned "what color is math" question, which also has no correct answer.
Is 371 a prime number?
Prime numbers are mainly important for finding common multiples and common denominators, which is surprising considering that they're largely NOT used once they get to fractions. The treatment gets ridiculously deep, and the one problem that made me ill was something like "is 371 a prime number?" Looks innocent enough, but the only obvious method is to take a calculator and divide 371 by every number smaller than 371. How long do you think it takes a child to divide by 370 different numbers on a calculator? By hand? A high school or college programming class might teach you how to write an efficient prime number finder in BASIC or Java. You only have to divide by a list of prime numbers up to the square root, but what 6th grader is going to know that?
Grade D: covers the topic, but you have to get to heck and back to get through some of the homework. It goes way too deep, and then largely skips the most important application of primes when lowest common denominators are left out of fractions later.
== Fractions ==
Bits and Pieces I goes up to comparing fractions. II covers arithmetic with fractions.
According to the CMP website and the parents letter, the students will learn how to use common denominators to add fractions. They will multiply tops and bottoms to multiply, and invert/multiply to divide. These are all standard methods, despite the usual aversion of standards-based books to standard methods.
No mention of common denominators in student book
However, neither student book index contains any reference to common denominators, only the term "equivalent fractions", which isn’t the same thing since only one pair of equivalent fractions can be used as a common denominator. It is interesting that one of the optional problems does touch on using least common multiple as denominators, (it's how you would find the lowest common denominator), it doesn't actually tell you to use this as a general method. There is no coverage of lowest common denominator at all, something every adult knows and uses as an english language idiom.
Fraction strips, benchmarks but not common denominator
Investigations cover using fraction strips (easy, just fold this piece of paper into 7 equal pieces, yeah right) and "benchmark" fractions like 1/2, 2/3 and 3/4 to compare fractions. Have you ever tried to fold a piece of paper into 7 equal pieces? It's very difficult. But not using a common denominator, which always works for any fraction, and is the standard method. Trying using fraction strips to compare fractions building a 787 at Boeing and see how long you keep your job.
Devise your own way to add fractions.
Investigations cover in a very round about way that equivalent fractions might be useful for adding fractions, but when it comes to looking up the index page for "algorithm for adding fractions", you get "write your own algorithm for adding fractions. Make it work for these samples. Make sure it works for every case". No working method is actually presented in the book to use with the problems, you have to use your own method.
Reviews by mathematically correct indicate that the first edition skipped dividing by fractions entirely. According to Elizabeth Phillips, even she said it was covered so poorly, she can understand and support any teacher that wanted to "supplement" the book. The second edition covers subtraction, multiplication and division, but the investigation uses common denominators to divide (very unusuasl), and mentions reciprocal without explaining how it might be useful in dividing fractions.
No methods are printed in the student book.
In short, no standard method for comparing fractions is presented at all. There is no coverage of the lowest common denominator, even though this is familar to just about all adults and is common used as an english phrase in social studies. There is only investigation based instruction for arithmetic. No reference is provided which explains ANY correct method for any operation, let alone the standard method.
Grade: F for 1st ed, D for 2nd ed.
== 6th graders play with blocks in the name of Geometry ==
An entire month booklet on Geometry is called Ruins of Montarek. That's a nice mathematical title? They're building structures out of cubes, like soma block puzzles,from various 3D views. There is a description of ziggurats (tall ancient towers) A Section on Isometric views (high school drafting, not in traditional math sequence in K12)
This is mainly useful for IQ tests, but not much else unless you're in a drafting course or reading diagrams like this at the car parts counter or appliance repair, and you really don't need training in isometric drawing reading. I worked this out when I did constructed my own 3D graphics in 12th grade in high school, it's normally done
in computer science 2nd year computer graphics course, which often not even offered.
Math for women and minorities
According to Elizabeth Phillips, she included this unit because she felt it was unfair that IQ tests showed that men did better of visio-spatial skills. One of the most common problems mentioned with CMP is that there is no way all of the units can be covered in a year, this is probably one of the most skipped units in the series I would predict.
Grade F: total waste of time.
A unit on circumference and area of circles that has LOGO programming (withno instruction) and no formula for 2 PI R or PI R SQUARED.
Another unit - Covering and Surrounding (Measurement)
This is a 6th grade unit culminating in, but not actually putting into the student text the formula for circumference and area of a circle.
Here's a question using the (famous?) Logo program
The logo instructions on the following page draw a polygon with so many sides it looks like a circle:
repeat 360 [fd 1 rt 1]
(this means repeat 360 times, step forward by 1, turn right by 1 degree. What you didn't figure that out? I couldn't figure it out without a LOGO manual either, and they showed us LOGO at MIT in the Artificial Intelligence lab in the 1970s. You were never taught how to program in LOGO? You obviously didn't get the same quality education as whoever developed this question!)
a) how many sides will it have (360 by the repeat statement)
b) what is the perimeter of the figure (each step is 1, so it will be 360)
c) how many steps would it take to go across the diameter of the figure? (if C = 2 PI D, then D = C / (2 PI))
Logo is not in the index. There is no instruction on how to install, run, or program in logo. Logo is virtually unknown outside of research projects in using computers in education in the late 1970s. Nowhere is it mentioned in the book that there are 360 degrees in a circle. According to wikipedia, most of the 170 version of Logo are no longer in use, which means it is largely obsolete, not a technology of the future like C# or Java. There is no instruction on what a turtle does after a 1 degree rotation.
Now here's where I did find a free windows logo http://www.cs.berkeley.edu/~bh/logo.html
Here's another investigation
"What happens to the enclosed area as a 60-cm perimeter loop is used to make polygons with more and more sides?" "If you have access to a computer and the use of the LOGO programming language, you might use the computer to draw those figures."
"What does the polygon resemble (ans - circle)"
- no program is provided or LOGO documentation is provided
- 6th graders are not expected to be computer graphics LOGO programmers
- NO SIMPLE LOGO program can be made to do this task, the previous LOGO program just won't do the job.
As an experienced graphics programmer who did a program like this in BASIC in 11th grade, and who saw LOGO at MIT, I didn't even know the solution off the top of my head. After consulting the internet for a LOGO manual, I've figured out that the solution is to divide 60 by the number of sides for a step size, and divide 360 by the number of sides for a turn angle.
For a triangle: repeat 3 [fd 20 rt 120]
OK can anybody else who has never done computer graphics come up with that solution? What student is going to know how to go on the internet, look up the Wikipedia LOGO article go the link, download Berkeley LOGO for windows, install it, and run the program? What teacher is going to know how to do it? According to Elizabeth Phillips, she wrote that she was exposed to LOGO when it was a popular experimental language (it's now largely obsolete for any purpose except for CMP or TERC), and included it just to increase the technology content, not because it's anything jr high students in the 2000s will ever need to know.
At the end of chapter summary, you might expect to find formulas, but instead you get:
"Write what would have you learned about calculating the circumfernce of a circle?"
- one "possible" answer: a number a little more than 3 called PI mutiplied by the radius times 2
At no point is the teacher supposed to actually tell the students what the answer is.
"... can you find the area of a circle"
- one possible answer PI time the radius squared. Is there another possible answer? According to the fuzzies, there are always more than one way to compute anything. Again, the teacher is not allowed to tell them, and thyey won't get it from their textbook. If they're smart, they can look it up on wikipedia.
Pi r squared is ONLY mentioned in the parent's letter and teacher edition as a "possible answer". There is NO formal introduction of either formula in the student book, the student has to "discover" it without the book revealing the "answer". All traditional books provide the formula for use with the problem sets. This characteristic of "standards based" books which are so "discovery based" they do not include the final standard methods because it "gives away the answer" that they must discover on their own.
What are the investigations like?
5.2 Get some measuring materials and measure the diameter and circumference of a objects. Make a table. Look for patterns and relationships. Can you find the circumference if you only know the diameter? The diameter knowing the circumference?
You can only prove the ratio pi is the same for all circles if you know that all circles are similar, differing only in scale. If you know this, you can prove all the ratios will be the same for all all circles. Going by measurements and noticing they are similar is only a guided lucky guess
The teacher isn't even given a proper explanation, only that a "possible" answer is the multiply or divide by 2 PI. Presumably there are other correct answers?
"You have discovered(?) that the circumference is a little more than 3 times the diamter. This special number is known as PI. "
Well, that's almost a useable formula, at least PI is explained in the book. But the direct formula C = 2 PI R IS NOT IN THE BOOK. "Can you find the circumference if you only know the diameter?" is the closest you get to a method in the student book.
The investigation for the area of a circle looks like this, and it's hilarious.
Draw a square.
Draw a circle inside the square
Split the square into four quarters with two lines.
Cut out the upper and left 3 slices that are outside the circle.
Chop the outside scraps into little bits (yes little bits)
Fill the remaining slice with the scraps, glue optional.
Note that if you fill it with all the scraps, there is still some space left between the little scraps.
That means you can take all of the area of 3 of the 4 radius squares, and the area is still a bit more.
Since a "bit more than 3" was PI the last time, students should "guess" PI is involved here too.
If you ask me that's the craziest way to waste time finding the way to get the area of a circle I've ever seen. It's worse than just counting out little area squares. Can you imagine cleaning up all the little 2mm sized scraps off the floor and desks?
The standard method of explaning this is in several of my older math books, including Saxon, you cut a circle into wedges, and stack them pointing up and down side-by-side, which gives you a bumpy parallelogram. If you use calculus-like thinking, you can argue that the tops and bottoms will be straight if they are very small. Then you get a rectangle that is R high by (2 PI R)/2 or PI r squared. No need to just make up some crazy cut and paste the scraps method that doesn't give any idea of where PI came from. But CMP doesn't even use standard methods for deriving formulas.
Based on that, the teacher is to somehow guide the students to "discovering" that the area is PI r squared, though the term "squared" really isn't defined as far as I could find. I didn't notice if they had even presented the idea of finding the area of a square multiplying one side by itself. With CMP you can seemingly assume that ANY correct answer is ommitted to prevent "cheaters" from looking up the correct method directly. There are many references in fuzzy math literature complaining that students just want to know how to find the right answer, not investigate, and it sounds like CMP was based on this common fuzzy complaint (we won't let the little monsters get the answer without an investigation! They might just look it up!)
Square roots? Solving Quadratic equations?
It gets really interesting because they go further and asks if you can find the radius if you know the area. There's a good reason most textbooks don't cover this in grade 6. It's a simple derivation. Algebra 1 says if a = pi r squared, then the square root of a is PI R, and divide by PI to get R. Never mind that's basically solving a quadratic equations in grade 7. Square root isn't in the index, and not usually taught until grade 7 as a whole number, and not in general with a calculator until grade 8. This is yet another example that I've observed time and time again in reform math textbooks and assessments is that kids that who have not mastered the basics are expected to walk in with the skills of an algebra or computer science graduate. TERC expects 2nd graders to use negative numbers to subtract, and the 2008 2nd draft of the Washington Revised Math Standards initially gave 2nd graders an algebra problem to solve a formula of the form y = ax + b and asked kindergartners to effectively multiply 2 x 5 x 3 before being formally being able to add. It's just called "problem solving" which just means having to solve a problem without being taught how to solve it.
The textbook never mentions PI r squared as a formula to use because students must develop their OWN algorithm. NO METHOD AT ALL is provided for students to use but their own. But it is in the teacheredition, and the letter to the parents states that students are expected to know this.
Grade D - some students will get it, but I'd be surprised if most figure out what all the investigation was actually about.
== DECIMAL MATH IN BITS AND PIECES ==
Bits and Pieces III follows non-instruction in fraction math for decimal operations. The parent's letter shows how to convert decimals to fractions with common denominators (.12 + .1 = 12/100 + 10/100), which is very awkward. Lining up the decimal points is briefly mentioned in the student book and teacher's book, but for some reason not in the parent's summary of what students should have learned. There are no examples showing how the standard method is used to actually solve a problem. The teacher's manual does give a 1 paragraph explanation of a method to add decimals, but only as a "possible answer" to the question asking the student to "write your own algorithm to add decimal numbers". The teacher's manual also lacks any complete explanation with examples for any of the 4 decimal operations. The investigations walk piece by piece, for example asking where and why you would place the decimal point for 2.1 x 3.1 based on 21 x 31, but does not actually state what the valid rules are.
Adding together decimal points to multiply and multiplying both sides by 10 to get an integer to divide are mentioned standard methods in the parent's letter, but I didn't see these spelled out in the book but "expected outcome" of surviving a tedious investigation.
Grade D: Scattered investigations that are supposed to lead to understanding of decimal math, but no solved examples or complete description of methods anywhere, not even in teacher's manual. Very poor coverage that will leave very few students practiced in standard methods. My kid evidently does have the standard methods down, but he's in the 95+ percentile and never had much problem with Everyday math where dozens of parents show up to complain that they didn't understand it. His teachers say that "no math book is perfect", and they don't have any particular problem with a textbook that essentially contains no explanations for any of the methods being taught as long a the teacher is competent.
== Algebra in 7th grade? ==
Connected Math's web page claims that linear equations are solved in 7th grade, which is pretty ambitious since in the 70s, that was 9th grade algebra 1, and you can still wait until college to take it. I recall first covering something like this in Jr high, but not formally doing it until Algebra 1. The proposed new WA standard also does it early, which is an NCTM, but not a classical goal. Homework on the web page shows real equations to solve, but not if there is any instruction to support it.
The book unit is "Moving Straight Ahead" and there is a unit on solving linear equations:
Here's a linear equation, and a diagram of it's graph:
A = 5 + 0.5d
is an equation in two variables which you can plot on a line.
If you know one, you can find the other.
if A = 10 then
10 = 5 + 0.5d
finding d is solving the equation for d
3.2 asks how to maintain equality of an equation if you
- add 5 to one side
- subtract 6 from one side
Of course they don't actually TELL you what the correct thing to do is, let alone show you anything.
In the next paragraph, "use your ideas to solve"
30 = 6 + 5x
7x + 5 + 5x
7x + 2 = 2.5x
Of course there are no step by step examples with an explanation - the student is supposed to discover this during the class time, maybe with the teacher showing how. Remember the teacher never actually explains anything in this series.
Then you get to write equations using pictures to represent bags of money and coins on either side which I didn't get.
After basically ONE OR TWO classroom days, you're presumed ready to solve any linear equation. In the old days, we spent a month or so getting this drilled into our brains step by step, but why waste so much time on this minor topic when we need to spend an entire book learning how to visualize building figures with cubes? I wonder why so many people complain that CMP graduates are awful at algebra? CMP is mentioned as a possible substitute for Algebra 1 since in another paper, it is claimed that half of students won't need A1, but what about the other half of kids?
CMP includes a detailed study book comparing test scores of students who take CMP vs those who take algebra 1, and it claims CMP students score better, but I can't see how that is possible given this 1-page treatment of algebra. There is NO systematic brief reference that tells you how to solve a general linear equation of the form y = ax + b. The study makes no note of complaints by parents and citizens all over the internet, or basic structural problems such as the lackof any reference to important standard formulas or methods.
Grade D: This short treatment might work for an adult, but almost no kids are going to get this in a day thrown into a years worth of other worthless topics.
== Overall Conclusion ==
Textbook series gets Grade D - it does incomplete and poor job of teaching basic arithmetic and algebra, Investigations do cover most standard methods eventually, but won't leave students with anything resembling a firm grasp of any of the topics. I would reserve F for text like TERC which do not contain any instruction in standard arithmetic methods.
CMP consistently leaves out the one most important thing you need to know about any topic from the student book because they evidently believe putting it in the book will "spoil" your discovery process. CMP sometimes assumes the same level of knowledge as a high school or computer science graduate in order to complete some of the discovery process, such as solving simple quadratic equations, or asking to write a LOGO program to draw a figure.
As far as I know, no one has previously published this startling conclusion that the student math book that purports to teach how to add fractions doesn't contain an explanation of how to add fractions, or anything else. I've talked to people devoted to fuzzy math who don't see any problem with textbooks that don't have any text of what they claim students are to learn, and that's just the problem when people don't notice such a fundamental ommision as pi r squared.
Anyone who has children in CMP should ask the district to tell the teacher to tell parents that the one important thing their kids should derive is NOT in the book, but in the back page of the parent's letter. Do NOT throw away the letter. The letter does not give a full explanation of any standard method, so you'll still need a simple reference book such as Modern Curriculum Press or look up on Wikipedia to actually find an explanation of how to do any standard mathematical method.