Friday, June 10, 2011

Math problems of the week: 8th Connected Math vs. enrichment math

For years I have turned a blind eye to what happens inside J's math class--so long as I get to decide what we do at home. Just recently I've learned what I've been missing. Without my lifting a finger or wagging a tongue, a minor miracle has occurred. The interim math teacher who took over mid-year appears to have been giving J some above grade-level classwork. Given what the alternative would look like (the school uses Connected Math), I'm all the more appreciative. 

Consider the final algebra problems in 8th grade Connected Math vs. the worksheet that J brought home three days ago:

I. The final four problems in the final algebra chapter in 8th grade Connected Math: "Frogs, Fleas, and Painted Cubes," p. 87:

1. What patterns would you expect for a quadratic function

a. in tables of (x, y) value pairs?

b. in graphs of (x, y) value pairs?

c. in equations relating x and y?

2. How are equations, tables, and graphs for quadratic relations different from those for

a. linear relations?

b. exponential relations?

3. For a quadratic relation in the form ax2 + bx + c, how can you tell whether the graph will have a maximum or minimum point?

4. What strategies can be used to solve quadratic equations such as 3x2 - 5x -3 and x2 + 4x = 7 using

a. tables of values of a quadratic function?

b. graphs of a quadratic function?

II. The last four problems in J's sheet: (source unknown)

Write the following functions in the form y = a(x + h)2 + k. Find the vertex.

y = x2 + 4x -9

y = x2 - 6x + 16

y = x2 - 8x - 1

y = x2 + x + 2

III. Extra Credit

Consider the likely performance of a language-delayed, high functioning autistic child on each problem set. How might estimates of mathematical ability in autistic children depend on whether the instrument involves Reform Math problems?


bky said...

Slightly related:

I have been homeschooling my 12 and 13 year olds for several years. Next year they go to public jr hi in 7th and 8th grade. My 12-year old is halfway through the Art of Problem-Solving algebra I book. It's very good. They want him to take 7th grade Connected Math. I want him to take 8th grade algebra (not CM !). My argument: he has successfully done Singapore 1-6 and is now halfway through a solid algebra curriculum. By Sept he will be almost done with it. So at least let him take Alg in school rather than Connected math 7, wait a year and then take algebra. They want everything standardized. Well, there is a test-out procedure. From what I can tell it's just a Connected Math grade 7 exit test. So if you know algebra and its prerequisites, but are not sure what a steam-and-leaf plot is, you're supposed to spend a year finding out. Current plan: teach him the delta between Singapore and CM over the summer and hope the test is reasonable.

Also, this CM grade 7 exit test is used to place my then-8th grader in the high or low algebra class. Which, given the topic list for the test, seems like a dubious proposition.

Anonymous said...


If your son has really done the AoPS algebra book, he shouldn't be in *either* CM7 or regular Algebra. You should probably push for whatever's on the other side of Algebra in your district.

-- tjb

Barry Garelick said...

CMP is horrendous. I'm familiar with the Frogs, Fleas and Painted Cubes book and wrote about a problem in it, in my article about "discovery learning"

Sally said...

@Barry Garelick:

After reading your article on experiences in your math ed courses, I am so sorry that you were given such misinformation. I am a Mathematics Specialist for a school district and a professor of math education at an university. It is always bothersome to me that these ideas are conveyed in so many university classrooms because the learning theory they describe is not at all accurate or reflective of the original intent. Constructivism is the way people learn, but is not synonymous with "discovery." Good teachers (like the ones you reported having as a student) are always helping students construct knowledge - make connections between what they know and new learning. But I have seen this done by teachers using just chalk, a blackboard, and some well-crafted questions. This can be done no matter which curriculum you use if you are knowledgeable about both math and how to teach so students learn. I would encourage you to gain a deeper understanding of the learning theories from the original theorists and apply them. A good book to read is "How Students Learn Mathematics" by the National Research Council.
BTW - there are many problems with CMP and not all of them are inherent in the curriculum itself. One major problem is that teachers using it don't understand the math they are teaching.. In such cases, how can they help students make connections that they themselves don't have? Questions from CMP that you posted re: comparison of quadratics to linear & exponential equations are important for students. Analyzing those functions in equation, tabular and graphical form helps students understand the mathematical structures of those function - similarities and differences. The questions about min and max were meant to help students start to understand beginning transformations of equations and the effect of a negative coefficient for a. But again, this will happen only if the teacher can even understand the content.

Barry Garelick said...


Thanks for your comment. My article shows that there are good and bad ways to effect "discovery" and I am in total agreement that "aha" experiences can be had via blackboard and chalk as I describe in the article. I disagree with your assessment of CMP. I know teachers who have taught from CMP who know their math very well. The structure of CMP is such that there is a lack of sequence. In the quadratics lessons in Frogs, Fleas, etc, students are not given experience with factoring prior to their having to use it. As such, they are confronted with factoring in a "just in time" fashion and are suddenly presented with having to master factoring to solve the problem(s) at hand. If factoring were taught in proper sequence with proper mastery, then teachers could build upon this successfully. The CMP unit on ratios is a complete mess, with students not given the necessary information or instruction in order to understand and master the topic successfully.

Here is a lesson from CMP on ratios per a video.