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A fundamental dilemma: the case of circles

May 15, 2011

The Connected Mathematics Project has always stood for coherence in units, and for telling a complete mathematical story. So we don’t do a little bit with fractions now and then a little bit three weeks from now. Instead we spend 4-6 weeks on fractions, decimals and percents-exploring equivalence, representations and models.

The Common Core State Standards presents a challenge to some of the story lines of our units. Consider area. In CMP, area and perimeter are dealt with in the sixth-grade unit Covering and Surrounding. We begin with irregular figures that are composed of whole numbers of square units, move to formalizing area of a rectangle and then use rectangles to develop area formulas for triangles, parallelograms and circles (as well as circumference of circles). All in one unit; all in sixth grade.

How many radius squares (shaded) fill the circle? More importantly, are students better served learning this in sixth grade with other area concepts, or in seventh as Common Core mandates?

But now…

Common Core, grade 6 geometry

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.

Common Core, grade 7 geometry

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

There are two schools of thought on the problem this presents.

1: Don’t worry about it. The argument goes like this: Do circles well at sixth grade. You’ll use area of circles at seventh grade, which will remind students of everything they need to know before it’s tested there. Don’t worry about it. Keep circles where they belong in the curriculum-with the area and perimeter of other plane figures.

2: Do as you’re told. The argument goes like this: Wait! Schools and districts will make decisions not just about which units to teach, but which parts of units to teach based on the Common Core Standards. Schools will skip circles at sixth grade in order to focus on what’s being tested there. If you don’t write a development of area of a circle into seventh grade, teachers will supplement and tell; this important topic won’t get done well.

Versions of this dilemma arise frequently in looking at CMP through the lens of Common Core. Volume of a cylinder is separated from volume of prisms. Volume of rectangular prisms with fractional side lengths is separated from volume of rectangular prisms with integer side lengths. Fractions and ratios are at one grade level; rates involving fractional units are at another. And on and on.

It would be helpful to hear from the field on this.

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From → Grade 6, Grade 7

5 Comments
  1. P. Mark Taylor permalink

    I think the dilemma is in the issue of control over curriculum decisions. There are many correct ways to do it. You could revamp CMP to highlight the connections that are inherent in the Common Core. Show us 1 correct way to make connections within the Common Core. Best to ride the wave taking advantage of the good things that are in it.

  2. Christopher permalink

    from a reader, via Facebook:

    I’d say CMP should be highlighting the Standards for Mathematical Practice, a very important piece where basically it blows away the competing middle-grades programs. Anyone can make a program that hits the Common Core bullet points for content. In this case, I say you’re doing it right — it will take less time to teach the overall material this way. You will need to make it clear to districts and teachers why you made this decision. I’m glad that in high school there is no grade-based decision making, although there are suggestions and the assessment decisions will probably be based on those suggestions.

    But the “hard part”, the focus on mathematical thinking skills, is where CMP should be focusing its efforts in this. (I also think this, not the content standards, is the place where Common Core can have some meaningful impact on the way math is taught and learned.)

  3. Christopher permalink

    Another loyal reader weighs in via Facebook:

    This is a dilemma but I do believe that the Common Core Standards will change in the near future to provide a better focus on concepts. I am biased — I prefer the current development of content.

    Our seventh grade teachers have always had to come back and review circles; our eighth grade teachers have always had to come back and review circles; our ninth grade teachers…Do you recognize the recursion???

    That said, it hasn’t taken long if the teachers can refer back to the work in sixth grade. That is the key. If they teach it “their way”, it becomes something new to learn.

  4. Jim Mamer permalink

    As a 6th – 8th grade teacher and user of CMP for many years my initial thoughts are “keep circles in 6th grade!” Asking a 7th grade student to “drop” into area again in seventh grade and to then construct the area of a circle based on the radius squares that cover is not going to be as developmentally strong as what we currently do….build the concept while we are in a 3-5 week unit on area… I have been guilty of leaving certain topics out at certain grade levels as we prepare for state testing only to be sooooo disappointed in my decision as I “drop” back into a topic months later and wonder how half my kids can look at me as if they have no clue what we spoke of those months earlier…Stay with a topic and get it then utilize in later grade levels to support the CORE standards as much as possible. If you don’t provide it within the curriculum, I guess I’ll have to do it myself. 🙂

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