Shaping the Standards: The Growth of Mathematical Ideas
This third article in a series reports on what proved to be one of the more challenging issues raised by the feedback to Principles and Standards for School Mathematics: Discussion Draft and how the writers used that feedback to create a more effective final document.
Although many comments addressed major issues, such as what role technology should play, other responses spoke to the heart of the Standards--what content should be included and how should it grow across the four grade bands of pre-K-2, 3-5, 6-8, and 9-12. Let us take a closer look at how the writers used readers' reactions about articulation of content to guide their revision of the draft. The Algebra Standard is used as an example.
The writers set out to show the growth of mathematical ideas across the grades through the use of ten common Standards. A draft statement of the Standard as it applies across the grades follows.
Mathematics instructional programs should include attention to algebra so that all students--
* understand patterns, relations, and functions;
* represent and analyze mathematical situations and structures using symbolic forms;
* use mathematical models to represent and understand quantitative relationships; and
* analyze change in various contexts.
Each grade band then includes expectations for each major area of the standard. For example, the 3-5 grade band drafted the following expectations for the first area--"understand patterns, relations, and functions:"
* Describe, extend, and make generalizatons, regarding geometric and numeric patterns.
* Represent and analyze patterns using words, tables, and graphs.
Many reviewers reacted positively to the use of common standards across the grades with more specificity at each grade band. For example, one reviewer stated, "A major positive feature of the new draft is the attention to longitudinal coherence, respecting the need to examine how big ideas grow over time." However, others thought that the draft had not always captured the necessary growth and highlighted these particular areas of concern:
* "Articulation across grade bands needs work; 9-12 doesn't match what happens in the other grade bands. The overviews don't address 9-12."
* "The one thing that we did notice is that ratio and proportion is a large topic in the 6-8 level. There is no discussion in 3-5 or pre-K-2 to lay the foundation."
* "There doesn't seem to be very much difference in the level of expectation from grades 3-5 to 6-8 in the geometry strand, and there is a big jump between grades 6-8 and grades 9-12."
As the writers laid out their plan to produce the final document, this feedback made it clear that more work was needed to ensure that mathematical ideas grow appropriately across the grade bands. To address this concern, the writers spent substantial time during the summer writing in cross-grade-band groups organized around the five standards that address mathematical content--number and operation; patterns, functions, and algebra; geometry and spatial sense; measurement; and data analysis, statistics, and probability. …