Envisioning a Standards-Based Classroom Environment
The information in Chapter 2 on recommendations for reforming curriculum, teaching, and assessment provides ideas on what a classroom informed by Standards-based principles should look like to reach students. Not surprisingly, creating coherent lessons that promote such reform is not easy, partly because acquiring a clear vision of the Standards for Mathematical Practices (SMP) and how they interrelate and connect to the content requires different ways of thinking, as well as practice, guidance, and time to evolve. Thus, teachers or curriculum writers must exercise caution against a limited vision of the Standards-based curriculum that might lead to superficial or misguided applications. As an example, consider the following lesson in an algebra class and ask, “How different are the teaching, instructional activities, and student participation from those in a traditional classroom?”
Example One: Reformed-Based Instruction?
The bell rings, and Nancy’s students enter class. They quickly sit in their assigned groups of four and take out their calculators. Nancy’s goal for the class is to have them model binomial multiplication with algebra tiles. She begins with a review of the properties of algebra tiles and their relationships to addition and multiplication of binomials and then gives each student a set of algebra tiles and a worksheet on multiplying binomials. Students decide who will tackle which problem, and the groups set to work. Nancy visits each group to monitor their progress.
This description includes many of the concepts that we associate with reform: The students are working in groups with manipulatives that include calculators, and the teacher monitors progress. How could the lesson not be reform based? Let us take a closer look.
In her discussion of the tiles, Nancy first defines the tiles: The length of the sides of the small square is 1 unit, and its area is 1 square unit; the larger square has sides of length x units, so its area is x2 square units. The rectangle has a length x units by a width of 1 unit, so its area is x square units (see Figure 3.1). She then shows how the tiles can be used to combine like terms in expressions such as 3x2 + 1 + 2x + 2x2 + 2x + 6, by just collecting like terms that are represented by the tiles: Collect 5 big squares, 4 rectangles, and 7 unit squares to get 5x2 + 4x + 7. Next she reviews how to multiply x(x + 3) by applying the distributive property and then representing each term by the corresponding tile. Hence, because x(x + 3) = x 2 + 3x, the product is represented by one large square and three rectangles.
A student asks, “Why do we have to use the tiles if we can get the answer by using the distributive property first anyway?” Nancy responds that this is just another way to do such problems. As she hands
FIGURE 3.1 Using the Distributive Law to Show the Use of Algebra Tiles
each student a sheet with exercises on binomial multiplication, Nancy instructs students to use the distributive property and then the algebra tiles to show the results of the distributive property. Students decide who will do which problem and begin working. Some use calculators to check their answers, and when most are finished, they wait for other students to finish working. Nancy visits each group, correcting any student errors and then assigning different students to put problems on the board.
Closer scrutiny shows that what looks like reformed teaching lacks key ingredients of reform. Let’s look at some of her strategies.