In pursuit of a focused and rigorous curriculum

This entry will be a bit different from others. Instead of providing my commentary on specific GPS statements, I want to talk about the challenge of creating a focused and rigorous school mathematics curriculum. According to the Executive Summary of the GPS online, the state Board of Education asked the Department of Education (DoE) to write a new standards (GPS) with four specific charges. Those charges were,
(1) The curriculum needs to be rigorous…
(2) The curriculum needs to be focused…
(3) The curriculum needs to be clearly understandable by teachers…
(4) Instruction needs to be student-centered…

So, the GPS is supposed to be standards for a focused and rigorous curriculum that is easily understood by teachers and implementable (is this a word?) through student-centered instruction. There are many questions we can ask about the GPS based on these charges, but today, I want to focus on a “focused and rigorous curriculum.” In fact, much of my attention will be on a “rigorous” curriculum and how difficult it is to actually create one.

What does a rigorous curriculum look like? In our everyday language, the word “rigorous” is often used interchangeably with the words like “difficult” or “challenging.” Clearly, we want a curriculum that is (appropriately) challenging to our students. However, speaking of mathematics and mathematics education, the word “rigor” or “rigorous” should also mean something else. When mathematicians use the word “rigor,” what they typically refer to is logical cohesiveness of their arguments. Thus, a rigorous mathematics curriculum must be logically cohesive and coherent. There must be logical sequencing of topics.

This idea seems to be so obvious and easy to accomplish, but it is not that simple because it requires us to know the standards very well. It is very easy to overlook something while creating a curriculum. A good example can be found in the Mathematics Framework, a document that was created by the DoE “to be models for articulating desired results, assessment processes, and teaching-learning activities that can maximize student achievement relative to the Georgia Performance Standards.” So, these documents are supposed to be illustrative of a rigorous curriculum. Unfortunately, there are some oversights.

For example, look at the “Quotient if Greater Than One” task found in the Grade 3 multiplication and division unit. The task gives the following expression: []/^ > 1 (it's supposed to be a box over a triangle), and ask students questions such as “If the dividend is 10, what is the largest number the divisor can be? Why is that true?” and “If the divisor is 6, what is the smallest number the dividend can be? Tell why.” In the discussion section, the Framework states the following:

“This activity requires that students prove their answers to be true. Just a simple numerical answer is not enough. Help students articulate their thinking with correct math vocabulary and make sure they understand the connection between division and fractions. Students should be able to write any division problem as a fraction and for any division problem that has a remainder they should be able to write the remainder as a fraction also.”

Although these goals are very important goals for our students, looking at these goals in Grade 3 is totally inappropriate within the GPS. In the GPS, understanding that a fraction indicates the division of the numerator by the denominator is a Grade 5 topic (M5N4a). Although it is not quite clear in the GPS when students should learn to express remainders as fractions, certainly we can’t expect them to understand this idea until they have made this connection between division and fraction. Thus, this task, though it may be perfectly appropriate and useful task in Grade 5, is inappropriate for Grade 3 because it requires students to use something they have yet to study. It is a good example how difficult it is to create a rigorous curriculum, that is not only challenging but also logically cohesive.