Yesterday I started writing the post below. When I went to develop an example that reinforced my point, I discovered that my example ended up being a counterexample to this argument. Rather than move on to a new post topic, I decided to share what I had learned with you. The original start to this post is in blue below. What I learned follows.
The organization I work for has been writing test items aligned to the Common Core State Standards for Mathematics for at least two years now. The majority of the items we develop are aligned to one standard from the Standards for Mathematical Content. Based on this experience, I am beginning to believe that the most interesting mathematics won’t be assessed if we continue to focus solely on the individual content standards.
We are successfully writing items that align to the standards and that assess important content skills and knowledge. Assessing these skills are an important, particularly when you are trying to provide teachers with information about what students still need to learn. However, to do more than that I’m beginning to believe that we should do two things.
- Assess the Standards for Mathematical Practice as well as the Standards for Mathematical Content.
- Combine content and practice standards in ways that illuminate interesting mathematics.
I believe that, by expanding the scope of what we’re trying to assess, not only will we be able to evaluate a student’s knowledge and skills but we’ll also be able to evaluate their application of these skills to solving practical, important, and interesting mathematics problems.
I intended to develop an example where the process of students graphing data, identifying a function that describes that data, and then using the function to answer questions in the context of the problem was spread out over several standards. However, what I found were several examples of standards that connect graph, function, and context together. Some examples are given below.
5.OA.02.03 – Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
6.EE.03.09 – Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
08.F.02.04 – Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
I still believe that some interesting mathematics can only be assessed by combining standards and practices together in meaningful ways. Now I need to find an example that works.
What do you think? Do you have an example of meaningful mathematics that can only be assessed by combining standards and practices together? Leave a comment with your thoughts and suggestions.
National Governors Association Center for Best Practices, Council of Chief State School Officers, (2010). Common core state standards for mathematics. Retrieved from website: http://www.corestandards.org/assets/CCSSI_Math Standards.pdf
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