I’m continuing my study of the book Focus on High School Mathematics: Reasoning and Sense Making. I’ve been going through the first four chapters again, just to get a second look at the material more closely.
I found this incredible nugget in the first example on page 22, and I’ve used it with several teachers in discussing the new CCSS Mathematical Practices, namely Attend to Precision and Reason Abstractly and quantitatively.
Yes, there’s the easy connection dealing with the significant digits and appropriateness of rounding, as well as “ clarify the correspondence with quantities in the problem.” In this example Student 3 supports his rationale for 200,000,000 being the appropriate surface area (SA) because of his understanding of the issues with the value of the radius, not only is there concern about the digits following the 4, there are concerns about the spherical nature of the Earth to begin with. I think his connection between the SA and radius is a simple component of this article but it illustrates to me one way of developing correspondences between values in the problem.
In the Reason Abstractly and quantitatively description, this example reinforces the “attending to the meaning of quantities, not just how to compute them.” Student 3 not only indicates a good understanding of significant digits, but also of the meaning of 12 x 16 = 192 which is then approximated to 200 million square miles. This simple example reinforces the power of good number sense in high school students.
This example is also a powerful indicator of how to make this happen in classrooms. The teacher is asking some simple questions that are not focused on the answer, not about the formula, but is about asking questions that move beyond the answer. It is important then that students understand that the answer is NOT the most important component of the problem, but justifying the answer and developing an answer that presents clear understanding of the problem is.
This can only happen in a classroom that values these expectations as part of the instructional routine. It will take intentional focus by the teacher to think about the routines that will need to be developed:
– Think alouds modeling the kind of thinking that goes into selecting the appropriate computational approach and answer
– Questions that push students to do more than find the right answer
– Student to student discourse where they practice this kind of justification and choice with each other before sharing out whole group to provide everyone the opportunity to develop their skills
– Routinely getting feedback from the teacher how they can justify their answer effectively not only verbally but in writing. (quick writes and exit slips)
– Assessments, both formative and summative, in which students fine tune their skills.
Later in chapter 4 (page 29) the authors slip a very simple but eloquent statement into the text, “Helping students make sense of these essential ideas [number and measurement] will ease their transition to more abstract topics.”
I know I should be talking about chapter 5, but I was so enthralled with example 1 that I had to revisit it. I found myself coming back to this example so often in my work that I couldn’t over look it. It is in these simple examples that I think we can best see what the CCSS Mathematics will look like in the coming months and years, and I promise to move forward in the book next time. I’ve already found an example that I think is of equal power (no fair peaking at example 8)!