DescriptionThe Common Core State Standards for Mathematics (2010) has identified eight varieties of expertise as Standards for Practice that students need to develop during their discovery and learning of mathematics in order to become mathematically proficient. These standards include:

•Make sense of problems and persevere in solving them.
•Reason abstractly and quantitatively
•Construct viable arguments and critique the reasoning of others
•Model with mathematics
•Use appropriate tools strategically
•Attend to precision
•Look for and make use of structure
•Look for and express regularity in repeated reasoning

Research has shown that students learn best when they actively discover mathematical concepts and it is essential that these standards for mathematical practice are included in this discovery process. In his article, Discovery Learning and Constructivism, Robert B. Davis writes, “Traditional school practice viewed ‘learning mathematics’ as a matter of learning, usually by rote, certain meaningless rules for writing meaningless symbols on paper in some very specific ways. The projects viewed ‘learning mathematics’ as a matter of building up, in your mind, certain powerful symbol systems that allow you to represent certain kinds of situations, and a matter of acquiring skill in creating such mental representations and in using them.” (Davis, 1990) Well-developed lessons should include activities that are facilitated in ways that address the above practices as means to develop a true and deep understanding of the complexities of mathematical concepts.

This analytic illustrates two of these standards in the work of 6th grade students at the Harding School in Kenilworth, NJ: Standard 1 - Make sense of problems and persevere in solving them and Standard 7 - Look for and make use of structure. The students were part of the Rutgers Kenilworth Longitudinal study, funded by the National Science Foundation directed by Robert B. Davis and Carolyn A. Maher. (Grant number: MDR-9053597)

Look for and make use of structure
Mathematically proficient students not only need to be able to find patterns, but be able to use these patterns to make generalizations when appropriate. Researcher Robert Davis introduced the symbols box and triangle to represent two variables. As an early introduction to functions, Professor Davis also introduced the notion of truth sets. He provided the students the following function tables for finding a rule:

Problem 1
□ ∆
0 1
1 3
2 5
3 7
4 9
5 11

Problem 2
□ ∆
0 5
1 7
2 9
3 11
4 13

Problem 3:
□ ∆
0 1
1 4
2 7
3 10

Problem 4:
□ ∆
0 7
1 17
2 27
3 37
4 47
5 57

Problem 5:
□ ∆
0 -2
1 8
2 18
3 28
4 38
5 48

Problem 6
□ ∆
0 1
1 2
2 5
3 10
4 17
5 26

The students were instructed to find the relationship between the numbers in the box column and the numbers in the triangle column, and to write an open sentence to explain the relationship. In the following events, Stephanie, Michelle, Ankur, Bobby, and Amy-Lynn find patterns among the numbers in the truth sets and make important discoveries about the relationship between the patterns and the numbers in the equations they write for each table.

Make sense of problems and persevere in solving them.
As is the case with nearly all problems one encounters in life, students of mathematics need to be able to understand the problem at hand, find a manner in which to approach the solving of the problem, and continue working, evaluating, and modifying the approach until the problem is solved. To gain mathematical proficiency students need to persevere, especially when challenged to solve increasingly complex mathematics problems. Through their perseverance students may deepen their understanding of concepts and develop a means to solve similar problems when they arise. The later events are selected to show Michelle and Ankur’s process of finding the equation for the particularly challenging function table in Problem 6, which is the first function that is not linear.

Davis, Robert B. Journal for Research in Mathematics Education. Monograph
Vol. 4, Constructivist Views on the Teaching and Learning of Mathematics (1990),
pp. 93-106+195-210