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- Capacity to Generalize: Connections for Future Mathematics Specialists

**CAPACITY TO GENERALIZE: CONNECTIONS FOR FUTURE MATHEMATICS SPECIALISTS**

__Dr. Maria A. Timmerman__

Longwood University

*Using the National Council of Teachers of Mathematics’ (NCTM, 2014) eight mathematics teaching practices, future mathematics specialists and teacher educators increased their ability to generalize about patterns, functions, and algebra. If teachers build few connections among multiple representations of algebraic concepts, supporting students in using representations to make mathematical conjectures and justify their mathematical thinking will not occur. During this session, teachers learned* *how to explore patterns in tables and make connections to graphs and equations. Details* *were shared for creating a new Longwood cohort of teachers at off-site locations who plan to earn a Master’s Degree applied to a Mathematics Specialist endorsement.*

Subject matter knowledge and pedagogical content knowledge are important constructs in the professional development of K-8 mathematics teachers. For teacher learning to occur, a deeper understanding of mathematics content knowledge they teach and how students construct this same knowledge needs to happen over time with opportunities to ‘try out’ NCTM’s (2014) teaching practices. Just as classroom teachers serve as mentors to students in strengthening their understanding of mathematics in meaningful ways, a mathematics teacher educator serves as a mentor to teachers enrolled in a mathematics specialist graduate program. By modeling new teaching practices, such as “elicit and use evidence of student thinking” and “facilitate meaningful mathematical discourse” (NCTM, 2014), teacher educators can guide and support teachers in their future role as mathematics specialists in a supportive learning community.

Russell, Schifter, and Bastable (2011) define mathematics as “a way of thinking that involves studying patterns, making conjectures, looking for underlying structure and regularity, identifying and describing relationships, and developing mathematical arguments to show when and why these relationships hold” (p. 2). Specific to the mathematics specialist courses, there is a dual focus of teachers examining their own understanding of mathematics and that of their students. Across grades K-8, students have multiple experiences exploring patterns using manipulative materials to make connections to numeric relationships. Blanton, Levi, Crites, & Dougherty (2011) state, “Understanding algebraic thinking deeply requires you not only to know important mathematical ideas but also to recognize how these ideas relate to one another” (p. 3).

One way to improve teacher learning related to making algebraic connections explicit is to provide teachers opportunities to solve problems and work collaboratively with representations they can use with students. Teachers need to experience a variety of concrete models, tables, graphs, and equations that facilitate their own and students’ construction of algebraic concepts. Participating in the process of *generalizing* engages teachers in identifying underlying structures and making sense of algebraic relationships (Blanton, et al., 2011). During this session, teachers explored course activities that are used in Longwood’s mathematics specialist courses to model and strengthen teachers’ understanding of algebraic ideas. With full implementation of the new Virginia Mathematics Standards of Learning (Virginia Department of Education, 2016) in fall 2018, the session focused on the patterns, functions, and algebra content standard in grades K-8. Teachers analyzed the vertical alignment of algebraic concepts, anticipating and recognizing students’ misconceptions and discussed teaching practices that could be used to address students’ limited understanding of algebraic thinking and connections among concepts.

For example, the teaching practice of “use and connect mathematical representations” (NCTM, 2014) engaged teachers in working together as a team and problem solved the growing table task (see Table 1). Teachers made connections among representations with the following task: For each table pattern, use pattern blocks (triangles, squares, or hexagons) to represent tables and counters for chairs to model one person sitting on each side of a table. Three people can sit around one triangular table. Build the next table using two triangular pattern blocks making sure one side of the first table connects to a complete side of the second table. Keep increasing by one the number of tables used in a straight row and model the number of people who can sit on each side of the growing table. Describe the pattern in words, make a table of input/output values (see Table 1), graph the data, and write an equation for the correspondence between the number of tables and the number of people that are seated. Use a correspondence (function) rule to find the number of people who can be seated around 50 connected tables.

Input: Number of triangle tables connected |
Output: Number of people seated at triangle tables |

1 |
3 |

2 |
4 |

3 |
5 |

Table 1: Triangle table pattern for growing table task.

An essential understanding of algebraic thinking in grades 3-5 consists of a progression of three different relationships: “In working with functions, several important types of patterns or relationships might be observed among quantities that vary in relation to each other: recursive patterns, co-variational relationships, and correspondence rules” (Blanton, et al., 2011, p. 51). Use an input/output (see Table 1) if students only focus on how the output values increase by +1, which is called a *recursive* pattern. When only a few triangle tables are connected, an output change can be quickly determined. Yet, if the 50^{th} row is needed, the value is not simple to find. If 50 triangular tables are connected in a row, 52 people can be seated at the growing banquet table. Moving to a higher-level pattern, students use a *co-variation* pattern, where both the input and output changes are stated for each row. For the triangular tables, a student recognizes that for each increase of one for an input value, there is an increase of one for an output value (see Table 1). Similar to the 1^{st} recursive pattern, if a large number of tables are used, it is time-consuming to find both the input and output values, as they are still calculated separately.

Lastly, a third-level pattern consists of using a *correspondence or function rule* where the relationship between an input value and an output value is found for any row of the data table, often called the *n ^{th}* term or figure. For the triangular tables, a correspondence rule can be generated: 1 times input + 2 = output, or, x + 2 = y. Using Table 1, an input value of 3 is multiplied by the coefficient, 1 times 3 = 3, and 2 is added, resulting in 3 + 2 = 5, as the output value. The correspondence rule is equivalent to the slope-intercept form of an equation, y = mx + b. Connections may be made to generating a graph for y = x + 2, where the y-intercept = 2 and the slope equals 1/1, meaning a change of y (rise of 1) over a change of x (run of 1).

As mathematics teacher educators, we need to ensure graduate programs continue to develop future mathematics specialists who serve as on-site resources for job-embedded professional development (Campbell, Ellington, Haver, & Inge, 2013). By modeling NCTM’s (2014) eight teaching practices in our graduate courses, we are sharing ways to mentor teachers in schools to improve mathematics instruction and students’ mathematical achievement. Solving problems with manipulatives, tables, graphs, and equations is just one example of mathematics specialists making connections to develop the ability to generalize. Making sense of tables of data prepares teachers and students to view algebraic reasoning explicitly as they solve practical problems.

**References**

Blanton, M., Levi, L., Crites, T., Dougherty, B.J. (2011). *Developing essential understanding of algebraic thinking for teaching in grades 3-5*. Reston, VA: National Council of Teachers of Mathematics.

Campbell, P. A., Ellington, A. J., Haver, W. E., & Inge, V. L. (2013). *The elementary mathematics specialists handbook*. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (2014). *Principles to actions: Ensuring mathematical success for all.* Reston, VA: Author.

Russell, S. J., Schifter, D., & Bastable, V. (2011). *Connecting arithmetic to algebra: Strategies for building algebraic thinking in the elementary grades*. Portsmouth, NH: Heinemann.

Virginia Department of Education (2016). *Virginia mathematics standards of learning*. Richmond, VA: Author.