Blog Post #1: Mathematical Practice Standards

CCSS.MATH.PRACTICE.MP2 – Reason Abstractly and Quantitatively

·         Students who reason abstractly and quantitatively are able to understand the meaning behind mathematical operations and problems, not just how to compute numbers into an answer.

·         Students are able to represent problems symbolically (i.e. algebraically, pictorially, etc.)

·         Students are able to contextualize the meaning of the symbolic representations of problems while performing calculations.

·         Students are able to use a variety of different mathematical operations in order to solve a problem.

 

Common Core Mathematics in a PLC at work (Grades 3-5)

                The mathematical practice standard that asks students to reason abstractly and quantitatively enables students to “make mathematics useable and useful” (Larson, et al. 2012). It is one thing for students to be able to perform calculations but another for students to be able to explain how they arrived at an answer as well as their reasoning behind performing the particular steps in solving the problem. Students need to have conversations with the teacher and with their peers in which they justify their thinking in solving problems. Students should consider examples and extensions to problems in order to fully understand the meaning behind the mathematical operations and processes. Teachers should seek to design and implement lessons that allow students to practice using and applying mathematical concepts and operations in multiple ways. Lessons should enable students to develop skills such as recognizing relationships among numbers, interpreting numbers within context, realizing the magnitude of numbers, and solving real-world problems using numbers.  Students will use these skills in all mathematical areas and in every grade level.

 

Establishing Standards for Mathematical Practice

                Teachers can develop student skills in reasoning abstractly and quantitatively by designing lessons that ask students to explain their reasoning to others as well as understanding the reasoning of others (Stephan, 2014). In the journal article written by Michelle Stephan, she describes the first lesson of the school year in which she begins teaching her students what she deems the “societal norms of the classroom”. The societal norms include 1) explain the reasoning to others, 2) indicate agreement or disagreement, 3) ask clarifying questions when they do not understand, and 4) attempt to understand the reasoning of others.  In this lesson, Stephan pairs students together and asks them to work collaboratively to solve a story problem. Then, students present the steps they took to solve the problem to the rest of the class. In these presentations, the presenter is held accountable for his/her explanation of the steps to the solution as well as their justification for why they performed those particular steps to solve the problem. The other students are held accountable for asking questions if they do not understand the justification along with re-explaining the presenter’s explanation in their own words. The teaching of the societal norms of the classroom is a yearly practice.  Teachers need to encourage students to think about mathematical problems conceptually so that they can explain why they performed the steps to solve the problem and what the calculations mean in terms of the problem and real-world application.

 

Stephan, M. L. (2014). Establishing standards for mathematical practice . Mathematics teaching in middle school , 532-538.

 

 

 

CCSS.MATH.PRACTICE.MP3 – Construct Viable Arguments and Critique the Reasoning of Others

·         Students utilize their prior mathematical knowledge in order to construct logical arguments.

·         Students make predictions or state their opinions about a problem and then use logic to reject or support their initial ideas or predictions.

·         Students are able to explain and defend their conclusions as well as communicate those conclusions to others.

·         Students are able to compare and contrast two or more possible arguments for a problem.

 

Common Core Mathematics in a PLC at work (Grades 3-5)

                The goal of the third standard for mathematical practice is for students to justify or explain the reasoning behind solving a problem to the teacher and to their peers. Additionally, students need to be able to critique their own reasoning and the reasoning of their peers to ensure comprehension. There are multiple ways to solve a mathematical problem and this standard allows students to understand the various methods their peers used to solve the same problem. Teachers need to create student-centered classrooms in which students feel comfortable sharing their answers with one another. Students need to be encouraged by the teacher to ask questions if they are confused about a classmate’s solution to a problem and to critique the work of their classmates if the reasoning does not make sense.

Advice for Mathematical Argumentation

                Mathematical argumentation consists of three steps: 1) make conjectures, 2) justify the conjectures, and 3) decide whether the conjectures are true or false. A conjecture is essentially a guess based upon prior knowledge. Teachers need to emphasize to students that a conjecture needs to go beyond what is known to include what is not known. Students should be encouraged to make multiple conjectures over the same problem and to avoid getting upset if a conjecture is proved to be false. Justification asks students to explain their reasoning behind their conjectures. There are three ways to justify a conjecture: 1) numerically, 2) visually, and 3) geometrically. Finally, conclusions determine whether the conjecture is true or false. Students need to critique the work of their peers and decide whether or not they agree or disagree with their classmates. Students who disagree can use counterexamples to prove the conjecture false. Once the entire class agrees on the validity of a conjecture, it should be recorded and used for future reference. Teachers need to encourage students to use mathematical argumentation in every lesson throughout the school year.

 

Knudson, J., Lara-Meloy, T., Stallworth Stevens , H., & Wise Rutstein, D. (2014). Advice for mathematical argumentation . Mathematics teaching in the middle school , 494-500.