The NCTM Problem-Solving standard discusses the necessity for
students to use their prior knowledge in order to find solutions to given
problems. Additionally, the standard describes the idea that students, as they
age, should be becoming proficient in selecting appropriate strategies to solve
problems. This is similar to the Standard for Mathematical Practice of making
sense of problems and persevering in solving them. This standard emphasizes the
idea of providing context to students when asking them to solve problems.
Students need to analyze the givens of the problem as well as the final goal before
beginning to solving the problem. Students then use their prior knowledge to
select the appropriate mathematical strategies to solve the problem accurately
and efficiently. While attempting to solve the problem, students are monitoring
their progress and checking their answers for reasonableness.
Problem Solving can also be related to the
standard of reasoning abstractly and quantitatively because the Standard for
Mathematical practice asks students to represent problems in multiple ways, to
understand the meaning behind the steps to solve the problem, and to be able to
explain why the steps were taken to solve the problem. Both standards require
students to analyze problems and to utilize their prior knowledge of
mathematical strategies in order to solve the problem. Problem solving requires
complex and flexible reasoning along with an understanding of the mathematical
principles being addressed in the problem.
Reasoning and Proof
In
order to construct a mathematical argument, students need to be able to
determine the mathematical concepts and content necessary to accurately back up
their ideas. Any counterarguments
against the claim of a peer are only valid if the student is able to provide mathematical
proof of why they disagree with the idea of a classmate. Constructing viable
arguments requires making predictions, justifying those predictions with
logical reasoning, and then sharing those ideas with others. Students need to
understand how to construct an effective argument using reasoning and
mathematical proof, so that their classmates can fully understand their claims and
learn from them.
The standard of making
sense of problems and persevering in solving them relates to reasoning and
proof because making sense of a problem requires reasoning and perseverance. Making
sense of problems requires students to monitor their own progress and to check
their answers to ensure that they are reasonable. Peers should be able to see
the steps that the student took to solve the problem and understand the
reasoning behind each step that was taken. Similarly, making sense of problems
requires collaboration and reasoning through problems can be more easily
understood with the assistance of others.
Communication
Reasoning abstractly and quantitatively requires students to
explain their reasoning behind solving problems to their peers. The ability to
effectively and clearly communicate one’s reasoning is an essential skill in the
mathematics classroom. Students need to be able to communicate orally and in
writing the steps to how they solved a problem using logic and mathematical
concepts. If peers are confused of the student’s meaning, the student also
needs to be able to explain in a different way to ensure that the communication
was effective.
The standard of attending to precision
asks students to communicate to others using clear and detailed language.
Students who are able to attend to precision can effectively communicate their
reasoning and arguments by specifying units, providing clear definitions, and
making accurate calculations. The more precise a student can be in his or her
explanations, the more likely that his or her classmates will understand and
learn from them. Inaccurate or unclear explanations will only seek to confuse
other students or to discount the credibility of the student’s reasoning.
Connections
Making sense of problems
provides students with the context to a problem. The context of a problem
allows students to see where they are going and where they just came from in
terms of mathematical concepts and how they relate to one another. Students are
making connections between their prior learning and what they are currently
studying. Connections allow students to see the subject of mathematics as a
constant continuation and buildup of ideas and concepts. Students will be more
likely to understand mathematical concepts if they can mentally visualize how
the concepts are related and how they fit into the mathematic umbrella.
Modeling with
mathematics asks students to use their prior knowledge to solve problems and
apply their knowledge to real world situations and problems. Essentially,
students are required to make connections between what they know and the real
world problem they are being faced with solving. Students are applying their mathematical
knowledge to the real world problem in order to solve it. Additionally,
students are asked to use tools, diagrams, graphs, etc. to aide them in solving
problems. Students must make connections between how they were taught to use
these tools and the situations in which each tool is the most effective.
Representation
The
standard of representation involves how students think about and write
mathematics. In today’s classrooms, pieces of paper and the chalk board are no
longer the only mediums in representing mathematics visually. Technology
provides both students and teachers with a myriad of ways to represent
mathematical concepts and ideas. However, students need to be instructed on the
appropriate use of these tools including their capabilities and their
limitations. Mathematical tools can be very useful in aiding student
understanding as long as students can manipulate them effectively.
Looking and making sure
of structure asks students to identify different ways of solving problems, to
look at problems from various perspectives, and to break down complicated
problems into more manageable steps. Representation plays a key role in how
students are able to accomplish this standard. Essentially, students are
analyzing how a problem is represented and asked to create new ways to
represent the same problem. Students are encouraged to locate patterns or
commonalities in representations and analyze those patterns to solve new
problems.
Very nice job! Thanks Sara:)
ReplyDelete