Problem-solving
is an essential skill for all students to have in not only a mathematics
classroom but in all subject areas. Problem solving consists of utilizing one’s
prior knowledge and experiences to determine the appropriate solution to a
problem. Teachers should take great care in selecting problems for students to
work through in the classroom. Appropriate problems are those that contain
real-world applications and challenge students to expand and connect their
prior knowledge to new ideas. Additionally, problem solving instruction should
not be separated from the rest of the mathematics curriculum but should be
fully integrated into all aspects of the math content. Students should be able
to identify specific strategies to use when problem solving and determine when
it is appropriate to use such strategies.
Reasoning and Proof
Reasoning
and proof are essential skills in not only mathematics but in all subject
areas. Student should be able to make conjectures and then justify those
conjectures using mathematical reasoning and evidence. These skills should be
taught consistently from early elementary school all the way through high school. Teachers need
to encourage students by asking questions that challenge students to think
deeper. Teachers should allow students to discover answers on their own or with
their peers, rather than stating claims and providing students with practice
problems. Students, especially in the younger grade levels, can explore
mathematical ideas and claims by working with manipulatives and other tools.
Eventually, students should be able to give both the answer to a problem along
with the reasoning why the answer is the correct one. Students should feel
comfortable sharing their ideas in the classroom in front of their peers.
Communication
Communication
is an aspect of the mathematics classroom that is often neglected or forgotten.
If students are given opportunities to communicate with one another about
mathematical concepts as well as to critique the ideas of their classmates,
they will expand their mathematical knowledge and take on responsibility for
their own learning. Instruction on how to mathematically communicate should
begin with early elementary students and continue throughout high school. By
the end of high school, students should be able to make clear and precise
mathematical arguments that can be easily understood by others. High School
students should be able to question and analyze the arguments of their peers by
using appropriate terminology and reason. The goal is for students to
eventually have communication as a natural part of their mathematical learning.
Connections
All
mathematics is connected, though it is frequently taught with strong divisions.
The knowledge and skills students gain in one year of schooling should continue
to build upon one another the following year.
When students learn to make mathematical connections, they learn not how
concepts but also how mathematics works as a whole subject. Teachers should
seek to help students discover mathematical connections throughout instruction
by avoiding teaching the subject in categories. Instead, teachers should build
connections in every lesson by asking students to think about their previous
learning and how it relates to current instruction. Students should be able to
see the connections between prior learning as well as to other subject areas
and daily life.
Representation
The
representation standard describes how we write mathematics and how we think
about mathematics. A mathematical representation can include diagrams, graphs,
symbols, etc. The choice of how to represent a problem is a major contributor
to whether the student will understand the problem and be able to find a
solution. They also serve to inform the teacher if the student truly
understands a concept or not. Students should be taught how to create several
different kinds of representations for a single problem so that they can chose
the best representation that makes the most sense to them. New forms of technology
have recently come available that can offer additional representations for
students to use to understand mathematical ideas and find solutions to problems.
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