What is math understanding?

Understanding a mathematics topic consists of having the
ability to operate successfully in three cognitive domains. The first
domain, knowing, covers the facts, procedures, and concepts students
need to know, while the second, applying, focuses on the ability of
students to make use of this knowledge to select or create models and
solve problems. The third domain, reasoning, goes beyond the solution
of routine problems to encompass the ability to use analytical
skills, generalize, and apply mathematics to unfamiliar or complex
contexts.
Knowing
Facility in using mathematics or reasoning about mathematical situations
depends on mathematical knowledge and familiarity with
mathematical concepts. The more relevant knowledge a student is able
to recall and the wider the range of concepts he or she has understood,the greater the potential for engaging in a wide range of problemsolving
situations and for developing mathematical understanding.
Without access to a knowledge base that enables easy recall of
the language and basic facts and conventions of number, symbolic
representation, and spatial relations, students would find purposeful
mathematical thinking impossible. Facts encompass the factual
knowledge that provides the basic language of mathematics, and the
essential mathematical facts and properties that form the foundation
for mathematical thought.
Procedures form a bridge between more basic knowledge and the
use of mathematics for solving routine problems, especially those
encountered by many people in their daily lives. In essence, a fluent
use of procedures entails recall of sets of actions and how to carry
them out. Students need to be efficient and accurate in using a variety
of computational procedures and tools. They need to see that particular
procedures can be used to solve entire classes of problems, not just
individual problems.
Knowledge of concepts enables students to make connections
between elements of knowledge that, at best, would otherwise be
retained as isolated facts. It allows them to make extensions beyond
their existing knowledge, judge the validity of mathematical statements
and methods, and create mathematical representations.

Behaviors Included in the Knowing Domain
1 Recall Recall definitions, terminology, notation,
mathematical conventions, number
properties, geometric properties.
2 Recognize Recognize entities that are
mathematically equivalent (e.g., different
representations of the same function or
relation).
3 Compute Carry out algorithmic procedures (e.g.,
determining derivatives of polynomial
functions, solving a simple equation).
4 Retrieve Retrieve information from graphs, tables,
or other sources.
Applying
Problem solving is a central goal, and often a means, of teaching mathematics,
and hence this and supporting skills (e.g., select, represent,
model) feature prominently in the domain of applying knowledge. In
items aligned with this domain, students need to apply knowledge of
mathematical facts, skills, procedures, and concepts to create representations
and solve problems. Representation of ideas forms the core of
mathematical thinking and communication, and the ability to create
equivalent representations is fundamental to success in the subject.
Problem settings for items in the applying domain are more
routine than those aligned with the reasoning domain and will typically
have been standard in classroom exercises designed to provide
practice in particular methods or techniques. Some of these problems
will have been expressed in words that set the problem situation in
a quasi-real context. Though they range in difficulty, each of these
types of “textbook” problems is expected to be sufficiently familiar
to students that they will essentially involve selecting and applying
learned procedures.

Problems may be set in real-life situations or may be concerned
with purely mathematical questions involving, for example, numeric
or algebraic expressions, functions, equations, geometric figures, or
statistical data sets. Therefore, problem solving is included not only in
the applying domain, with emphasis on the more familiar and routine
tasks, but also in the reasoning domain.
Behaviors Included in the Applying Domain
1 Select Select an efficient/appropriate method
or strategy for solving a problem where
there is a commonly used method of
solution.
2 Represent Generate alternative equivalent
representations for a given mathematical
entity, relationship, or set of information.
3 Model Generate an appropriate model such as
an equation or diagram for solving a
routine problem.
4 Solve Routine Problems
Solve routine problems, (i.e., problems
similar to those students are likely to
have encountered in class). For example,
differentiate a polynomial function, use
geometric properties to solve problems.

Reasoning
Reasoning mathematically involves the capacity for logical, systematic
thinking. It includes intuitive and inductive reasoning based on
patterns and regularities that can be used to arrive at solutions to nonroutine
problems. Non-routine problems are problems that are very
likely to be unfamiliar to students. They make cognitive demands over
and above those needed for solution of routine problems, even when
the knowledge and skills required for their solution have been learned.
Non-routine problems may be purely mathematical or may have reallife
settings. Both types of items involve transfer of knowledge and
skills to new situations, and interactions among reasoning skills are
usually a feature. Problems requiring reasoning may do so in different
ways. Reasoning may be involved because of the novelty of the
context or the complexity of the situation, or because any solution to
the problem must involve several steps, perhaps drawing on knowledge
and understanding from different areas of mathematics.
Even though many of the behaviors listed within the reasoning
domain are those that may be drawn on in thinking about and
solving novel or complex problems, each by itself represents a valuable
outcome of mathematics education, with the potential to influence
learners’ thinking more generally. For example, reasoning involves
the ability to observe and make conjectures. It also involves making
logical deductions based on specific assumptions and rules, and justifying
results.

Behaviors Included in the Reasoning Domain
1 Analyze Investigate given information, and select
the mathematical facts necessary to solve
a particular problem. Determine and
describe or use relationships between
variables or objects in mathematical
situations. Make valid inferences from
given information.
2 Generalize Extend the domain to which the result
of mathematical thinking and problem
solving is applicable by restating results
in more general and more widely
applicable terms.
3 Synthesize/
Integrate
Combine (various) mathematical
procedures to establish results, and
combine results to produce a further
result. Make connections between
different elements of knowledge and
related representations, and make
linkages between related mathematical
ideas.
4 Justify Provide a justification for the truth or
falsity of a statement by reference to
mathematical results or properties.
5 Solve
Non-routine
Problems
Solve problems set in mathematical
or real-life contexts where students
are unlikely to have encountered
similar items, and apply mathematical
procedures in unfamiliar or complex
contexts.
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