Appendix B: Assessing Mathematical Reasoning
2
- Jumps in and starts "doing stuff"
without a plan and without carefully
considering how chosen activities
(and/or information) might
contribute to a solution
- Gets immersed in a sub-goal and
forgets how it contributes to the
larger problem.
3
- Identifies relevant information.
- Develops a general plan and/or
considers how chosen activities
might contribute to the solution of a
larger problem
- May use “brute force” methods to
solve problems; does not look for
more efficient methods.
4
- Identifies relevant information and
defines assumptions.
- Develops a plan, modifies it as
needed; simplifies if possible
- Summarizes/organizes work in
progress to develop more effective
strategies.
- Uses more than one strategy, and
checks to make sure the answers are
consistent.
- Assumes similarities to prior
problems/situations hold true
(overgeneralizes).
- Identifies connections to other
problems / situations, but does not
explore connections more deeply
- Considers how a given problem or
situation is like/unlike others.
- Seeks out potential biases that may
make a certain strategy or procedure
seem plausible.
- Tests random cases
- Systematically tests a variety of
cases, but does not look for counterexamples.
- Jumps to conclusions without
sufficient evidence (e.g. "It looks
like all odd numbers will work,
because all the ones I've tested so
far work").
- Systematically tests a variety of
cases; looks for counter-examples.
- Organizes information in a way that
makes highlights patterns
- Develops partial explanations for
why patterns occur (e.g. “It might
have something to do with…
because….).
- Systematically tests a variety of
cases, including special and extreme
cases; looks for counter-examples.
- Proves the solution OR proves
conjectures that could contribute to
a complete solution (e.g. "All odd
numbers can be expressed as the
sum of consecutive #s, because....")
- Chooses an inappropriate way to
model a given situation.
- Assumes final answer is correct if
calculations are correct; does not
consider impact of modeling
choices on conclusion(s).
- Chooses an inappropriate way to
model a given situation.
- Describes how poor modeling
choices impact conclusion(s).
- Chooses an appropriate way to
model relevant aspects of a given
situation (e.g. controls and/or takes
into account appropriate variables).
- Considers how choices re: how to
model a situation might affect
accuracy of conclusion(s).
- Chooses one or more appropriate
ways to model relevant aspects of a
given situation; modifies as
necessary
- Considers how choices re: how to
model a situation might affect
accuracy / confidence in
conclusion(s).
Modeling
Inductive & Deductive
Reasoning
Analogical
Reasoning
Problem Solving
1
- Doesn't know how to start. Relies
on others for direction.
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Number
Measurement
Geometry
- Defines objects inaccurately
Sporadically explores the properties
of numerical objects / relationships
to see the effects of chosen
manipulations; does not hold
constraints constant
- Defines objects accurately, but
without necessary conditions
- Chooses inappropriate instruments,
intervals, and/or degrees of
precision; describes how these
choices impact conclusion(s)
- Systematically explores the
properties of numerical objects /
relationships to determine what
types / degrees of change are
possible with chosen constraints
- Defines objects accurately and
sufficiently, but not minimally
- Uses properties of numbers to
explain why certain relationships
hold true under particular conditions
- Defines objects according to
minimum necessary conditions
- Chooses appropriate instruments,
intervals, and degrees of precision.
- Considers potential effect(s) of error
on conclusion(s)
- Uncritically accepts given
geometrical definitions /
relationships; does not consider
parameters within which those
definitions hold true
- Defines objects inaccurately
- Randomly or sporadically explores
the properties of geometrical objects
/ relationships to see the effects of
chosen manipulations; does not hold
constraints constant.
- Defines objects accurately, but
without necessary conditions
- Generalizes quantitative
relationships with vague referents
that do not adequately distinguish
different aspects of the situation
being described
-
- Systematically explores the
properties of geometric objects /
relationships to determine what
types / degrees of change are
possible within chosen constraints
(without measuring).
- Defines objects accurately and
sufficiently, but not minimally
- Clearly expresses quantitative
relationships in general form;
distinguishes relevant constants and
variables
- Chooses appropriate instruments,
intervals, and degrees of precision
- Makes reasonable adjustments to
account for error
- Identifies degree of potential error
in conclusion
- Uses structural properties to explain
why certain relationships hold true
under particular conditions
- Defines objects according to
minimum necessary conditions
- Gathers irrelevant data
- Gathers relevant data, but represents
and/or organizes it in such a way
that relevant trends are difficult or
impossible to discern
- Reduces data to highly questionable
values based on personal opinions
about what seems right
- Identifies factors that may affect
results, but does not account for
them when interpreting results
- Chooses inappropriate instruments,
intervals, and/or degrees of
precision
- Does not consider how choices
might impact conclusion(s)
Statistics
Algebra
- Uncritically accepts given
numerical definitions /
relationships; does not consider
parameters within which those
definitions hold true
- Defines objects inaccurately
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- Gathers, represents, and organizes
relevant data
- Uses relevant characteristics of a
data set to identify representative
values, but overlooks important
factors
- Identifies factors that affect
personal level of confidence in
results
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- Clearly expresses quantitative
relationships in general and
simplified form; distinguishes
relevant constants, variables, and
types of variation
- Manipulates algebraic
representations to obtain new and/or
needed information and/or to
demonstrate equivalence
- Gathers, represents, and organizes
data such that relevant trends and /
or comparisons are highlighted
- Chooses and justifies appropriate
method(s) to reduce data set to
representative values
- Makes reasonable adjustments for
factors likely to affect results
- Chooses and justifies appropriate
method(s) to express degree of
uncertainty
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Complex
Systems
Communication.
Technology
- To be developed….
- To be developed….
- To be developed….
- To be developed….
- Methods / insights are almost
impossible to decipher from written
or oral presentation of work; may be
scattered all over the place in the
manner they emerged with no
attempt to re-organize and clarify
-
-
-
-
- Uses appropriate technology (e.g.
- Develops a deep understanding of
other people's strategies (e.g. listens
carefully, attempts others’ methods,
asks questions).
- Presents ideas in a clear and
organized manner; invites others to
participate
- Seeks out software features that allow
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spreadsheets, dynamic geometry
software) to systematically explore
problem spaces
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the user to define and manipulate
problem spaces in particular ways and to
perform repetitive tasks efficiently (I
wonder if there’s a way to make the
spreadsheet….”)
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