Algebra
Dimension
Description
Students with problem solving and strategic competence in algebra build from the
mathematical structure of arithmetic patterns towards using the variable to generalize the
pattern by using tables, graphs, and diagrams to better understand the relationship.
Problem Solving /
Strategizing
Proficient problem solvers reflect on their process and are willing to rethink their approach. In
working with students we want to consider, what can students learn from strategies that do
work and strategies that don’t work?
Examples
1) A student understands that the variable is the generalizer of
possible solutions to a problem and is able to create an
equation based on the context of the problem.
2) A student will compare different representations of a
relationship (e.g. – table, graph, equation, diagram) in order to
determine which will be most helpful in solving the problem.
3) A student will take data from a problem and create a
representation that will help him or her to explain and make
predictions about the relationship captured in the data.
Note: Within this dimension of the rubric there is recognition that there are multiple correct
strategies a student can take when solving a problem.
Reasoning is the capacity for logical thought, and justification. Students reason through
problems and offer evidence in support of their ideas.
Within algebra students reasoning and proof relies heavily on their ability to use evidence to
justify their thinking. This includes testing, and justifying conjectures in algebra through the
understanding of generalized arithmetic ideas.
Reasoning & Proof
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1) A student will discuss why 3−2 is equal to 2 using arithmetic
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patterns.
2) A student can discuss how to find the rate-of-change using
multiple representations and explaining how they are
connected (e.g. – a student can talk about how using a table to
find slope is connected to slope in a graph).
3) A student can talk about why his or her solution to a problem
makes sense through justifying their process.
Students that exhibit an exemplary ability to reason and prove construct arguments that
follow logical progressions that require no inference.
Note: Within this dimension of the rubric the word argument is used to broadly incorporate
the entirety of the student’s response to a given question.
Communication
Students organize and display their mathematical thinking through different representations
including: words, tables, graphs, equations, and diagrams to explain their process and their
reasoning. Students will often analyze the mathematical thinking strategies of others and use
the language of mathematics to express mathematical ideas precisely.
1) A student can communicate a solution to a problem through a
narrative, a clear algebraic progression, or a visual/graphical
representation.
2) A student can communicate why a procedure makes sense
based on conceptual understandings. This explanation can be
through words and/or algebraically.
Students who communicate well leave minimal gaps for the reader to fill in or interpret.
3) They communicate their thinking coherently to peers, teachers and
others through written or oral presentations.
Mathematical Thinking Dimensions
Common Core Mathematical Practice Standards
Problem Solving / Strategizing
MP1 – Mathematically proficient students start by explaining to
themselves the meaning of a problem and looking for entry points to
its solution. They analyze givens, constraints, relationships, and goals.
They make conjectures about the form and meaning of the solution
and plan a solution pathway rather than simply jumping into a solution
attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its
solution. They monitor and evaluate their progress and change course
if necessary.
- Perseverance is hard to measure
- CCS doesn’t talk about perseverance in the
description of MP1 at all, but rather the qualities of
a proficient problem solver.
MP2 – the ability to decontextualize—to abstract a given situation and
represent it symbolically and manipulate the representing symbols as if
they have a life of their own, without necessarily attending to their
referents—and the ability to contextualize, to pause as needed during
the
manipulation process in order to probe into the referents for the
symbols involved….Quantitative reasoning entails habits of creating a
coherent representation of the problem at hand; considering the units
involved; attending to the meaning of quantities, not just how to
compute them; and knowing and flexibly using different properties of
operations and objects.
MP6 – They calculate accurately and efficiently
MP7 – Mathematically proficient students look closely to discern a
pattern or structure.
Reasoning and Proof
- Reasoning is a larger notion than a viable
argument.
- In HS our central goal is to develop a broad notion
of proof (see Prof. Wu) where argument is
embedded within the idea of proof.
- Critiquing the reasoning of others primarily takes
place in the classroom and thus is not primarily
something exhibited in a task.
Communication
- Communication is essential to all disciplines.
Problem solving/strategizing, reasoning,
representations, and mathematical connections are
essential to mathematical communication.
- We notice that the creators of the common core
used the word communicate in different Practices
and we felt it could be combined into one strand.
MP3 – Mathematically proficient students understand and use stated
assumptions, definitions, and previously established results in
constructing arguments. They make conjectures and build a logical
progression of statements to explore the truth of their conjectures.
They justify their conclusion…They reason inductively about data,
making plausible arguments that take into account the context from
which the data arose.
MP1 – Mathematically proficient students can explain correspondences
between equations, verbal descriptions, tables, and graphs or draw
diagrams of important features and relationships, graph data, and
search for regularity or trends.
MP3 – communicate them (the justification of their conclusions) to
others
MP6 – They try to use clear definitions in discussion with others and in
their own reasoning. They state the meaning of the symbols they
choose, including using the equal sign consistently and appropriately.
They are careful about specifying units of measure, and labeling axes to
clarify the correspondence with quantities in a problem.