Wisconsin Standards for Mathematics 93
intercept have meaning related to the context. Quantitative reasoning also entails knowing and flexibly using different properties of
operations and objects. For example, in middle school, students use properties of operations to generate equivalent expressions and use the
number line to understand multiplication and division of rational numbers.
Math Practice 3: Construct viable arguments, and appreciate and critique the reasoning of others.
6-8 Mathematically proficient middle school students understand and use assumptions, definitions, and previously established results in
constructing verbal and written arguments. They understand the importance of making and exploring the validity of conjectures. They can
recognize and appreciate the use of counterexamples. For example, using numerical counterexamples to identify common errors in algebraic
manipulation, such as thinking that 5 – 2x is equivalent to 3x. Conversely, given a pair of equivalent algebraic expressions, they can show that
the two expressions name the same number regardless of which value is substituted into them by showing which properties of operations can
be applied to transform one expression into the other. They can explain and justify their conclusions to others using numerals, symbols, and
visuals. They also reason inductively about data, making plausible arguments that take into account the context from which the data arose.
For example, they might argue that the great variability of heights in their class is explained by growth spurts, and that the small variability of
ages is explained by school admission policies.
While communicating their own mathematical ideas is important, middle school students also learn to be open to others’ mathematical ideas.
They appreciate a different perspective or approach to a problem and learn how to respond to those ideas, respecting the reasoning of others
(Gutiérrez 2017, 17-18). Together, students make sense of the mathematics, asking helpful questions such as “How did you get that?” “Why is
that true?” and “Does that always work?” that clarify or deepen everyone’s understanding. Mathematically proficient students are able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in
an argument, explain what it is. Students engage in collaborative discussions, drawing on evidence from problem texts and arguments of
others, follow conventions for collegial discussions, and qualify their own views in light of evidence presented. They can present their findings
and results to a given audience through a variety of formats such as posters, whiteboards, and interactive materials.
Math Practice 4: Model with mathematics.
6-8 “In the course of a student’s mathematics education, the word ‘model’ is used in many ways. Several of these, such as manipulatives,
demonstration, role modeling, and conceptual models of mathematics, are valuable tools for teaching and learning. However, they are
different from the practice of mathematical modeling. Mathematical modeling, both in the workplace and in school, uses mathematics to
answer big, messy, reality-based questions” (Bliss and Libertini 2016, 7).
Mathematically proficient middle school students formulate their own problems that emerge from natural circumstances as they
mathematize the world around them. They can identify the mathematical elements of a situation and generate questions that can be
addressed using mathematics (e.g., noticing and wondering). Middle school students can see a complicated problem and understand how that
problem contains smaller problems to be solved. They are comfortable making assumptions as they decide “what matters”. Mathematically