As described in previous posts, the Math Practices are the backbone of what it means to work as a mathematician. The practice standards outline the habits that we try to develop in our math learners to support the development of conceptual understandings across the domains. Whether are students are working in number sense, operations, algebraic thinking, geometry, measurement, data, or integrating these, the habits used to successfully make sense of problems, think mathematically and communicate understanding are critical to developing our students as mathematicians.
In each unit, we have focused on a few of these, so that all practices are developed throughout the school year. In unit 4, we focused on the last 3 of the 8 Practice Standards, numbers 2, 4, and 6. Each are described in more detail below.
Reason abstractly and quantitatively: Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: 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.
More generally, Mathematical Practice #2 asks students to be able to translate a problem situation into a number sentence (with or without blanks) and, after they solve the arithmetic part (any way), to be able to recognize the connection between all the elements of the sentence and the original problem. It involves making sure that the units (objects!) in problems make sense.
Model with mathematics: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas.
Mathematically proficient students can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
Attend to precision: Mathematically proficient students try to communicate precisely to others. 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.
They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
~Common Core State Standards
As you work with your child in math, consider how you can show your own practices in thinking mathematically, and help them to communicate their practices clearly to you. As you review math assessments with your child, notice how they have shown these practices and how they might continue to grow their habits of thinking and communicating as mathematicians.
We are continuing to experiment with tools for assessing and communicating your child’s growth and practices in math with you and with them. Taking time to review your child’s work with them and discussing these practices can only support the work we are doing here at school. This support is certainly appreciated!