Mathematical Practices

In March, I attending a training session given by the College Board on the new 2019 AP Calculus Course and Exam Description (2019 CED). I was impressed by the copious other materials the College Board had prepared for the roll-out that will be available at summer institutes. Among these was Mathematical Practices. The MPACs (Mathematical Practices) from the 2016 CED have been revised and condensed from six down to four. In both forms they summarize how mathematicians work, think, and communicate. Therefore, they outline what students need to learn and do when learning mathematics.

The Practices are summarized on page 13 – 14 of the 2019 CED and discussed in detail in the “Developing the Mathematical Practices” chapter (p. 214 – 220) where, included with each of the skills, are Key Questions, Sample Activities, and Sample Instructional Strategies. Each unit in the 2019 CED starts with a short discussion of the Mathematical Practices that apply to that unit.

While the Practices are listed with examples specifically for the AP Calculus courses, they really apply to the entirety of a student’s mathematical learning and thinking from grade school on. If your school district has a Math Vertical Team, an ongoing discussion of the Practices is certainly an appropriate topic. Otherwise, share them with the teachers from the lower grades and sending schools. They are relevant at all grade levels.

One thing you can do to help students with the Practices is to make and keep them aware of them. Put them on a poster in the room. Make a handout of pages 13 and 14 for the front of their notebook. Refer to them whenever you use one of the items on the list.

The practices are these. (I have slightly edited them to remove the numbering and the calculus-specific examples.) My thoughts and comments are below the quotes.

Practice 1: Implementing Mathematical Processes – Determine expressions and values using mathematical procedures.

  • Identify the question to be answered or problem to be solved.
  • Identify key and relevant information to answer a question or solve a problem.
  • Identify an appropriate mathematical rule or procedure based on the classification of a given expression.
  • Identify an appropriate mathematical rule or procedure based on the relationship between concepts or processes to solve problems.
  • Apply appropriate mathematical rules or procedures, with and without technology.
  • Explain how an approximated value relates to the actual value.

The first Practice really describes the problem-solving process. This Practice is applicable throughout a student’s study of mathematics from grade school on.

The first two bullets while marked as “not assessed [on the AP Calculus exams]” are the beginning of the problem-solving process. The next two are how you start the work of problem solving, and the fifth applies to carrying out the rules and procedure you’ve decided upon. The last needs to be considered whenever your answer is not exact – which may be most of the time.

Practice 2: Connecting Representations – Translate mathematical information from a single representation or across multiple representations.

  • Identify common underlying structures in problems involving different contextual situations.
  • Identify mathematical information from graphical, numerical, analytical, and/or verbal representations.
  • Identify a re-expression of mathematical information presented in a given representation.
  • Identify how mathematical characteristics or properties of functions are related in different representations.
  • Describe the relationships among different representations of functions ….

Multiple representations, often called the “Rule of Four”, help one see and delve deeper into mathematical situations. Graphs, tables, and symbolic expressions representing the same thing show different ways of expressing and understanding mathematical ideas. Expressing the relationships in words by writing, talking, discussing, and arguing about them helps students understand and internalize the mathematics (see Practice 4). Technology is invaluable in doing this.

All four should be considered in every situation and for every concept. Sometimes one is more informative and useful than the others, other times a different perspective sheds additional light on the concept. And, once again, this should be done from the beginning of a student’s mathematical career.

Practice 3: Justification – Justify reasoning and solutions

  • Apply technology to develop claims and conjectures.
  • Identify an appropriate mathematical definition, theorem, or test to apply.
  • Confirm whether hypotheses or conditions of a selected definition, theorem, or test have been satisfied.
  • Apply an appropriate mathematical definition, theorem, or test.
  • Provide reasons or rationales for solutions and conclusions.
  • Explain the meaning of mathematical solutions in context.
  • Confirm that solutions are accurate and appropriate.

Technologies (in the broad sense of anything other than paper and pencil: blocks, beads on wires, and other manipulatives in grade school, to computer programs, spreadsheets, CAS, and Oh BTW, graphing calculators) are an increasingly important tool for mathematicians. Technology should be incorporated at all grades and levels. Students should learn how to use them no only to do and check their work, but also to explore mathematics and discover mathematical ideas (even if these are already known to more advanced students).

Definitions and theorems formalize the results of mathematical exploration and point the way to other discoveries. Students should become familiar, not just with a few theorems and definitions, but with the structure of them and relationships between them (converses, inverses, and contrapositives). They need to know that if the hypotheses are true, then the conclusion is true. They need to be able to show (confirm) that the hypotheses are true before they apply a theorem or definition to a given situation.

In early grades, stating theorem formally is not always necessary or desirable. Still, students should be aware that there are certain rules (which after all are theorems) and they may be used only when appropriate. I’ve often told students that in real life you can do whatever you want unless there is a law saying you can’t, but in mathematics you can’t do anything unless there is a law that say you can.

Part of the problem-solving process in Practice 1 should include making sure your result makes sense in context. That means student mathematicians need to understand the meaning of their results and be able to confirm that the work and the solution are accurate and appropriate. Explaining this verbally to other and in writing, a communication skill from Practice 4, is a way to do this. This can be does at all grade levels.

The previous MPACs from the 2016 CED list “Students can … analyze, evaluate, and compare the reasoning of others.” (MPAC 6f.) At all levels, this is one way to have students confirm and explain their results and understanding.

Practice 4: Communication and Notation – Use correct notation, language, and mathematical conventions to communicate results or solutions.

  • Use precise mathematical language.
  • Use appropriate units of measure.
  • Use appropriate mathematical symbols and notation
  • Use appropriate graphing techniques.
  • Apply appropriate rounding procedures.

As we’ve all learned early in our teaching careers, after teaching a topic two or three times we understand it much better. We see the fine points and appreciate the connections. It was that communication, the teaching of it, that helped us understand it. Activities where students communicate help them understand as well.

The items under Practice 4, are important because communication with others orally and in writing will help your students learn and understand mathematics. To use the language of mathematics, students need to know the structure of mathematical reasoning (return to Practice 3 – theorems and definitions), and the tools for doing so (notation, units, etc.). At all grade levels, students should practice in communicating and using the language and notation – this will help them learn.

Take a good look at the Mathematical Practices and incorporate them into your thinking and teaching. Help your students look at what they are doing, to look at the big picture. It will help with the details.


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