Standards for Mathematical Practice: Cheshire Cat’s Grin, Part Two

This is Part Two of a three part article [Part One] which provides the description of each of the Standards for Mathematical Practice as written in the Common Core math standards. It discusses aspects of each SMP that can be interpreted along conventional or traditional approaches to math teaching and contrasts this with how each one may be implemented under the math reform interpretation.

by Barry Garelick

SMP 3: Construct viable arguments and critique the reasoning of others.

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 are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also 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. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

The skills described in this SMP are a necessary part of learning mathematics, and the standard is an appropriate one for students who have gained the understanding, vocabulary, and mathematical tools by which they can conduct such analyses. The analysis and arguments expected of students, therefore, must be appropriate to the grade level.  In lower grades, students are still developing the analytic tools and vocabulary by which to express mathematical ideas and arguments.   In K-5, therefore it is appropriate to have students observe a problem being worked, identify if the problem is being done correctly, and if not, explain what is being done wrong.

In higher grades such as pre-algebra and above, students now have the tools to express mathematical ideas symbolically and also have a greater mathematical vocabulary.  Analyzing arguments and mathematical reasoning can now be done by being able to express the mathematical ideas symbolically and reason and draw conclusions from their manipulation.  In geometry, analysis of arguments is very important since that subject requires students to prove propositions and theorems.

The danger of this SMP is that in early grades, an emphasis on argumentation and understanding may eclipse the importance of learning basic skills, and problem solving procedures.  Students in early grades would be expected to make arguments beyond just recognizing why an approach to a problem was wrong. The SMP states that “elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.”  But this is requiring arguments to be made with inefficient tools that are better made in the later grades when studentshave the tools to generalize in a formal manner.  Again, this SMP assumes that making such arguments, albeit inefficiently, creates the habit of mind of logic.  The SMP states that students will “reason inductively with data.”  Thus, as in SMP 2 discussed above, students will carry with them a grade school level of inductive reasoning that will not serve them well in higher level math courses.

SMP 4: 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. They 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.

Modeling is admittedly a trendy term, but it generally means solving problems by representing a situation mathematically, and then solving it.  Using addition and proportions in early grades to solve problems as stated in the SMP has merit and is the approach taken with traditional mathematics teaching.

The traditional approach generally holds that there is one right answer. Such answer can be a set of numbers, called a “solution set”.   The reform approach to math extends the traditional approach by including open-ended and ill-posed problems in the belief that textbook problems are too “nice”. The fact that the textbook provides the data students need to solve the problems is viewed is an educational detriment which will not prepare students for the “real world” of having to find things out for themselves.  These beliefs lead to providing students with messy problems that are said to duplicate the types of problems that are encountered in the “real world” of problem solving where there is “more than one right answer”.

Thus, students are given problems where there is supposedly “more than one” right answer. For example, a problem may say that some children are given $40 to buy supplies for a party for 10 kids.  The problem lists a number of things that they could buy.  The students are asked to decide what to buy, but not go over the limit of $40. Educators don’t realize that mathematicians would define a merit function that codifies the personal choices. There are then mathematical solution techniques they use to find the one solution that meets their requirements. This is a known class of problems, but the math reform approach holds that by having students come up with multiple solutions, they are teaching students to think like mathematicians.  A mathematician would view the problem as having one optimal solution.

SMP 5: Use appropriate tools strategically

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

While spreadsheets and calculators are useful tools that students should learn how to use, mathematical proficiency goes beyond these tools, whether a student can use such tools “strategically” or not.  The SMP states that “mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator”.  In fact, mathematically proficient high school students should know how to graph functions by hand, by knowing the formulae and graphical representations of  conic sections, rational functions, exponential/logarithmic, and periodic functions.  In addition, proficiency includes the knowledge of how such functions are translated and shaped.

Being able to identify external mathematical resources on the internet is useful, but an emphasis on Googling for information at the expense of solving difficult and challenging problems is misguided at best.  The SMP’s opening statement that  “mathematically proficient students consider the available tools when solving a mathematical problem” should be interpreted to mean that at the high school level, the emphasis should be on applying knowledge of mathematical procedures and deductive reasoning–not which calculator or computer program would be best suited for solving a problem.

Part One is available here, and Part Three will be published on Friday, April 26.