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

This is Part Three of a three part article [Part One, Part Two] 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 6: 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.

Being able to calculate accurately and to judge the degree of precision appropriate for a problem is an important skill as is using correct units of measure and labeling axes correctly.   This SMP also seems to be about providing explanations of one’s work; that is being able to show one’s work on a problem in such a way that others can follow how it was solved.  Showing the mathematical steps is for many if not most math teachers an explanation that “attends” to precision.  Students in early grades do not have the language ability to express such an idea which to them is innately obvious and therefore hard to express. Thus, a sensible way to interpret this SMP for the early grades, say K-6 is to let the math “do the talking”, which was previously known as “showing your work”.

Writing an explanation for one’s reasoning is another matter, however.  Many students asked to provide written explanations of their reasoning are stymied as to how to explain what mathematics does quite economically and efficiently.  They often respond: “But that’s what I just did,” or “It just is.”

Admittedly there is an advantage to learning how to express mathematical ideas in words.  Such skill is an essential part of constructing mathematical proofs, and therefore an asset to have in geometry and other math courses.  Thus, if learning to write a mathematical explanation is a desired goal, students should be instructed in how to do so rather than assuming that 1) students automatically know how to do this if they truly “understand” or 2) such a goal is efficiently achieved by students engaging in group discussions to learn the technique from each other.

For example, consider the following problem: “The length of a rectangle is twice the width.  If the length were increased by 3 units and the width by 2 units, the area would be increased by 34 square units.  Find the length and width of the original rectangle.”  A student may readily solve this by representing the problem as (2w + 3)(w + 2) = 2w2 + 34, where w and 2w are the width and length of the original rectangle.  To provide instruction on how to explain reasoning, the teacher could ask a student who has solved the problem to work the problem at the board, and ask the student questions.  “How did you represent the length and width?  What do 2w + 3 and w + 2 represent?  Why did you multiply them?  What does the expression 2w2 + 34 represent?”  The teacher can also show how diagrams are part of the explanation as well as words.  Students receiving such instruction and doing this routinely once or twice a week in class, as well as providing such explanation for one or two problems in homework assignments will learn.

Expecting that students will learn such technique by working in groups and having discussions with other students is unrealistic. But the view of many reformers however, is that despite a student getting the right answer to a problem, the moment a student stops doing all the intermediate steps/algorithms and/or fails to explain in words how he or she solved the problem, then he or she is using a “trick” or “rote memorization” to jump to the end result, and/or lacks true “understanding” of the mathematical concepts involved. Such a view is inaccurate and unfair. Setting up the equations to solve complex problems requires a great degree of understanding. It entails understanding what the problem is asking, as well as how to express what’s going on in the problem mathematically.

SMP 7: Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

While observation, awareness and recognition of patterns is necessary in mathematics, it is not sufficient.  Some may interpret the SMP in this way, however and conclude that the habits of mind for pattern and structural recognition can and should be developed outside of the context of the material being learned—that is the vehicle which produces the patterns and structure in the first place.  For example, drawing auxiliary lines in geometry is important, but makes sense when students are given instruction in how that is done, and in the context of conducting proofs or solving problems.

Mathematics demands mastery of foundational steps in order to build upon them. As such, it is relentlessly linear.  The reason a coherent, sequential, efficient, and exercise-rich curriculum works is that the brain requires a great deal of repetition over time to consolidate learning in long term memory.  Without such a foundation, students will not be prepared to solve new and complex problems.  Proficiency is also unlikely to come about in a “problem-based learning” setting, in which a problem is posed that may require certain procedures and skills in order to solve the problem—such as factoring.  Having students learn the procedures on an “as needed” or “just in time” basis is ineffective.  Students need to master the skills in order for such procedures to be applied to problems.  Pattern and structure recognition alone won’t do it.

SMP 8: Look for and express regularity in repeated reasoning

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

It is important to make use of repetition in understanding the derivation of a rule.  While this can be done in a direct and efficient manner of instruction, the write up of this SMP can be interpreted as advocating a discovery type approach.  I.e., “By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3”.  Initial guidance about slopes and how to use them in determining if points are on a line can effectively build a foundation for solving more difficult systems later on.  Students can be given problems such as figuring out the slope, as an introduction and means for understanding the derivation of the point-slope form of a line (y1-y) = (x1-x)m.  But expecting all students to discover this is a result of working through checking whether points are on the line through a specific point and slope (e.g., (1,2) with slope 3) is unrealistic, as is the expectation that students will discover what repeating decimals are on their own.  Students can still be mathematically proficient even if he or she is provided an explanation. And in fact, once initial instruction and worked examples are provided, homework problems can be scaffolded in difficulty so that students are required to apply the basic information in situations that vary from the initial problem.

Conclusion

Implementing the SMPs using the straightforward and traditional techniques discussed above are what some math teachers have done for years.  On the other hand, those promoting reform-based practices are fearful that more traditional practices will lead to what they believe is an unsatisfactory outcome that they call “skills-based math”.

Based on articles in newspapers on how the SMPs are being interpreted, it is probably not inaccurate to say that the SMPs and the content standards themselves will continue to be implemented along the lines of the reform agenda.  SMPs will be pointed to as justifying the teaching of math in a “just in time” manner, and will foster bad habits of mind. The result will, in my opinion, leave many students with the task of finding the cat that is producing a confounding and puzzling grin.

Read Part One and Part Two of Standards for Mathematical Practice: The Cheshire Cat’s Grin.