When I speak to people about their memories of math class growing up, many of them go on to describe a classroom culture that valued accuracy, speed, and correct answers. Some thrived in this environment, enjoying the feeling of being fast and competing to get high grades on tests. But for those who were not naturally good at memorizing or computing quickly, this environment could lead to self-doubt, anxiety, and a disassociation from math as a subject altogether. Oftentimes this insistence on exactness in calculations and answers come from a desire for precision -- one of the eight mathematical practices outlined in the Common Core. In this post we'll unpack what Mathematical Practice 6 is all about (and what it's NOT about), and how to avoid the common pitfalls associated with it.
Defining MP6
The short phrase "attend to precision" has a lot packed into it. To “attend” means to pay attention to, be attentive of, or heed. “Precision” in its plain English use is usually associated with “exactness” but in mathematics we can extend its definition to include clarity, accuracy, and rigor. We want students to be accurate and exact in their calculations, yes, but even more so, we want to encourage clarity in communication. Both of these ideas are incorporated in Mathematical Practice 6: Attend to Precision.
The idea of clarity applies to both our internal mathematical thinking and reasoning as well as our external communication of that thinking, in the use of specific mathematical language, definitions, symbols, and units. Our everyday language and conversational patterns often rely on context and shared human understanding; it would be obnoxious to tell someone to underline-segment a word on their paper, or to hand us the book with the title “A Tale of Two Cities” that is currently positioned on your lap rather than just “hand me the book”. In math, however, an argument follows a logical sequence of ideas that build on each other and must be communicated without room for ambiguity or misunderstanding.
Clarity in communication, both written and verbal, means:
Avoiding vague phrases like “the function”
Labeling axes and using appropriate units
Using precise definitions
Defining variables carefully (m=M&Ms vs. m=the cost of a pack of M&Ms)
Distinguishing between everyday and mathematical uses of a word
This last point about different uses of a word is worth unpacking further, as it is the source of much student confusion and an easy avenue for teachers to have an impact. Consider the word “add”. In plain English, the word “add” always means to make more, whereas in math, adding could mean making more, making less, or no change. Similarly, the use of the word “or” in everyday language is used in its exclusive form, as in one or the other, McDonalds or Burger King. In math, we tend to use the inclusive definition, one or the other or both. Other words like “difference”, “line”, “curve”, “function”, “sequence”, and “series” all have different definitions when used conversationally versus when used mathematically.
Another important application of words having multiple meanings is in teaching students proof, a notoriously challenging skill for students to master. The instruction to “prove that the diagonals of a rectangle are congruent” could be interpreted by a student to mean that they should show some specific rectangle with congruent diagonals (as in, show an example of a rectangle that…) if they are interpreting “a” in the plain English sense. However, the mathematical interpretation is what we are really after: “Prove that the diagonals of any rectangle are congruent” which means that one specific example is no longer sufficient. We must be explicit with students and encourage discussion around how the meaning of a word as it is used in math class might differ from its conventional, everyday meaning.
Now let’s unpack some other facets of precision: accuracy and rigorous reasoning.
Accuracy means:
Using symbols (like the equal sign) appropriately
Calculating efficiently and with correct outcomes
Being able to produce exact answers: 16π square units, rather than 50.3 square units, or a hypotenuse of length 3√2 inches
Considering all solutions (If x^2=25, then x=±5, not just 5)
Rigorous reasoning means:
Examining claims and making explicit use of definitions
Being metacognitive about the thoroughness and reasonableness of a solution
Will this always work?
Are there counterexamples?
What flaws in the logic need to be revised?
Does this pattern apply only to a certain subset of items?
What specific conditions produce this result?
Specifying the conditions under which a rule works and doesn’t work (using the SAS triangle congruence theorem only when the congruent angle is included between the two sets of congruent adjacent sides, or knowing when to apply each convergence test)
Considering domain restrictions (when should a graph consist of a set of discrete points rather than a continuous curve?)
Sometimes the level of precision that is demanded of students can feel stringent, inflexible, and harsh, pushing students away from the field of mathematics. We will discuss this more later in the section about the dark side of precision. However, acknowledging that there are downsides to precision brings us to an important mathematical skill: determining the degree of precision that is appropriate for the problem context.
What Level of Precision Is Required for This Problem?
The Common Core describes MP6 by stating that students should be able to “express numerical answers with a degree of precision appropriate for the problem context”. This idea is repeated in the mathematical practice on modeling (MP4) and in many other frameworks. In all these cases, it is implied that students will make decisions about the level of accuracy that is fitting for the problem, based on both its usefulness and the reasonableness of being able to perform the calculation. I find this skill to be wildly underrated in most math classrooms.
Take the “accurate to 3 decimal places” rule in AP Calculus. Of course it’s helpful to have a guideline when reporting calculator-generated answers, but if we spent half the time we usually spend talking about truncating vs. rounding answers and pleading with our students to avoid intermediate rounding and instead asked students to consider what level of precision is necessary or helpful for a given problem, students might be better served in the long run. For example, knowing that people are entering an auditorium at a rate of 18.247 people per hour offers no additional (practical) information than knowing that people are entering an auditorium at a rate of 18.2 people per hour (or, to be even more contentious, at a rate of 18 people per hour). On the other hand, using an exponential regression equation and preserving only one decimal point of accuracy in the parameters will produce vastly different results when making a prediction using the model. In this situation, it makes sense to keep 3 or more decimal places.
As with anything in life, applying a hard and fast rule is easy, actually thinking about what a situation requires is hard. The real world is full of messy, ambiguous, and complex mathematics, and hard and fast rules hardly ever work. Precision, while important, needs to be developed in tandem with an ability to estimate, to determine if an answer is reasonable, and to think critically about when precision will pay off and when it won’t.
The Dark Side of Precision
There is beauty in precision but there is also a danger of propagating a narrow mathematics that focuses only on accuracy and correct answers, rather than creativity, reasoning, number sense, and real-world application. What do I mean by this? First, many students struggle with number sense because they are too focused on the calculation (the “how”) rather than on the elements of the problem itself (the “what”). For example, when presented with a line with a y-intercept of 24 and a slope of -4/5 and asked to find the y-value when x=10, all Algebra 1 teachers will be familiar with the slew of wrong answers that are likely to be produced based on incorrect fraction calculations. If we let go, temporarily, of the need for an exact answer, students might be more willing to reason about their answer conceptually. The y-value when x=10 will certainly be less than 24 since the slope is negative and x=10 is to the right of x=0. The y-value will also be a bit above 14, since we are subtracting just less than 1 from the output for each horizontal run of 1 unit. This kind of thought process requires conceptual reasoning around linear relationships and an understanding of the fraction of -⅘ as being a value close to, but not as small as -1.
Second, most of the numbers we use in our day-to-day lives are estimates, such as when we estimate how long it will take to do a Costco run, or when we estimate how wide the couch is in our new apartment. In Jo Boaler’s newest book “Math-ish”, she makes the case for “ish-ing” in math, or encouraging students to estimate answers to develop number sense and encourage conceptual thinking. She argues that this could transform people’s approach to mathematics as they learn to see mathematics from a big picture perspective, which encourages students to play, jump in, and engage and also “softens the sharp edges of mathematics” (p. 113).
How Does the EFFL Model Support MP6?
An important aspect of our teaching model, Experience First, Formalize Later, or EFFL for short, is that mathematical vocabulary and notation is added during the debrief in the margin notes. This occurs after students have had the opportunity to wrestle with the ideas in activities that leverage the strategies, language, and reasoning that students already have available to them.
Precise mathematical vocabulary and generalized strategies are layered onto what students are already thinking and talking about. This consolidation phase is critical to the lesson but with intentional teacher moves, we can slowly dial up the formality, instead of cranking up the heat all at once. At least a part of this formalization happens by the end of the class period, but other parts may take until the end of the week, or even sometimes until the end of the unit.
The goal is not to ditch the formality and precision, but to reconsider when it happens. By starting with the big picture concept instead of the fine details we give students a foundation on which the later details can land (like when we wait to use the terms “sine”, “cosine”, and “tangent” until after students have calculated the ratios of the red, blue, and green sides of a set of similar triangles).
This is why we call the work students do in groups their “rough draft” and then use the debrief as a chance to update and revise the draft. Adding precise terms and notation increases the clarity of the mathematics, rather than diminishing or discrediting the students’ previous work.
As we work to help students develop the mathematical practice of attending to precision, let’s keep in mind that precision is most powerful when it is coupled with big picture ideas, and when we strive for clear communication because we have thinking and reasoning to communicate at all!
Want to learn more about the other mathematical practices? Check out the other blog posts in our series!