Saturday, May 3, 2014

Doing Math vs.Thinking Mathematically: What's the Difference?

There's a lot of talk about the dilemma and discord surrounding "Common Core Math", which actually is a misnomer (see my blog "This Is Not the Frustrated Facebook Father's Math - Or Yours Either").  Primarily, the discussions have been pedagogical and political. However, now even performers such as Louis C.K. are engaging in the discussion (Actually, it's surprising more celebrities or comedians are not espousing their opinions about the Common Core.)

We've been warned that the college and career standards our states have adopted and the assessments that will test our students on these standards will expected them do math differently than what they and even we educators and our students parents have experienced.  In fact, this comedic monologue and tune from Tom Lehrer actually sums up how the term "new math" and how it has been presented not only to us educators and our students but also the general public.

However, is it that are students are expected to do math differently or rather think differently about mathematics?

We've also been told this is a "new math", which brings some baggage since the term has been used so loosely and incorrectly over the years every time there's a new educational initiative or reform.

However, is there new math to learn to do or are there new expectations for demonstrating and communicating mathematical thinking?

What's the difference?

Doing math is an operation. It's about arithmetic and applying mathematical procedures such as addition, subtraction, multiplication, division, estimation, and measurement to solve an algorithmic or story problem correctly and successfully.  It's all about the reproducing and applying facts and procedures to achieve or attain that correct answer because, in the end, that's all that mattered - get the correct answer!

What Is Mathematics?
Thinking mathematically is an art - specifically, as Lockhart (2002) states, "the art of explanation.  It's about actively developing deeper knowledge, understanding, and awareness of mathematical concepts, practices, and processes - more specifically, analyzing how, evaluating why, and creating new ways of thinking about and using mathematics.  It focuses on deeper understanding of procedural knowledge, deeper thinking about conceptual knowledge, and deeper awareness of how mathematics can address, handle, settle, or solve real world issues, problems, and situations.

Our primary concern should be that we do not confuse doing math with thinking mathematically.  Doing math is about using math to answer algorithmic questions and solve story or word problems.  Thinking mathematically is about how analyzing how and evaluating why mathematical concepts, practices, and processes are used to address math problems and creating new ideas, procedures, and ways of thinking about math.

What Is Mathematical Thinking?
Mathematical thinking correlates to the process standards identified by the National Council of Teachers of Mathematics (2000).  Students should be challenged and engaged to apply mathematical concepts, practices, and processes to solve problems in mathematics and in other contexts.  They should demonstrate their ability to think mathematically by developing and evaluating mathematical arguments and proof.  They should communicate their depth of knowledge of mathematics using oral, written, creative, and technical expression to explain how they achieved and attained their responses and results.  They should realize and establish connections between mathematics and the real world.  They should represent mathematical phenomena with an algorithm or formula that explains how and why and determines relationships.

Mathematical Thinking in Problem-Based Learning
The core of mathematical thinking is problem solving.  However, the problems presented to students should represent actual real world issues, problems, and situations that would require students to engage in doing math and thinking mathematically rather than a story problem that has students do math in a real world context.  With the adjacent story problem, the problem is not the difference in how many bouquets are sold but rather what is the least amount of bouquets they must sell the subsequent month in order achieve their goal or quota for the quarter.  With this problem, not only are students learning how to subtract but also how subtraction is used in business (financial literacy).  They are also learning academic vocabulary (quota), subject-specific terminology (2nd quarter, which is how time is measured in business). 

Mathematical Reasoning and Proofing
Students should also be challenged and engaged in mathematical reasoning that has them achieve responses and prove the results they have attained.  Take a look at this routine algorithmic problem involving dividing fractions. Students are expected to do the math by reproducing and applying the facts and procedures in order to find the single correct answer - in this case, 1/2.  Students would be challenged and engaged to think mathematically if they were expected to evaluate why 1/2 is the answer and analyze how the result is attained.

Communicating Mathematical Thinking
Mathematical thinking also challenges and engages students to communicate their depth of knowledge as well as demonstrate higher levels of thinking.  For example, instead of providing students with a list of algorithmic problems to solve, provide the students with an open-ended, text dependent question that challenges and engages them to defend and justify their mathematical thinking.  Not only will students learn that math is not just about finding the answer or solution but also being able to explain how responses and results are attained.

Cognitive Rigor Questions for Mathematical Thinking
Communication has become as essential in mathematics as counting, calculation, and computation.  We educators can challenge and engage students to demonstrate and communicate mathematical thinking not only by expecting them to solve mathematical problems and explain their processes by providing them open-ended, text dependent questions in which the solution is provided and they have to analyze how the answer was attained and evaluate why the outcome is correct or incorrect.

Communicating Mathematical Thinking
To encourage communication in mathematics, provide students with a format that allows them to engage in technical writing in which they first identify what the problem is and the mathematical concept, practice, or process that is being addressed in the first paragraph, list the procedures or steps they took to solve the problem, then defend and justify their responses and results using the facts and procedures they reproduced and applied.

Communicating and Connecting Mathematical Thinking
The new assessments students will be taking during the 2014-2015 academic year will expect them not only to solve mathematical equations and problems but also use verbal language to explain their reasoning and proofs.  On the PARCC and Smarter Balanced assessments, some of the questions will look like this one to the right, challenging and engaging students to express how and why a response is correct using oral, written, creative, or technical expression rather than reproduce and apply facts and procedures to attain the result.  

Connecting Mathematical Thinking to Real World Situations
The mathematical problems presented should also show students the connection between academic mathematical concepts and the real world.  Connecting mathematical thinking fosters and supports students' ability to engage in data analysis and interpretation.  It also allows students to use other forms of academic thinking and process such as the scientific method or historical analysis and research to interpret statistical information and draw their own conclusions.

Representing Mathematical Thinking
Students should also learn how to create and use representations to organize, record, and communicate mathematical ideas as well as use representations to model and interpret physical, social, and mathematical phenomena.  In the accompanying example, the algorithmic formula for speed represents how long it would take me to drive from my house in Phoenix, Arizona, to Disneyland in Anaheim, California, (357 mi / x hrs.) if I drove 65 miles per hour (65 mi / 1 hr).  The problem also allows the student delve deeper into the problem by changing the variable measures (e.g. the average speed from 65 mph to a faster or slower speed).

If there is a "new math" our students are learning, it's that thinking mathematically through problem solving, defending and justifying response and results through reasoning and proofing; analyzing how and evaluating why those responses and results were achieved or attained using oral, written creative, and technical communication; establishing connections between mathematical concepts and ideas as well as between math and the real world, and using concrete representations to model and interpret not only mathematical but also physical and social phenomena are just as essential as doing the math to find those responses or results.

So how can we get our students to think mathematically as well as do the math?  Use these problems as a guide to create mathematical experiences that encourage students to think critically, creatively, and deeply.  Connect each mathematical concept or idea to a real world issue, problem, or situation and have them examine and explore how math can be used to explain what happens in life.  Show, don't tell, how math is an essential concept they will use in some aspect of their personal and professional lives.  Make math real, and encourage them to think about as well as do the math.

The true challenge may not to be the children but rather us adults who were most likely not expected to think as deeply about math.  We were just expected to do the math just as we were taught to get the correct answer.  However, math is no longer about correct and incorrect answers.  It's also about how responses and results can be defended, explained, and justified using mathematical thinking.

How can we have the adults develop deeper knowledge, understanding, and awareness of the new expectations for math?  At your next staff development or parent meeting, present this quote and the corresponding questions to the adults in your audience.  Not only will they hopefully develop depth of knowledge about the new expectations for mathematics but also the importance and value of going beyond calculating and computing to communicating, connecting, and considering mathematics.

What Is Mathematics?

-- E.M.F.



  1. This a perfect comparison about the two sides of mathematics: not just the practice of algorithms and application but also the rich metacognitive tensions at play in art of thinking mathematically. I will be referencing this page frequently! Thank you! (by the way, there is a typo of "soling" instead of "solving" in the fourth paragraph from the end *grin*)

  2. In may ways, "thinking mathematically" (which I define as something akin to "mathematical thinking") is no different than "thinking philosophically" in the sense that both are exercises in critical thinking. Of course, philosophy and math have a shared past in logic ... and Aristotle (who was more of a biologist than a mathematician; Plato was the mathematician) was the founder of logic.

  3. Fine, but with the abundance of technical aids, perhaps we should reduce focus on simple procedural math (e.g. long division). Or do these complicated algorithms actually teach the students something about maths?

    In any case, I think it is very important that teachers contemplate WHY we want the students to learn whatever we are teaching them.

  4. All I know is that this school year my focus on WHY math works and not just HOW to do it allowed my students to increase significantly in their standardizes assessment scores. My fifth grade class had a passing rate of 75%. That same group of students had 100% passing on the state assessment this academic year. The difference was in how they learned the language of math and why the math works the way it does. They had plenty of HOW to do math, but they had no understanding of why they had to do the things they did.

  5. Great article. I am a homeschool mom and for the past 3 years I have been holding a math club in my home. Not only do I try and prepare the middle schoolers for various math contests, but I also try and get them to have fun with math.
    The big disconnect between their knowledge of how to "do" math and a real understanding of how math can be applied to problems is quickly apparent. Most students who are achieving As will struggle to solve a simple problem if they don't know what sort of problem it is (ie if they aren't told this is a ratio problem, or percentage problem etc). They have no idea where to start and how to analyze a problem and determine how math can help solve it.
    I feel that this would be so much easier if it was integrated into daily life by parents and teachers with children from a young age. For example, on shopping trips get kids to figure out which size of a product is the most economical, or what the new price of an item on sale is. Baking with my kids gave me a chance to teach ratios and on a long car trip, when the kids asked "How much longer till we get there?" I told them to watch for the next distance board and check the speed of the car and figure it out.

  6. "Doing Math and Thinking Mathematically"

    I like the ring of that phrase since it describes what the next iteration of the Common Core standards for math should become, to my experience as a now-retired instructor of physics, math, and electronics technology at a technical college.

    As the Francis article says, "Our primary concern should be that we do not confuse doing math with thinking mathematically. Doing math is about using math to answer algorithmic questions and solve story or word problems."

    While the article goes on to address the mathematical thinking portion of the statement, I would like to address the unspoken, forgotten part, which is at the heart of the goals of the next iteration of the Common Core standards, I feel.

    I would like to focus on the part that describes what we mean as "Doing Math". In that regard, I would like to propose the following axioms as "self-evident truths" for high-school math graduation standards:

    - ALL students perform pencil-paper computations of dollar and cent problems involving whole and decimal numbers, and common fractions.

    - ALL students perform measurements using a variety of instruments in US Customary and metric units, including linear and angular geometric measures, as data for further computational inputs.

    - ALL students perform computations involving comparisons of numbers, including: inequalities, ratios, rates, percents, percent change, as well as basic statistical computations.

    -ALL students perform formula calculations involving the substitution of quantitative values in place of literal symbols, using all versions of typical life and workplace formulas. (In other words, the three formulas for an area computation are provided, rather than solved.) Reference materials are provided, and may be used during tests.

    - ALL students perform calculations involving powers and roots, especially as applied to geometric uses.

    - ALL students perform calculations involving signed numbers, including: inequalities, statistical histograms, and 2- and 3-axis coordinate systems.

    - ALL students manipulate typical geometric and money formulas, using techniques to isolate a given unknown.

    - ALL students utilize reference materials and calculators (except for the whole and decimal computations and common fractions mentioned above), including testing.

    - NO students should need to use the words "algebra" or "proof" in the performance of the above tasks.

    These generalized statements are provided as the essential topics for the Next Generation CCSS, in my opinion (NG-CCSS?). They include what EVERY high school graduate NEEDS TO KNOW to be prepared for life and work in this 21st Century. They include nothing of what SOME people SHOULD know if they wish to go beyond to pursue additional training for skilled and professional careers. And these standards have nothing to do with what a few believe is NICE TO KNOW about appreciating mathematics in many ways.

    Thus, these are the elements of "Doing Math", imho.

  7. doing math and thinking mathematically is so awesome....I think this will help some students love math

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