The Elusive Search for Balance
By Matt Larson, NCTM PresidentFebruary 20, 2017In a recent President’s blog post on the need to make homework comprehensible, I referred to the Fordham Institute Report, Common Core Math in the K–8 Classroom: Results from a National Survey. The report offers another interesting finding: “The math wars aren’t over.” The authors of the report observe, “The Common Core math standards seek to bring a peaceful end to the ‘math wars’ of recent years by requiring equal attention to conceptual understanding, procedural fluency, and application (applying math to real-world problems). Yet striking that balance has not been easy. We see in these results several examples of teachers over- or underemphasizing one component to the detriment of another” (p. 6).I found this statement particularly striking. This over- or underemphasis may be less a function of independent teacher actions in the classroom than a result of teachers doing their best to interpret and implement what they find in their curricular materials, which, as the Fordham report indicates, are aligned (or not) with the Common Core State Standards to varying degrees. The over- or underemphasis of a particular component may also reflect teachers’ trying hard to comply with mandates at the district or school level. It is critical to appreciate that the over- or underemphasis phenomenon is not a new one. Mathematics teachers in the United States today are just the latest generation of U.S. educators to be caught in a 200 plus–year pendulum swing between an overemphasis of rote practice of isolated skills and procedures and an overemphasis of conceptual understanding, with their respective overreliance on either teacher-directed or student-centered instruction. It all began in 1788 (the same year that the U.S. Constitution was ratified), when Nicolas Pike published the first major U.S. mathematics textbook, entitled Arithmetic. The process that Pike recommended for teachers was to state a rule, provide an example, and then have students complete a series of practice exercises very similar to the example. If that teaching process sounds familiar, it is probably because that was the way you experienced math instruction as a student yourself. This process became the U.S. script for teaching mathematics and is deeply embedded in our culture—expected by the vast majority of students and parents alike.In the 1820s the pendulum swung for the first time when Warren Colburn published a series of texts, including Colburn’s First Lessons: Intellectual Arithmetic, Upon the Inductive Method (1826). Colburn recommended that teachers use a series of carefully sequenced questions and concrete materials so that students could discover mathematical rules for themselves. By the 1830s the pendulum was swinging back with the publication of the Southern and Western Calculator (1831) and The Common School Arithmetic (1832), which once again emphasized direct instruction of rules and procedures, taught the “good old fashioned way.”The late 1950s and 1960s were the era of “New Math.” Proponents of new math worked to make the pendulum swing from rote learning to discovery teaching approaches that emphasized developing students’ understanding of the structure of mathematics, how mathematical ideas fit together, and the reasoning (or habits of mind) of mathematicians. The 1970s and 1980s saw the pendulum swing in the opposite direction yet again as these two decades became known as the “back to the basics” era, with a focus again on procedural skills and direct instruction.In 1989 NCTM gave birth to the standards-based education reform effort with the release of Curriculum and Evaluation Standards for School Mathematics, and subsequently NCTM followed up this transformative publication with a series of other standards publications, culminating in Principles and Standards for School Mathematics (2000). By the mid-1990s over 40 states had created state math standards or curriculum frameworks consistent with the NCTM standards. But by the late 1990s the pendulum began to swing back to the basics when the “math wars” erupted in California and then spread across the country as parents and others demanded a renewed emphasis on procedural skills and direct instruction.And that brings us to the latest perceived pendulum swing: the Common Core State Standards and the associated myriad of misinformation and misinterpretations surrounding them, as well as the historic and seemingly inevitable pushback that now benefits from and is fueled by social media. So what should a mathematics teacher caught up in these historic and continual pendulum swings do? My advice: Seek balance.In many ways it seems as though we live in a world that is out of balance—pushed to extremes—that has “lost the middle” in various ways. To move mathematics teaching and learning forward, we have to resist the urge to be pushed to extremes. We have to do our part to break the historic cycle of pendulum swings. As Hung-Hsi Wu, professor emeritus of mathematics at the University of California–Berkeley, wrote nearly two decades ago, “Let us teach our children mathematics the honest way by teaching both skills and understanding.” This is essentially what the Common Core authors argue when they state, “[M]athematical understanding and procedural skill are equally important” (National Governors Association and Council of Chief State School Officers 2010, p. 4). Over a decade ago the National Research Council (NCR) published Adding It Up (2001), which promoted a multifaceted and interwoven definition of mathematical literacy:
Procedural fluency and conceptual understanding are often seen as competing for attention in school mathematics. But pitting skill against understanding creates a false dichotomy … Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. By the same token, a certain level of skill is required to learn many mathematical concepts with understanding, and using procedures can help strengthen and develop that understanding. (p. 122).
We need to return to, promote, and implement in our classrooms the NRC definition of mathematical literacy. The goal of mathematics education is not complicated. We want students to know how to solve problems (procedures), know why procedures work (conceptual understanding), and know when to use mathematics (problem solving and application) while building a positive mathematics identity and sense of agency. How? Why? And when? These questions are the very essence of rigor in the Common Core. We can ask ourselves simple reflective questions at the end of each lesson and over the course of a unit:
If the answer to all four questions is yes, that is more than likely a good sign. If we stay focused on all four learning goals, and resist (or dodge) the pendulum swings in any one direction while steadily building students’ positive mathematics identity, perhaps the constant criticism that moves the pendulum will stop. When we stay the course and let students engage, learn, and develop their understanding, skills, and abilities to use mathematics, our students will be the beneficiaries of our efforts to find and maintain that balance.
For more information, see Balancing the Equation: A Guide to School Mathematics for Educators and Parents.
Matt & The Other Seekers of the Elusive Search for Balance,
Thanks for sparking a great discussion. I do agree that we jump back and forth from skills to understandings but also see that we jump into and out of technology and other innovative explorations. We had some great logic growing out of the programming era (Logo, Basic Programming, Calculator programs, spreadsheet explorations, etc ...) By and large those innovations have been disappeared from the current high school mathematics curriculum.
We explored some great innovative ways in thinking with number bases, historical number systems, fractals, probability, chaos theory and other exploratory topics (yes - even the crazy modern math). I suspect the intensity of the demands of CCSM at the high school have stolen time for lots of these.
We still have materials that support the structure of mathematics as a developmental and proof-based system. Paul Forester's reference to his chapter on quadratics was fun to revisit. It was one of my favorite chapters to teach. That orderly development of the quadratic formula is a classic.
I hope we get more time in teaching where mathematics teachers can independently add fun exploratory, challenging, innovative topics to their curriculum. That component is illusive but critical for developing diversity of thinking in America!
Excellent article. As both a student through high school and 3 universities and a math teacher now for 18 years, I have experienced some of the swings. There is another, equally confusing activity in math teaching that also needs balance, I believe. That is the area of assessment. For many years, I have helped students prepare for various standard tests. Most notably, the ACT and SAT. I have come to conclude that these tests are fundamentally flawed in that they emphasize speed and memory. 85 % of takers do not finish the tests and the average scores are the equivalent of a D grade. The tests do not really show college readiness, their stated purpose, but sort out the supposed top 15% of HS graduates. I appeal to NCTM to investigate, report and recommend a better approach.
Love the historical background provided to this important issue as well as a very practical solution: Seek balance. Thank you.
Paul Foerster responds to Matt Larson’s “The Elusive Search for Balance,” 2/20/17
Matt is right on target with this posting. He pleads for avoiding extremes in the teaching of mathematics. In this response I present the ways I have followed (or perhaps led) a middle path in my 50 years of active teaching (and six years in retirement) of high school mathematics, authoring five nationally-used textbooks, and presenting at hundreds of NCTM-type conferences and workshops. First, some background information to tell you “where I am coming from.”
After earning an undergraduate degree in chemical engineering I served four years as an engineering duty officer with the Navy’s Nuclear Propulsion Program. In 1957 when I started, the program was new enough that there were problems to be solved that no one had ever solved before. “Back to basics” meant that in order to make intelligent applications, one must have a gut understanding of underlying theory. And “understanding” does not necessarily mean “proving theorems.”
Upon release from active duty in 1961 I spend a spring and summer in college earning a teaching certificate, then landing a position teaching in high school. My aim was to instill critical thinking skills in students’ minds in keeping with my engineering experience. Indeed, teaching is much like any other engineering project. You start with certain “raw materials,” have specifications about what the “product” should be able to do, and must work within certain time and budgetary constraints.
Upon being assigned temporarily to teach mathematics I found I could accomplish these objectives through that subject. So I stuck with mathematics instead of switching my prior first choice, chemistry. I have been fortunate to teach in a school system where teachers are given the freedom to innovate, without requiring “every teacher to be on the same page every day,” whether the students are ready or not.
Typical algebra “word problems” did not reflect my experience with engineering problems. So I started writing problems for my own students in which “the variables really vary,” not just stand for unknown constants. Because students found such problems challenging, I wrote narrative materials to go along with the problems. The materials use wording the students had come up with as they struggled for understanding, and that I insured were mathematically correct.
Over the 50 years of teaching I wrote 150 to 200 “Explorations” to accompany each text, Algebra I, Algebra II, Trig, Precalculus, and Calculus. These can be used by students working in cooperative groups or individually, to discover things about a topic before a more formal classroom presentation.
The textbooks, now published by Pearson/Prentice Hall and by Kendall Hunt, are used by teachers who enjoy a challenging program that blends both understanding of mathematical structure and facility with computation—a middle of the road approach such as advocated by Professor Larson. They are mathematics books with applications, not applications books with mathematics.
There is also a focus on correct language. Mathematics can be thought of as “applied English.” Every equation you write is a sentence with a subject, a verb (the “=” sign) and an object (or predicate nominative, for purists). Students must be able to comprehend the given problem, “do” it, then present the results in a way that the person posing the problem can understand. The books were written to enlighten my students, rather than to impress my teaching colleagues. This fact makes the materials particularly suitable for home-schooling. The second-person singular “you” is used to focus on the student, rather than the more common “we.”
Along the way I have picked up a number of “sayings,” some original, others borrowed. A group of students and teachers in Houston dubbed these sayings “Foersterisms.” Here are some that relate to the topic of this posting.
“In order to make intelligent applications, you must understand the theory.”
“You can’t afford to spend all of your creative energy on tasks that should be routine.”
“One who knows how can always find work. But one who know why will boss the jerk.” (Overheard at an NCTM Annual Meeting in Cincinnati.)
“Mathematics is applied English.”
“Do you count off for spelling?” “No, unless you get it wrong.”
“No one will believe you understand the mathematics if you can’t even spell the words.”
“Mathematics, well taught, is difficult.” (The late Julius Hlavaty, former NCTM president.)
Please feel free to communicate with me on this or related topics.
Paul A. Foerster
Teacher Emeritus of Mathematics
Alamo Heights High School, San Antonio
Thanks, Matt. Nice.
One way to build in this balance is to develop tasks that use the interplay between technical fluency and what NCTM has called ``mindful manipulation'' to come to some general principle. For example, the SSS theorem in geometry says that a triangle is determined by the lengths of it 3 sides. So the area of the triangle should be computable from its side-lengths. Forget about formulas, try it with numbers:
(*) Find the area of a triangle whose side-lengths are 13, 14, and 15(*) Find the area of a triangle whose side-lengths are 12, 17, and 25(*) Find the area of a triangle whose side-lengths are 12, 17, and 26.
This kind of etude is designed to encourage the ``abstract regularity'' practice. If you work the problems (without trig) and delay the evaluations (use fractions, not decimals) you find that there's a certain rhythm to what you are doing and a certain form to the intermediate steps. Noticing this regularity is a habit that's so important to develop throughout K--12.
And you get quite a bit of computational practice here (especially the last one), but there's a bigger goal---the process that is common to all the calculations, stated precisely, is a start to an answer to the question of finding the area of a triangle whose side-lengths are a, b, and c.
And then mindful manipulation rears its head again. The general expression for what you did is a candy store of structural patterns. Students with an eye for structure in expressions can get all the way (or almost all the way) to Heron's formula.
So, I guess I see it more of a synergy than a balance.
Thanks Al. Synergy is a nice way to frame the issue. Matt.
Two issues of imbalance I often see.
First, there are teachers and school leaders who seem to have a rather hostile view toward the standard algorithms. I think it is perfectly possible to help children re-invent the standard algorithms based on their conceptual understanding of operations and our numeration system. There is no need for us to be avoiding the standard algorithms or helping students develop them.
Second, "problems" do not have to be "real-world" problems. When students have learned how to add 2 two-digit numbers - maybe using the standard algorithm - figuring out whether or not the same idea would work with 2 three-digit number is a mathematical problem worth investigating. It is perfectly fine to pose a word problem that involves the addition of 2 three-digit numbers as the motivation, but teachers should help students realize that the mathematical problem they are tacklying by trying to solve the problem is how to add 2 three-digit numbers.
I also agree that too often curricular materials are far from adequate.
I totally agree Tad! Thanks for sharing this point of view!
It seems like in the grand rush to create real-world problems, we have lost the thread that we need to create interesting real-world problems. Two of the components of interesting problems are complexity and challenge. In reviewing common core texts, most problems fail these two hurdles. In many cases, the complexity and challenge reside in the semantics and verbal presentation of the problems, and not in their mathematics Another component is relevance, which is sorely lacking. These issues are particularly challenging in middle school and high school, as students become more sophisticated as they become older.
When I was in high school, my math courses taught us the tools (we used Dolciani et al.) and our science teachers provided the real-world problems. Most of the science courses had prerequisites that tied them to particular math courses (Algebra for Biology, Algebra II for Chemistry, and PreCalc for Physics). The interesting real-world problems were the labs (complex, challenging, and relevant).
In my opinion, what we need to be doing is not trying to ensure that we are incorporating real-world problems into our math courses (where the good problems end up looking like science labs). Instead, we should be ensuring that level appropriate use of mathematics is incorporated into other parts of the curriculum. Working with chemistry students to help them understand the mathematics is much more productive and rewarding than the out-of-context real-world problems that we are currently generating.
I truly admire how Adding it Up (2001) listed 5 components of mathematical proficiency.
—comprehension of mathematical concepts, operations, and relations
—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
—ability to formulate,represent, and solve mathematical problems
—capacity for logical thought,reflection, explanation, and justification
—habitual inclination to see mathematics as sensible, useful, worthwhile,coupled with a belief in diligence and one’s own efficacy.
Powerful mathematics teachers work daily to provide those 5 components to a core of big ideas. It helps to remember that the important procedural skills for todays students span the gamut from mental math to use of technology!
Thank you Matt! This will help teachers to more deeply understand that there is not always a "right" way. We have to think about the needs of our students and how we can best help them to understand and be successful with the mathematics they are learning.
Thanks for the four questions! They help remind me of my focus and let me know how I'm doing. I love teaching common core , as my students are much more excited about math and confident in their skills when they see how math works in their every day lives. It is easier for global thinkers like myself to see how all the pieces of mathematical thinking and learning come together, but is still a challenge to put it into practice. I appreciate the balance reminder!
Love it! Balance is definitely key! I would propose that a teacher who sees math as a pencil and paper subject, thus relying on the materials (aligned or not...) found in comsumable math workbooks, is not acheiving the desired balance.
I fully agree. Question: is there a preferred ORDER for teaching understanding, fluency, and application? It seems a logical order would be a brief glimpse/introduction of application, then understanding, then fluency, and finally circle back to application to pull it all together - that is when, WHY, HOW, WHEN. This formula works well for the sciences. Shouldn't it apply to math too? I'd love to hear what others think.
Michael, et al. –
Good question. I find that the answer varies with the topic. There are at least two for which I even have the students memorize the way to pronounce it before they have any idea about what it means.
One is the quadratic formula. I have Algebra I students write the formula (not at first telling them that they are learning only the “then” part!) and say it chorally: “x equals the opposite of b plus or minus the square root of the quantity b squared minus 4ac, all divided by the quantity 2a.” Over the next few days they solve equations whose solutions look more and more like this quadratic formula, e.g. (x + 3)^2 = 7. They finally learn the “If ax^2+bx+c=0” part.
Another is the epsilon, delta definition of limit (on six lines):
“L is the limit of f(x) as x approaches c
“if and only if
“for any positive number epsilon, no matter how small,
“there is a positive number delta such that
“if x is kept within delta units of c, but not equal to c,
“then f(x) stays within epsilon units of L.”
Here again the students are given examples that lead them to understand what the various intricacies of the definition mean, in particular, the italicized words.
Other topics are more amenable to other sequences, depending on how the instructor find it convenient to present them.
I think the question of ORDER is answered by "it depends." However, I'm thinking that it depends more on the STUDENT than the TOPIC. Probably ANY topic can be taught in ANY order - understanding before fluency, fluency before understanding, or whatever. But I suppose some students respond better to certain orders than others. Personally, I remember needing to grasp fluency before I really understood a concept, but I suspect I was in the minority. Looking at it objectively, it seems that since understanding serves as a foundation, it should be introduced first as Ryan alluded.
I interpret Matt's "interwoven" comments below to mean that understanding, fluency, and application are taught together over a period of at least a few days. I don't think he means that they need to be taught literally SIMULTANEOUSLY on any given topic. I'm not even sure if this is possible or advisable. So given that they are not taught simultaneously, the teacher must pick an intial order. When tutoring, one can follow the student's lead and see what he/she responds to best. But in a classroom, the teacher needs to lead in a pre-determined order. Not an easy thing to do if some of your students need conceptual guidance first while others need procedural guidance first.
Note that Matt is ALSO saying that the order, for example, could be 1) understanding, 2) fluency, 3) understanding, 4) fluency, 5) application, 6) fluency, 7) understanding, etc... It is the skillful teacher that knows what to apply when. I usually, START with understanding, and if I find that the student is not connecting, I might cut understanding short and see how the student responds to fluency. If there is a better response to fluency, then I would try to connect it with understanding.
I don't think there is a preferred order. As the authors of Adding it Up wrote, "the most important observation we make about the five strands [of proficiency] is that they are interwoven and interdependent" (p. 5).
Love the article and agree that the concept of interwoven and interdependent is most important. However, the 6th Math Teaching Practice in Principles to Actions states that procedural fluency is built on a foundation of conceptual understanding on page 42. Additionally, "NCTM and CCSSM emphasize that procedural fluency follows and builds on a foundation of conceputual understanding, strategic reasoning and problem-solving."
Thanks Ryan. I don't believe the NRC definition of mathematical literacy is at all inconsistent with the recommendations in PtA. My interwoven comment was meant to imply that problem solving does not have to wait until conceptual understanding and procedural fluency are in place. All of mathematics and teaching and learning should be based on a foundation of conceptual understanding. Finally, the interwoven comment is in the context of a learning progression that might take place over a unit, year, or beyond.
Thanks for clarifying! The power behind both Principles to Actions and Adding it Up is immeasurable, and the impact that those seeking to understand them has been incredible for moving forward. Coupling those documents with the structure of the CCSSM and groups like the #MTBoS makes this an amazing time to be part of NCTM and the math education community. Thank you for leading!
Now THIS would be an excellent area for math education research! I agree that the END result should feel interwoven and interdependent to the student. However, when you are learning math (or anything else for that matter) there is a lot to learn! I'm having difficulty understanding how it could be better to learn it all at once instead of one step at a time. Wouldn't it be best for the profession to build separate tools for teaching understanding and for teaching fluency and have the student become proficient in both in which ever order works for them? After they have internalized both, they can integrate them with application. This is the approach that works best for me personally, but I have no clinical evidence to back me up. I think it would be a great study.
Thanks Matt. Another thoughtful, balanced and insightful piece. You continue to give us lots to think about and advocate for.
Wow! This is just the message we need to spread to parents, educators, and policy makers! Bravo, Matt. And a big Thank You!
Thank you, Matt. The historical perspective reveals the challenges that teachers face in their daily work to maintain the balance.
Thanks. good article.