# Standards for Mathematical Practice: The Cheshire Cat’s Grin

This is Part One of a three part article 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

The Common Core Standards for math are a set of guidelines written for both math and English language arts under the auspices of National Governors Association and the Council of Chief State School Officers. Where they are adopted, the Common Core standards will replace state standards in these subject areas, establishing more common ground for schools nationwide.

The Standards of Mathematical Practice (SMP) are a part of the Common Core math standards.  On the surface, and to those unaware of underlying concerns and issues, the SMPs appear reasonable.   It’s doubtful that any single mathematician or math teacher would disagree with anything in them, in principle.  They are process standards, which address the “habits of mind” of mathematics that are tied to the content standards.   The term “habits of mind” comes up repeatedly in discussions about education — and math education in particular.

Habits of mind make sense when the habits arise naturally out of the material being learned.

Thus, a habit such as “Say in your head what you are doing whenever you are doing mathwill have different forms depending on what is being taught. In elementary math it might be “One third of six is two”; in algebra “Combining like terms 3x and 4x gives me 7x”; in geometry “Linear pairs add to 180, therefore 2x + (x +30) = 180”.  Similarly, in fifth or sixth grade, students can learn to use the distributive property to multiply 57 x 3 as 3 x (50 + 7). In algebra, that habit expresses itself more formally: a(b + c) = ab + ac.

But developing “habits of mind” outside of the context of the material being learned is like the Cheshire Cat of Alice in Wonderland.  Such approach forces students to consider a grin well before they are presented with the cat associated with it.  Based on statements made by school officials and others in education, it appears that their interpretation and implementation of Common Core math standards in general, and the SMP in particular follow the tenets of the math reform ideology that has gained momentum over the last two decades.  In addition to the “habits of mind” approach, the education establishment also emphasizes collaborative group work and inquiry-based learning.   A glance at the agendas of professional development seminars that are being given to teachers on implementing Common Core spend much if not the majority of time on the SMP rather than the content standards themselves.   In fact, the connection between the SMP and the content standards is made clear in the Common Core standards document itself:

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily.  (See  Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content: http://www.corestandards.org/Math/Practice)

The tension between “understanding” and procedural fluency is one of several significant tensions between two philosophies in math teaching which for lack of better terminology, I will call the “traditional” mode and the “reform math” mode.  The tensions between the two groups who practice and advocate each type have come to be known as “the math wars”.  The unrelenting search for “understanding” in the teaching of mathematics often trumps the procedural skills and problem solving techniques that lead to such understanding in the first place.

This article provides the description of each SMP as written in the Common Core math standards:  (http://www.corestandards.org/Math/Practice).   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.

SMP 1:  Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

The SMP writeup describes a problem solving mind-set as well as a variety of problem solving strategies that students should have. It is important to realize that the goal of this SMP comes about after years of experience and practice.  The ability to solve problems and think mathematically develops over time.  Problem solving cannot be taught directly; rather, it is based on mastery of many basic skills.  (See (http://www.ams.org/notices/201010/rtx101001303p.pdf )

Requisite for learning how to solve problems is an explanation of how specific types of problems are solved using worked examples and practice with routine problems.  A set of problems can then escalate in difficulty through careful scaffolding: i.e., by changing aspects of the problem so that students must apply their knowledge of the basic procedure to new forms of the problem.  In this way homework is not just a set of repetitive “exercises”.  Students progress from simple routine problems to those which increase in complexity and are non-routine.   The non-routine problems can then be extended into even more challenging problems.  Such challenging problems should definitely be given but students must be able to use prior knowledge of skills and procedures in solving them.  The goal of math teaching is to provide sufficient opportunities to apply skills and knowledge so that students know how to turn “problems” into routine exercises.

While the approach described above is a sensible and effective interpretation of this SMP, the reform math ideology that is dominating Common Core implementation is likely to reject it.  That philosophy is to regard math as some sort of magical thinking process.  It holds that “understanding” the problem and seeing the big picture is math, while the mechanics of problem solving are just a rote afterthought.  Worked examples and routine problems are generally disparaged as “non-thinking” and “routine achievement”. The reform approach usually manifests itself as giving students a steady diet of “challenging problems” in an effort to build up a problem solving habit of mind that is sometimes referred to as “sense-making”.  Such approach does not accomplish this, however.  Instead, the constant pursuit of “challenging problems” stands in the way of developing fluency with certain classes of problems and building on what one already knows.

The description of this SMP also states that students “consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.”  Such strategy comes from Polya’s classic advice on problem solving.  Students are told Polya’s rules for problem solving at early ages before such rules even make sense.  Polya intended his approach for upper level high school, and college students.  For lower grade students, Polya’s advice is not self-executing and has about the same effect as providing advice on safe bicycle riding by telling a child to “be careful”. For younger students to find simpler problems, they must receive explicit guidance from a teacher–i.e., the teacher often must provide the simpler problem for the student to then use as a template for solving the more difficult one.

SMP 2:  Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Like most of the SMPs, this one is a habit of mind.  The SMP promotes developing habits of mind used in abstract and quantitative reasoning.  It is not directly teachable, however.  Rather, it arises from the practice and mastery of specific mathematical skills and procedures.  Thus, one way to interpret this SMP is to provide students with sufficient instruction and practice in complex, multi-step problems that are appropriate to the class in which they are given.

While abstract and quantitative reasoning are important goals of algebraic thinking, the SMP opens itself up to the prevalent belief in the reform math camp that students can be taught various algebraic habits of mind outside of an actual algebra course.   An example of this type of thinking can be seen in a certain type of problem presented to students in early grades.  For example, the students are shown pictorial problems like black and white beads in a numbered series of growing sequential patterns.  The problem shows the first three patterns and asks students to predict the number of white beads in pattern 5, say.  Students in fifth grade have not yet learned how to represent equations using algebra.  Also, the problem is more of an IQ test than an exercise in math ability. Furthermore, such problems ignore the deductive nature of mathematics. An unintended habit of mind from such problem is to encourage inductive type reasoning.  Students then learn the habit of jumping to conclusions once they identify a pattern, thinking nothing further needs to be done.

Presenting problems outside of a pre-algebra or algebra course which require algebra to solve will likely result in clumsy attempts at solutions that may or may not lead to algebraic thinking. Algebraic thinking is not inherent at such a stage. But there is a big transition that students of these methods will have to make when moving to high school math which is still mostly taught traditionally. Students who use the inductive grade school understandings for the simple part simply can’t make the leap to complex. They see no need to learn actual “algebra” for easy problems because the old understanding works and they can do the problems in their heads.  They cannot, however, solve 2/3(6x + 24) = -3(x – 1) in their heads.  Many such students give up in frustration.

Parts Two and Three will be published on Wednesday, April 24 and Friday, April 26.

Monday
04 22, 2013