Common Core State Standards for Mathematics: Does It Add Up or Down? Part 3

Standards for Mathematical Practice

The Common Core State Standards for Mathematics Table of Contents includes two types of standards. First listed are Standards for Mathematical Practice. Second listed are Standards for Mathematical Content. Before we explore the Standards for Mathematical Practice (SMP), let’s make a distinction between the SMP and Content standards. The SMP are process standards. They are a part of the CCSS. Most states have had similar process standards. As process standards, the SMP are probably as good as any others.

This table comes from slide 43 of a presentation at the Washington State School Directors Association conference in Nov. 2011. There are over 300 content standards in K-8. This above table presents one content standard and one of the Standards for Mathematical Practices.

Here are the eight Standards for Mathematical Practice.

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

The US Coalition for World Class Math provided Comments on the Common Core Standards for Mathematics June 2010 K-12 Final. The introduction to those comments starts off:

1. Introduction

The Common Core State Standards lead off with Standards for Mathematical Practice.

The introduction to the Standards reads:

The standards for mathematical practice rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

To the casual observer, these words sound reassuring. For those who have been involved in the debate over how best to teach mathematics for the last two decades, this paragraph is extremely disturbing. NCTM’s process standards have been interpreted and implemented so as to downplay the importance of procedures and algorithmic involved in the debate over how best to teach mathematics for the last two decades, this efficiency in the name of “understanding”. It also favors finding more than one way to arrive at an answer that usually can be arrived at very simply in one way, and of eschewing word problems that provide the data that students will need to solve the problem in the belief that finding the data by themselves builds better problem solvers. We believe that the allegiance to the principles of the NCTM standards and ideology in Adding it Up will manifest itself in a student-centered, inquiry-based approach to math. We set out below attributes of the standards that are particularly weak and which lend themselves to such educational philosophy. As such, these standards in our opinion will diminish, not enhance, the mathematical proficiency and knowledge of students in K-12.

Possibly a little too esoteric but the concerns expressed regarding these standards manifesting in more student-centered, inquiry based approach to math are being realized.

Let’s take a close look at one of the SMP.

SMP 6. . Attend to precision.This sounds like they are calling for computational accuracy. One needs to look further and detect the nuance of emphasis in the narrative that gives more information about the meaning of this standard. Here is that narrative:

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. (bold and color added for emphasis)

I have bolded the only phrase addressing accurate calculations. This does not come until the fifth sentence of the narrative. In the first sentence, “try to communicate precisely” is given a position of greater importance. While I am glad the writers thought to add “calculate accurately” into this standard, this standard appears to have more to do with attending to communicating with precision than calculating with precision or calculating accurately. Seven sentences… six related to communication, one about calculating accurately. How important is calculating accurately? What if the process of an inaccurate calculation is communicated precisely? And are you comfortable driving over that bridge or flying in that plane knowing that the engineers had great ability at communicating precisely about their inaccurate calculations?

The focus on communication was a problem with the old Washington math standards and other state standards influenced by the NCTM standards. This looks much the same, just a tad more sophisticated. Hey, it sure does sound great though. I’ll take a dozen… oh, there are only 8… that’s okay, I’ll still take a dozen.

Write or Wrong?
The focus on communication in the SMP may, in part, be the source or justification for the emphasis in asking students to explain the process they use. The ability to explain may be given greater importance than getting the right answer. For many math problems, the work students show should be explanation enough and is a great indicator of understanding. There has been a shift in math it seems. Answers to straight-forward math problems used to be either right or wrong based on their being a correct answer to the problem. That no longer seems to be the case and an answer is deemed to be right if a group of students reach consensus about it.

Publishers of poor math textbooks/programs, professional development programs, and the SMARTER Balanced Assessment Consortium are emphasizing the Standards for Mathematical Practices (SMP) rather than the content standards. This emphasis on the SMP will influence local school district math textbook adoptions. The misguided but deliberate emphasis on the SMP rather than the content standards simply renders the CCSS as set of complex standards akin to the NCTM standards. The SMP flew in under the radar and few people were concerned about them. This emphasis will not serve the students across the country well.

The emphasis placed on the Standards for Mathematical Practice supports a constructivist approach. This approach is typical of “reform” math programs to which many parents across the country object. Programs like Investigations and Everyday Math are able to claim they address the CCSS SMP. Publishers of reform programs are aligning their programs with the CCSS Standards for Mathematical Practice. The adoption and implementation of the CCSS will not necessarily improve the math programs being used in many schools.

The emphasis on the SMP is driving professional development, textbook development, textbook selection and adoption, and assessment development. As a result of this emphasis, the SBAC may resemble a super sophisticated WASL rather than an actual assessment of student math skills.

Many math professional development programs for school administrators and teachers are focusing on the SMP. What is taking place in your local school district?

A couple of years ago I attended a meeting of math teachers at a middle school. None of the teachers had yet heard of the Common Core State Standards. The school principal was in attendance and was excited to share information about a seminar he attended the previous week. This principal distributed a one-sheet handout with the eight Standards for Mathematical Practice to the math teachers. He told the math teachers that these were the Common Core State Standards for Mathematics and this is what the school and the district will begin to focus on for math instruction. He gave no mention or indication of any awareness of content standards. These Standards for Mathematical Practice are what the school leadership has been told are the math standards and it is what leadership is telling teachers. Furthermore, administrators will evaluate teachers’ ability to deliver instruction on the SMP. Are the administrators in your local schools receiving similar training?
For those who find the CCSS math content standards to their liking, and possibly an improvement over their old or current state math standards, I urge you and everyone else to beware of the deliberate emphasis on the Standards for Mathematical Practice.

Here are links to a couple of articles related to the emphasis on the SMP.

http://www.theatlantic.com/national/archive/2012/11/a-new-kind-of-problem-the-common-core-math-standards/265444/

http://www.educationnews.org/education-policy-and-politics/the-pedagogical-agenda-of-common-core-math-standards/

 

Here are hotlinks to:

Common Core State Standards for Mathematics:  Does it Add Up or Down?  Part 1

Common Core State Standards for Mathematics:  Does it Add Up or Down?  Part 2

 

 

Standards for Mathematical Practice (Part III)

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.

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 “does 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 written explanation is a desired goal, students should be instructed in how to do so rather than 1) assuming that students automatically know how to do this if they truly “understand” or 2) that such 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.

Originally published at Education News, republished with permission of the author.

Common Core Standards for Mathematical Practice (Part II)

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

SMP 3: Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

The skills described in this SMP are a necessary part of learning mathematics, and the standard is an appropriate one for students who have gained the understanding, vocabulary, and mathematical tools by which they can conduct such analyses. The analysis and arguments expected of students, therefore, must be appropriate to the grade level.  In lower grades, students are still developing the analytic tools and vocabulary by which to express mathematical ideas and arguments.   In K-5, therefore it is appropriate to have students observe a problem being worked, identify if the problem is being done correctly, and if not, explain what is being done wrong.

In higher grades such as pre-algebra and above, students now have the tools to express mathematical ideas symbolically and also have a greater mathematical vocabulary.  Analyzing arguments and mathematical reasoning can now be done by being able to express the mathematical ideas symbolically and reason and draw conclusions from their manipulation.  In geometry, analysis of arguments is very important since that subject requires students to prove propositions and theorems.

The danger of this SMP is that in early grades, an emphasis on argumentation and understanding may eclipse the importance of learning basic skills, and problem solving procedures.  Students in early grades would be expected to make arguments beyond just recognizing why an approach to a problem was wrong. The SMP states that “elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.”  But this is requiring arguments to be made with inefficient tools that are better made in the later grades when students have the tools to generalize in a formal manner.  Again, this SMP assumes that making such arguments, albeit inefficiently, creates the habit of mind of logic.  The SMP states that students will “reason inductively with data.”  Thus, as in SMP 2 discussed above, students will carry with them a grade school level of inductive reasoning that will not serve them well in higher level math courses.

SMP 4: Model with mathematics

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Modeling is admittedly a trendy term, but it generally means solving problems by representing a situation mathematically, and then solving it.  Using addition and proportions in early grades to solve problems as stated in the SMP has merit and is the approach taken with traditional mathematics teaching.

The traditional approach generally holds that there is one right answer. Such answer can be a set of numbers, called a “solution set”.   The reform approach to math extends the traditional approach by including open-ended and ill-posed problems in the belief that textbook problems are too “nice”. The fact that the textbook provides the data students need to solve the problems is viewed is an educational detriment which will not prepare students for the “real world” of having to find things out for themselves.  These beliefs lead to providing students with messy problems that are said to duplicate the types of problems that are encountered in the “real world” of problem solving where there is “more than one right answer”.

Thus, students are given problems where there is supposedly “more than one” right answer. For example, a problem may say that some children are given $40 to buy supplies for a party for 10 kids.  The problem lists a number of things that they could buy.  The students are asked to decide what to buy, but not go over the limit of $40. Educators don’t realize that mathematicians would define a merit function that codifies the personal choices. There are then mathematical solution techniques they use to find the one solution that meets their requirements. This is a known class of problems, but the math reform approach holds that by having students come up with multiple solutions, they are teaching students to think like mathematicians.  A mathematician would view the problem as having one optimal solution.

SMP 5: Use appropriate tools strategically

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

While spreadsheets and calculators are useful tools that students should learn how to use, mathematical proficiency goes beyond these tools, whether a student can use such tools “strategically” or not.  The SMP states that “mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator”.  In fact, mathematically proficient high school students should know how to graph functions by hand, by knowing the formulae and graphical representations of  conic sections, rational functions, exponential/logarithmic, and periodic functions.  In addition, proficiency includes the knowledge of how such functions are translated and shaped.

Being able to identify external mathematical resources on the internet is useful, but an emphasis on Googling for information at the expense of solving difficult and challenging problems is misguided at best.  The SMP’s opening statement that  “mathematically proficient students consider the available tools when solving a mathematical problem” should be interpreted to mean that at the high school level, the emphasis should be on applying knowledge of mathematical procedures and deductive reasoning–not which calculator or computer program would be best suited for solving a problem.

Originally published in Education News (republished with permission of the author)