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.

by Barry Garelick

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: The Cheshire Cat’s Grin. 

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He is co-founder of the U.S. Coalition for World Class Math.

The post Standards for Mathematical Practice: Cheshire Cat’s Grin, Part Three appeared first on Education News.

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

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.

Part One is available here, and Part Three will be published on Friday, April 26.

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He is co-founder of the U.S. Coalition for World Class Math.

The post Standards for Mathematical Practice: Cheshire Cat’s Grin, Part Two appeared first on Education News.

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 math” will 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.  And yet, this is how the SMP are being interpreted.  Based on statements made by school officials and others in education, it appears that the Common Core math standards in general, and the SMP in particular are following the tenets of the math reform ideology that has gained momentum over the last two decades.  In fact, 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 )

Such explanation plays into an ongoing interpretation of the Common Core standards that downplays the importance of procedures and algorithmic efficiency in the name of “understanding”. The unrelenting search for “understanding” in the teaching of mathematics has often trumps the procedural skills and problem solving techniques that lead to such understanding in the first place.  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”.

Among those in the reform math area, there has been a push to interpret the SMPs along reform math ideologies that push certain mathematical “habits of mind” outside of the context in which such habits are learned, as well as a predominate use of collaborative group work and inquiry-based learning.  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.

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He is co-founder of the U.S. Coalition for World Class Math.

The post Standards for Mathematical Practice: The Cheshire Cat’s Grin appeared first on Education News.

by Barry Garelick

Chances are good that most students in high school today know that the sum of the measures of angles in a triangle equals 180 degrees. Unfortunately, chances are also good that most high school students today cannot prove that proposition.

Geometry as taught today is for the most part lacking in the most important aspect of the subject: Proofs. Prior to 1980, most if not all high school geometry classes were very much proof-based. While there are those who bemoan the teaching of K-12 math as being mired in “computational” and “procedural” aspects of math while ignoring the larger beauty of what mathematics is about, it is ironic that when it comes to geometry, the true mathematical nature of the subject is largely ignored.

A glance at the geometry textbooks that are typically used in high schools today reveals that the problems students are given in such courses require one or two proofs  that are not very challenging in a set of problems devoted to the application of theorems rather than the proving of propositions.  Most of the problems presented in these textbooks require students to apply various theorems and definitions to find the lengths of line segments and angles.  Typical courses in geometry are lacking in proof-based problems; instead, they contain many problems in which missing angles or segments are indicated as algebraic expressions. For example, opposite sides of a quadrilateral that is identified as a parallelogram may be labeled x + 2 and 2x – 6; the student is asked to find the length of the segments. This problem requires knowledge of the properties of a parallelogram leading to the conclusion that the two segments of interest are congruent.  The two sides, expressed as x + 2 and 2x – 6 then lead to the equation x + 2 = 2x – 6.   Figure 1 shows another example of a problem that does not require formal proof.

Figure 1:

This problem requires students to know and apply that the sum of the angles in a triangle equals 180 degrees, and to know what are linear pairs of angles, and that they sum to 180 degrees. From this, students can piece together information and compute angle R.

While the types of problem discussed above constitute a form of proof (requiring applications of theorems and definitions combined with deductive reasoning to justify the necessary computation), such problems do not fully develop the skills necessary to develop a logical series of statements that constitute proof.  In contrast, consider a problem that requires a student prove a particular proposition, such as shown in Figure 2:

Figure 2:

This problem does not require any numerical calculation. It requires knowledge of theorems of parallel lines in a plane and properties of isosceles triangles.

Defeating the Purpose of a Geometry Course

To limit the number of challenging and substantial proofs in a geometry text defeats the purpose of a course in geometry.  Geometry differs from other math courses the student has had up through algebra.  There is little disagreement that problem sets are the heart and soul of a mathematics course.  Students learn the knowledge or skills of mathematics by solving problems that incorporate such knowledge  The problems in a geometry course that require proofs of propositions are not only an application of the theory, but a part of it.  If done right, the study of geometry offers students a first-rate and very accessible introduction to the nature and techniques of logical argument and proof which is central to the spirit of mathematics itself.  As such, a proof-based geometry course offers to students—for the first time—an idea of what mathematics means to mathematicians, and how it is used.  Also, unlike algebra and pre-calculus, since geometry deals with shapes, it is easier for students to visualize what it is that must be proven, as opposed to more abstract concepts in algebra.

To make matters worse, many of the modern day geometry textbooks are now in written in a disorganized manner: a hodgepodge of topics that do not follow in the heirarchy of postulates, definitions and theorems that make up a traditional geometry course.  The theorem that the sum of the angles in a triangle is always 180 degrees is sometimes presented before it is actually proven.  Also, it is not unusual to see that coordinate geometry (that is, graphing figures on a grid) is presented almost immediately and mixed in throughout the course with the traditional Euclidean geometry. Coordinate geometry is often used as the basis for proofs of theorems about triangles, using tools such as the distance formula, slope, and midpoint formula. This is done long before the Pythagorean Theorem is presented and (sometimes) proven.  Coordinate geometry should be introduced only after presentation of the Pythagorean Theorem. This is because the coordinate grid is based on a network of horizontal and vertical lines that are perpendicular to each other; the formula for finding the distance between two points on the grid is based on the Pythagorean Theorem.

Today’s geometry textbooks are also written with an eye to being relevant to students, and therefore contain “real world” type problems which are, by and large, of a computational nature.  This development is ironic considering the complaints of those who have advocated for reforms in math education.  Such math education advocates claim that math as traditionally taught fails to teach true mathematical understanding because it is mired in computational and procedural aspects.  They claim that such approaches ignore the larger beauty of what mathematics is about and, as evidenced in the Common Core Standards for Math, believe that students in lower grades (K-6) must “understand” the conceptual underpinnings of procedures.

A Topsy Turvy Approach to Mathematical Understanding

The focus on understanding in the lower grades, and the dearth of proofs in geometry seems to be a rather topsy-turvy approach. From a mathematical perspective, both understanding and procedural fluency are important.  But in the early years, most students progress with procedural fluency to build up their level of problem solving efficiency and comfort, which in turn allows them to better understand the conceptual underpinnings.  Some students learn these in the early grades, while most others gain the conceptual understandings later, particularly when they have the powerful tools of representing arithmetic operations in algebraic symbols. Taken to extremes, the emphasis of understanding over procedure in early grades can yield absurd results. Consider a student in 5^th grade who is able to solve the problem of how many 2/3 oz servings of yogurt are in a 3/4 oz container of it.  The Common Core standards’ embodiment of the “math is not just about computation” philosophy would judge such student to not “understand” fractional division if he/she can’t explain the invert and multiply rule. It is therefore perplexing if not frustrating that when it comes to geometry, the true mathematical nature of the subject is largely ignored.

One would think that the more rigorous treatment of geometry would be favored not just for the mathematical structure and logic, but also for the boost it gives to problem solving ability.  A frequent criticism of the advocates for math reform of traditionally taught math courses are that students work problems for which they already know the procedure for solving.  That is, the solution of a problem can be found by repeating a method that the student has learned, and thus using an “algorithmic procedure”.  The criticism goes that students do not learn how to apply their prior knowledge to new and non-routine problems and for which they are not able to rely on worked examples.  Yet, requiring students to prove geometric propositions would address this criticism.  Proofs do not lend themselves to specific procedures.  Rather, students must apply their knowledge of theorems, definitions and postulates to perform the proof successfully.  They are forced to ask what needs to be shown for the proposition to be true?  And from that they must work backwards to see the sequence of statements (referencing the appropriate theorems and definitions to verify their truth) that produces a logical demonstration of what it is that must be proven.

The Delay of Mathematical Maturity

I learned from a geometry book in the old SMSG series that prevailed as part of the 60′s new math. While the 60′s new math’s abstract and formal approach had disastrous results for the lower grades, the texts produced for high school were a different story. The SMSG Geometry book was written primarily by Edwin Moise (a first-rate mathematician) and Floyd Downs (a high school math teacher).  The book eventually went into commercial production and is still available (Geometry, by Moise and Downs).  The book is structured so that each theorem presented is proven using only theorems that have been proven previously.  Thus, while some theorems presented could easily be proven using the theorem that the angles in a triangle always sum to 180 degrees, the proofs are presented only in terms of what came before.  This highly structured approach taught me (and I assume many others) about the logical structure of mathematics and the nature of proof, which served as an important foundation for subsequent courses in math that I took as a mathematics major.

People roll their eyes when they hear about proofs because they may recall the “two column method” of proof: give a statement, and give a reason for every statement about the proof. The two-column method is used as an introduction to proofs to initiate students to the method of rigor and to force them to think about every statement made in a proof. The initial experience of the two-column proof is a basic training in how the heirarchy of the definitions, postulates and theorems that have been presented are used to prove something new.   Students learn to ask “Can I make such a statement?  How is this statement justified?”  After a few weeks of such method, students are allowed to produce narrative types of proof.  Having earned their “stripes” through a basic training of “rigor”, students are rewarded by not having to give a reason for things that have now become obvious (e.g., drawing in a diagonal in a quadrilateral no longer needs to be justified because “any two points determines a straight line segment”).  Following this progression of learning, students are placed on a path to a new level of mathematical maturity which entails being able to tell the difference between what is obvious and what needs to be justified.  It is a form of mathematical understanding, which entails the skills of logic and structured argument.

The level of mathematical maturity such understanding brings with it has been lost for two decades and counting. Whether it returns under the new Common Core math standards to be implemented in 45 states is an open question.  From what I’m seeing so far, the implementation of the standards is turning out to be a matter of interpretation, and that interpretation appears to be the same emperor with the same wardrobe.  My hope is that the standards allow those teachers who believe in the importance of the proof-based geometry course in high school to teach it, and that they have the proper textbooks with which to do it.

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California.

The post The Modern Day High School Geometry Course: A Lesson in Illogic appeared first on Education News.

by Barry Garelick

The idea of whether algebraic thinking can be taught outside of the context of algebra has attracted much attention over the past two decades.  Interestingly, the idea has recently been raised as a question and a subject for further research in a recent article appearing in American Mathematical Society Notices which asks,  “Is there evidence that teaching sense making without algebra is more or less effective than teaching the same concepts with algebra?” I sincerely hope this request is followed up on.

The term “habits of mind” comes up repeatedly in discussions about education — and math education in particular. The idea that teaching the “habits of mind” that make up algebraic thinking in advance of learning algebra has attracted its share of followers. Teaching algebraic habits of mind has been tried in various incarnations in classrooms across the U.S.

Habits of mind are important and necessary to instill in students.  They make sense when the habits taught arise naturally out of the context of the material being learned. Thus, a habit such as “Say in your head what you are doing whenever you are doing math” will 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”; in calculus “Composite function, chain rule, derivative of outside function times derivative of inside function”.

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 is extended to a more formal expression: a(b + c) = ab + ac.

But what I see being promoted as “habits of mind” in math are all too often the teaching of particular thinking skills without the content to support it. For example, a friend of mine who lives in Spokane directed me to the website of the Spokane school district, where they posted a math problem at a meeting for teachers regarding best practices for teaching math.

The teachers were shown the following problem which was given to fifth graders.  They were to discuss the problem and assess what different levels of “understanding” were demonstrated by student answers to the problem:

Not only have students in fifth grade not yet learned how to represent equations using algebra, the problem is more of an IQ test than an exercise in math ability.  Where’s the math?  The “habit of mind” is apparently to see a pattern and then to represent it mathematically.

Such problems are reliant on intuition — i.e., the student must be able to recognize a mathematical pattern — and ignore the deductive nature of mathematics.  An unintended habit of mind from such inductive type reasoning is that students learn the habit of inductively jumping to conclusions.  This develops a habit of mind in which once a person thinks they have the pattern, then there is nothing further to be done.  Such thinking becomes a problem later when working on more complex problems.

Presenting problems like the button problem above prior to a pre-algebra or algebra course will likely result in clumsy attempts at solutions that may or may not lead to algebraic thinking. Since the students do not have the experience or mathematical maturity to express mathematical ideas algebraically, algebraic thinking is not inherent at such a stage.

Specifically, one student answered the problem as 1 x (11 x 3) + 1, which would be taken as evidence by some that the child is learning the “habit” of identifying patterns and expressing them algebraically.  Another student answered it as 4 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 34.

Rather than establishing an algebraic habit of mind, such problems may result in bad habits.  It is not unusual, for example, to see students in algebra classes making charts for problems similar to the one above, even though they may be working on identifying linear relationships, and making connections  to algebraic equations. By making algebraic habits of mind part of the 5th-grade curriculum in advance of any algebra, students are being told “You are now doing algebra.”  By the time they get to an actual algebra class, they revert back to their 5th grade understanding of what algebra is.

In addition, the above type of problem (no matter when it is given) is better presented so as to allow deductive rather than inductive reasoning to occur.

“Gita makes a sequence of  patterns with her grandmother’s buttons.  For each pattern she uses one black button and several white buttons as follows: For the first pattern she takes 1 black button and places 1 white button on three sides of the black button as shown.  For the second pattern she places 2 white buttons on each of three sides of one black button; for the third 3 white buttons, and continues this pattern.  Write an expression that tells how many buttons will be in the nth pattern.”

The purveyors of providing students problems that require algebraic solutions outside of algebra courses sometimes justify such techniques by stating that the methods follow the recommendations of Polya’s problem solving techniques.  Polya, in his classic book “How to Solve It”, advises students to “work backwards” or “solve a similar and simpler problem”.

But Polya was not addressing students in lower grades; he was addressing students who are well on their way to developing problem solving expertise by virtue of having an extensive problem solving repertoire — something that students in lower grades lack.  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.

As an example, consider a student who stares blankly at a problem requiring them to calculate how many 2/15 mile intervals there are in a stretch of highway that is 7/10 of a mile long.  The teacher can provide the student with a simpler problem such as “How many 2 mile intervals are there in a stretch of highway that is 10 miles long?” The student should readily see this is solved by division: 10 divided by 2. The teacher then asks the student to apply that to the original problem. The student will likely say in a hesitant voice: “Uhh, 7/10 divided by 2/15?”, and the student will be on his way.  Note that in this example, the problem is set in the context of what the student has learned — not based on skills or concepts to be learned later.

Giving students problems to solve for which they have little or no prior knowledge or mastery of algebraic skills is not likely to develop the habit of mind of algebraic thinking.  But the purveyors of this practice believe that continually exposing children to unfamiliar and confusing problems will result in a problem-solving “schema” and that students are being trained to adapt in this way.  In my opinion, it is the wrong assumption.   A more accurate assumption is that after the necessary math is learned, one is equipped with the prerequisites to solve problems that may be unfamiliar but which rely on what has been learned and mastered.  It would indeed be amazing if we could teach students algebraic thinking skills devoid of the content that allows such thinking to occur. I tend to believe, however, that a proper study of this will show what many have known since the time of Euclid: there is no such royal road.

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California.

The post Developing the Habits of Mind for Algebraic Thinking appeared first on Education News.

by Barry Garelick

Mathematics education in the United States is at a pivotal moment. At this time, forty-five states and the District of Columbia have adopted the Common Core standards, a set of uniform benchmarks for math and reading. Thirty-two states and the district have been granted waivers from important parts of the Bush-era No Child Left Behind law. As part of the agreement in being granted a waiver, those states have agreed to implement Common Core. States have been led to believe that adoption of such standards will improve mathematics and English-language education in our public schools.

My fear (as well as that of many of my colleagues) is that implementation of the Common Core math standards may actually make things worse. The final math standards released in June, 2010 appear to some as if they are thorough and rigorous. Although they have the “look and feel” of math standards, their adoption in my opinion will not only continue the status quo in this country, but will be a mandate for reform math — a method of teaching math that eschews memorization, favors group work and student-centered learning, puts the teacher in the role of “guide” rather than “teacher” and insists on students being able to explain the reasons why procedures and methods work for procedures and methods that they may not be able to perform.

I base my opinion on what I see being discussed at seminars on how to implement the Common Core.  The emphasis in such forums is not on the content standards, but on the 8 Standards of Mathematical Practice (SMP).  The SMP are a slight repackaging of  the National Council of Teachers of Mathematics (NCTM’s) process standards.  And while some maintain that “process” doesn’t mean the same as “practice”, from what I see, process is still trumping content.  The popular interpretation of SMP is a pedagogical agenda that features student-centered and inquiry-based approaches. The practice of “making sense of mathematics” sounds great on paper.  But what it means to those of the thoughtworld of the education establishment is what is also called “habits of mind” in which students are taught habits of analyzing problems long before they have learned the procedural knowledge and content that allows such habits to develop naturally.  They are called upon to think critically before acquiring the analytic tools with which to do so.  More precisely, they supposedly are acquiring the analytic tools by being given problems to solve and learning via their groups and exploration (with teachers “facilitating”) and being forced to learn the techniques in a “just in time” basis.  Such a process while eliminating what the edu-establishment views as tedious “drill and kill” exercises, results in poor learning and lack of mastery.

In addition, both the SMP as well as the content standards themselves are predicated on a belief that conceptual understanding MUST precede procedure.  Evidence that this belief will have widespread implementation is seen in a recently published document that provides guidance to publishers on criteria for aligning textbooks to the standards. ( See http://www.corestandards.org/assets/Math_Publishers_Criteria_K-8_Summer%202012_FINAL.pdf )  Two of the writers of this guidance document –Phil Daro and William MacCallum–are the lead authors of the math standards.  The document  states that “conceptual understanding needs to underpin fluency work,” or that “[sufficient] fluency can be practiced in the context of applications.” (Found on page 11 of the referenced document.) It is untrue that conceptual understanding “needs” (implying it always does) to underpin fluency. Often it does, often it does not..Understanding and procedure work hand in hand; sometimes students learn procedure before understanding the concept.

While the math standards may be an improvement over existing standards in some states, they are still largely deficient.  Members of the U.S. Coalition for World Class Math have addressed the content standards in comments submitted to CCSSO and NGA.  These comments are on the web site of the U.S. Coalition for World Class Math: (http://usworldclassmath.webs.com/U.S.%20Coalition%20for%20World%20Class%20Math%20Comments%20on%20June%202010%20CCSSI%20Math%20Standards.pdf)

Ze’ev Wurman has also written extensively about these standards in a report published by the Pioneer Institute.  (See http://www.pioneerinstitute.org/pdf/common_core_standards.pdf )  He is an executive in the high tech industry in Silicon Valley and was a member of the 2010 California Academic Content Standards Commission that evaluated the suitability of Common Core’s standards for California. He served as a Senior Policy Adviser with the Office of Planning, Evaluation, and Policy Development at the U.S. Department of Education from 2007 to 2009. I echo his concerns with the content standards as summarized below:

– Common Core replaces the traditional foundations of Euclidean geometry with an experimental approach. This approach has never been successfully used in any sizable system; in fact, it failed even in the school for gifted and talented students in Moscow, where it was originally invented. Yet Common Core effectively imposes this experimental approach on the entire country, without any piloting.

– Common Core excludes certain Algebra II and Geometry content that is currently a prerequisite at almost every four-year state college. This effectively redefines “college-readiness” to mean readiness for a nonselective community college, as a member of the Common Core writing team acknowledged in his testimony before the Massachusetts Board of Elementary and Secondary Education.

– Common Core fails to teach prime factorization and consequently does not include teaching about least common denominators or greatest common factors.

– Common Core fails to include conversions among fractions, decimals, and percents, identified as a key skill by the National Research Council, the National Council of Teachers of Mathematics, and the presidential National Advisory Mathematics Panel.

– Common Core de-emphasizes algebraic manipulation, which is a prerequisite for advanced mathematics, and instead effectively redefines algebra as “functional algebra”, which does not prepare students for STEM careers.

More specifically, at the K-8 grade span:

– Common Core does not require proficiency with addition and subtraction until grade 4, a grade behind the expectations of the high-performing states and our international competitors.

– Common Core does not require proficiency with multiplication using the standard algorithm (step-by-step procedure for calculations) until grade 5, a grade behind the expectations of the high-performing states and our international competitors.

– Common Core does not require proficiency with division using the standard algorithm until grade 6, a grade behind the expectations of the high-performing states and our international competitors.

– Common Core starts teaching decimals only in grade 4, about two years behind the more rigorous state standards, and fails to use money as a natural introduction to this concept.

– Common Core fails to teach in K-8 about key geometrical concepts such as the area of a triangle, sum of angles in a triangle, isosceles and equilateral triangles, or constructions with a straightedge and compass that good state standards include.

There is already evidence that book publishers’ revisions to texts that align with the standards are highly likely to be “inquiry-based”.  (See http://www.virtual-strategy.com/2012/08/22/houghton-mifflin-harcourt-offers-math-expressions-common-core-2013-support-national-scien  Of note is this statement : “With special emphasis on Mathematical Practices and Mathematical Progressions, teachers create an inquiry-based environment and encourage constructive discussion.”)   Discovery and group learning approaches to math have had poor results when they have been used in classrooms across the country.  The Common Core math standards will in effect be a  national mandate for reform math.  I do not believe they will be good for this country.

Barry Garelick has written extensively about math education in various publications including Education Next, Educational Leadership, and Education News. He recently retired and has obtained his credential to teach math (middle school/high school) in California.

The post The Pedagogical Agenda of Common Core Math Standards appeared first on Education News.

Detroit's Mumford High School, demolished in July, 2012.

The Destruction of Education and the Preservation of Inequity

By Barry Garelick

From the fall of 1964 through June 1967, I attended Mumford High School in Detroit.  .   (I wrote about it previously here: http://www.educationnews.org/commentaries/156298.html.) My brother and sister also attended Mumford and graduated in 1962 and 1958.   As I write this, Mumford is in the process of being demolished.  The demolition is part of the $500 million Detroit Public Schools Capital Improvement Program started in 2009.   Because of a shrinking student population, the state is in the process of shutting down almost half the schools in Detroit.  The school district is attempting to sell the shut down schools for redevelopment, an effort that has earned over $10 million since 2009.

But although Mumford is being destroyed, a new version of the high school has been built on what had been the original Mumford’s athletic field.  The new school cost $52 million to build, secured by a bond issue in a city where almost half of its schools have been shut down.   From the web page for the new Mumford, it is described as a “LEED Silver Certified state-of-the-art facility and will offer academic core areas, a high-tech media center, modern science laboratories, a courtyard student quad, and a community health clinic.”  Mumford is one of 15 schools that has been taken out of the Detroit Public School system and assigned to Michigan’s Educational Achievement Authority–a statewide district set up for managing struggling schools.  The new Mumford will open in the fall.

Mumford High, 1967.

For those of us who grew up in a city that was at one time vibrant and had one of the best school systems in the country, it has been difficult to watch its decay.  We have seen the closing and destruction of automobile plants, the magnificent J. L. Hudson’s Department Store building, Tiger Stadium, and other landmarks with no effort made to preserve even a fraction of their historical significance. But even sadder for me is to watch the destruction of my old high school, once one of the top schools in the city.  It  is one more icon, a school built in 1949 with distinctive blue tile and an art deco architecture, which with proper maintenance and care could well have lasted a few more decades.

It is fairly easy to dismiss the decline of education in Detroit on its failing economy and its accompanying social ills and even for some to make sweeping accusations about the low cognitive ability of Detroit’s student population.  But the decline of education is not limited to Detroit.  It has been happening across the U.S. for quite some time, although not as catastrophically as in Detroit.  The vision of education in this country for many decades has been education and equity for all.  In fact, the history of the equity problems and their solutions in the U.S. has its parallels in the history of education in Detroit.  It is a story of how the fight to eliminate inequity in education has actually increased it.

The educational system in the U.S. has historically pitted many groups against each other— skin color was not the only determinant. Children from farms rather than from cities, and children of immigrants, for example, were often assumed to be inferior in cognitive ability and treated accordingly.

Mirel (1993) (an education professor at University of Michigan who has done much research in the history of the Detroit Public School system) points out that during the depression of the 30’s in Detroit as in the U.S as a whole, the push in education was to keep graduation rates steady and prevent drop outs.  With the deficit of work, the thinking was that it was better to have young people in school than out with nothing to do.  The general curriculum in Detroit at that time was experiencing a large number of failure rates.  To keep the students off the streets, courses were made easier.  “Descriptive” science courses were introduced in lieu of lab-based courses and focused on useful topics such as how vacuum cleaners worked.  “Relevance” was the watchword just as “engagement” is now.   Topics such as traffic safety were woven into classes such as civics, and schools offered courses in personal standards, focusing on topics such as diet, dress, etiquette and personal hygiene.  Girls were offered courses on “Appearing to Advantage, “Homemaking”, “Use of Leisure Time” and “Bride and Trousseau”. 

This pattern continued long past the depression, past World War II and into the 50’s and 60’s, during which time the influx of African Americans from the south into Detroit continued because of the well-paying factory jobs.  Curricula in high schools had evolved into four different types: college-preparatory, vocational (e.g., plumbing, metal work, electrical, auto), trade-oriented (e.g., accounting, secretarial), and general.  Students were tracked into the various curricula based on IQ and other standardized test scores as well as other criteria.  By the mid-60’s, Mirel (1993) documents that most of the predominantly black high schools in Detroit had become “general track” institutions that consisted of watered down curricula and “needs based” courses that catered to student interests and life relevance. Social promotion had become the norm within the general track, in which the philosophy was to demand as little as possible of the students.

During the period of the 50’s and early 60’s, Mumford had been one of the premier high schools in Detroit, competing with Cass Tech—a magnet school in the downtown area that had admission requirements—for the academic achievements of its students.  Mumford was located in northwest Detroit which was a mix of working class, middle and upper middle class neighborhoods.  Though predominantly Jewish, the northwest section was also home to upper middle class African Americans who attended Mumford (Graham; 1999).

Northwest Detroit started to experience the phenomenon of white flight starting in the late fifties.  The flight began in the twenties, starting in a more central area of Detroit—an area that would be the site of the riots to occur in the summer of 1967—that had also been predominantly Jewish in the 20’s through the 40’s. The flight’s trajectory continued and by the 60’s included northwest Detroit where I was living, and would continue to the suburbs.  The demographic of Mumford was changing as well.   According to Mirel (1993), 22 percent of black students were in the general track by 1967 with whites making up only 2 percent of the general track despite it being one of the most academically oriented high schools in the city.

After graduation, most boys who were in the general and vocation tracks in Detroit were drafted and sent to Viet Nam.  Boys in the college prep track who went on to college, on the other hand, were given a draft deferment status which students in the general and vocational tracks did not receive.  The role of the draft policies at the time of Viet Nam played no small part in contributing to the recognition of inequity between white and blacks.  One key manifestation of the tension occurred in April of 1966 when the students of Northern High School—largely black at that time—staged a massive walkout in protest of the principal’s decision to ban an editorial in the school paper that protested the “inferior education” and lack of college prep courses at Northern.  Northern had been a premier academic institution in the 20’s, 30’s and 40’s.  With the influx of blacks into the high school, Northern had become a general track school.

The summer of 1967 brought the riots that ultimately escalated the white flight to the suburbs, and brought in its wake racial rifts from which some say Detroit has never recovered.  In the years shortly following the riots, there were efforts to decentralize and desegregate schools—met with protest from the white communities that would be affected.  A friend of mine attended Cooley High School (located a few miles away from Mumford). In 1968 he witnessed a race riot that occurred at the school in April of that year, shortly after Martin Luther King was assassinated. He described it as follows:

“It really was a travesty, particularly in that the kids fighting were the ones in tech programs, being streamed into Viet Nam.  Cooley was half black, half white and a powder keg as long as I was there.   I went back to visit a year after I graduated — the white exodus occurred and Cooley had become almost entirely black.  Achievement was up, violence was down — it had transformed into a 50′s middle class school.”

My friend’s description is apt in a way that he may not have realized when he wrote it.  To his eyes, achievement looked like it was on the rise as it did in many schools.  Change had in fact come to Northern to address the protest of a few years back and it came to Cooley, Mumford and other predominantly black schools in Detroit. But the change came in the form of more “relevance”, focusing on black history, black culture, the arts, dance, and creative writing.  The feeling was that the outlying suburban schools (where the whites had now fled) had a college prep curriculum that was “white-based” and did not reflect the needs, values or interests of the black community.  As Mirel (1993) states, “it was a matter of community control—not curriculum reform.”

The reforms in the Detroit schools were consistent with reforms brought about across the U.S. during the 60’s and 70’s, by the prevalent radical critics of schools at that time.  These reformers brought accusations of sadistic and racist teachers, said to be hostile to children and who lacked innovation in pedagogy.  “Traditional” schooling was seen as an instrument of oppression and schools were recast in a new, “hipper” interpretation of what educational progressivism was supposed to be about.  In moving away from the way things were, the education establishment’s goal was to restore equity to students rather than maintaining the tracking that created dividing lines between social class and race.   The end product however was a merging of general track with college prep with the result that college prep was becoming student-centered and needs-based with lower standards, and less homework assigned.  Classes such as Film Making and Cooking for Singles were offered, and requirements for English and History courses were reduced if not dropped.  Social class and race was no longer a barrier for such classes as evidenced by the increasing numbers of white students who began taking them.  With the requirements for graduation being diminished in the “general” track as a result of the student-centered fad, this track saw an increase in students from 12 percent in the late 60’s to 42 percent by the late 70’s. (Ravitch, 2003).

Currently, high schools have an honors/AP track, and a general track.  The general track consists of less rigorous courses and represents a lower level of education.  Qualification for the honors/AP track starts before high school, and in elementary and middle schools, there is also a two-tier system.  The higher tier (starting at about third grade) is the gifted and talented track.  In general, this track provides teacher-directed traditional instruction at or above grade level.  For students who do not qualify (and the criteria for qualifying vary, with differing definitions of what constitutes “giftedness”), they are placed in classes with students of varying abilities.  As such, they are subjected to a one-size-fits-all curriculum that involves student-centered group activities, and project-oriented approaches to learning.  This educational approach is guided by an undying faith led by the educational establishment that all we need to do is teach students how to learn, how to think critically and that facts and content are things they can look up (on Google) on a “just in time” basis whenever they really need to know.   For many students, the approach is a guarantee that they will not be able to handle honors or AP courses in high school.

The elimination of inequity and tracking, therefore, has evolved to a two track system starting in lower grades and continuing on in high school.  Those students not considered “smart enough for the gifted programs” are consigned to the lower track.  This may not be the case if they get the skills and knowledge they need from elsewhere (tutors, learning centers, parents).  But in low income areas, this is probably not very common.  It certainly is not common in Detroit.

In the meantime, there is a push for more technology in the schools, as if SmartBoards in every classroom and an iPad for every student is the answer. The tearing down of Mumford and the building of a state-of-the-art high school are indicative of what education has now become. As in Detroit, the U.S. as a whole has long observed the demolition of education in the US; helpless to do anything about it, and given promises of great progress and changes for the future.

Barry Garelick has written extensively about math education in various publications including Education Next, Educational Leadership, and Education News. He recently retired and has obtained his credential to teach math (middle school/high school) in California.

References

Graham, Lawrence Otis.  1999.  Our Kind of People.  HarperCollins; New York.

Mirel, Jeffrey; David L. Angus.  Equality, Curriculum, and the Decline of the Academic Ideal: Detroit, 1930-68; History of Education Quarterly, Vol. 33, No. 2 (Summer, 1993), pp. 177-207

Ravitch, D.  (2003) The Test of Time.  Education Next. Spring.  Available at: http://media.hoover.org/documents/ednext20032_32.pdf

The post The Destruction of Education and the Preservation of Inequity appeared first on Education News.

By Barry Garelick

In the never-ending dialogue about math education that has come to be known as the “math wars”,  proponents of reform-based math tend to characterize math as it was taught in the 60’s (and prior) as “skills-based”.   The term connotes a teaching of math that focused almost exclusively on procedures and facts in isolation to the conceptual underpinning that holds math together.  The “skills-based” appellation also suggests that those students who may have mastered their math courses in K-12 were missing the conceptual basis of mathematics and were taught the subject as a means to do computation, rather than explore the wonders of mathematics for its own sake.

Without delving too far into the math wars, I and others have written that while traditional math may sometimes have  been taught poorly, it also was taught properly.   In fact, a view of the textbooks in use at that time reveal that they provided both procedures and concept.  Missing perhaps were more challenging problems, but also missing from the reformers’ arguments is the fact that not only are procedures and concepts taught in tandem  but that computational fluency leads to conceptual understanding.  (See http://www.psy.cmu.edu/~siegler/r-jhnsn-etal-01.pdf )

John Woodward, currently Dean of the School of Education at the University of Puget Sound, is one such person who refers to traditional math teaching as “skills-based” in various papers he has written (as well as in a personal communication to me).  I was therefore interested to learn that he chaired the panel that wrote “Improving Mathematical Problem Solving in Grades 4 through 8″  which was published by the Department of Education’s “What Works Clearinghouse”. [1] (http://ies.ed.gov/ncee/wwc/pdf/practice_guides/mps_pg_052212.pdf)   Upon going through the guidance, I was heartened to see that the panel recommends whole class instruction, defining terms so that students are not thrown off by unfamiliar vocabulary, and helping students recognize and articulate mathematical concepts and notation.

The recommendation of whole class instruction is admittedly a step in the traditional direction, as opposed to reform methods such as problem-based learning in small groups, facilitated by a teacher who refrains from direct/explicit instruction.   As if to ensure that such a step is not interpreted as advocating a purely “skills-based” approach to teaching math, the report is careful to recommend that whole class instruction include presentation of non-routine as well as routine problems.  Non-routine problems are those for which there are not predictable approaches suggested by the problem, or worked-out examples that apply to them.

There is no argument from me or others in the traditional camp that students benefit by being given both routine and non-routine problems.  It is important to recognize, however, that routine problems are prerequisite for solving the non-routine ones.  And while students certainly should be given challenging non-routine problems, they must be able to be solved using prior knowledge of skills and procedures.

The necessity of prior knowledge is something that reformers tend to dismiss.  A prevalent belief among math reformers is that just as students develop problem solving habits for routine problems, a similar “habit of mind” development occurs for solving non-routine problems.  And in fact, it appears that based on an example of a non-routine problem included in their report Woodward and the other panel members are thinking along the “habits of mind” route.  In the problem, the student is asked to find the value of an angle as shown below:

The problem is described as “likely non-routine for a student who has only studied simple ge­ometry problems involving parallel lines and a transversal.”   This is true but the authors fail to completely characterize why students would find it non-routine. The problem is solved by drawing in a line that is not shown, called a “supplemental line”.  If the students have had no prior knowledge in supplemental lines and how they are used in proofs, the problem is non-routine not because of its newness, but because they lack the prior knowledge and skills needed to solve the problem.

The figure below shows how drawing in a supplemental line to extend an existing one creates a transversal where there wasn’t one before.  At the top parallel line, the supplementary angle to 155 is easily calculated as 25.  The transversal now makes it obvious that the supplemental angle of 70 is an alternate interior angle and is the second angle in the triangle formed by the supplemental line.  Since angle x is an exterior angle to the triangle, it is the sum of the two remote angles 70 and 25, or 95.

The report does not make clear for what grade level the non-routine problem is being presented. I assume that since the report is for math taught in grades 4-8, that this problem would be for eighth graders.  While an appropriate way to introduce how to use supplementary lines in proofs and solving problems (followed by explicit and systematic instruction in the technique) the report makes no mention of using it in this fashion.  Without the knowledge of drawing supplemental lines, students are at a significant disadvantage in trying to solve the problem.  Teachers guiding the student would ultimately give hints about supplemental lines, and would provide the needed knowledge in a “just in time” basis.   The new knowledge acquired in such fashion may show the student how to proceed, but does not develop any kind of habit of mind.

In another chapter of the report (on how teachers can provide prompts to help students solve problems), they give an example of a problem in which, again, students do not have the proper tools to solve it efficiently. In particular, they pose the following problem:  Find five different numbers whose average is 15. They then give an example of the type of “prompts” teachers can give students to help them solve it.

They describe a student who is picking numbers, adding them and dividing by 5. The teacher notices that the student has some numbers bigger than 15 and some smaller and through questioning, gets the student to observe that they can’t all be greater than 15, nor all smaller than 15 because then the average would be greater than or smaller than 15. The student says “Some have to be bigger and some smaller. I guess that is why I tried the five numbers I did.” The teacher responds: “That’s what I guess, too. So, the next step is to think about how much bigger some have to be, and how much smaller the others have to be. Okay?”

In essence the teacher is helping the student develop a more efficient way to do guess and check which is an inherently inefficient process. The problem would be a good one for a pre-algebra or algebra class in which students have had some instruction in expressing words algebraically.  Rather than present this problem to students who lack algebraic knowledge or skills, it could be presented to pre-algebra and algebra students.  Then, rather than prompting the student to do an inefficient method efficiently, the teacher could prompt the student by asking what is an average, and whether the problem tells us what the sum of the five numbers is. Since the problem does not provide the sum, the student can be prompted to express the unknown sum as “x”, thus setting up a way to express the average using algebraic symbols. Since the sum is divided by how many numbers are summed, an equation of x/5 = 15 is obtained.  Early students of algebra know how to solve the one-step equation to obtain 75. Now it is much easier to then find five different numbers that average 15, since the student now only needs to find 5 different numbers that sum to 75.

People may object to my criticisms here by saying that the recommendations  of including non-routine problems and of guiding students via prompts are very reasonable and sound.  I agree; they are.  But despite the authors’ willingness to enter into discussions of traditional modes of teaching where the edu-establishment has been reluctant to go before, the examples I have discussed here belie a general cautiousness.  It is as if they are afraid of an outcome that will be their recurring nightmare:  Skills-based math.   And so they fall back on their conceptions of “habits of mind”.  The authors probably believe that they have taken significant steps to meet the traditionalists half way.  It is probably more accurate to say that their best intentions are driven by an agenda that will continue to teach math in a “just in time” manner, and will foster bad habits of mind. They have obsessed over the simplest good ideas to the point that they become bad ones.

Barry Garelick has written extensively about math education in various publications including Education Next, Educational Leadership, and Education News. He recently retired and has obtained his credential to teach math (middle school/high school) in California.

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Only 20 percent of fourth graders correctly calculated the answer to 314 x 12 on the 2004 National Assessment of Education Progress (NAEP) exam. Eighth graders’ performance was also disturbing: a question asked for the length of a line segment above a ruler, with one end at the 2 cm mark and the other at the 7 cm mark. Only 58 percent of eight graders got it right; and it was multiple choice. On the international front, anyone following how U.S. fourth and eighth graders fare in international tests in math (called TIMMS) have by now noticed that U.S. has come in about 14th or 15th, and that Asian countries top the list (Singapore is number one).

To put the issue of math education in context, one has to understand the prevailing attitude toward math education in this country. Two years ago, at a packed conference on math education, Jim Milgram–a math professor from Stanford (now retired)–presented the following story problem which, he noted, seventh grade students in Russia are expected to solve:“Two people left their villages at sunrise and walked, each to the other’s village at constant speed. They met at noon and the first arrived in the others’ village at 4:00 PM while the second arrive at 9:00 PM. What time was sunrise?”

At this, a man sitting behind me huffed “Who cares?” This is a fairly typical reaction. Many people believe that U.S. students do not perform well in math because they are not taught how to apply it to real-world, relevant problems.  That the problem is challenging was of no concern to the commenter.

Sentence first—verdict afterward

In May, 2005, the National Council of Mathematics Teachers (NCTM) in a statement which appeared in the Washington Post echoed the exact same “Who cares?” sentiments as the disgruntled man at the conference:

“For generations, mathematics was taught as an isolated topic with its own categories of word problems. It didn’t work. Adults groan when they hear ‘If a train leaves Boston at 2 o’clock traveling at 80 mph, and at the same time a train leaves New York…’ Whatever problems and contexts are used, they need to engage students and be relevant to today’s demanding and rapidly changing world.”

NCTM is a large organization based in Reston, Virginia which exerts considerable influence over how math is taught in this country. In 1989, NCTM published a set of curriculum and evaluation standards for math, and revised them in 2000. Some states have relied on these standards in framing their own. Such standards de-emphasize learning basic skills, is light on content and heavy on context-based learning otherwise known as “real life math”. Cathy Seeley, current president of NCTM is critical of math texts and programs that tell students “here’s the rule, now do the problem” and says there is too much “teacher instruction” in the U.S. NCTM’s topsy-turvy approach to teaching math is more like “Here’s the problem, you figure out the rules needed to solve it”—an approach alarmingly similar to the Queen’s declaration at Alice’s trial in Alice in Wonderland: “Sentence first–verdict afterward.”Some real life problems

Here’s an example of a real life problem which can be found on NCTM’s very own web site in the section called “Illuminations”:

“Suppose you have saved $63. You find a used video game system that you would like to buy. The seller is asking $180. You earn $10 a week doing odd jobs. How long will it take you to earn enough money to buy the game?”

While this type of problem has been around for years, NCTM’s suggestions for how to “explore” the problem in class is what’s different. They explain that adults typically subtract 63 from 180 and divide by 10. While this would be a preferred approach for students to have mastered by the 5th or 6th grade–the grade level for this activity–NCTM describes with particular pride a student entering 63 into the calculator (no apology was offered for calculators being used here), then adding his first week’s allowance, then the second, third, and so forth until the display showed that he had at least $180 (12 weeks).

NCTM explains: “Allowing students the freedom to use strategies that are intuitively obvious to them helps them to feel more comfortable in the problem-solving process. At some stage it also helps them appreciate the efficiency of standard algorithms.” NCTM does not discuss when this stage will occur. One would hope that it occurs quickly so that the calculator-aided counting-on-fingers method can be supplanted with the more efficient method that students in Japan and Singapore have mastered by the third grade.

A Word from NCTM

Recently, when asked why U.S. students suffer from an inability to perform complex reasoning and mathematical assignments compared to students overseas, Cathy Seeley (then the president of NCTM) responded: “We’re not doing as much problem-solving of that type as we need to be.” In another instance, she said “We can definitely learn lessons from Singapore, Japan and China. But we have to look beyond their textbooks to determine what these lessons are.”

Even a faithful NCTM adherent would not fail to notice that in Singapore’s textbooks, problems require multi-step solutions that are considerably more complex than what we expect US students to solve at that grade level. From a sixth grade Singapore textbook: 3/5 of Mary’s flowers were roses and the rest were orchids. After giving away ½ of the roses and 1/4 of the orchids, she had 54 flowers left. How many flowers did she have at first?

Looking beyond the textbook as Cathy suggests allows NCTM to throw the baby out with the bath water, and to reject problems that are good by saying “It’s not the text, it’s the teaching.” In fact, in Japan, Singapore and Russia, they do teach math differently. They teach it correctly. They teach content. They teach skills and facts as a foundation upon which understanding will be built. They teach like they used to in the U.S.

Alice’s retort

In Alice in Wonderland, Alice tells the royal family “Who cares for you? You’re nothing but a pack of cards!” There are many packs of cards at work in education. It starts with education schools that propagate the philosophy that knowledge must be top down, rather than “skills-based”. Boards of education, school districts, departments of education and of course NCTM follow the ed school lead and have become packs of cards. The result is that math education is almost content free. Anyone who disagrees with such philosophy is wrong and told “Off with your head”. Over and over, as parents, teachers, and world-class mathematicians protest how math is being taught, and tell school boards and administrators the type of content students should be mastering, they are viewed as trespassers in Wonderland. Story problems are met with groans, proclaimed not to be real life, and dismissed with a mighty “Who cares?”

“Who cares is not the point,” Jim Milgram says. “Let me give you an example of a problem that people had better care about since it will affect their very lives. Design a robot arm to select and lift items off an assembly line and place them on a second line correctly positioned for a second robot to work on them. There is no chance in hell that someone can do this if they can’t do the Russian problem about the two villagers.”

Until reform math is recognized for the pack of cards it is, the influence of NCTM and their followers will continue. The wake of this influence engulfs our children, many of whom do not know how to multiply two-digit numbers without a calculator, nor how to use a ruler.

Barry Garelick has written extensively about math education in various publications including Education Next, Educational Leadership, and Education News. He recently retired from the federal government and has completed his requirements for a credential to teach math (middle school/high school) in California.

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By Barry Garelick

In a well-publicized paper that addressed why some students were not learning to read, Reid Lyon (2001) concluded that children from disadvantaged backgrounds where early childhood education was not available failed to read because they did not receive effective instruction in the early grades. Many of these children then required special education services to make up for this early failure in reading instruction, which were by and large instruction in phonics as the means of decoding. Some of these students had no specific learning disability other than lack of access to effective instruction. These findings are significant because a similar dynamic is at play in math education: the effective treatment for many students who would otherwise be labeled learning disabled is also the effective preventative measure.

In 2010 approximately 2.4 million students were identified with learning disabilities — about three times as many as were identified in 1976-1977. (See http://nces.ed.gov/programs/digest/d10/tables/xls/tabn045.xls and http://www.ideadata.org/arc_toc12.asp#partbEX). This increase raises the question of whether the shift in instructional emphasis over the past several decades has increased the number of low achieving children because of poor or ineffective instruction who would have swum with the rest of the pack when traditional math teaching prevailed. I believe that what is offered as treatment for math learning disabilities is what we could have done—and need to be doing—in the first place. While there has been a good amount of research and effort into early interventions in reading and decoding instruction, extremely little research of equivalent quality on the learning of mathematics exists. Given the education establishment’s resistance to the idea that traditional math teaching methods are effective, this research is very much needed to draw such a definitive conclusion about the effect of instruction on the diagnosis of learning disabilities.¹

Some Background

Over the past several decades, math education in the United States has shifted from the traditional model of math instruction to “reform math”. The traditional model has been criticized for relying on rote memorization rather than conceptual understanding. Calling the traditional approach “skills based”, math reformers deride it and claim that it teaches students only how to follow the teacher’s direction in solving routine problems, but does not teach students how to think critically or to solve non-routine problems. Traditional/skills-based teaching, the argument goes, doesn’t meet the demands of our 21st century world.

As I’ve discussed elsewhere, the criticism of traditional math teaching is based largely on a mischaracterization of how it is/has been taught, and misrepresented as having failed thousands of students in math education despite evidence of its effectiveness in the 1940’s, 50’s and 60’s. Reacting to this characterization of the traditional model, math reformers promote a teaching approach in which understanding and process dominate over content. In lower grades, mental math and number sense are emphasized before students are fluent with procedures and number facts. Procedural fluency is seldom achieved. In lieu of the standard methods for adding/subtracting, multiplying and dividing, in some programs students are taught strategies and alternative methods. Whole class and teacher-led explicit instruction (and even teacher-led discovery) has given way to what the education establishment believes is superior: students working in groups in a collaborative learning environment. Classrooms have become student-centered and inquiry-based. The grouping of students by ability has almost entirely disappeared in the lower grades—full inclusion has become the norm. Reformers dismiss the possibility that understanding and discovery can be achieved by students working on sets of math problems individually and that procedural fluency is a prerequisite to understanding. Much of the education establishment now believes it is the other way around; if students have the understanding, then the need to work many problems (which they term “drill and kill”) can be avoided.

The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of the educationists on the panel, that “the way mathematicians learn is to learn how to do it first and then figure out how it works later.” This sentiment was also echoed in an article written by Keith Devlin (2006). Such opposition has had limited success, however, in turning the tide away from reform approaches.

The Growth of Learning Disabilities

Students struggling in math may not have an actual learning disability but may be in the category termed “low achieving” (LA). Recent studies have begun to distinguish between students who are LA and those who have mathematical learning disabilities (MLD). Geary (2004) states that LA students don’t have any serious cognitive deficits that would prevent them from learning math with appropriate instruction. Students with MLD, however, (about 5-6% of students) do appear to have both general (working memory) and specific (fact retrieval) deficits that result in a real learning disability. Among other reasons, ineffective instruction, may account for the subset of LA students struggling in mathematics.

The Individuals with Disabilities Education Act (IDEA) initially established the criteria by which students are designated as “learning disabled”. IDEA was reauthorized in 2004 and renamed the Individuals with Disabilities Education Improvement Act (IDEIA). The reauthorized act changed the criteria by which learning disabilities are defined and removed the requirements of the “significant discrepancy” formula. That formula identified students as learning disabled if they performed significantly worse in school than indicated by their cognitive potential as measured by IQ. IDEIA required instead that states must permit districts to adopt alternative models including the “Response to Intervention” (RtI) model in which struggling students are pulled out of class and given alternative instruction.

What type of alternative instruction is effective? A popular textbook on special education (Rosenberg, et. al, 2008), notes that up to 50% of students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction. This idea is echoed by others and has become the mainstay of RtI. What Works Clearinghouse finds strong evidence that explicit instruction is an effective intervention, stating: “Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review”. Also, the final report of the President’s National Math Advisory Panel states: “Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities, as well as other students who perform in the lowest third of a typical class.” (p. xxiii). The treatment for low achieving, learning disabled and otherwise struggling students in math thus includes some of the traditional methods for teaching math that have been decried by reformers as having failed millions of students.

The Stealth Growth of Effective Instruction

Although the number of students classified as learning disabled has grown since 1976, the number of students classified as LD since the passage of IDEIA has decreased (see Figure 1). Why the decrease has occurred is not clear. A number of factors may be at play. One may be a provision of No Child Left Behind that allows schools with low numbers of special-education students to avoid reporting the academic progress of those students. Other factors include more charter schools, expanded access to preschools, improved technologies, and greater understanding of which students need specialized services. Last but not least, the decrease may also be due to targeted RtI programs that have reduced the identification of struggling and/or low achieving students as learning disabled. .

Having seen the results of ineffective math curricula and pedagogy as well as having worked with the casualties of such educational experiments, I have no difficulty assuming that RtI plays a significant role in reducing the identification of students with learning disabilities. In my opinion it is only a matter of time before high-quality research and the best professional judgment and experience of accomplished classroom teachers verify it. Such research should include 1) the effect of collaborative/group work compared to individual work, including the effect of grouping on students who may have difficulty socially; 2) the degree to which students on the autistic spectrum (as well as those with other learning disabilities) may depend on direct, structured, systematic instruction; 3) the effect of explicit and systematic instruction of procedures, skills and problem solving, compared with inquiry-based approaches; 4) the effect of sequential and logical presentation of topics that require mastery of specific skills, compared with a spiral approaches to topics that do not lead to closure and 5) Identifying which conditions result in student-led/teacher-facilitated discovery, inquiry-based, and problem-based learning having a positive effect, compared with teacher-led discovery, inquiry-based and problem-based learning. Would such research show that the use of RtI is higher in schools that rely on programs that are low on skills and content but high on trendy unproven techniques and which promise to build critical thinking and higher order thinking skills? If so, shouldn’t we be doing more of the RtI style of teaching in the first place instead of waiting to heal reform math’s casualties?

Until any such research is in, the educational establishment will continue to resist recognizing the merits of traditional math teaching. One education professor with whom I spoke stated that the RtI model fits mathematics for the 1960s, when “skills throughout the K-8 spectrum were the main focus of instruction and is seriously out of date.” Another reformer argued that reform curricula require a good deal of conceptual understanding and that students have to do more than solve word problems. These confident statements assume that traditional methods—and the methods used in RtI—do not provide this understanding. In their view, students who respond to more explicit instruction constitute a group who may simply learn better on a superficial level. Based on these views, I fear that RtI will incorporate the pedagogical features of reform math that has resulted in the use of RtI in the first place.

While the criticism of traditional methods may have merit for those occasions when it has been taught poorly, the fact that traditional math has been taught badly doesn’t mean we should give up on teaching it properly. Without sufficient skills, critical thinking doesn’t amount to much more than a sound bite. If in fact there is an increasing trend toward effective math instruction, it will have to be stealth enough to fly underneath the radar of the dominant edu-reformers. Unless and until this happens, the thoughtworld of the well-intentioned educational establishment will prevail. Parents and professionals who benefitted from traditional teaching techniques and environments will remain on the outside — and the public will continue to be outwitted by stupidity.

Source: U.S. Department of Education, National Center for Education Statistics (2011). Digest of Education Statistics, 2010 (NCES 2011-015), Chapter 2.

Barry Garelick has written extensively about math education in various publications including Education Next, Educational Leadership, and Education News. He recently retired from the federal government and has completed his requirements for a credential to teach math (middle school/high school) in California.

¹This article focuses on math teaching and learning, but the same pedagogical issues arise in history, science, and English Language Arts (ELA), including grammar, spelling, composition, reading comprehension and literature.

References

Devlin, Keith. (2006). Math back in forefront, but debate lingers on how to teach it. San Jose Mercury News. Feb. 19.

Geary, David. (2004). Mathematics and learning disabilities. J Learn Disabil 2004; 37; 4

Lyon, Reid (2001), in “Rethinking special education for a new century” (Chapter 12) by Chester Finn, et al., Thomas B. Fordham Foundation; Progressive Policy Inst., Washington, DC.
Available via http://eric.ed.gov/PDFS/ED454636.pdf

Rosenberg, Michael S., Westling, D.L., McLeskey, J. 2008. Special Education for Today’s Teachers: An Introduction. Columbus: Pearson, Merrill Prentice Hall.

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