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“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. I hope research in this area is indeed conducted.”

Full text posted at Education News:  http://www.educationnews.org/k-12-schools/developing-the-habits-of-mind-for-algebraic-thinking/

Developing the Habits of Mind for Algebraic Thinking

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 fourth or fifth 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.

Presentating 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. I hope research in this area is indeed conducted. I hope it proves me right.

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.

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Thanks to Barry Garelick for permission to post his article here.

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Here are highlights from a great article written to anyone who doesn’t understand what the problems are with common core math:  Full text: http://betrayed-whyeducationisfailing.blogspot.com/2013/01/common-core-leading-districts-to-adopt.html

In the article, Laurie Rogers, Washington educator, explains the term “student-centered” and shows why Common Core’s “student-centered” math is failing us.  She writes that:

“Many of America’s public schools have incorporated “student-centered learning” models into their math programs. An adoption committee in Spokane appears poised to recommend the adoption of yet another version of a “student-centered” program for Grades 3-8 mathematics.

It’s critically important that American citizens know what that term means.

… Student-centered learning has largely replaced direct instruction in the public-school classroom. It was pushed on the country beginning in the 1980s by the National Council of Teachers of Mathematics, the federal government, colleges of education, and various corporations and foundations. Despite its abject failure to produce well-educated students, student-centered learning is coming back around, again pushed by the NCTM, colleges of education, the federal government and various corporations and foundations.

Despite the lack of supporting research for the approach, trillions of taxpayer dollars were spent on implementing it across the nation. Despite its grim results, trillions more will be spent on it via the Common Core initiatives….

Student-centered learning is designed to “engage” students in discussion, debate, critical thinking, exploration and group work, all supposedly to gain “deeper conceptual understanding” and the ability to apply concepts to “real world” situations. New teachers receive instruction in student-centered learning in colleges of education, and their instruction in the approach (i.e. their indoctrination) continues non-stop at state and district levels.

The popularity of student-centered learning in the education community rests on: a) constant indoctrination, b) ego, c) money, and d) the ability to hide weak outcomes from the public.

Ask yourself this: How does one actually quantify “exploration,” “deeper conceptual understanding” and “application to real world situations”? How do we test for that? We can’t, really, which helps explain why math test scores can soar even as actual math skills deteriorate.

With student-centered learning, teachers are not to be a “sage on the stage” – they are to be a “guide on the side.” Students are to innovate and create, come up with their own methods, develop their own understanding, work in groups, talk problems out, teach each other, and depend on their classmates for help before asking the teacher. Student-centered learning is supposed to be a challenge for teachers, whereas direct instruction is considered to be too easy (basically handing information over to students on a silver platter).

Ask yourself this: How much learning can be done in a class with 28 students of different abilities and backgrounds, all talking; a teacher who guides but doesn’t teach; and classmates who must teach each other things they don’t understand? How do students get help with this approach at home? What happens to students who don’t have a textbook, don’t have proper guidance, and don’t have any help at home? Direct instruction does make learning easier; that’s a positive for it, not a negative. Learning can be efficient and easy. How is it better to purposefully make children struggle, fail and doubt themselves?

…In student-centered learning, student discussion and debate precedes (and often replaces) teacher instruction. “Deeper conceptual understanding” is supposed to precede the learning of skills. But placing application before the learning puts the “why” before the “how,” thus asking students to apply something they don’t know how to do. How does that make sense?

In student-centered learning, it’s thought to be bad practice to instruct, answer student questions, provide a template for the students, teach efficient processes, insist on proper structure or correct answers, or have students practice a skill to mastery. It’s OK for a class to take all day “exploring” because exploration supposedly promotes learning, whereas efficient instruction is supposedly counterproductive. Children are supposed to “muddle” along, get it wrong and depend on classmates for advice and guidance. Struggling is seen as critical to learning. Getting correct answers in an efficient manner is seen as unhelpful.

Ask yourself this: How can “efficient” instruction be counterproductive? Math is a tool, used to get a job done. Correct answers are critical, and efficiency is prized in the workforce. Quick, correct solutions reflect a depth of understanding that slow, incorrect solutions do not. Students do not enjoy struggling and getting things wrong. For children, struggle and failure are motivation killers.

The focus of a student-centered classroom is on supposed “real-world application.” (My experience with “real-world application” is that it’s typically a very adult world rather than a child world, and that now, it’s also a political world with a heavily partisan focus.)

Ask yourself this: How does it help children to be enmeshed in an adult world of worries, prevented from learning enough academics, and basted in a politically partisan outlook? (It doesn’t help them, but it suits adults who want a certain kind of voter when the students turn 18.)

All of this is at the expense of learning sufficient skills in mathematics…

Gaps in perception:

•Proponents of the “student-centered” approach see themselves as hard workers, suffering with opponents who are stuck in the 18th century. The “deeper conceptual understanding” that they believe they foster in students seems more important to them than building math skills that consistently lead to correct answers.
•Proponents of direct instruction see the students’ weakening self-image and poor skills, and we view the student-centered approach as limiting and even unkind. Math skills and correct answers are the point of math instruction, and we don’t believe students can have “deeper conceptual understanding” if they lack procedural skills.

Proponents of student-centered learning like to call their approach “best practices,” “research-based,” “evidence-based,” and so on, but no one has ever provided verifiable, replicable proof that student-centered learning works better than direct instruction as a method for teaching math. There is actually a wealth of solid evidence to indicate the contrary.

… the stated mission of Spokane’s adoption committee is to “deeply” align to the Common Core. (Not to choose a curriculum that will – oh, I don’t know – lead students to college or career readiness?) In supporting their stated mission, committee members asserted that the Common Core was vetted by “experts,” so they believe the initiatives will produce internationally competitive graduates. They provided no data, no proof, no solid research or studies for their belief. And they can’t because there aren’t any. The Common Core initiatives are an obscenely expensive, nation-wide pilot of unproved products.

Welcome to public education: Another day, another experiment on our children, except that this time, there is strong evidence that this experiment – a rehashing of the last experiment – will again fail. Try telling that to education and political leaders. No one seems to see the evidence. When you tell leaders about it or show it to them, no one seems to care. Meanwhile, many of those leaders get tutoring or outside help for their own children. (FYI: I have never seen a professional tutor use the “student-centered” method to teach math to any child.)

The Spokane adoption committee’s mission of “deep” alignment to the Common Core has caused them to choose to pilot – you guessed it – several sets of new (and unproved) materials that are distinctly more “student-centered” in their approach, heavy on words and discovery, and light on actual math.

Kicked to the bottom of their preferences were proved and rigorous programs favored by homeschooling parents and tutors, including Saxon Mathematics* and Singapore Math*. Saxon got my own daughter almost all of the way through Algebra II by the end of 8th grade, most of that without a calculator. When I asked my email list and various online contacts for their preferences, the majority picked Saxon over every other math program, and by a wide margin.

But a member of the Spokane adoption committee – a district employee – told me the Saxon representative called Saxon “parochial” and that the publisher initially refused to send Saxon to Spokane because it was unlikely to be adopted. (“Parochial” means provincial, narrow-minded, or “limited in range or scope.”) Do you believe the Saxon rep would call his product narrow-minded and limited in scope? Saxon is efficient, thorough, clear and concise. If there is a stronger K-8 math program out there, I don’t know of it. Naturally, the Spokane adoption committee does not want Saxon.

One of the programs the committee did choose to pilot is Connected Mathematics, a curriculum already being used in Spokane, one of the worst programs on the planet, excoriated for decades by mathematicians from border to border and from coast to coast. The district employee assured me the committee is hiding nothing from the public, but the committee didn’t mention to the public that it is again piloting Connected Mathematics. They don’t seem to see its failure. They love its focus on student-centered learning. The devastation it wreaks on math skills appears to matter naught to them…”

Full text here:  http://betrayed-whyeducationisfailing.blogspot.com

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Thanks to Laurie Rogers for her research and her blog.

* Note:  Both Saxon math and Singapore math are now being rewritten to “align” with Common Core.  Only the older texts may be really trustworthy. -Christel