My First Educational Treatise and What I Learned

My mother was my first critic of my first educational treatise, written when I was a senior in high school. During my junior year, I had tried unsuccessfully to get transferred out of a geometry class taught by a teacher whose reputation had preceded her for years. In retrospect, she was a very bright woman who had emotional problems and spent the class talking about this, that and anything other than about geometry and left it to us to read the book, do the problems, and make presentations on the board.  At that time, however, I wanted someone who taught.  The math department head was more than familiar with her problems but when I asked for a transfer had told me that 1) I wasn’t a teacher and 2) I had only had her for three days; thus: how could I judge?

My father (who was not familiar at all with how school politics worked) tried to intercede on my behalf but was outmaneuvered by the high school bureaucracy/double talk. I managed to survive the semester with her, and the next fall when I was a senior, I decided to do the student body of the high school a favor and wrote a little pamphlet called “A Manual for Personal Objectors”.

The title was a take on the “Manual for Conscientious Objectors” which was required reading in the mid-60’s for those who were trying to escape the draft (and being shipped to Viet Nam) on the basis of religious and other grounds. I felt that trying to get out of a bad teacher’s class was as hard as getting out of the draft.  I typed my treatise up on ditto masters (which was like a mimeograph that used that awful smelling solvent, and the print came out purple) that my mother had, since she was a teacher).  I typed it in “landscape” mode so I could fold it over and bind it like a pamphlet. My mother ran off the pages at her school and because two-sided pages were unheard of for dittos, I then had to paste pages together to get a back-to-back, real pamphlet look. My mother helped me to glue them and we put the pamphlets together.

I started selling the pamphlets at school for 25 cents a piece which got me in a little bit of trouble at school, but that’s another story. One day my mother got mad at me for something I said to her (probably complaining about something I didn’t like about something she did) and she let me have it. It was one of those “After all I’ve done for you” type rants, and once she got going there was no stopping her.  Among the things she mentioned was running off the dittos and helping me glue them together.

I knew at that point that I had lost this battle, but she wasn’t done by a long shot.  “And another thing,” she said.  “How do you think I felt when I read what you wrote about how teachers’ grading policies are unfair and you said ‘This is especially true of English teachers.’ ”  She was an English teacher.

There was only one thing I could say, and I said it through tears: “I didn’t mean you, Mom.”

I often think of this when I hear complaints from students about a teacher as happened in a school where I taught. I was one of two math teachers for the seventh and eighth grades at the school—we had both been hired at the same time. I was friendly with the other math teacher and she would often tell me of the frustrations she was facing in her classes.  I knew that we didn’t see eye to eye on math education, given that she was a fan of Jo Boaler, and she believed that memorization eclipsed understanding.

In that particular school, I was available for tutoring students prior to the first period, and some of her students would occasionally come in for help. In one case, a boy was having difficulty with proportion problems, such as 3/x = 2/5. I showed him how it is solved using cross multiplication (without explaining why cross multiplication works—I was after just getting him through the assignment). I ran into the boy’s father during the summer, and he made a point to thank me for helping his son.  I found this odd since I had only tutored him maybe one other time.  “He was getting D’s and F’s on his test and since you helped him, he started getting A’s and he ended up with an A- in the class.”

From what I knew about the teacher, I suspected that she offered little instruction, expected students to collaborate and discover, and make connections. As it turned out, there were many complaints to the school from parents and she was fired that year. I recall her telling me that she was not going to be rehired, that parents were complaining about her, and that she felt as if she were the victim of a witch hunt.

Since she had read one of the books I wrote, I’m fairly certain she knew we were on opposite sides of how math should be taught. But she never said anything about my methods, nor I about hers. I enjoyed the autonomy the school gave me to teach as I wished so I kept my thoughts to myself. I was friendly with her, and my wife and I had her over a few times.

There are those who might say it is my duty to speak up about how a teacher teaches. But in a school, we are in a fragile situation. There are vast differences in teaching philosophies within the teaching profession, but you have to work and get along with fellow teachers as well as the people in power. The tightrope I walk is remaining loyal to how I believe math should be taught, while finding the common bond with the other teachers and the administration. And more importantly, realizing that many teachers are victims of the indoctrination of ed school group-think which has dominated the education profession for many decades. In the meantime, I’ll continue to write my criticisms of math education—being careful to call out the group-think and its perpetrators.

 

 

Misunderstandings about Understanding, Dept.

What do we mean by “understanding” in math? I gave a talk about this at the researchED conference in Vancouver. I have included an excerpt from my talk, and added some commentary at the very end which is designed  1) to further elucidate the issues and 2) to infuriate those who disagree with my conclusions.

Understanding Procedures

One doesn’t need to ‘deeply understand’ a procedure to do it and do it well. Just as football players and athletes do numerous drills that look nothing like playing a game of football or running a marathon, the building blocks of final academic or creative performance are small, painstaking and deliberate.

Many of us math teachers do in fact teach the conceptual understanding that goes along with an algorithm or problem solving procedure. But there is a difference in how novices learn compared to how experts do. Requiring novices to retrieve understanding can cause cognitive overload. Anyone who has worked with children knows that they are anxious to be able to solve the problem, and despite all the explanations one provides, they grab on to the procedure. The common retort is that such behavior comes about because math is taught as “answer getting”. But as students acquire expertise and progress from novice to expert levels, they have more stored knowledge upon which to draw. Experts bundle knowledge around important concepts called “neural links” which one develops in part through “deliberate practice”.

Furthermore, understanding and procedure work in tandem. And along the pathway from novice to expert, there are times when the conceptual understanding is helpful. But there are also times when it is not.

It’s helpful when it is part and parcel to the procedure. For example, in algebra, understanding the derivation of the rule of adding exponents when multiplying powers can help students know when to add exponents and when to multiply.

When the concept or derivation is not as closely attached such as with fractional multiplication and division, understanding the derivation does not provide an obvious benefit.

When the Concept is Not Part and Parcel to the Procedure

One common misunderstanding is that not understanding the derivation of a procedure renders it a “trick”, with no connection of what is actually going on mathematically. This misunderstanding has led to making students “drill understanding”. Let’s see how this works with fraction multiplication.

Multiplying the fractions  is done by multiplying across and obtaining But some textbooks require students to draw diagrams before they are allowed to use the algorithm.

For example, a problem like  is demonstrated by dividing a rectangle into three columns and shading two of them, thus representing  of the area of the rectangle.

Fig 1

The shaded part of the rectangle is divided into five rows with four shaded.  This is 4/5 of (or times) the 2/3 shaded area.  The fraction multiplication represents the shaded intersection, giving us 4 x 2 or eight little boxes shaded out of a total of 5 x 3 or 15 little boxes: 8/15 of the whole rectangle.

Fig 2

Now this method is not new by any means. Such diagrams have been used in many textbooks—including mine from the 60’s—to demonstrate why we multiply numerators and denominators when multiplying fractions. But in the book that I used when I was in school, the area model was used for, at most, two fraction multiplication problems. Then students solved problems using the algorithm.

Some textbooks now require students to draw these diagrams for a variety of problems, not just the fractional operations, before they are allowed to use the more efficient algorithms.

While the goal is to reinforce concepts, the exercises in understanding generally lead to what I call “rote understanding”.  The exercises become new procedures to be memorized, forcing students to dwell for long periods of time on each problem and can hold up students’ development when they are ready to move forward.

On the other hand, there are levels of conceptual understanding that are essential—foundational levels. In the case of fraction multiplication and division, students should know what each of these operations represent and what kind of problems can be solved with it.

For example: Mrs. Green used 3/4 of 3/5  pounds of sugar to make a cake. How much sugar did she use?  Given two students, one who knows the derivation of the fraction multiplication rule, and one who doesn’t, if both see that the solution to the problem is  3/4 x 3/5, and do the operation correctly, I cannot tell which student knows the derivation, and which one does not.

Measuring Understanding

 Given these various levels of understanding, how is understanding measured, if at all?  One method is by proxies involving procedural fluency and factual mastery but which involve some degree of mathematical reasoning.

Here’s an example. On a multiple choice placement test for entering freshmen at California State University, a problem was to simplify the following expression.

image

In case you’re curious, here’s the answer:   (y+x)/(y-x)

This item correlated extremely well with passing the exam and subsequent success in non-remedial college math. Without explaining one’s answer, simply recalling the arithmetic properties of fractions along with being fluent in factoring was enough for a reasonable promise of mathematics success at any CSU campus.

In short, the proxies of procedural fluency demonstrate the main mark of understanding: being able to solve all sorts of variations of problems. Not everyone needs to know the derivation to understand something at a useful—and problem solving—level.

Nevertheless, those who push for conceptual understanding, lest students become “math zombies”, take “understanding” to mean something that they feel is “deeper”. In a discussion I had recently with an “understanding uber alles” type, I brought up the above example of fraction multiplication and the student who knows what the fraction multiplication represents. He said “But can he relate it back to what multiplication is?”  Well, that’s what the area model does—is it necessary to make students draw the area model each and every time to ensure that students are “relating it back” to what multiplication is?

They would probably say it’s necessary to get a “deeper” understanding. My understanding tells me that what is considered “deeper” is for the most part 1) not relevant, and 2) shallower.

 

IN CASE YOU’RE INTERESTED: The entire talk can be obtained here: It is the PowerPoint slides, which if viewed in notes format contain the script associated with each slide.

 

 

 

 

Principal Gladhand, Dept: Coexistence in The Land of Oz and Kansas

With the Common Core annual testing coming up in California, Principal Gladhand’s weekly missive to parents brings good tidings about how his school is dealing with it.

He starts with the age-old premise that tests don’t matter:

We work hard to ensure our students are learning not because we want them to do well on any given test, but because the learning is important. We don’t
work to “teach to the test” or take excessive practice tests to “get our students used to” taking standardized tests. This isn’t teaching for mastery, it’s just teaching about testing.”

He goes on to boast how at his school, a formative assessment approach is used, so that tests don’t really matter. And in so doing, he can’t resist taking a swipe at memorization. And who can blame him when the edu-world around him feels that in this digital age, we can just Google information and focus on “higher order thinking skills/critical thinking/problem solving” etc.

We work to focus on “Mastery, not Memory.” This is why many teachers have re-take policies or offer students multiple modalities through which students can show what they know. Many of our teachers are switching to mastery-based grading as well; a student can have a low grade until they’ve mastered the content. Once they have mastered the content, the low grades change to reflect mastery.

Ignore for the moment that re-taking a test until one does well on it is in essence a form of practice and memorization. Focus instead on Principal Gladhand’s idyllic world, where neither tests nor grades matter–one simply has to remember the “Mastery not Memory” mantra. This is tantamount to clicking one’s heels three times and saying “There’s no place like home” and voila!  You’re transported back to Kansas, while retaining the sights, wonders and groupthink of his Land of Oz.

This allows him to function in his private Kansas where once again tests do matter–particularly the state Common Core test, (the SBAC which California’s Dept of Ed has renamed CAASP to facilitate their private Land of Oz).  And in fact he steps seamlessly from one domain to the other:

The CAASP (or SBAC) test is a way of measuring how our students are doing in meeting the state standards. We know this is but one small snapshot of how our students are doing on any given day, but it is a fairly good representation of how well our students have mastered what we wanted them to. The SBAC is based on the state standards all of us are using in our classrooms. The tests are not timed, adjust to the abilities of the students, and are written in such a way as to give us a good look at our students reasoning and thinking rather than just simple regurgitation of facts.

This is a masterpiece of what George Orwell called “double-think”.  He covers all the bases. Tests are just a snap-shot, but they’re OK if they measure mastery of what is covered by Common Core standards, and do not rely on memory at all. And so having said this, Principal Gladhand makes his final plea in the form of a proclamation from his pulpit in the Land of Oz:

We will be encouraging all of our students to work their hardest to show what they have learned this year. The information we gain from standardized testing shows us where we, as teachers, may need to offer more instruction or practice and allows us the data we need to improve the quality of education here.

He doesn’t forget, however, that he resides in his private Kansas, and concludes with this:

We feel our students are going into this test well-prepared and ready to show us what they know. Please encourage your child to try hard each day, eat a good breakfast, and get plenty of sleep this week!

Final translation: If there are any low scores, it’s on the teachers and the parents.

 

More from the annals of Ed School, Dept.

In my Educational Psychology class, I gave a presentation on constructivism, showing the difference between minimal guidance, and guided instruction, and evidence that inquiry-based approaches are ineffective. The professor lauded me with praise afterward and said it really got her thinking, plus she really was intrigued with Singapore Math (which I used as examples of explicit and guided instruction).

A few minutes later, I overheard her saying to a student in the class:

“Direct instruction works well in the short term but there’s research that shows that over the long term, students who were taught with discovery learning retained more. They also did better on standardized tests over the years than students taught with direct instruction.”

But I got an A on the presentation.

 

UPDATE:  I contacted the professor to ask what particular research she was referring to.  Her response follows:

There isn’t just one study that shows this – there are several studies in different contexts. You can read about them synthesized here in the National Academies Press book, How People Learn  – https://www.nap.edu/catalog/9853/how-people-learn-brain-mind-experience-and-school-expanded-edition

This book can be downloaded, which I have just done. Will let you know if I find anything of interest; stay tuned.

From the Annals of Ed School, Dept.

A new series that comprises a collection of things heard and overheard in ed school, uttered by students and professors alike. Ed school is the place where discarded and discredited psychological theories go to thrive.

Student:  Is the horse in George Orwell’s “Animal Farm” an example of cognitive motivation (i.e., motivated to achieve mastery) vs social cognitive
(performance appearance).

Professor: Yes, the horse exhibited cognitive mastery.

For those of you who haven’t read Animal Farm, or have forgotten it, the horse was forever criticized by the “collective” and responded to the criticism with “I will do better.” Not quite sure if I agree this is cognitive motivation or milieu control but as I’ve said before, there are no wrong answers in ed school.

Unintended Consequences of Teaching “Habits of Mind” for Algebraic Thinking

(This is a modified version of an article that appeared in Education News on January 28, 2013. )

 

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 was  raised as a question and a subject for further research in an 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 someone follows up on this question.

The term “habits of mind” comes up repeatedly in discussions about education — and math education in particular.  Teaching algebraic habits of mind outside of and in advance of a proper algebra course 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:

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. Another drawback is that very few if any students in fifth grade have   learned how to represent equations using algebra.

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. For example, 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.  An unintended habit of mind from such inductive type reasoning is that students learn the habit of jumping to conclusions.  This develops the habit of mind in which a person thinks that discovering a pattern is the solution and nothing further needs to be done.  Such thinking becomes a problem later when working on more complex problems.

The purveyors of providing students problems that require algebraic solutions outside of algebra courses occasionally 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 who are on the novice end of the novice-expert spectrum of learning.  He was addressing students who are well on their way to developing problem solving expertise by virtue of having an extensive problem solving repertoire.  For lower grade students, Polya’s advice is not self-executing. Telling younger students to “find a simpler version of the problem” has about the same effect as providing advice on safe bicycle riding by telling a child to “be careful”.

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.

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.

Count the tropes, Dept.

With respect to the poster below, questions abound.

Why is scaffolding only for “diverse learners”? And what IS a diverse learner? And must classroom routines be co-created? Why is teacher authority a bad thing?

Why are students now called “learners”? There really is no need to invent a whole new vocabulary. It may make you think you’re important, but most people see through it. And in the schools I’ve taught, we talk about the students or kids.

And what is with “learner agency”? It used to be “ownership” which was bad enough–now we have “agency”. Of course it pertains to “self-directed learning”. Nothing wrong with students doing things on their own but they do have to receive instruction somewhere along the line. Novices are not experts–but the chart makes no allowance for where a student (or learner) might be on that spectrum.

Finally there’s “productive struggle”. Yes, students should have to stretch beyond initial worked examples. But if they’re struggling, give them some help. Though the chart makers would cringe at this, sometimes students are ready to absorb a direct answer to a question. If so, then just tell them!Edugraphic

Edu-Soap Operas, Dept.

The first in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder” is now appearing at Truth in American Education.

Here’s an excerpt:

Various Narratives, Growth Mindsets, and an Introduction to One of my Parole Officers

If you are reading this, you either have never heard of me and are curious, or you have heard of me and have pretty much bought into my “narrative” of math education.

I tire of the word “narrative” (almost as much as I tire of the word “nuance”) which I see in just about everything I read nowadays. I thought I’d charge it rent, so to speak, since it seemed appropriate for the teaching experiences I’m about to describe. I’m currently teaching seventh and eighth grade math at a K-8 Catholic school in a small town in California. Prior to that, I taught seventh and eighth grade math for two years at a K-8 public school in another small town in California, which is where I will start this particular narrative.

It is a one-school district so superintendent and principal were always close by. After receiving praise from the superintendent both formally and informally, I received a lay-off notice. Such notices are common in teaching, with the newest teachers receiving such notices and usually getting hired back in the fall.  Nevertheless mine was final.

It is tempting to make my termination fit various narratives pertaining to the kind of teachers the teaching would like to see less of. Specifically teachers like me who choose to teach using explicit instruction; who use Mary Dolciani’s 1962 algebra textbook in lieu of the official one; who believe that understanding does not always have to be achieved before learning a procedure; who post the names of students achieving the top three test scores; who answer students’ questions rather playing “read my mind” type of games in the attempt to get them to discover the answer themselves, and attain “deep understanding”.  However logical, compelling and righteously indignant such narrative might be, my termination will have to remain a mystery.

 

Read the rest here.  

The So-Called “Instructional Shifts” of the Common Core and What They Mean

Long-winded Introduction/Preamble

The San Luis Coastal Unified School District is in the central coast area of California. It includes schools in San Luis Obispo and the nearby towns of Morro Bay and Los Osos. The district, under the direction of the current superintendent, follows the trend of  teaching that adheres to constructivist-oriented approaches; i.e., inquiry type lessons, with teachers facilitating rather than teaching. The math text used starting in middle school through sophmore year in high school is CPM, an inquiry-based program.

The district is so much beholden to this philosophy that part of the interview procedure for teaching jobs entails giving a mini-lesson to students, which is in turn rated according to criteria in the Danielson Framework.  The Danielson Framework is (according to their website) “a research-based set of components of instruction, aligned to the INTASC standards, and grounded in a constructivist view of learning and teaching.”

In other words, traditional-minded teachers need not consider applying for a teaching job in the District.  (For followers of my writings, I wrote about two teaching assignments in the District in “Confessions of a 21st Century Math Teacher”.)

I am always curious about math teaching positions that are advertised in the District.  As of this writing, two positions are open. The application always asks for the same essay-type question which I’ve always found intriguing: “Describe your knowledge of the shifts occurring in Common Core State Standards.”

The “shifts” in the Common Core State Standards (CCSS) are not something that are stated as standards. Rather, people who subscribe to the view that the CCSS are game changing, refer to the change of the game as the “shifts”–a change in how math is being taught because of the standards themselves.

Inside Common Core’s “Instructional Shifts”

The “shifts” in math instruction are discussed on Common Core’s website. There are three shifts defined: 1) Greater focus on fewer topics,  2) Coherence: Linking topics and thinking across grades, and 3) Rigor: Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity.

The first shift is a nod to the notion that previous standards and what they covered resulted in curricula “a mile wide and an inch deep” which has been the prevailing criticism of how math has been taught for the past several decades.  It suggests that math has/is taught “without understanding” and succumbs to rote memorization.

The second shift is another attack on how math has been taught, stating that there has been no connection between mathematical ideas, and that topics are taught in isolation–again “without understanding” and using rote memorization techniques.

Which brings us to the third shift, “rigor” to which I want to devote the most attention and focus.  The website translates “rigor” as “Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity.” The site also mentions that students should attain fluency with core functions such as multiplication (and by extension, multiplication of fractions): “Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.”  Again, a nod to the notion that before Common Core, math was taught as a set of procedures “without understanding” using, yes, rote memorization.

This shift has been interpreted and implemented by having students use time consuming procedures that supposedly elucidate the conceptual underpinning behind things like multidigit multiplication, fraction multiplication and other topics.

I learned of the connection between these “instructional shifts” and the current practice of drilling understanding in a conversation I had with one of the key writers and designers of the EngageNY/Eureka Math program. EngageNY started in New York state to fulfill Common Core and is now being used in many school districts across the United States. I noted that on the EngageNY website, the “key shifts” in math instruction went from the three on the original Common Core website  to six. The last one of these six is called “dual intensity.” According to my contact at EngageNY, it’s an interpretation of Common Core’s definition of “rigor.” It states:

Dual Intensity: Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year.

He told me the original definition of rigor at the Common Core website was a stroke of genius that made it hard for anti-intellectuals to speak of “rigorous” in loosey-goosey ways. He was able to justify EngageNY/Eureka’s emphasis on fluency. So while his intentions were good—to use the definition of “rigor” to increase the emphasis on procedural fluency—it appears he is taking the reformist party line of ensuring that “understanding” takes precedence and occurs before learning the standard algorithms or procedures.

In our discussion, I pointed to EngageNY’s insistence on students drawing diagrams to show place value in adding and subtracting numbers that required regrouping (a.k.a. “carrying” and “borrowing”—words now anathema in this new age of math understanding). I asked if students were barred from using the standard algorithm until they acquired “mastery” of the pictorial procedure.

His answer was evasive, along the lines of “Of course we want students to use numbers and not be dependent on diagrams, but it’s important that they understand how the algorithms work.” He eventually stated that Eureka “doesn’t do standard algorithms until students know the prerequisites needed to do them.”

Thus, despite Common Core’s proclamations that the standards do not prescribe pedagogical approaches, it appears their definition of “rigor” leaves room for interpretations that conclude understanding must come before procedure.

What Does This All Mean?

What this means for me is that I do not subscribe to this philosophy. I believe it is injurious to students and defeats the purpose of providing understanding by burdening their overloaded working memories.

I am essentially providing this essay as a public service to anyone who is thinking of applying for the various teaching positions in the San Luis Coastal USD.  If you do apply for the positions, resist the temptation to provide a link to this page when they ask you about the Common Core shifts.

But I think you knew that going in.