Quebec Dominates in Math, Here Is Why Ed Reformers Should Pay Attention

The Canadian and Quebec Flags
Via Wikimedia Commons (CC-By-SA 4.0)

Paul Bennett had an interesting piece in Policy Options, a public forum run by the Institute for Research on Public Policy, a Canadian think tank located in Montreal, Quebec.

He notes that Quebec dominates the rest of Canada in math and have done so for many years. In spite of that, the other Canadian provinces don’t want to emulate Quebec’s success. 

He highlights a study conducted by British Columbia’s Ministry of Education into Quebec’s success. I wanted to highlight a couple of the findings that he writes about.

The first finding is that Quebec has a clearer philosophy and sequence. Bennett writes:

The scope and sequence of Quebec’s math curriculum is clearer, demonstrating an acceptance of the need for clarity in setting out a progression of content and skills focused on achieving higher levels of achievement. The 1980 Quebec Ministry of Education curriculum set the pattern. Much more emphasis in teacher education and in the classroom was placed upon building sound foundations before progressing to problem solving. Curriculum guidelines were much more explicit about making connections with previously learned material.

Quebec’s grade 4 curriculum made explicit reference to the ability to develop speed and accuracy in mental and written calculation and to multiply larger numbers as well as to perform reverse operations. By grade 11, students were required to summon “all their knowledge (algebra, geometry, statistics and the sciences) and all the means at their disposal…to solve problems.” “The way math is presented makes the difference,” says Genevieve Boulet,a professor of mathematics education at Mount St. Vincent University with prior experience preparing mathematics teachers at the Université de Sherbrooke.

Did you catch that? A clear scope and sequence was key, but not only that, an emphasis was placed on building sound foundations before tackling problem-solving. 

Now compare that to Common Core. We’ve noted Common Core’s Math Standards:

  • Delay development of some key concepts and skills.
  • Include significant mathematical sophistication written at a level beyond understanding of most parents, students, administrators, decision makers and many teachers.
  • Lack coherence and clarity to be consistently interpreted by students, parents, teachers, administrators, curriculum developers, textbook developers/publishers, and assessment developers.  Will this lead to consistent expectations and equity?
  • Have standards inappropriately placed, including delayed requirement for standard algorithms, which will hinder student success and waste valuable instructional time.

Bennett then notes Quebec uses stronger math curriculum:

Fewer topics tend to be covered at each grade level in Quebec, but they are covered in more depth than in BC and other Canadian provinces. In grade 4, students are generally introduced right away to multiplication, division and standard alogrithms, and the curriculum unit on measurement focuses on mastering three topics — length, area and volume — instead of six or seven. Concrete manipulations are more widely used to facilitate comprehension of more abstract math concepts. Much heavier emphasis is placed on numbers and operations as grade 4 students are expected to perform addition, subtraction and multiplication using fractions.

Fewer topics, they go in depth and students are introduced right away to standard algorithms. Common Core puts conceptual understanding before they master practical skills. Barry Garelick wrote about this in The Atlantic in 2012:

Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them “understand” the conceptual underpinnings.

Yet Quebec does not do this.

Canadian provinces are wise to emulate Quebec’s success in math, but we in the United States would be as well. 

Explicit Instruction in Math Can Reduce the Achievement Gap

A retired math teacher says he knows how Seattle Public Schools’ achievement gap in math can be reduced.

The Seattle Times published a guest op/ed by Ted Nutting who taught math in the school district for 17 years. He writes:

In mathematics, American students do poorly by international comparison. This has been true for decades, and it is due in large part to the weakness of math instruction here.

If Seattle Public Schools ever hopes to eliminate its gaps in achievement between students of different racial backgrounds, it must address that problem.

I taught math in the Seattle schools for almost two decades. In my experience, what works is explicit instruction. That means explaining concepts in a clear, straightforward way, showing each student how to use them and following up with lots of practice – including rigorous tests.

Some may find this method old fashioned. But you can see explicit instruction at work in three Seattle middle schools where the achievement gap is shrinking. Mercer International, Aki Kurose, and David T. Denny International — where students of color are the majority — post solid math scores and are narrowing the achievement gap much more than other schools.

A study, “Middle Schools that Narrow the Opportunity Gap in Math,” prepared last year by district staffers Anna Box and Marni Campbell, points this out. Seventh graders at each of these schools have shown continued progress on the state test, sometimes surpassing citywide proficiency rates. Until recently, all three schools scored well below the city average.

What a surprise! Teaching kids the straightforward way to solving problems and then drilling it until they know it works? I mean I’m utterly shocked that works because we’ve been told the exact opposite of that from those who have pushed reform math into the classroom and then doubled down on it with Common Core.

Now, I just wonder if this could possibly work anywhere else other than Seattle?

Read Mr Nutting’s entire piece here.

Incredibles 2 Trailer Has a Veiled Common Core Math Reference

I just watched the extended trailer for Incredibles 2 and laughed… HARD at a veiled Common Core math reference. They didn’t explicitly say Common Core, but it’s clear that is what they were talking about.

Mr. Incredible finds himself in the role of a stay-at-home dad while his wife, Elastigirl, is off doing her superhero thing. He is helping his oldest son, Dash, with his homework. Dash holds up a textbook that says “New Math for Life.”

Dash says, “That’s not the way you are supposed to do it, Dad. They want us to do it this way.”

Mr. Incredible replied, “I don’t know that way, why would they change math? Math is math.”

Dash replies, “Because, it’s ok Dad.”

Mr. Incredible continues to rant, “MATH IS MATH!”

This isn’t much different than what many parents have experienced at the kitchen table trying to help their kids with “the new way” of doing math.

The Assessments Are Rigged

We all know that polls can be skewed and that ‘what everybody knows’ may not be so. Similarly, assessments and assessment data can be gathered, used, and presented in various ways to feed an agenda.  Just because a child is said to be proficient on a state assessment doesn’t mean he or she actually is ‘proficient’ in the way parents want him or her to be.

When I was in school, my teachers would give us tests to help figure out how much of what they were teaching we had actually learned.  Then, the state stepped in and started giving assessments to make sure teachers were teaching what the state wanted them to teach.  And now?  We’re told the assessments are great, but we are just supposed to trust.  We can’t see the assessment questions.  The algorithms (mathematical formulas) determining which questions come next or whether you have a higher or a lower score are kept secret. The State Boards of Education or the assessment vendors, themselves, can move and change the ‘proficiency’ levels at will.

We take it on faith when a student passes a math assessment it means the student is proficient.  Is it possible to rig an assessment?  Not only is it possible, but it’s also being done all the time.  I have four examples of how the assessments are and have been manipulated to provide different results than most people expect.  This is being done without oversight, without insight into what is occurring, and certainly without permission from parents.

The first example is assessing not just what a student is supposed to know but making them do the problem in a particular way. Ask yourself, does this create a disadvantage for a child who knows the math facts but hasn’t been shown a particular way of doing things?

This problem is an example of a Common Core Math Standard from First Grade:

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).   

This question doesn’t just assess whether a student knows how to do an addition word problem, but it assesses whether a student has been trained on the Making Ten Strategy as outlined in the standard.  Could a student solve 8+6 without knowing the Making Ten Strategy?  Yes, of course.  Does using the Making Ten Strategy indicate critical thinking?  Or does it simply indicate a student has been instructed in this strategy?  Would you be able to succeed as a mathematician without learning this Making Ten Strategy in First Grade? Have you successfully used addition in your life without thinking about the Making Ten Strategy?

Many parent complaints about Common Core Math come from having to show the various methods for getting the answer or having to explain why an answer is correct.

Parent:“When I was in school, we did it this way.”

Child: “I have to do it this other way or it will be marked wrong.”

One mother asked her child’s teacher if he could simply do the standard algorithm on all his math homework because the multiple strategies were causing him stress.  The teacher said if he didn’t learn the strategies, he wouldn’t do well on the state assessment.  Once the mother indicated her child would not be taking the assessment, the teacher readily agreed to give credit for just the standard algorithms.  The reason for the multiple methods?  To do well on the assessment.

A review written in 2011 by Dr. Stephen Wilson of Johns Hopkins University states the following about the Common Core SBAC test (then under development).  He says, “It appears that the assessments will focus on communication skills and Mathematical Practices over content knowledge.”

Furthermore, “Mathematical Practices, or what was usually called ‘process’ standards in most states, do little more than describe how someone pretty good at mathematics seems to approach mathematics problems. As stand-alone standards, they are neither teachable nor testable. Mathematics is about solving problems, and anyone who can solve a complex multi-step problem using mathematics automatically demonstrates their skill with the Mathematical Practices, (whether they can communicate well or not).”

In short, we see Dr. Wilson’s concerns demonstrated in the above example: the process of getting the answer is of greater importance than the actual mathematical abilities most people think the assessment should be assessing.

A second example comes from Utah’s SAGE (end-of-year) sample assessment for Third Grade. This question is supposed to assess a deeper understanding of division than simply asking if a child knows the answer to 12 ÷ 4. Unfortunately, in creating a more convoluted problem, the assessment question can be solved without knowing anything more than how to count and how to write a division problem. Division facts, themselves, are not necessary.

There are lots of kids who can divide things equally by putting them in different boxes without knowing 12 ÷ 4 = 3.  Supposedly, by dragging the stars and dragging the numbers, you are assessing higher-order thinking.  But what you are really assessing is the child’s familiarity with the software interface, the format of the problem, and whether they can count and relate counting to division.  But they don’t have to know 12 ÷ 4 = 3.

Would a child who knows her division facts be able to do this problem anyway?  Most likely.  However, it is also true this question doesn’t distinguish the child who does know her math facts from the one who does not.

A third example has to do with reading comprehension.  It dates back to the 1980’s but illustrates that what is on an assessment and how it is asked can be used to manipulate and ‘direct’ a student’s thought processes.  I quote Dr. Peg Luksik who worked for Pennsylvania’s Department of Education.  From her video :

‘A sample question said: “There’s a group called the Midnight Marauders and they went out at night and did vandalism. I (the child) would join the group IF…”

“…my best friend was in the group.”

“…my mother wouldn’t find out.”

There was no place to say they would not join the group. They had to say they would join the group.’

Dr. Luksik states that while this was listed as a citizenship assessment, the internal documents stated, “We’re not testing objective knowledge. We are testing and scoring for the child’s threshold for behavior change without protest.”

Additionally, Dr. Luksik discusses another state’s Reading Assessment question: “If you found a wallet with money in it, would you take it?”

She asked, ‘Do you read better if you say “yes”? Or do you read better if you say “no”? Or were they assessing a child’s honesty on a state assessment with their name on it…?’

Clearly, these are examples of assessment questions that were not assessing either citizenship or reading as you and I would define them.

And finally, before a single Utah student took the state’s SAGE assessment in 2014, the head of state assessments warned local school board members that student test scores were going to drop by 10 or 20 points.  He also stated there was no way to correlate the previous test results with the SAGE results.  So, how did he know this?  The point was they knew what the target proficiency rate was.  Utah was looking for a proficiency rate in the 40’s.  And as they went through the process of setting those proficiency scores, they did so after the first round of testing. Then they modified the scoring to make sure the result fell within that 40% range*.  So, in one year, did Utah kids lose 20 points of knowledge?  Or does it simply mean the Powers That Be decided only 40% of the kids got to be labeled ‘proficient’ regardless of what they actually knew?

The only sure way of knowing an assessment is truly measuring academic content and grading it appropriately requires transparency with the assessment questions, the assessment methodology, and independent verification procedures.

Instead of wondering how kids are doing on state assessments and whether a school is “good” based on the assessment scores, we need to be asking what are these assessments supposed to be measuring and how do we know they really are measuring what they claim?

*Alpine School Board Study Session Audio September 23, 2014, Additional Media->Study Session @ 45 minutes.


Massachusetts’ New Standards Are Still Inferior to Pre-2010 Standards

The Pioneer Institute released a report co-written by Mark Bauerlein, R. James Milgram, and Jane Robbins this week that reviews Massachusetts new academic standards. You don’t have to guess at their general opinion when you see the title – Mediocrity 2.0: Massachusetts Rebrands Common Core ELA & Math.

The report outlines how K-12 education in Massachusetts declined after they replaced their superior pre-2010 academic standards with Common Core:

How has the move from excellent standards and tests to Common Core and its aligned tests worked out? One of the best ways to answer that question is to rely on the NAEP assessment (the so-called “nation’s report card”), which is administered every two years in reading and math to a sampling of fourth- and eighth-graders in every state. Between 2011 and 2015 (the Common Core era), Massachusetts was one of 16 states in which NAEP reading scores actually fell, and one of 39 states in which NAEP math scores fell. From 2013 to 2015 alone, Massachusetts scores declined in three of the four testing categories.

Evidence of a decline in the performance of Massachusetts students is also observable on the SAT. Since 2006, those scores have dropped by nine points in reading, 10 points in math, and 15 points in writing. Thee writing decline, especially, suggests that the reorientation of English class from classic literature to the “informational texts” of Common Core may be bearing bitter fruit.

Massachusetts in 2016 changed its assessment to an MCAS-PARCC hybrid. They also started on a review and revision of their standards which included Common Core.

They note the new language arts and literacy framework still has the same weakness that Common Core had, it lacks domain knowledge:

Apart from the verbal skill deficiencies that high-school students in Massachusetts fail to overcome during their years in the classroom, the great danger of the current English Language Arts curriculum is that students leave high school with meager domain knowledge. If the standards that are to guide the curriculum do not broach the actual, specific subject matter of the discipline, then the education of students in English falls short. Students may acquire certain skills—the current standards are broken up into Reading, Writing, Language and Speaking/Listening, which each have their skills side— but their knowledge of literature, language, and criticism never develops.

We raise the issue because this is what we see in the 2010 standards and even more so in the new ones. The skills elements in the four areas are solid, but not the knowledge areas.

They note there are four major drawbacks to the new standards:

  1. There is an absence of philology (and therefore of phonetics, lexicology, and references to historical events).
  2. The new framework lacks English and world literary history.
  3. The new framework displaces important civic-literary historical writings
  4. It denies of one of the prime instructions that English used to claim, namely, the recognition of the great, the good, and the mediocre.

They then looked at the math standards:

This analysis focuses on the two major areas that students need to learn in grades one through eight: basic arithmetic, and perhaps somewhat surprisingly, ratios, rates, percents, and proportions…

….The finding was that—aside from a tiny number of added phrases that do not impact the mathematical content in the arithmetic, ratio, rate, percent, and proportion standards in any way—the new document is identical to the, clearly failed, previous one.

Before they provided an analysis they wanted to state that there is no such thing as 21-Century Mathematics:

Before the main analysis can be presented, it is necessary to discuss the idea promulgated by proponents of the Common Core that there is such a thing as 21st- mathematics, such that the mathematics learned by students even 30 years ago is now obsolete. Their claim is that this 21st-century math is focused on problem-solving so that the main focus of instruction should be on the generalized subject of problem-solving.

The truth is radically different. ere is no such generalized subject, and the main objective of math has always been on its use as a crucial tool in solving problems not only in mathematics but in the sciences and any other precisely de ned subject of human endeavor. But in practice, one finds that before problem-solving can begin in any area, the person attempting it has to know as much as possible about that area and the mathematics that most likely will be necessary….

….Even the mathematics that was developed over 2,000 years ago is as essential (and correct) today as it was then. But there are two subjects in mathematics that have become far more important today than they were previously: 1) algorithms and computers, and 2) statistics and data analysis. therefore, these subjects should be covered adequately in the current document—which, of course, is not only not the case, but is as far from actually happening as possible.

Their analysis of the new math standards came to a troubling conclusion:

By eighth grade, the new Massachusetts math standards are at least three full years behind actual expectations in countries such as Korea, China, Japan, Singapore, and the other highest-achieving countries in the world in the most important mathematics the students are expected to learn. Further, if these standards continue to be faithfully followed for the rest of these students’ K–12 experience, the students will be even more than three years behind.

Read the whole report below:

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Look to China for Math?

The New York Times reported that the United Kingdom was investing heavily in textbooks used in schools in Shanghai, China.

Amy Qin writes:

Educators around the world were stunned when students in Shanghai came first in their international standardized testing debut, in 2010, besting their counterparts in dozens of countries in what some called a Sputnik-like moment.

Now, some British schools will try to replicate that success by using translated textbooks that are otherwise all but identical to those in public elementary schools around Shanghai.

Starting in January, teachers in England will have the option of using “Real Shanghai Mathematics,” a series of 36 textbooks translated directly from Chinese into English. The only difference? The renminbi symbols will be replaced by British pound signs.

She notes that the Shanghai style is very similar to the method used in Singapore and then writes:

The teaching method, known as the “mastery” approach, is based on the idea that all students can succeed in learning mathematics when given proper instruction. Whereas teachers in the West might describe a concept and then assign problems for students to solve individually, the mastery method is more interactive. Teachers frequently pose questions to students who are then expected to precisely explain both solutions and underlying principles in front of their classmates.

Students learn fewer concepts under this approach, which allows them to go into those concepts in greater depth. For fractions, for example, teachers might ask students to apply the underlying principle “part of a whole” in different contexts, making use of pictorial representations and other visual techniques to explore the abstract idea. Ideally, only when the entire class has demonstrated understanding or “mastery” of one concept does the teacher move to the next.

While Common Core advocates claim that Singapore’s math curriculum is similar to Common Core, such as an Achieve report that The Huffington Post reported on in 2013:

The standards don’t lead to a complete Algebra I course until high school, unlike in other high-achieving countries. An analysis by Achieve, a nonprofit organization that has supported the Common Core, found that Singapore’s math curriculum was similar to Common Core, but that in Singapore, students more quickly reach a higher level of math proficiency.

Like Shanghai (and Singapore) they also claim that Common Core uses fewer concepts:

When the Common Core standards were developed, policymakers looked to the success of other high-performing countries, including countries that scored well on the Trends in International Mathematics and Science Study. Remember: Singapore is consistently at the top.

It’s no surprise Common Core standards mirror several Singaporean approaches, including a narrower focus with greater depth. But to better align to the standards in each state, Brillon said Singapore Math Inc. introduced new textbooks last year.

The fact remains that Common Core Math focuses on skills, not concepts, unlike what is seen in Shanghai and Singapore.

In fact, there isn’t much alignment at all as a study from the American Educational Research Association found:

In mathematics, there are data for Finland, Japan, and Singapore on eighth-grade standards; alignments to the U.S. Common Core are .21, .17, and .13, respectively. All three of these countries have higher eighth-grade mathematics achievement levels than does the United States. The content differences that lead to these low levels of alignment for cognitive demand are, for all three countries, a much greater emphasis on “perform procedures” than found in the U.S. Common Core standards. For each country, approximately 75% of the content involves “perform procedures,” whereas in the Common Core standards, the percentage for procedures is 38%. Differences for the other five levels of cognitive demand are not as uniform across countries. However, none of the three countries puts as much as 1% of its content emphasis on “solve nonroutine problems,” whereas Common Core puts 4.5% of its content emphasis there. Clearly, these three benchmarking countries with high student achievement do not have standards that emphasize higher levels of cognitive demand than does the Common Core. Marginal distributions for coarse-grain topics are quite similar between each of the three benchmarking countries and the U.S. Common Core.

While there is a lot about the Chinese educational system that I would not want the United States to adopt, we would, perhaps, benefit from taking a closer look at how the Chinese teach math. While we may not want to completely adopt Shanghai math like the UK we should have a conversation about whether Common Core is taking the United States in the right direction.

I would submit that it is not.

Drilling “Rote Understanding”

Over the last several years, the press and television have publicized many parents’ frustration with how math is being taught in the lower grades.  On the internet, videos abound with examples of how procedures such as addition and subtraction are being taught to students using convoluted methods that are leaving students and parents baffled as to 1) how to do the procedure and 2) angry that the standard methods are delayed.  (This video is one of many examples of parent concern over how math is taught under Common Core.)

The current interpretation of Common Core by publishers, instructional coaches, professional development vendors, and other educational entities, maintains that teaching the standard methods (known as standard algorithms) for various procedures too early can eclipse the conceptual underpinning of why the algorithms work, and can lead to students being confused.  A video by one instructional coach argues that teaching only procedures 1) has only worked for a small group of students and 2) that the reason students have a hard time with math is “No one taught them to understand the concepts and why we’re doing what we’re doing.  We didn’t teach them how to think; we just taught them how to ‘do’ and execute…”  The premise stated by this coach and others, contains the usual mischaracterization that procedures were taught in a void without contextual understanding. He also maintains that Common Core’s main focus is on “understanding”. This article explores this notion, and how and why Common Core is interpreted and implemented in the ways we are seeing.

A case in point

A case in point has presented itself in my recent work with a group of fifth graders in need of math remediation at the school where I teach.  The students were doing exercises from their textbook on multiplying fractions.  Instead of applying the standard method (or algorithm) in which numerators are multiplied by numerators and denominators multiplied by denominators, students first had to draw diagrams for each and every problem.

The diagrams I speak of have been used in many textbooks as a means to motivate the particular procedure for multiplying fractions.  Such diagrams use the area of a square as the means to illustrate what multiplication of fractions represents, and why one multiplies numerators and denominators.  For example, a problem like 3⁄4  ×  2⁄3 is demonstrated by dividing a square into three columns, and shading two of them, thus representing  2/3 of the area of the square.  Then the square is divided into four rows, with three of them shaded–this is 3/4 of the area of the square.  Where the two shaded areas intersection therefore represents 3/4 of 2/3 of the square. The intersection of the two yields 6 little boxes shaded out of a total of 12 little boxes which is 6/12 or 1/2 of the whole square. This is done as the reasoning—the conceptual understanding—behind multiplying numerators and denominators.

The students see what 3/4 of 2/3 means in this model in terms of area of a square.

Nothing New Under the Sun

This was the explanation used in my old arithmetic book from the 60’s (and in other textbooks from that time and earlier times thus belying the notion that traditionally taught math ignores understanding and focuses only on rote memorization.)

Source: “Arithmetic We Need” by Brownell, Buswell, Sauble; 1955.

The method used in my old textbook is also the method used in Singapore’s math textbooks. It is an effective demonstration of what fraction multiplication represents and why one multiplies numerators and denominators. In Singapore’s textbooks (as in mine), students are asked to use the area model for, at most, two fraction multiplication problems. Then students are let loose to solve them using the algorithm.

But in the current slew of textbooks claiming alignment with the Common Core, after the initial presentation of the diagram to show what fraction multiplication is, and why and how it works, students are then required to draw these type of diagrams for a set of fraction multiplication problems.  The thinking behind having students draw the pictures is supposedly to “drill” the understanding of what is happening with fraction multiplication, before they are then allowed to do it by the algorithmic method.

The approaches to math teaching in the lower grades in schools is a product of many years of mischaracterizing and maligning traditional teaching methods. The math reform movement touts many poster children of math education. Their views and philosophies are taken as faith by school administrations, school districts and many teachers – teachers who have been indoctrinated in schools of education that teach these methods.

Such topsy-turvy approaches to math education have been around for more than two decades, but the interpretation and implementation of Common Core have made them more popular.  To compensate for what reformers believe is a lack of understanding, the teaching of mathematics has been structured to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations.

What ultimately happens is that these exercises in understanding simply become new procedures, which small children attempt to learn and memorize because that is what many small children do.  On top of all that is that these methods are not efficient and very confusing, resulting in frustration and feeding into children’s dislike of math—something this method was supposed to cure.

The Instructional “Shifts” of Common Core: The Source of Much of the Hidden Pedagogy 

Where is this interpretation coming from? One possible source are the “shifts” in math instruction that are discussed on the website for Common Core.  One of the shifts called for is “rigor” which the website translates as: “Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity”.  Further discussion at the website 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.”

I learned of the connection between these “instructional shifts” talked about at the Common Core website, 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 and is now being used in many school districts across the US.) On the EngageNY website, the “key shifts” in math instruction went from the three that were on the original Common Core website (Focus, Coherence and Rigor) to six.  The last one of these six is called “dual intensity” is, according to my contact at EngageNY, an interpretation of Common Core’s definition of “rigor” and 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 that 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. And, in fact he was able to justify the emphasis on fluency in the EngageNY/Eureka math curriculum.  So while his intentions were good (using the definition of “rigor” to increase the emphasis on procedural fluency) it appears to me that he may have been unwittingly co-opted to make sure that “understanding” took precedence.

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 (aka “carrying” and “borrowing”–words that are 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 would appear that in their definition of “rigor” they have left room for interpretations that understanding must come before procedure.

“Understanding” Coexists with Procedural Fluency

Understanding and procedure work in tandem. Sometimes understanding comes first, sometimes later. As evidenced by EngageNY/Eureka Math, and other programs making inroads in school districts across the US, the interpretations of Common Core have resulted in an “understanding first, procedure later” approach.  That    interpretation makes it appear as if both sides have reached common ground.  Reformers can now say “You see? We’re not against drills”—provided such drills are drilling understanding.

The major problem with this approach is that not all students take away the understanding that the method is supposed to provide. Some get it, some don’t. And while it may work to give the adults who design such programs a mental visualization, they’ve had the advantage of many years of math experience (and brain growth) that students in 5th, 6th and even 7th and 8th grades do not have.

Students are forced to show what passes for understanding at every point of even the simplest computations. This drilling of understanding approach undermines what the reformers want to achieve in the first place. It is “rote understanding”: an out-loud articulation of meaning in every stage that is the arithmetic equivalent of forcing a reader to keep a finger on the page, sounding out every word, every time, with no progression of reading skill.

The Seductive Nature of a Bad Idea

The scary part about all of this is how easy it is to get swept in to the recommended methods.  I was working with the fifth graders and insisting that they draw the diagram to go along with each problem, when midway through the period I realized that I was forcing them to do something that I felt was ineffective.  The next day, I announced to them that instead of them having to do the rectangle diagrams, they could just do the fraction multiplication itself.  While my decision was met by cheers from the fifth graders, I couldn’t help feeling guilty in spite of my own beliefs. I  imagined reformers shaking their heads in dismay, believing that I was leading the students down the path of ignorance, destined to be “math zombies”.

The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. These beliefs are now extending to well-meaning mathematicians who had publicly opposed much of the reform philosophy. The math reform movement, in trying to overturn students “doing but not knowing” have unwittingly created a new poster child. While the reformers believe the new poster child represents one with “deep understanding”, they have instead created a child for whom “understanding” foundational math is not even “doing” math.

The Destructive Capacity of David Coleman

David Coleman announces the SAT redesign.

David Coleman announces the SAT redesign.

Since Common Core architect David Coleman took over as president of the College Board, the scandals or at least embarrassments have come fast and furious (see here and here). The latest is a Reuters investigation, reported by EdWeek, that discovered the College Board’s vaunted redesign of the SAT math section erects even more hurdles to students who traditionally score lower anyway (low-income and minority students). This is because the new math section focuses more on reading than actually working math problems, so a student who is good at math but less so at reading will score lower on math than he would have under the traditional SAT design.

The problem is the new SAT’s alignment with the Common Core national standards. The Common Core math standards are based on the idea that knowing math is insufficient; a student must be able to read a tome and apply math skills to the supposed “real-life” problem it presents. (The engineers who brought the Apollo 13 astronauts home on a crippled spacecraft somehow managed to apply their antiquated math education to a real-world problem, but pay no attention to that.) While the text-heavy approach may work for strong readers, turning a math test into a reading test creates unnecessary problems for students who traditionally don’t score as well on the SAT anyway.

From reviewing internal emails, Reuters discovered that College Board officials “knew of the potential problem with the word-heavy math questions because outside academics raised the issue as they reviewed items while they were being developed.” And in a confidential 2014 test run, only about half the students even finished the math section.

Maybe Coleman and Co. intended to correct the problem, but apparently they never got around to it. Or maybe they’re so invested in the Common Core ideology of “deeper conceptual understanding” that they simply don’t care.

So assume the situation of an immigrant student (call him Carlos) whose family speaks Spanish at home. Assume he’s a math whiz but still struggles with English, because he’s been in the country only five or six years. With the old SAT he might have performed poorly on the verbal portion but scored an 800 on the math. With the new test, he’ll perform poorly on both. Well done, Mr. Coleman.

Not only will Carlos suffer from this ideological redesign, but his school may as well. This is because the new fed-ed bill, the Every Student Succeeds Act, allows states to replace their high-school achievement tests with the SAT. The ramifications of the redesign are thus troubling both for Carlos and for honest accountability for schools.

Dr. Sandra Stotsky raised a related concern years ago about Common Core math, in that case with respect to children in the early grades. Since Common Core applies the word-heavy approach across K-12, young children are also expected to read paragraphs rather than simply grasp math calculations. This means, Dr. Stotsky warned, that many little boys might struggle with math even if they have a gift for numbers – because little boys are generally less verbal than little girls. Johnny might be proud that he can work math problems more quickly than anyone in the class, but don’t worry, Common Core will beat that sense of accomplishment out of him.

Common Core theorists call this “productive struggle.” Normal people might call it academic malpractice. By all means, let’s extend it to teenagers as well.

The Common Core realignment of the SAT math section will hurt low-income students in other ways. In a Pioneer Institute report, mathematician James Milgram and testing expert Richard Phelps explained that aligning the SAT with Common Core essentially converts it from a test predicting college success to one that simply measures high-school achievement. These experts pointed out that an achievement test is less effective at identifying students with significant STEM (science, technology, engineering, and math) potential who attend schools with inferior science and math programs. And the Common Core math standards – which stop with an incomplete Algebra II course – will ensure that many schools, especially those serving low-income students, will have such inferior programs.

Michael Cohen, a prominent developer of and cheerleader for Common Core, testified several years ago that we won’t know the full effects of Common Core “until an entire cohort of students, from kindergarten through high school graduation, has been effectively exposed to Common Core teaching.” Having already lowered national test scores, increased the achievement gap, driven excellent teachers out of the profession, and now wrecked the SAT, it looks like Common Core is ahead of schedule. Mr. Cohen underestimated the destructive capacity of Mr. Coleman.

Math Professor Makes Excellent Point About New Way of Assessing Students

Photo credit: Bartmoni (CC-By-SA 3.0)

Photo credit: Bartmoni (CC-By-SA 3.0)

Wayne Bishop, a professor of math at Cal State University Los Angeles, wrote an op/ed in the San Gabriel Valley Tribune that is a must read.

Bishop first addressed the problem of shifting assessments from multiple choice to making tests much more verbal, especially where math is concerned.

There is a widely held misconception that multiple-choice tests are misinforming because it is “easy for students to guess answers.” This fact ignores the reality that all students are in the same boat, with strong students having a better opportunity to demonstrate what they know.

As described by the officials, the new test requires students to answer follow-up questions and perform a task that shows their research and problem-solving skills. Nice as this sounds, reality is that it makes the mathematics tests far more verbal. Any student with weak reading and writing skills is unfairly assessed. That is especially problematic for English learners.

Low socio-economic Latino kids will be further burdened in demonstrating their mathematics competence, and Chinese or Korean immigrants who are a couple of years ahead mathematically (as was my daughter-in-law when she immigrated as a fifth-grader from Korea) will be told their mathematics competence is deficient. Absolutely absurd. Mathematics carried her for a couple of years until her English became good enough for academic work in other subjects.

I encourage you to read the entire piece, it’s worth the read.

HT: Barry Garelick

Blame the Textbooks for Poor Common Core Implementation!

Photo credit: World Economic Forum (CC-By-SA 2.0)

Gates funds the standards, funds reviews of the standards, and now funds reviews of the textbooks.
Photo credit: World Economic Forum (CC-By-SA 2.0) reviewed five high Common Core-aligned math textbooks in their first round of reviews and found only one textbook was “aligned.”

  • College Board – nope.
  • Houghton Mifflin Harcourt – nope.
  • Pearson – nope.
  • Carnegie Learning – partial credit for “focus and rigor,” but nope.
  • The CPM Learning Program was the only textbook deemed “Common Core-aligned”

Pearson wasn’t happy with the review because obviously this isn’t good for the bottom line.

They wrote:

Our analysis of the EdReports evaluations of Pearson Integrated High School Mathematics Common Core ©2014 shows that the EdReports evaluations continue to be plagued by inaccuracies, misunderstandings of program instructional models, misinterpretations of the both the intent and the expectation of the Common Core State Standards for Mathematics and the Publisher’s Criteria, and a lack of understanding of effective curriculum development and pedagogy. Pearson Education and its authors consider the EdReports evaluation an incomplete, invalid, and unreliable reporting of the quality of the program and of its alignment to the expectations of the CCSS-M.

This group recently said all of the K-8 math textbooks reviewed were not “Common Core-aligned.”

Look here is all you need to know about They received just shy of $1.5 million in 2015 from the Gates Foundation (by way of the Rockefeller Philanthropy Advisors, Inc.) for operating support “to enable them to build their core priorities of publishing reviews of instructional materials, and to grow their operations and capacity to include teacher feedback of such materials.”

See if all the textbooks are bad then they can blame the poor implementation of Common Core on the textbooks, not the standards themselves.  They have already started that narrative. See teachers just need better resources, not new standards… Nothing to see here folks, just ignore the clear conflict of interest.