Conceptual vs. Procedural Math at Mastery charters


“Maths” by Chris de Kok is licensed under CC BY 2.0

There was an interesting recent the74 piece on a Philadelphia charter organization, Mastery, which takes low performing schools and works to “turn them around.”


Embedded within this article is the implication that a shift to a focus on the teaching of conceptual math, rather than “rote” procedural teaching, led to a swift downturn in math scores.

“So this year, the network began reintroducing teaching techniques that had been a staple at Mastery schools for years, while seeking a middle ground between no excuses and restorative practices. It’s a ‘journey of trying to find out what’s the right mix,’ Gordon said.

Specifically, the network is reintroducing procedural math instruction, which focuses on rote instruction like memorization and repetition.”

It seems worth digging into this supposition a bit more.

Is Mastery’s downturn in math scores due to the failure of conceptual math in general as a pedagogical approach? Or is it a failure of the network to attract and train teachers who can teach this type of math more effectively?

Or is it a failure in the assessments that were used as a reference? Or was it that conceptual math takes longer to “stick” and pay dividends? Or was it a failure of the curriculum they used to move in a more conceptual direction? . . .

Special Education: Inclusion or Specialized Intervention

“Street Crowd” by Théophile-Alexandre Steinlen (French (born Switzerland), Lausanne 1859–1923 Paris) via The Metropolitan Museum of Art is licensed under CC0 1.0

A fair amount of academic literature calls for clinical, evidence-based models of intensive intervention for students with disabilities in K-12. Yet in the field, there is limited effective implementations of such interventions.

Models such as Response to Intervention (RTI), multi-tiered support systems (MTSS), and standard, evidence-based protocols and programs all make complete sense when you learn about them. But there’s also a problem with these interventions: they are based on clinical frames of implementation, as in a trained clinician in the given model or protocol delivers the intervention in a prescribed manner.

The daily reality of a K-12 school, however, is far from clinical. Opportunities to deliver prescribed interventions, whether in a small group or in the ideal of a 1:1 setting, are few and far between. Moreover, opportunities to be trained in such interventions are few and far between. One is certainly not trained in any given intervention in any traditional education program.

The very model of a self-contained classroom, a class in which students with more severe disabilities are separated from their peers, relies upon this clinical ideal. And again, in isolation, as an ideal, it makes perfect sense. Let’s separate out the kids with greatest of needs so we can provide them with individualized, supportive instruction.

Similarly, within an inclusive classroom, district leaders continually speak about and prescribe the need to move away from a one-teach, one assist model to a parallel, station, or team teaching model. Or they speak of the need to “differentiate” and “individualize” instruction.

Idealized models that make perfect sense and sound great, but that rarely play out that way on the ground.

A Division Between Inclusion and Specialized Intervention

There is some scholarly debate about this. Fuchs et al, in a 2010 paper, “The ‘Blurring’ of Special Education in a New Continuum of General Education Placements and Services,” provides a useful delineation into two camps they term IDEA and NCLB. The IDEA group advocates for a top-down (i.e., replicable), linear, and time-sensitive process with fewer tiers of instruction, which serves both prevention and a more valid method of disability identification. They believe in evidence-based programs at Tier 1, the strength of standard protocols in Tier 2 and Experimental Teaching for Tier 3 intervention. They believe in the importance of a distinct special education program.

On the other hand, the NCLB group focuses on a problem-solving approach based on standards. “Whereas special education remained a distinct entity in reform making in the 1980s and 1990s, many in the NCLB camp today are advocating for obscuring, smearing, dimming, and confusing special education by blurring it into general education. In their plans—however implicit—special education vanishes in all but name (and maybe in name as well).”

Research suggests that the standard-protocol approach is superior to problem solving in accelerating the progress of children with serious learning problems. However, the authors acknowledge that “because there are insufficient numbers of such protocols in many academic areas and in the higher grades, and because ‘the school bus arrives every morning,’ many practitioners may have little choice but to rely on some variant of problem solving.”

In a more recent paper in 2015, “Inclusion Versus Specialized Intervention for Very-Low-Performing Students: What Does Access Mean in an Era of Academic Challenge?“, Fuchs and other authors again examine the split between those who push for a problem-solving, inclusionary approach vs. that of “specialized intervention” approach and advocate for a focus on explicit, specialized intervention delivered in a separate setting.

Here’s a couple of provocative quotes from this paper that struck me:

“. . . access cannot be assumed even when inclusive instruction reflects state of-the-art accommodations and support. Instead, only evidence of adequate student outcomes demonstrates that access to the curriculum has been accomplished. In fact, the present analysis indicates that such access is sometimes more satisfactorily achieved under a service delivery arrangement that occurs outside the physical space of the inclusive program and using instructional methods that differ from the inclusive program. All this argues for a definition of access to the general educational curriculum that is based on empirical evidence of adequate learning— regardless of the setting in which or the instructional methods by which that learning is achieved.”

“…it is not possible to ignore students’ foundational skill deficits if progress toward CCSS is to be realized. For example, to demonstrate meaningful improvement with informational text, specialized intervention must address very low performers’ decoding, word recognition, and vocabulary deficits, and this often requires out-of-level foundational skills instruction. Therefore, although reconceptualizing access as empirical demonstration of learning, schools must also recognize that the access mandate often requires schools to provide out-of-level instruction to meet students’ needs for accessing the grade-level curriculum.”

Yet I don’t agree with the authors that putting in place explicit instructional intervention programs will solve all the problems they’ve identified with inclusionary practices. You can place my own professional stance as firmly within the “NCLB” camp outlined above. Schools are not clinics, and unfortunately, special education teachers and other personnel in school buildings are rarely, if ever, trained in the delivery of specific interventions.

In fact, I think the issue of either strong inclusionary instruction or specialized intervention comes down to the same fundamental issue: there is a general lack of instructional capacity and expertise in most schools, in addition to a general lack of curricular coherence and vision.

Either way, we certainly need to rethink how we are putting in place supports for students who struggle the most and assessing whether those supports are actually effective.

My argument, however, is to place our primary and immediate focus on establishing coherent and rigorous curriculum and expectations for all students. I thus argue for inclusion and a problem-solving approach.

A recent article in Education Next,”Reforming Remediation” neatly encapsulates the rationale for this inclusionary argument. Students placed directly in college-level statistics did far better than their counterparts in remedial classes.

While that example is focused on a higher education setting, we can find parallels in K-12 by looking at access to Advanced Placement (AP) or International Baccalaureate (IB) programs, or to difficult academic subjects such as Latin. Disadvantaged students rarely have the opportunity to experience such rigorous curriculum. Yet when they do, as Bronx Latin teacher Peter Dodington put it, “The combination of a difficult topic and a well-ordered, step-by-step curriculum allows even otherwise weak students to succeed, and gives them a new understanding of their own strengths and talents.”

If we raise our expectations and the rigor and coherency of our curriculum, then we will see more educational benefit for all students. The dire reality of poor teacher training and knowledge of the content they teach is a significant problem, but a stronger school-wide curricular program can help to assuage this.

I strongly believe in the need for specialized interventions for students who require the most support. But how can we put in place effective interventions when a strong and well-implemented core curriculum is not present?

Let’s address the foundations first before moving to the clouds.

Experience is Relational and Subjective

Because neither ‘memory banks’ nor ‘representations’ of stimuli exist in the brain, and because all that is required for us to function in the world is for the brain to change in an orderly way as a result of our experiences, there is no reason to believe that any two of us are changed the same way by the same experience. If you and I attend the same concert, the changes that occur in my brain when I listen to Beethoven’s 5th will almost certainly be completely different from the changes that occur in your brain. Those changes, whatever they are, are built on the unique neural structure that already exists, each structure having developed over a lifetime of unique experiences. (Bold added)

–Robert Epstein, “The Empty Brain” on Aeon

Epstein’s challenge to the “information processing” model is highly relevant to education and worth considering the implications of. I would pair this reading with the quantum theory of QBism: QBism challenges the notion of an “objective reality,” suggesting instead that reality lies in the eye of the beholder.

Smörgåsbord: The Chaos Begins. The Work of Education Continues

What is there to say about the rude awakening that shook cosmopolitan, progressive minded Americans and the world?

Well, here’s a few positive spins on it:

  1. Trump is a chaos monkey that will assist us in building a better democracy by forcing us to re-establish the original balance of power our founders intended. (This would require the Republican party to pull up their Big Boy pants and actually govern.)
  2. The one thing that united Clinton and Trump campaigns was a commitment to investing in infrastructure — and it is the one thing Democrats are already reaching across the aisle to work on.

What does it mean for education politics and policy?

  1. Rick Hess: Who the heck knows?
  2. Chad Aldeman: NCLB will suddenly look really good to Democrats, on hindsight. And you can kiss any education related investment goodbye.
  3. Elizabeth Green: Education reformers will pivot their attention to long-neglected rural and rust-belt communities.
  4. Neerav Kingsland: Charter proponents need to recognize the populist appeal of local, traditional public schools and thus address fears that public schools will be harmed by charter expansion.
  5. Matt Barnum: If Trump actually wants to follow through on his anti-Common Core rhetoric, he’d paradoxically have to wield federal power.

What relation does this election have to knowledge or the lack thereof?

  1. Rick Kahlenberg: Civics and democratic values need to be explicitly taught. (But Andrew Rotherham and Doug Lemov are angry about the anti-choice aspect of his piece)
  2. Problems with our democracy are due to lack of knowledge. For that, we can blame schools.
  3. George Thomas: In our shift to populism, we’ve lost the educative purpose of a representative democracy as envisioned by Madison.
  4. “Trump was not elected on a platform of decency, fairness, moderation, compromise, and the rule of law; he was elected, in the main, on a platform of resentment.”
  5. An additional bonus of a knowledge-rich curriculum is that it can help kids do better on tests.
  6. Some are blaming Facebook and social media for the segregation of our attention from those who could challenge our “crony beliefs”.
  7. Three reasons to teach a knowledge-rich curriculum: cognitive, socio-cultural, and economic.

A Balanced Complexity

An interesting relationship to consider:

  1. A balanced complexity of ecosystem sounds = environmental health
  2. A balanced complexity of brain activity = mental health














If you’re interested in the concept of self-organized criticality or networks, more here:

Success can’t be measured by one or two numbers

“Whenever you make huge decisions about complex situations based on one or two numbers, you’re headed for disaster — especially when those numbers can be gamed.”

—Mark Palko and Andrew Gelman, “How schools that obsess about standardized tests ruin them as measures of success” on Vox

We’ve questioned Success Academy’s “success” on this blog before. These statisticians bring a new lens to that question.

I don’t want to denigrate the good work that Success Academy teachers and students are doing. There are practices and systems well worth replicating and investigating in these schools. But Eva Moskowitz’s political framing and marketing of her schools as the solution to poverty is problematic.

Even stable relationships are steeped in conflict

By Charlesjsharp (Own work) [CC BY-SA 4.0 (, via Wikimedia Commons

“We like our black-and-white narratives, with clear heroes and villains. The very term symbiosis has been twisted so that its original, neutral meaning – ‘living together’ – has been infused with positive spin, and almost flaky connotations of cooperation and blissful harmony. But evolution doesn’t work that way. It doesn’t necessarily favour cooperation, even if that’s in everyone’s interests. And it saddles even the most harmonious relationships with conflict.”

—Ed Yong, “Microbes have no morals” on Aeon

Now think about education politics. Or school relationships between staff and admin, or adults and students.

There’s no right way out of a riptide


Everything we’ve done points to the fact that there’s not one single message that works. Sometimes swim parallel is great, sometimes it doesn’t work. Same for floating.”

It’s a view that the Surf Life Saving Australia has taken, too. After working with Brander, they’ve updated their messaging. Rips are a complex, dynamic hazard and the multitude of variables—swimming ability, current strength, circulation, wave size—make the threat nearly impossible to solve with one-size-fits-all advice. No single “escape strategy” is appropriate all the time, the group now says, and lifeguards in Australia currently recommend combining the advice from both MacMahan’s circulation concept and traditionalists like Brewster. If you’re not a strong swimmer, stay afloat and signal for help; if you can swim, consider paddling parallel to the beach toward breaking waves—though be mindful of the potential circulating current. “All responses,” the group concedes, “have their pitfalls.” [Bold added]

—David Ferry, “Everything You Know About Surviving Rip Currents Is Wrong” on Outside

While this quote may refer to rip tides, I think the concept of dynamic complexity and variability could just as easily apply to schools and school systems. As we’ve explored here before, in the face of complexity, sometimes you’ve just got to try multiple strategies until something sticks. And sometimes,  you’ve just got to pick something and stick to it. It is not always so easy to predict what will lead to a breakthrough.

Incremental Change

“Silvia Abbato, the district’s superintendent, said she could not pinpoint any one action that had led to the better scores. . . . 

“It’s not something you can do overnight,” Ms. Abbato said. “We have been taking incremental steps everywhere.”

—Motoko Rich, Amanda Cox, and Matthew Bloch, “Money, Race, and Success: How Your District Compares” in The NY Times

Principles for Robustness

“Snow Crystal” — Wilson Alwyn Bentley

In an article in Harvard Business Review, “The Biology of Corporate Survival*” the authors lay out some principles for managing complex adaptive systems. While framed for a business audience, these principles could be applicable to the complex adaptive system (or, perhaps more aptly, the “complexicated” system) of a school.

The authors delineate a set of principles for robustness into structural features, and managerial levers:

Structural Features

  • Heterogeneity (Diversity)
  • Modularity
  • Redundancy

Managerial Levers

  • Expect surprise, but reduce uncertainty
  • Create feedback loops and adaptive mechanisms
  • Foster trust and reciprocity

How might these principles apply in a school?

I’ll leave that to you to contemplate, but for the record, I’ll note that most public school managers typically do quite poorly in reducing uncertainty and in fostering trust.


* “The Biology of Corporate Survival – Harvard Business Review.” 2015. 15 Apr. 2016 <>