Scaffolding vs Differentiation

My recent post on scaffolds and success criteria seemed to be useful to folks, so I thought I should share something more on scaffolding that can help to further clarify the term.

Scaffolding and differentiation are both words frequently thrown around in schools, often interchangeably and without precision. But there’s a clear distinction between the two that must be made, most especially as teachers are increasingly pressured to “differentiate” their lessons by school and district leadership with little guidance and concrete models.

So what is differentiation?

Differentiation asks teachers to plan different tasks or learning experiences for different groups of students, with the idea being that we can better meet the needs of diverse learners.

So for example, if we had our “high” student, our “medium” student, and our “low” performing student, we might provide a different text for each one in order to ensure they would be reading something at their “level.”

But there’s a problem with this. If the so-called “low” students only ever receive less complex and challenging expectations, texts, and learning experiences, they will continue to perform at a lower level.

This is why too many of our kids end up in “remedial” classes if and when they make it to college. This is why too many students consigned to a segregated “self-contained” program rarely even make it to graduation.

There’s another problem with differentiation. It demands quite a bit from a teacher to design (at least) 3 separate tasks or resources for any given lesson — time and energy that could perhaps be more effectively spent elsewhere.

To acknowledge these problems is not to say that differentiation can’t be applied effectively nor that it is universally the wrong thing to do. Structures for guided reading, for example, draw from this model and can be very effective in a school and classroom that have developed the necessary systems and routines. But these problems are big enough that they should give us strong pause before mindlessly evaluating and chiding teachers about whether they are adequately “differentiating” their lessons.

What is scaffolding, and how is it different?

The concept of scaffolding shifts how we approach meeting the needs of diverse learners.

We may have students coming into a learning experience with differing levels of knowledge, ability, or background, but rather than providing them with something different, we instead consider how we can provide the scaffolding necessary to ensure they can work together in grappling with a common task or text.

This is a shift offered by common standards, which demand a shared set of expectations for all students. For our students who may struggle in meeting these more rigorous demands, we must consider how to scaffold the concepts, procedures, and environment to support their engagement in practice with the texts and learning experiences that can enable them to meet those expectations.

This is certainly not an easy thing to do, either. But if I had to take a pick about what is going to give me the most bang for my buck, designing access and practice with common vocabulary, tasks, and texts is where I would ask educators to more wisely invest their time and energy.

For more on what that scaffolding should look like, take a look at my prior post, Scaffolding and Success Criteria.

Smorgasbord: Summer in the City

By Juhanson [GFDL ( or CC-BY-SA-3.0)

The school year in NYC just ended on the 28th; summer school begins this week. I’ve been facilitating training for summer school programs the last few days so have been pretty busy, but I’ve still got a few ed-related links worth reading compiled for you.

I’m waaay late to this, but this NTCQ report on what teachers need to learn in ed programs is excellent

I’ve had this PDF sitting on my desktop since January and just finally got around to perusing it.

The 6 strategies, based on extensive research, are deceptively simple. And yet, barely any of them, aside from asking probing questions, are covered in most teacher prep textbooks or courses.

Here’s the 6 strategies:

  • 1. Pairing graphics with words.
  • 2. Linking abstract concepts with concrete representations.
  • 3. Posing probing questions.
  • 4. Repeatedly alternating problems with their solutions provided and problems that students must solve.
  • 5. Distributing practice.
  • 6. Assessing to boost retention.

These would be a good focus for any sustained PD for a school. And I would argue that numbers 5 and 6 would be the biggest bang for your buck if you looked across the curriculum of a school.

What do most teacher prep textbooks focus on? Stuff like, “How teachers and students should organize themselves (e.g., inquiry learning, direct instruction, or cooperative learning).”

You know what? That’s what most professional development focuses on, too.

National Council on Teaching Quality, Learning About Learning

Ideas for how to change entrenched misconceptions in education (and beyond)

“We think advocates of learning science should be more curious about why teachers believe what they believe, including learning styles.”

Why mythbusting fails: A guide to influencing education with science, Deans for Impact

Joe Kirby presents important ideas on how to distribute practice and assess to boost retention

If your school is committed to addressing those 6 points outlined above, Joe Kirby has some good advice that not only will promote better student learning, but furthermore reduce your workload.

Three Assessment Butterflies, Pragmatic Education

Amanda Ripley forwards a wonderful idea for promoting tolerance and understanding diversity: cultural exchanges within the US

Over the last year, I spent a lot of time thinking about the problems of segregation across New York state, and something I found under-explored and often trivialized by integration advocates was how to address rural and urban divides, as well as other students who are geographically isolated in urban areas.

Plus, it must be said that the reality is that there isn’t enough affluent or white kids in our public schools to spread around, if we’re going to start counting beans. So we need to look at more than only getting kids of different backgrounds in the same schools, though that can go a long way. We also need to look at how we can bring kids together in other ways.

Kentucky is bringing people of all ages together through a Rural-Urban Exchange. This is something other states should emulate, most especially for our children.

“It’s harder to demonize someone once you’ve stayed in their homes and shared meals and stories together.”

America, Meet America: Getting Past Our Toxic Partisanship, Wall Street Journal

How did this DC public school get to 100% college acceptance?

System-wide support and money.

“a strong support system within D.C. public schools made it a reality. Staff tracked students, often working side by side with them to apply for college in the library. It also took a lot of money. Grants, donations and district funds took kids on college tours, and the school incentivized students with pep rallies, T-shirts and free food.”

Every Senior Applied To College At This Washington, D.C., High School, NPR

And a reminder that “transforming” schools is incredibly hard

Well-reported, and as Eliza Shapiro (Politico reporter who’s an ace journalist herself) stated on Twitter, “nothing cute, no triumphant narrative, just proof of how hard it is to change schools.”

The Fight for Fairmount Park Elementary, Tampa Bay Times

Revisiting the ‘Parents Involved” SCOTUS case and its unrealized potential for racial integration

“for all the obstacles confronting a rebooted school desegregation movement, the legal path towards integration still lies mostly open”

‘Parents Involved,’ A Decade Later, American Prospect

Rural youth are leaving (and pushed to leave) for greater opportunities

“Researchers have found that the hollowing-out of heartland communities is the result of a push-pull phenomenon: Ambitious students are drawn to the attractions and opportunities of major cities, but they’re also encouraged to leave by teachers and parents who see college as a chance to escape stagnation.”

When College Grads Don’t Come Back Home: New Numbers Show a Widening Urban-Rural Education Divide, the 74

If rural communities want to keep their youth, they must engage them

“Their creative thinking and problem solving can advance communities when given the opportunity.”


And here’s an example of that

“I was taking a class called Latina Leadership Initiative — that class is all about empowerment. When I saw that I had access to these programs to bring them to Perry, I kind of let go of [my] shyness and said – this is something I need to do.”


Bronx Borough President Ruben Diaz with a great finale to a commencement speech

“Oh, by the way. That young lady I was chasing? Twenty-eight years later, she’s still my wife.”

Bronx borough president to high school grads: ‘Start breaking the mold of what the face of techies look like’, Chalkbeat NY

We may be jumping the gun on ‘microaggressions’

We’re already incorporating “microaggressions” into trainings here in NY (the mandated 6-hour DASA trainings), but we may be jumping ahead of any solid evidence that backs up the concept.

A psychologist argues for a moratorium on use and training around the term until more evidence is gathered.

Microaggressions?, Aeon

Climate change will increase inequality in the US

“The ‘hidden costs’ of carbon dioxide emissions are no longer hidden, since now we can see them clearly in the data,” said Jina, a postdoctoral scholar in the department of economics at the University of Chicago. “The emissions coming out of our cars and power plants are reshaping the American economy. Here in the Midwest, we may see agricultural losses similar to the Dustbowl of the 1930s.”

Climate change damages US economy, increases inequality, ScienceDaily

Scaffolding & Success Criteria

What is a scaffold, anyway?

While working on a series of workshops about scaffolding, I came to a revelation about what the term really means. It’s one of those words, like “differentiation,” so ubiquitous in the field it seems to mean almost anything. But such generality can easily lead to some big misconceptions in actual application. I’m going to share my learning here in the case it may be useful and help others to avoid some of those missteps.

In order to truly understand scaffolding, I think you need to be able to answer this question:

Why should a scaffold used during a lesson align to the success criteria of a unit of study?

Furthermore, I think you need to understand what it takes to master a parachute landing fall at the army jump school in Fort Benning, Georgia.

Mastering a Parachute Landing Fall

For a reading that serves as a great basis for a team discussion on rigor, assessment, or scaffolding, I urge you to read Chapter 4 pg. 69 – 71 of Make It Stick by Peter C. Brown, Henry L. Roediger III, and Mark A. McDaniel. Read the whole darn book while you’re at it, of course; it’s all good. But I found this particular chapter especially relevant to scaffolding.

Since I obviously can’t reproduce the chapter for copyright reasons, I’ll give you an executive summary. At the army jump school in Georgia, mastering a parachute landing fall is a necessity. The jump school supports and drives recruits to mastery of this extremely complex and difficult skill within three weeks. How do they do this?

First, recruits initiate their practice in a gravel pit. They receive demonstrations of different falls. Then they practice falling and they receive feedback and they practice some more.

Next, recruits move to a short platform a couple feet off the ground. They practice jumping off the platform and executing the falls they had mastered in the gravel pit.

The challenge is ratched up. Recruits move to a zip line and practice falling from a higher drop that’s propelling them in different moving directions. They can control when they drop off the line.

You see where this is going. The recruits move to a platform 12 feet off the ground. They put on gear and jump down a mock chute, connected to a zip line. But this time the instructor pulls the cord and introduces the element of uncertainty and surprise. Recruits now have to be able to demonstrate a PLF according to chance, simulating the variables of an actual fall.

Finally, they move to a 34 foot tower. This is the final step of demonstrating mastery before boarding a moving airplane and engaging in a real-world application of the skill.

There’s some aspects of this narrative that illuminates effective scaffolding:

    • Each step provides practice and feedback on a component skill that requires mastery before moving on
    • Practice grows increasingly complex and difficult
    • At no point is the practice easy
    • Practice serves simultaneously as performance-based assessment


There’s a term that the authors of Make It Stick introduce that is useful for this progression of increasingly rigorous steps: desirable difficulty.

Desirable Difficulty, Academic Rigor, and Scaffolding

Desirable difficulty, a term coined by psychologists Elizabeth and Robert Bjork, is a way to describe short-term impediments that lead to stronger learning. At the army jump school, you can see desirable difficulty in action, and it highlights key aspects of effective scaffolding and how it connects to academic rigor.

At each successive step of training, the recruits are engaged in practice at a level of desirable difficulty required for them to master each successive component skill. Once they’ve mastered that skill or set of skills, they are then ready to move to the next level.

This is important to highlight for a few reasons. One is that it’s clear that at no level is the work and practice “easy.” A common misapplication of scaffolding is that it makes work easier for a student. This might make the teacher and student feel better about themselves, but it does long-term damage to student learning.

But a well-designed scaffold should not make a task or concept easy. Rather, it should provide the right level of impediment and challenge for the level of practice in the skills or concepts required to move forward.

This means that instructors can have extremely high expectations for students, as the army jump school has for its recruits, while providing well-structured practice and guidance that will lead to achievement that matches those expectations.

It also means instructors must be crystal clear about the component skills and practice that will build successively and sequentially to mastery.

Scaffolding as Performance-based Formative Assessment

“It’s one thing to feel confident of your knowledge; it’s something else to demonstrate mastery. Testing is not only a powerful learning strategy, it is a potent reality check on the accuracy of your own judgement of what you know how to do. When confidence is based on repeated performance, demonstrated through testing that simulates real-world conditions, you can lean into it.”

Make It Stick

There’s another aspect of scaffolding that is really interesting to consider from the jump school example: effective scaffolding is a performance-based form of formative assessment. Formative assessment, for those of you not up on the ed jargon, simply means testing that occurs during learning. This is in opposition to summative assessment which takes place at the end as an evaluative measure, and is usually accompanied with a grade. In the jump school example, the summative assessment would be executing the parachute landing fall from a plane.

A well-designed scaffold, therefore, engages a student in the practice of a skill that informs the instructor whether the student is ready to move on. This should be immediately visible and clear, enabling the instructor to provide ongoing feedback as the student engages with the scaffold.

Now let’s return to our original question and bring this back to the classroom:

Why should a scaffold used during a lesson align to the success criteria of a unit of study?

Success Criteria

Let’s break this down. What the heck are “success criteria”?

Success criteria are what you use to assess whether you’ve achieved the goals for learning in a unit of study. You’ll typically see these as a rubric or a checklist. The criteria are directly aligned to standards or expectations for learning for the subject and grade.

As an example, let’s say I’m an ELA teacher and I wanted to assess a third grade student’s ability to determine a central idea of a text. The Common Core Standard for Reading Literature states that by the end of third grade, kids should be able to “Determine the main idea of a text and explain how it is supported by key details.

So an example of a few success criteria I might use to assess that student’s progress towards the standard could be:

☐ I can distinguish between the important and unimportant details in a text.
☐ I can determine the details most essential to understanding a text.
☐ I can combine key details to determine a main idea.

Let’s look one of these criteria: “I can distinguish between important and unimportant details in a text.”

Many students will struggle with this abstract skill (not least because they don’t know enough about whatever they’re reading in order to do so . . . but that’s a whole other post). So they will need some type of scaffold to assist them in getting started on this.

So what could such a scaffold be?

One possible scaffold could be to support them in first distinguishing between details that are merely interesting, such as details that an author gives to make the text more engaging to read, from details that are central to understanding the topic of the text.

We might create some type of graphic organizer or chart to support students in practicing this with a text, and of course, we’d probably model it and do it together as a whole class before having students practice it in groups or pairs, then we’d ask individual students use it on their own.

Some students may be ready to just make a T-chart in their notebooks, while other students may need some more guided practice with a handout. Some students may need manipulatives, such as cut outs of both interesting and important details, in order to get started and to feel success before they are ready to move to greater abstraction.

But notice something about my description: the scaffold is less about a graphic organizer, chart, or manipulative, and more about the practice of a specific skill component: clarifying the difference between interesting and important details.

In other words, the point of a scaffold like this isn’t really about the thing — it’s about the thinking that students are training their minds to do through the application of the scaffold.

A scaffold should therefore provide the thinking practice that a student needs to master the criteria for success.

If we just told students to distinguish between the important and unimportant details in an informational text, some might be able to do so, and some will not. The point of the scaffold, in this example, is to train students who don’t yet see it to become aware of the difference between details provided by an author that are merely interesting, versus details that are important to understanding the topic.

Eventually, those students should no longer need a scaffold. They’ll internalize the concept and be able to apply it without thinking. A few students may never need such a scaffold at all. That’s the differentiation piece. If they don’t need it, they shouldn’t be practicing it.

We may think of something like a stepladder or the scaffolding on a building when we use the word “scaffolding.” Or you might think of a bike with training wheels, or a parent holding the bike as the child learns to pedal.

The model of a bike with training wheels is probably closer to the way we should think of what a scaffold means in instruction. We want to shift our mental model of what a “scaffold” is away from it being a tool that merely makes a task easier, to a process or activity that engages a student in the practice that they can experience success with, while on the road to mastery.

What’s the difference? Some students will need to practice a whole bunch using a scaffold before they get it. A few students may not need it at all. But the expectation is that all students will be expected to achieve that mastery.

Which leads us to another realization about what function a scaffold serves. If a scaffold is directly aligned to the success criteria in a unit of study, then it serves not only as a form of practice to achieve mastery, but it furthermore serves as method of formative assessment for both students and teachers. It provides performance-based, task-based feedback on whether or not a student has achieved the success criteria.
So a scaffold does not mean making learning easy. It doesn’t mean giving kids a shortcut so they can reach something they will never be able to reach again. It’s about having rigorous expectations and demanding that students practice in a way that will enable them to achieve those expectations.

A New Definition of Scaffold

A scaffold provides opportunities for performance and practice of the component content and skills that a student requires to achieve success in a unit of study.

Characteristics of Scaffolds

  • Scaffolds are smaller components of a complex task or skill
  • Scaffolds are at the right level of “desirable difficulty” for practice.
  • Scaffolds are not “easy”
  • Scaffolds must be mastered at each step along the way. Students shouldn’t move along until they have mastered each component
  • Scaffolds serve as performance-based formative assessment


Smorgasbord: 3rd week of May 2017: Segregation, CTE, Curriculum, and the One Straw Revolution


On the 63rd anniversary of Brown v. Board of Ed, worth reviewing last year’s GAO study findings

“[Segregated] schools, investigators found, offered disproportionately fewer math, science and college-prep courses and had higher rates of students who were held back in ninth grade, suspended or expelled.

What’s more, GAO investigators found, public charter schools, a key strategy in improving education for such students, may take minority and poor students from larger more diverse public schools and enroll them into less diverse schools.

Overall, investigators found, Hispanic students tended to be “triple segregated” by race, economics and language.”

GAO study: Segregation worsening in U.S. schools, USA Today

A panel on desegregation offers insight

Jill Bloomberg: “So there were lots of questions about safety, which are really very coded questions about race and racism. We assured them that their kids would be fine.”

David Goldstein: “We would create these little Shangri-Las of these beautiful little high-performing schools that were diverse and all that. Meanwhile, all the rest of the schools got squat. And that wasn’t our plan, so we went districtwide.”

And a comparison of integration to broccoli.

Upper West Side parents gather to tackle middle-school integration, Chalkbeat NY

As in Staten Island, so in the US

When it comes to Staten Island’s North Shore, as in many other areas of our society, “We make judgements about a whole community without ever walking in the door.”

Equity for North Shore schools still a work in progress, SILive

High school admissions changes in the works for NYC may promote diversity

This is good to hear. But we’re going to need to look at zoning and the elementary school level if we’re really going to fight segregation.

Chancellor: ‘We’re reconsidering how some enrollment is done’ in high schools, Chalkbeat NY

Great data visualizations and background on segregation in Indiana

Examining the Cross-Roads

Vacations (or the lack thereof) highlight class divisions

“school vacations can highlight disparities and fracture the sense that students are equal in the ways that matter most”

This piece also points to an often under discussed aspect of school integration: it takes a lot of work to ensure kids (and staff) are interacting with one another’s differing experiences and perspectives in a constructive manner.

I know as an advocate of integration myself, I don’t usually even bring this up because the very first step: just getting kids physically (or even virtually) into the same classrooms and schools is hard enough in and of itself. But it’s an essential piece. Just getting kids together is only half the battle. Curriculum, conversational protocols, academic interventions, and social-emotional support then needs to be firmly in place.

Kids’ Vacations Highlight School Segregation, motto

Student voices on segregated schools

“My reality is gym lockers with brown rust.

My reality is the suffocating phenomenon of poverty present on a daily basis.”

‘I am a product of the South Bronx’: One student on how the city’s high school choice process failed her

“Education was my only hope for redefining my life. But it seemed like the bar was always set out of reach for people like me, and most of our time was spent elevating ourselves to reach the bar instead of figuring out how to surpass it.”

‘I didn’t realize that an A in Harlem was not the same as an A in a majority-white high school’: One student’s discovery

Jeb checks the NY Times

“Florida’s McKay Scholarship Program simply gives parents options if their children are stuck in the wrong learning or social environment for their unique needs. It is not a condemnation of public schools or a seal of approval for private schools. In fact, the McKay program includes public school choice as well.”

What the Media Is Getting Wrong About Florida’s Push to Help Students With Disabilities, th74

Recent research on wrap-around services brings to light our goals for public services

MDRC has recent research that brings into question the impact of the “community school” model — if we assume that raising test scores is the goal of providing such additional services.

But as a community school advocate notes:

“The services themselves are, of course worthwhile — don’t we all agree that having kids who have access to mental and physical health care, regular nutritious meals, and quality, safe afterschool and summer programs is inherently a good thing?”

Let them eat cake?

Community schools are expanding — but are they working? New study shows mixed results, the 74

Strange things are afoot at Deborah Meier’s school

What exactly is the problem going on here? Too progressive? Not progressive enough? More to explore here, for sure.

East Harlem Elementary Principal Is Out After a Yearlong Fight, NY Times

The need for a progressive agenda for the working class

“Democrats need a comeback strategy, and the American working class needs an ally. The solution to both problems can be the same: a muscular agenda to lift up people without four-year college degrees.”

How Democrats Can Get Their Mojo Back, NY Times

Was the high-profile LA school board president race determined by the negativity bias?

So there’s this cognitive bias called the “negativity bias.”

Aside from the vast funds that were plowed into this race, I wonder whether that played a role? Here’s a description from the74 that suggests it did:

“Zimmer campaigned on a platform that the district is improving, pointing to rising graduation rates. Melvoin campaigned on the premise that the district was failing and the board needed to act with more urgency to improve student achievement and address its financial situation.”

Education Reformers Sweep Los Angeles’s School Board Elections, Setting Up Pro-Charter Majority, the74

North Carolina Representative Virginia Foxx takes a stand for CTE

“In order for these people to thrive, we must do everything we can to change the way people think about CTE, shifting the narrative from a Plan B option to a Plan A option”

Career and Technical Ed Should be ‘Plan A,’ Foxx Says as House Takes Up Perkins Bill Today, the74

And the House takes a bipartisan stand for CTE

A nice moment of positive legislation in the midst of the chaotic destruction the GOP has been nurturing in DC.

Career and Technical Education Overhaul Bill Approved by House Ed. Committee, ED Week

Sure would be nice to see this piece of legislation on school infrastructure get bipartisan support . . .

“The legislation has six other Democratic lead co-sponsors in the House, but no Republican lead co-sponsors.”

Oh, and “The IES survey also found it was an average of 44 years since the construction of the main instructional building at schools.”

School Infrastructure Spending Plan Introduced by House Democrats, ED Week

Professional development should be based on the curriculum

“We argue the need to take the important but often overlooked step of organizing teachers’ professional learning around the curriculum materials they are using with their students.”

Makes sense to me. I go into schools to support ELA teachers, and the only way my work is able to have any traction is by supporting implementation of a curriculum.

But there’s more to it than this. Which curriculum? Why? A school needs to coalesce around its vision for what skills and knowledge it wants students to graduate equipped with — and then align their curriculum to that vision.

Instead, I see schools teaching something just because they think they are supposed to. (“Why are you teaching these texts?” “Because I’m told to.”) And getting weird directives from their bosses, such as that EL (EngageNY) or CodeX are a “reading” curriculum, then adding Teacher’s College units as the “writing” curriculum. These kinds of misunderstandings become embedded into the scheduling: a teacher is teaching EL lessons for 3 days a week, and TC lessons for 2 days a week.

If you are an ELA teacher, then you know how incredibly difficult it is just to implement one ELA curriculum with fidelity, let alone two completely different and unaligned ones.

In other words, the problem isn’t just that curriculum is detached from PD — it’s that curriculum is detached from school and district leadership and the structures and schedules they enforce.

In Washington, D.C., a Road Map for Reinventing Professional Development in Schools, the 74

And there’s mounting evidence that a coherent curriculum is an effective method for improving outcomes. Like some of us have been saying all along . . .

“There are no silver bullets in education. But a growing body of both empirical and real-world evidence makes a compelling case that curriculum is a key component of student success.”

A Compelling Case for Curriculum, US News

BASIS schools exemplify what a coherent and rigorous curriculum can do

BASIS is also doing some really interesting practices worth emulating:

Student notebooks as sources of communication and data between teachers and parents. . .

“Many schools create an online grade portal that allows parents to see how their children are performing. BASIS doesn’t. Any information about grades comes to parents because their kids have shown them the contents of their planner, which contains test scores, homework assignments, and notes to see the teacher after school for help.”

Building empathy and understanding of diverse perspectives through it’s Global Classroom Project:

“…which connects kindergartners in different BASIS schools virtually to help them learn about one another. In one project, the children exchanged pictures of their local grocery stores so they could compare them. They also sent the Shenzhen school a video of second-graders sharing a Lunar New Year greeting in Mandarin.”

BASIS: Inside the Acclaimed School Network That’s Blended Together the World’s Best Education Practices, the74

More sunlight = higher test scores

One of the central tenets of this blog are that some of the most basic contextual factors are overlooked in schooling, and here’s one that’s so basic but clearly powerful: starting school later results in better test performance.

More sunlight, more fresh air, more greenery. The best method for improving test scores? Very well may be.

Sunshine Improves Test Scores, The Atlantic

As in ecosystems, so in schools?

Look to the soil for our future.

A geomorphologist and author’s book, “Growing a Revolution: Bringing Our Soil Back to Life,” outlines the drastic difference that healthy soil management practices can make, and the common practices that good soil management entails.

Not surprising at all that conventional methods (monoculture, frequent tilling) ain’t good for soil.

If you are interested in this kind of stuff, there’s a book written long before this one with the word “revolution” also in the title, in which the author lays out the philosophy and practice of no-till farming: “The One Straw Revolution” by Masanobu Fukuoka.


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? . . .

On Threshold Concepts and Experiences


In a recent post, “On Knowledge and Curriculum,” we reviewed a few disruptive ideas from cognitive psychologist Daniel Willingham, with the most incendiary implication being:

a school needs to come to a consensus on the topics, texts, vocabulary, and concepts that are most essential to know within and across each academic domain and carefully sequence and reinforce those concepts across classrooms and grades.

So how can a school embark upon this quest? In this post, I will attempt to provide some guiding ideas and protocols for this work.

How Do We Reinforce Knowledge?

First off, a few guiding documents to equip you with the cognitive principles of affirmative testing, which are essential to reinforcing knowledge over time:

—Annie Murphy Paul, “Affirmative Testing” (she has designed an entire e-course around these concepts!)

—Deans for Impact, “The Science of Learning


How Do We Determine the Knowledge that is ESSENTIAL?

Folks are going to disagree about this, including the “experts,” so ultimately, this determination should be made collaboratively within a school (and beyond). The key is that the school comes to a consensus on this essential knowledge, then teachers carefully sequence it across the curriculum and quiz it repeatedly in a low stakes manner.

There’s a useful frame, known as threshold concepts, for drilling down to this “essential knowledge” within a specific academic domain. Threshold concepts come out of higher ed academia, and it’s admittedly a bit esoteric in the literature, but I think it’s a useful lens with practical implications. Threshold concepts are very much related to Wiggins and McTighe’s “big ideas,” but with a few interesting twists.

I first stumbled over the threshold of these concepts in blogs from UK educators, to whom I’m indebted for starting me on this journey:

—Alex Quigley, “Designing a New Curriculum: What are your ‘Big Ideas?’

—Joe Kirby, “One scientific insight for curriculum design” (he also sums up research on affirmative testing really well here)

What we’re really trying to get here is that 20% of the knowledge that is most essential to understanding an academic domain in a specific grade. Here’s a frame for this:


How can threshold concepts help us to determine that 20% of essential knowledge within a specific academic domain? I decided to review some of the literature for further clues:

—Ray Land, Jan Meyer, and Jan Smith,”Threshold Concepts within the Disciplines” (Land and Meyer are the rockstars who originated the concept)


—James Rhem, “Before and after students “get it”: threshold concepts” (a useful overview)

This idea of a transformation of understanding that is essential for progressing deeper into the academic content is really interesting.


—James Atherton, “How do people ‘get it’?” (another useful overview)

There are many other characteristics that were identified, but they don’t all seem very useful in a practical sense for K-12. I think the three outlined above are the most relevant and applicable.

—Tracy Fortune, Priscilla Ennals, and Mary Kennedy-Jones, “The Hero’s Journey: Uncovering Threshold Barriers, Dispositions, and Practices Among Occupational Therapy Students


I love this idea of viewing a student’s passage through a threshold concept as akin to a hero’s journey. In considering this journey, what are the troublesome obstacles and bottlenecks that student will face? The idea of “bottlenecks” comes thanks for the link immediately below. In thinking through this, I also think we need to acknowledge that bottlenecks may not be purely conceptual — they can also be procedural, in the form of skills required to complete academic tasks, as well as social-emotional (this can be a tremendous and often unaddressed barrier for many kids).

In this sense, then, we can expand the notion of crossing a threshold to not solely refer to concepts, but furthermore experiences. As educators, we seek to design experiences in which students engage in an academic form of a hero’s journey, learning to overcome barriers and gain the intellectual accomplishment of mastering skills and knowledge.

—Joan Middendorf & David Pace, “Decoding the Disciplines: A Model for Helping Students Learn Disciplinary Ways of Thinking” (they offer a useful protocol that I’ve drawn from below)

OK, So How Do We Discover These Threshold Concepts and Experiences?

But we still need some kind of process for distilling away all the cruft and getting down that 20% of the most essential knowledge within a content and grade.

Here in the US, we have a general list of skills we use as guidance in the form of state standards. And as I’ve done with the Common Core standards, we can do a deeper analysis to begin unpacking what that knowledge might be.

But this can still be at a pretty abstract level, and we want this to be relevant to classroom teaching. By focusing on the topics and texts that will be studied, we can make this more concrete.

Because English Language Arts is my specific area of expertise, I’ve focused my efforts in this area, especially since this content area is probably the most difficult to pin down in terms of a progression of knowledge.


There’s two ways we could utilize this protocol: 1) unit of study, or 2) over the entire school year.

1) Consider the topics and/or texts that will be taught.

2) What will be the product or products that students will be expected to create that can demonstrate their mastery of learning? (This product could also be a performance).

3) What are the primary modalities that this product is focused upon? In literacy, of course we’re focused on all modalities, but it helps for a department to focus upon the one they consider most essential.

4) Now consider the standards that your district adheres to. For the Common Core, they are helpfully broken up by modality, so turn to that modality. Then, narrow down which specific, few standards you will primarily be targeting.

5) What are the bottlenecks, most especially those that are conceptual and specific to this content, that students will encounter?

6) Evaluate the list of items you have generated. Do they fit the criteria of a threshold concept or experience? Are they transformative, integrative, and troublesome? If not, they may not be essential.

7) You don’t have to do this, but I find that at this step it can be useful to phrase the threshold concept in the form of a message or lesson, akin to a theme statement.

For example, for an upcoming professional learning session I’m working on about supporting struggling middle school readers, I’ve identified the following threshold concepts:

  • Students that struggle with reading comprehension also often struggle with a lack of academic and world knowledge. An English Language Learner can also be understood as native English speakers that do not understand the language of math, science, social studies – i.e. academic, formal, domain specific language.
  • A teacher must work through a task/text in order to identify key takeaways, key vocabulary, and potential barriers to learning, regardless of whether a curriculum is provided.
  • All learners can be engaged in reading and comprehending complex academic texts through well-designed activities, tasks, and resources.
  • An environment in which a student feels safe to take risks in front of peers is a prerequisite for learning — most especially for struggling readers.
  • Learners should be explicitly equipped with strategies and mindsets for when they encounter challenging vocabulary on their own.

It’s important to note that threshold concepts will vary completely depending on any teacher’s specific set of knowledge, perspectives, and interest, and I think that’s OK. What’s most important is that once these most essential concepts and experiences have been identified and voiced, they will not only help to focus that teacher’s instruction on what they feel is most important, they will serve as a basis for arriving at a consensus as a department and as a school.

Here’s a few really basic examples at a unit level of study:

Now that threshold concepts and experiences have been identified, here’s the really hard part:


This is where the rubber hits the road. This is the part that is so very contrary and disruptive to the norms of public education.


I hope some of these resources in this post are useful to your work. The slides outlined above and the protocol are accessible and downloadable here:

On Knowledge and Curriculum


Forget the presidential debates, this is more important.

I attended a PD today that featured cognitive psychologist Daniel Willingham. It’s pretty rare that my employer, the NYCDOE, offers professional learning that has someone presenting from the academic research realm, so when I saw this was happening, I jumped on it.

I’ve been following Willingham’s articles in American Educator and have read Why Kids Don’t Like School, and I’ve been convinced of the importance of a strong curriculum and building knowledge for a while now, but it was nice to get a direct and clear reminder of what really matters in literacy, especially when that message is so very rare.

There were a lot of great ideas and takeaways on motivating children to read at home and on reading comprehension in general, but there were two main points that especially struck me during his presentation (the summary and phrasing is my own):

  1. Knowledge needs to be “in the mind” in order for reading comprehension to be effortless, rather than a problem-solving struggle to determine meaning.
  2. We can only build the broad, world knowledge required for literacy through a carefully sequenced and structured curriculum.

If you take these points to be accurate, then the implications are quite revolutionary in comparison to the regular practices of most schools and districts.

Let’s break down why this runs so counter to the norm.

Here’s what would need to happen:

  • If knowledge must be “in the mind” (rather than on Google) than that means the knowledge considered worth studying must be reinforced and revisited, tested, interleaved, sequenced, and spaced throughout a school’s curriculum.
  • That means across classrooms and across grades.
  • Therefore, a school needs to have come to a consensus on the topics, texts, vocabulary, and concepts that are most essential to know within and across each academic domain.
  • That means that each teacher (or at the very least, a department head or team) will have to have invested a substantial amount of time, both individually and collaboratively, into studying those texts and topics themselves in order to know how to design a learning environment, projects, activities, field trips, and interim assessments that will provide the access to and reinforce that knowledge for all students.
  • This would of course be accompanied by adjusting the curriculum periodically based on an analysis and reflection on interim assessment data and student work.

Sound pretty straightforward? No. Here’s the norm in most schools:

  • What most prioritizes a school’s focus are external assessments, such as state tests. ELA tests in this vein consist of random passages of text that are meant to focus on isolated reading skills devoid of knowledge. Therefore, what is taught and focused upon are the practice of skills devoid of knowledge.
  • That’s what constitutes an ELA curriculum for many schools.
  • A teacher is either not provided a curriculum, or is provided a curriculum but no support, or is provided a curriculum and support but the curriculum is not oriented around sequentially building knowledge.
  • Even when a curriculum might be provided and might be relatively well-crafted (this is a rarity, and if you know of such a curriculum, tell me. I can name two. Maybe three), I have yet to have seen any curriculum that still does not require a teacher to revise and adjust it substantially based on the needs of their students, the circumstances of their school or classroom, or their own particular style and knowledge.
  • Thus, in those rare schools where there is even a coherent curriculum “in place,” the point made above about investment of time still holds. A substantial amount of time needs to be spent in designing and continually molding the school around and in support of that knowledge embedded within the curriculum.
  • Most of what is taught in different classrooms in a school has little coherency across a school.
  • What is a taught in any given classroom is rarely reinforced via low stakes quizzing across an entire school year.

See the problem? From what I’ve seen in much of the professional development sessions and focus of schools and districts is a focus on individual teacher strategies and practices. But let’s get real. If a school does not come together to determine and design it’s mission around the knowledge and skills it will teach sequentially and systematically, then there will be little impact.

For ideas on how a school might begin to do this work, check out my next post on this topic: On Threshold Concepts & Experiences

Smörgåsbord: A Smattering of Politics and Science for Your Columbus Day Weekend


Ed Politics and Policy

The UFT establishes a hypocritical stance towards NY State’s proposed revisions of the Common Core standards. Mulgrew says the revisions don’t go far enough. But the UFT supported the original version of the standards, then later criticized the state’s implementation of them. So what’s the problem, Mulgrew? The standards themselves, or the implementation? Or maybe it’s politics that’s the real problem here, eh.

Speaking of politics, the plot thickens in the Bronx, where an assistant principal has accused DOE officials and a state assemblyman of collaborating to prevent low income children from attending an overcrowded Riverdale public school. First, the superintendent rapidly resigned, then the school’s principal got demoted. Now, a finger is pointed at a top NYCDOE official.

In moving forward, one way the NYCDOE could show a stronger commitment to equity and school diversity is by supporting Manhattan’s District 1 in its efforts to implement a “controlled choice” model.

Speaking of lawsuits, a Detroit lawsuit challenges local control in public education by claiming that to be able to read and write is a constitutional right.

And the Supreme Court will examine just how much educational benefit a student with a disability should be expected to derive from access to a “free and appropriate education.” (That’s FAPE, for my fellow SPED heads out there.) Currently, it’s a pretty low bar.


Some rough news for those of us who do professional development with teachers, as well as for those of us who are proponents of strong content knowledge: in a study of math teachers that were provided training,”teacher participation in the 93 hours of PD did not have a positive impact on student achievement,” even though the teacher’s knowledge of the math content improved.

We avoid people who give us critical feedback, to the detriment of our performance.

A useful primer and cheat sheet on cognitive biases.

“If all the kids in the world were to line up for a race, the average American child would finish at the foot of the field” (47 out of 50, to be exact).

So maybe we need to give all of our kids standing desks.

More education results in better outcomes. But “differences between educated and less educated people may be partially due to residual genetic and socioeconomic confounding.”

The more a mother’s voice imprints in a child’s brain, the better at communication that child will become.

Reading literature may not improve theory of mind.

Teaching cursive should go the way of the dodo. Writing in manuscript is just fine.

Groups in which all members talk about the same amount perform better.

People can be tricked into arguing for positions that are not their own.


Providing context for concepts may hinder transfer

“One implication of this pattern of results is that as an instructor designs their instructional materials, they should be asking themselves whether they are trying to optimize their students’ demonstrated mastery of the material itself or their ability to transfer their understanding to new materials. . . .

Simply adding richer meaningful content to in-class examples may make intuitive sense, and may have immediately obvious benefits in terms of student engagement and comprehension. But as the results of our experiments make clear, these short-term benefits seem to come at the cost of students’ long-term ability to apply their knowledge. If educators are to take advantage of these inherent benefits, they will need to give careful consideration to how such examples are designed and used together in order to plan the most effective instruction.”

—Samuel Day, Benjamin Motz, and Robert Goldstone, “The Cognitive Costs of Context: The Effects of Concreteness and Immersiveness in Instructional Examples” on frontiers in Psychology (H/T Greg Ashman)

Common Core Task Force Hearing

I went to Governor Cuomo’s Common Core Task Force hearing yesterday evening at La Guardia Community College to share my views on the Common Core Standards. This is the first public hearing I’ve attended, and I hadn’t even thought what I would say until I was traveling down from the Bronx. I jotted down my thoughts in my journal, and here’s essentially what I said (though not quite so coherently):

We need to acknowledge the challenges that have arisen in the implementation and adoption of these standards, much of which have come as a result of hasty efforts to tie teacher evaluation and student testing to the standards before educators and parents were provided an adequate opportunity to fully understand and work with them.

But let’s not lose sight of the opportunity that they have provided us.

The new, higher standards provide an opportunity for all students to engage in a productive struggle with rigorous and meaningful concepts, skills, and texts.

This focus on a higher bar for all students has shifted our focus of instruction to “how can we provide access to all learners to this challenging, essential content–and this is pivotal for students historically denied access to college and career ready skills and knowledge.

Having a more coherent set of standards has furthermore provided an opportunity for all teachers to engage in developing aligned curriculum, materials, and resources not only across classrooms and grades within a school, but furthermore across districts and states. I have been fortunate to work with teachers across our nation in developing aligned materials and curriculum, as well as across my city. And I have seen how the standards have provided opportunities at my school to vertically align content and skills across the grades and across subjects within a grade.

As we move forward, let’s not lose sight of these opportunities as we review the standards within New York State.

I stayed for the entire hearing so I could listen to the other speaker’s testimonies, and found all of them interesting. I especially liked Council member Dromm’s opening testimony — he’s a former educator, and he gets it. There was some of the typical anti-Common Core rhetoric. Many speakers were united in their condemnation of the current form of NY State tests. However, Arva Rice of the NY Urban League offered what I thought to be an important defense of testing for the purpose of surfacing inequity. One MORE caucus retired teacher wove history into his testimony, referring back to Ocean-Hill Brownsville to make the case for a culturally relevant curriculum. There was a solid base of testimony from teachers on how the higher standards have provided benefit to the students they serve, including one educator who has witnessed the positive impact higher standards has had on students most in danger in dropping out of school. There was also a common thread on how there is a real need to empower teachers to collaborate and provide them the time to do so around the standards. A UFT representative made several great points about concerns with the standards, especially in regards to the need to continue to teach literature in ELA (a point I strongly agree with). And there was one guy wearing a gold hard hat who eloquently provided an out-of-this-world sermon on the need to teach our children respect (not sure what the link was to Common Core, but kudos to him for his gold sneakers and for coming out to expound).

The vibe from the Common Core Task Force members who were there, Assemblywoman Catherine Nolan and educator Kishayna Hazelwood, was that they already had a few of their own biases, but that they were interested in learning more. They seemed interested especially (unsurprisingly) in how they could change testing. There was a very strange and tense moment where Assemblywoman Nolan first verbally accused a teacher of showing disrespect during the parent’s testimony (for “smiling”), and when the teacher denied that she had done this, Nolan physically got up from behind the table and stepped right up to the teacher to say menacingly, “I saw you smile.”

First of all, I spoke with this teacher, a 20 year veteran teacher in the Bronx, and she informed me that she had not been smiling derisively, but had rather been making an unrelated connection in her mind between something said earlier. Second of all, even if she HAD been smirking, the parent testifying did not see this, and Ms. Nolan’s response was unnecessarily aggressive. I’m sure she was trying to make the point that she valued all perspectives and would stand up to ensure they were heard, which is great, but the manner in which she did this was inappropriate.

But I appreciated both Nolan’s and Hazlewood’s engaged questioning of various speakers, as it showed they were interested in learning more and valued the testimony of different perspectives and that people had given their time to come to the hearing. I hope they continue to hear from across the state on how the higher standards have provided greater opportunities for teachers and students to engage in meaningful content, while also hearing that teachers need more time to engage with the standards.

Here’s a couple of other accounts of the event: