Private Schools Work to Build Diverse Teaching Staffs

Education Week

As student bodies at private schools have become more diverse, changes in the teaching corps have been lagging behind, and independent schools sometimes struggle to attract and retain teachers of color.

Ethnic and gender diversity has been the topic of several sessions at the National Association of Independent Schools’ annual conference taking place in Boston this week.

About 30 percent of the students enrolled in NAIS member schools are students of color, up from almost 22 percent a decade ago. By comparison, only 15 percent of faculty members are nonwhite, although that number has risen from around 11 percent in 2004.

Faculty members of color are often expected to be the support system for students of color, and that’s no small task when you consider the disparity in their numbers, according to a panel of deans who spoke on the issue at the NAIS conference.

The extra work can lead to burnout for teachers of color, making retention more challenging, and compounding already disappointing recruitment results.

“Often you ask people of color, why did you accept the job? They talk about who they met while they were interviewing, and did they actually see people like themselves,” said Linda Griffith, the dean of community and multicultural development at Phillips Academy in Andover, Mass. She’ll soon start a new job at the school heading up equity and inclusion initiatives, a position that was just created.

Having staff members and resources dedicated to equity and diversity is a major component to attracting and retaining faculty from minority groups, Griffith said.

Phillips Exeter Academy in New Hampshire has been experimenting with some innovative ways to increase diversity among its faculty with a fair amount of success, according to its dean of faculty, Ron Kim.

The school started hiring teachers as it found good candidates, instead of waiting for a position to open up on staff. To accommodate the new hires, the school has them split a teaching load with a senior faculty member who then also doubles as a mentor.

Phillips Exeter Academy also created a dissertation fellowship for minority doctoral candidates in fields such as English and mathematics. The school gives fellows a stipend, an apartment on campus, and access to school facilities while they finish writing their dissertation. There’s no requirement to teach at the school afterwards. The benefit for Phillips Exeter Academy comes simply from the fact that offering the fellowship puts the school in touch with dozens of highly qualified people of color who are going to be looking for jobs soon.

Both Griffith and Kim say their schools’ efforts are paying off: The number of racial and ethnic minorities on Phillip Exeter Academy’s faculty has grown from 10 to 18 percent, while about half of Andover’s new hires are from diverse backgrounds.

Why Our Children Don’t Think There Are Moral Facts


George Washington, depicted here taking the oath of office in 1789, was the first president of the United States. Fact, opinion or both?

George Washington, depicted here taking the oath of office in 1789, was the first president of the United States. Fact, opinion or both?Credit via Associated Press

What would you say if you found out that our public schools were teaching children that it is not true that it’s wrong to kill people for fun or cheat on tests? Would you be surprised?

I was. As a philosopher, I already knew that many college-aged students don’t believe in moral facts. While there are no national surveys quantifying this phenomenon, philosophy professors with whom I have spoken suggest that the overwhelming majority of college freshman in their classrooms view moral claims as mere opinions that are not true or are true only relative to a culture.

A misleading distinction between fact and opinion is embedded in the Common Core.

What I didn’t know was where this attitude came from. Given the presence of moral relativism in some academic circles, some people might naturally assume that philosophers themselves are to blame. But they aren’t. There are historical examples of philosophers who endorse a kind of moral relativism, dating back at least to Protagoras who declared that “man is the measure of all things,” and several who deny that there are any moral facts whatsoever. But such creatures are rare. Besides, if students are already showing up to college with this view of morality, it’s very unlikely that it’s the result of what professional philosophers are teaching. So where is the view coming from?

A few weeks ago, I learned that students are exposed to this sort of thinking well before crossing the threshold of higher education. When I went to visit my son’s second grade open house, I found a troubling pair of signs hanging over the bulletin board. They read:

Fact: Something that is true about a subject and can be tested or proven.

Opinion: What someone thinks, feels, or believes.

Hoping that this set of definitions was a one-off mistake, I went home and Googled “fact vs. opinion.” The definitions I found onlinewere substantially the same as the one in my son’s classroom. As it turns out, the Common Core standards used by a majority of K-12 programs in the country require that students be able to “distinguish among fact, opinion, and reasoned judgment in a text.” And the Common Core institute provides a helpful page full of links to definitions, lesson plans and quizzes to ensure that students can tell the difference between facts and opinions.

So what’s wrong with this distinction and how does it undermine the view that there are objective moral facts?

First, the definition of a fact waffles between truth and proof — two obviously different features. Things can be true even if no one can prove them. For example, it could be true that there is life elsewhere in the universe even though no one can prove it. Conversely, many of the things we once “proved” turned out to be false. For example, many people once thought that the earth was flat. It’s a mistake to confuse truth (a feature of the world) with proof (a feature of our mental lives). Furthermore, if proof is required for facts, then facts become person-relative. Something might be a fact for me if I can prove it but not a fact for you if you can’t. In that case, E=MC2 is a fact for a physicist but not for me.

But second, and worse, students are taught that claims are eitherfacts or opinions. They are given quizzes in which they must sort claims into one camp or the other but not both. But if a fact is something that is true and an opinion is something that is believed, then many claims will obviously be both. For example, I asked my son about this distinction after his open house. He confidently explained that facts were things that were true whereas opinions are things that are believed. We then had this conversation:

Me: “I believe that George Washington was the first president. Is that a fact or an opinion?”

Him: “It’s a fact.”

Me: “But I believe it, and you said that what someone believes is an opinion.”

Him: “Yeah, but it’s true.”

Me: “So it’s both a fact and an opinion?”

The blank stare on his face said it all.

How does the dichotomy between fact and opinion relate to morality? I learned the answer to this question only after I investigated my son’s homework (and other examples of assignments online). Kids are asked to sort facts from opinions and, without fail, every value claim is labeled as an opinion. Here’s a little test devised from questions available on fact vs. opinion worksheets online: are the following facts or opinions?

— Copying homework assignments is wrong.

— Cursing in school is inappropriate behavior.

— All men are created equal.

— It is worth sacrificing some personal liberties to protect our country from terrorism.

— It is wrong for people under the age of 21 to drink alcohol.

— Vegetarians are healthier than people who eat meat.

— Drug dealers belong in prison.

The answer? In each case, the worksheets categorize these claims as opinions. The explanation on offer is that each of these claims is a value claim and value claims are not facts. This is repeated ad nauseum: any claim with good, right, wrong, etc. is not a fact.

In summary, our public schools teach students that all claims are either facts or opinions and that all value and moral claims fall into the latter camp. The punchline: there are no moral facts. And if there are no moral facts, then there are no moral truths.

The inconsistency in this curriculum is obvious. For example, at the outset of the school year, my son brought home a list of student rights and responsibilities. Had he already read the lesson on fact vs. opinion, he might have noted that the supposed rights of other students were based on no more than opinions. According to the school’s curriculum, it certainly wasn’t true that his classmates deserved to be treated a particular way — that would make it a fact. Similarly, it wasn’t really true that he had any responsibilities — that would be to make a value claim a truth. It should not be a surprise that there is rampant cheating on college campuses: If we’ve taught our students for 12 years that there is no fact of the matter as to whether cheating is wrong, we can’t very well blame them for doing so later on.

Indeed, in the world beyond grade school, where adults must exercise their moral knowledge and reasoning to conduct themselves in the society, the stakes are greater. There, consistency demands that we acknowledge the existence of moral facts. If it’s not true that it’s wrong to murder a cartoonist with whom one disagrees, then how can we be outraged? If there are no truths about what is good or valuable or right, how can we prosecute people for crimes against humanity? If it’s not true that all humans are created equal, then why vote for any political system that doesn’t benefit you over others?

Our schools do amazing things with our children. And they are, in a way, teaching moral standards when they ask students to treat one another humanely and to do their schoolwork with academic integrity. But at the same time, the curriculum sets our children up for doublethink. They are told that there are no moral facts in one breath even as the next tells them how they ought to behave.

We can do better. Our children deserve a consistent intellectual foundation. Facts are things that are true. Opinions are things we believe. Some of our beliefs are true. Others are not. Some of our beliefs are backed by evidence. Others are not. Value claims are like any other claims: either true or false, evidenced or not. The hard work lies not in recognizing that at least some moral claims are true but in carefully thinking through our evidence for which of the many competing moral claims is correct. That’s a hard thing to do. But we can’t sidestep the responsibilities that come with being human just because it’s hard.

That would be wrong.

Justin P. McBrayer is an associate professor of philosophy at Fort Lewis College in Durango, Colo. He works in ethics and philosophy of religion.

Teachers Share How Black History Month is Taught in Schools

In Anthony Marshall’s first-hour class on Monday, students were doing research about Booker T. Washington and W.E.B. Du Bois.

“With whom do you agree?” the assignment on the classroom white board reads. “Who had the best plan for equality?”

Marshall, a teacher at Booker T. Washington High School, said he wanted his students to look into the two activists’ backgrounds to help them understand how they developed their philosophies. At the end of the assignment would come a debate, with students assigned to take either side.

Marshall teaches Advanced Placement U.S. history, and he also teaches a course focused on African-American studies. He refers to it as “U.S. history from an African-American perspective.”

The course covers subjects such as the civil rights movement and the Tulsa Race Riot, but it also takes a look at less known facets of black history in the U.S., including the contributions of African-Americans in every stage of the nation’s history.

Posters in Marshall’s class include the text of the Emancipation Proclamation and a list of African-American inventors and scientists.

Marshall, who organized several events at his school in February in honor of Black History Month, said he does not believe black history is being adequately taught in Oklahoma classrooms.

“If history were taught the way it should be taught, you would not need a separate class called black history,” he said.

Marshall said the fact that there is a need for his specific class shows that “there is a void in how we teach.”

Marshall said that when he began working at Booker T. in 2006, he and a few community activists worked to re-introduce a black history course in Tulsa Public Schools. The course is still an elective in the schools that offer it, and many schools throughout the area do not offer a course dedicated to black history.

He said so far, the teaching of black history has been pushed by individual “bold” teachers who seek to bring this perspective of history to their students.

Earl Derkatch, a history teacher at Owasso High School and chair of the social studies department, said his students are nearly finished studying the civil rights movement.

Though he agrees that more attention could be given in history courses to black history, as well as Native American history and the history of other ethnic groups, he said today’s textbooks are better than they have been in the past in terms of inclusion.

Derkatch has taught history for more than 40 years.

“The way textbooks are written today, they’ve interwoven in a lot of black history,” he said.

Attention is given to the contributions of African-Americans in everything from the Spanish-American War to the Vietnam War.

But Derkatch still has to bring in outside material to supplement what is provided in the textbooks.

During Black History Month, he brings in the works of African-American authors. When the class is talking about economics, he includes the writings of the notable black economist Thomas Sowell. If it’s time for discussions about pre-colonial times, Derkatch brings in books about the contributions of black individuals.

Derkatch’s and Marshall’s goals are the same: To show students the lesser-known contributions from minority communities, specifically those of African-Americans.

Cori Williams, a senior in one of Marshall’s classes, said the course has opened her eyes to aspects of history she’d never known before.

For example, she knew about the slave trade and that slaves were brought the America on crowded ships. She just never knew the extent of the ill treatment of them on those ships, nor exactly how packed they were.

“I think the black perspective is really important because we’re a part of this nation as well,” she said.

For senior Darian Walters, the class is interesting because it is not focused on the traditional aspects of black history, such as slavery and the civil rights movement.

“This is like, black people have played a part in everything in America,” he said.

Derkatch says he makes an effort to make all groups of students in his class feel included historically, whether its his African-American, Native American or Hmong students.

“They get a sense of belonging and pride when they feel included,” he said.

Pauline Harris, human rights coordinator for Tulsa Public Schools, has taught African-American studies at Tulsa Community College in past years.

Though she felt that not all students were exposed to the subject properly in high school, she said schools are doing better now than in the past.

She’s said workshops have been conducted at TPS on how to teach a variety of topics, including black history.

Harris also said Oklahoma’s academic standards — which the state has reverted to after Common Core was repealed last year — include the teaching of the Tulsa Race Riot, which in the past had not been taught by all schools.

She said textbooks now are more diverse and inclusive.

“We’ve made great strides,” Harris said.

But both Derkatch and Marshall say more can be done.

Derkatch said one of the problems with survey classes is that there is too much material to cover and not enough time to cover it. That is why it is difficult to include so many perspectives or contributions of all ethnic and racial groups in a single course, he said.

Money shortages also pose a problem when considering adding special electives to cover the extra material, he said.

Marshall said it is important for all students to be able to learn about American history from an African-American perspective.

“Race is one of the biggest issues in our country,” he said.

If Your Teacher Likes You, You Might Get A Better Grade


FEBRUARY 22, 2015
The Teacher's Pet

LA Johnson/NPR

Were you ever the teacher’s pet? Or did you just sit behind the teacher’s pet and roll your eyes from time to time?

A newly published paper suggests that personality similarity affects teachers’ estimation of student achievement. That is, how much you are like your teacher contributes to his or her feelings about you — and your abilities.

“Astonishingly, little is known about the formation of teacher judgments and therefore about the biases in judgments,” says Tobias Rausch, an author of the study and a research scientist at the Otto-Friedrich-Universität Bamberg in Germany. “However, research tells us that teacher judgments often are not accurate.”

This study looked at a group of 93 teachers and 294 students in eighth grade in Germany. Everyone took a short test to establish basic features of their personalities: extraversion, agreeableness and the like.

They gave the students reading and math tests too, sharing the test items with the teachers. Then they asked the teachers two questions: How good is this student compared to an average eighth grader? How well will this student do on this test?

In other words, the first question asked the teacher to give a global judgment; the second asked for a task-specific judgment.

The study found that when teachers and students were peas in a pod, the teachers overestimated the students’ general abilities. Conversely, students who were dissimilar from their teachers were judged less positively.

But when the judgment was grounded to a specific test, the effect disappeared.

This finding is maybe not that surprising. But it’s important for two reasons.

First, there’s concern that teacher bias of many kinds may unfairly hold back groups of students.

For example, a recent study from Israel showed that teachers gave girls lower grades on math tests when they knew their gender. And lots of researchers have looked at the importance of having teachers who share the racial and socio-economic backgrounds of their students.

If teachers give students who are similar to them better grades, or even just maintain higher expectations of those students, what does that do for the students who don’t look or act like their teachers?

Second, as Rausch, the co-author, points out, this study points to the importance of balancing teachers’ “holistic” evaluations with standardized assessments, or at least assessments that aren’t graded by a student’s own teacher.

Rausch also says it might be a good idea to spend more time training teachers to notice their biases. “The best way to control for is probably raising teachers’ awareness concerning the way they assess their students’ competencies and their awareness concerning typical judgment biases and tendencies,” he says. Human judgment, after all, is only human.

Straight Talk for White Men

SUPERMARKET shoppers are more likely to buy French wine when French music is playing, and to buy German wine when they hear German music. That’s true even though only 14 percent of shoppers say they noticed the music, a study finds.

Researchers discovered that candidates for medical school interviewed on sunny days received much higher ratings than those interviewed on rainy days. Being interviewed on a rainy day was a setback equivalent to having an MCAT score 10 percent lower, according to a new book called “Everyday Bias,” by Howard J. Ross.

Those studies are a reminder that we humans are perhaps less rational than we would like to think, and more prone to the buffeting of unconscious influences. That’s something for those of us who are white men to reflect on when we’re accused of “privilege.”

White men sometimes feel besieged and baffled by these suggestions of systematic advantage. When I wrote a series last year, “When Whites Just Don’t Get It,” the reaction from white men was often indignant: It’s an equal playing field now! Get off our case.

Yet the evidence is overwhelming that unconscious bias remains widespread in ways that systematically benefit both whites and men. So white men get a double dividend, a payoff from both racial and gender biases.

Consider a huge interactive exploration of 14 million reviews on RateMyProfessors.comthat recently suggested that male professors are disproportionately likely to be described as a “star” or “genius.” Female professors are disproportionately described as “nasty,” “ugly,” “bossy” or “disorganized.”

One reaction from men was: Well, maybe women professors are more disorganized!

But researchers at North Carolina State conducted an experiment in which they asked students to rate teachers of an online course (the students never saw the teachers). To some of the students, a male teacher claimed to be female and vice versa.

When students were taking the class from someone they believed to be male, they rated the teacher more highly. The very same teacher, when believed to be female, was rated significantly lower.

Something similar happens with race.

Two scholars, Marianne Bertrand and Sendhil Mullainathan, sent out fictitious résumés in response to help-wanted ads. Each résumé was given a name that either sounded stereotypically African-American or one that sounded white, but the résumés were otherwise basically the same.

The study found that a résumé with a name like Emily or Greg received 50 percent more callbacks than the same résumé with a name like Lakisha or Jamal. Having a white-sounding name was as beneficial as eight years’ work experience.

Then there was the study in which researchers asked professors to evaluate the summary of a supposed applicant for a post as laboratory manager, but, in some cases, the applicant was named John and in others Jennifer. Everything else was the same.

“John” was rated an average of 4.0 on a 7-point scale for competence, “Jennifer” a 3.3. When asked to propose an annual starting salary for the applicant, the professors suggested on average a salary for “John” almost $4,000 higher than for “Jennifer.”

It’s not that we white men are intentionally doing anything wrong, but we do have a penchant for obliviousness about the way we are beneficiaries of systematic unfairness. Maybe that’s because in a race, it’s easy not to notice a tailwind, and white men often go through life with a tailwind, while women and people of color must push against a headwind.

While we don’t notice systematic unfairness, we do observe specific efforts to redress it — such as affirmative action, which often strikes white men as profoundly unjust. Thus a majority of white Americans surveyed in a 2011 study said that there is now more racism against whites than against blacks.

None of these examples mean exactly that society is full of hard-core racists and misogynists. Eduardo Bonilla-Silva, a Duke University sociologist, aptly calls the present situation “racism without racists”; it could equally be called “misogyny without misogynists.” Of course, there are die-hard racists and misogynists out there, but the bigger problem seems to be well-meaning people who believe in equal rights yet make decisions that inadvertently transmit both racism and sexism.

So, come on, white men! Let’s just acknowledge that we’re all flawed, biased and sometimes irrational, and that we can do more to resist unconscious bias. That means trying not to hire people just because they look like us, avoiding telling a young girl she’s “beautiful” while her brother is “smart.” It means acknowledging systematic bias as a step toward correcting it.

Applying Mindfulness to Mundane Classroom Tasks

By Abby Wills, FEBRUARY 5, 2015


Day after day, I entered my psycho-physical education classroom at New Roads Middle School, primed and prepared to facilitate an inspiring lesson full of mindful practices. And day after day, I felt subtle disappointment as students stumbled in rowdy from recess and fumbled their way through the necessary task of moving the eight full-size tables and 25 chairs aside to make space for our work. A chore that could take well under five minutes was eating up our class time and starting our class period with a raucous energy.

I tried assigning students to particular desks as they walked through the door. This helped establish greater participation, but the noise level was still an issue. I tried whispering to students as they entered the room to set a quiet tone. The noise level receded, but the chaos and clatter of tables and chairs banging into each other was still an obstacle to the peaceful experience I intended to create for our class.

I even tried arriving very early and moving the furniture myself, at which point I realized I was over-handling the situation. I was taking full responsibility, rather than trusting my students with their fair share. In short, I wasn’t applying my pedagogy to transition time. Instead, I was trying to completely manage the transition so that we could get to the lesson sooner.

As good ideas often do, this one landed like a ton of bricks — it’s the journey, not the destination.

Calming the Waters

That night I spent my lesson planning time in contemplation on my cushion, rather than at my computer. “How can I integrate the necessary classroom housekeeping into the content of our curriculum?” I listened. I waited. Aha!

The next morning, I rounded up eight colorful plastic bowls from my kitchen and packed them along with a large jar of water. Upon arrival at my classroom, I placed a half-full bowl of water on each table. I placed the portable whiteboard near the entrance of the room so that as every student walked in, he or she would read this message:


Then I stood at the back of the room and observed as each student’s mind stopped for a brief moment. Novelty led to curiosity, and as they carried on with the regular task of moving desks, the mundane was transformed into a simple exercise of mindful movement and peer cooperation.

The tables seemed to float on thin air. Not a single bang or drop. Voices were hushed as eyes focused on the colorful bowls of water that were ready to splash if not well attended.

After my students quietly and mindfully prepared the room, without spilling a drop of water, I congratulated them and asked them to reflect on how the classroom felt different on that day compared with the usual way we conducted the furniture task. I wrote their responses on the board. In their own words:

  • More calm
  • Peaceful
  • Organized
  • Less crazy
  • Together
  • Focused
  • Balancing
  • Not a drag.

I was pleased at how attuned my students were to the emotional difference associated with this minor change in protocol.

The next day, students entered the room and inevitably asked, “Where are the bowls?” “Go and see,” I replied. On each table was a piece of paper on which I’d written their very words from the previous day describing the positive difference they felt when applying mindfulness to an ordinary task. Students were pleased to see their own words instructing the activity.

Again, our time together began with a sense of ease.

The following day, I put nothing on the tables. When students inquired, I asked them to visualize a sleeping baby jaguar in the center of each table. I watched students hold each other accountable for being very quiet indeed, both vocally and physically. For several class periods to follow, students were eager to offer suggestions for what would be imagined on the desks while moving them — a marble, a bucket of acid, a tower of cupcakes.

A Habit of Mindfulness

Before this theme began to wear thin, I changed course by asking students to sit on top of the tables, usually a prohibited action. We talked in depth about the choices we always have in how we do things, even when we do not especially want to do them (like homework). I explained that my ideal wish would be for our school to afford a specialized room for our mindful practice, one with heated wooden floors, always-clean cushions, and lots of empty space. I asked students to imagine what their ideal room might look and feel like. They shared truly fantastic visions of rooms surrounded by waterfalls, equipped with fiber-optic lighting and aromatherapy misters. I asked them to name the qualities each of their ideal visions encompassed. Then I encouraged students to embody those qualities while we put the tables aside that day.

From that day forward, preparing the room mindfully became a part of our class culture. And on days when our mindfulness was distracted, I simply observed and did not give in to the impulse to manage. I allowed students to feel the difference, and therefore their behavior was self-limited by natural consequences. This experience of mindfulness, sparked by a genuine desire for coherence, set the stage for our learning through sitting mindfulness practice, mindful movement, and conscious breathing.

In my work with teachers, I encourage honest, fearless contemplation on what is and is not working in the classroom. This simple exercise can help you discover the mundane moments and tasks in your own classroom that are just waiting for your creativity to transform them into mindful learning opportunities.

Make math concrete with digital fabrication

Make math concrete with digital fabrication

By Kimberly Corum and Joe Garofalo 2/3/2015

For too many students, doing mathematics means just plugging numbers into a memorized formula to get an answer. And because they don’t understand the formulas they’re using, they often fail to use the right one.

Take a look at Isaac’s work below, for example. He is a fifth grade student who tried to find the surface area of a rectangular prism by incorrectly adapting a previously memorized formula for calculating perimeter. He calculated two times the length plus two times the width (2L + 2W) and tried to account for the height by multiplying it by 4, then adding it to the previous sum. Unfortunately, Isaac is not alone in this type of approach. Students who use formulas by rote may never come to see mathematics as sense-making and may never understand the formulas they use. And there are so many formulas to memorize! Teachers who prematurely introduce students to formulas risk denying them opportunities to develop the necessary conceptual foundations for mathematical understanding.


One way to help students like Isaac understand surface area is to present it conceptually using manipulatives. Physical manipulatives — concrete objects such as blocks, chips, unifix cubes and geoboards — are already a common staple of mathematics classrooms. Virtual manipulatives — computer images created by, for example, computer-aided design (CAD) software, Geometer’s Sketchpad or Flash programs — are also becoming more and more prevalent in classrooms. And now, with the increased availability of 3D printers and CAD software in schools across the country, digital fabrication gives teachers an opportunity to incorporate both physical and virtual manipulatives into the same lesson.

What is digital fabrication?

Digital fabrication is the process of creating a physical object from a digital design developed on a computer. You can transform your classroom computers into personal fabrication systems that your students can use to create both virtual and physical manipulatives by installing CAD software and hooking them up to a 3D printer, a die cutter or even a standard inkjet or laser printer.

CAD software, such as FabLab ModelMaker, Fab@School Maker Studio and Autodesk 123D Design, allows students to design, rotate, transform and measure 3D solids. After creating these virtual manipulatives on screen, they can print the shape’s corresponding net on cardstock, vinyl or other types of material. Finally, they can use a die cutter to cut the outer edges of the net and perforate the fold lines to make construction of the physical 3D solid easier, or they can just cut it out by hand.

2_modelmaker-screenshot die-cut-manipulative

Both virtual manipulatives (shown in the photo on the left) and physical manipulatives (printed out as a net, on the right) can help students make sense of surface area.

How we used digital fabrication to teach surface area

Our fifth grade students’ used their own personal fabrication systems — which included MacBooks, FabLab ModelMaker, Canon inkjet printers and Silhouette die cutters — to complete the following lessons during their digital fabrication unit:

Initial exploration. Students looked at different models of 3D solids and identified key attributes, such as the number of faces, the shapes of the different faces, etc.

Introduce hardware/software. Students created their own one-inch cardstock cubes, which gave them the opportunity to explore the software. They designed their solids, measured different attributes, looked at different views, and printed physical models from their virtual manipulatives.

Define surface area. We asked students what they thought surface area might mean, focusing on the relationship between surfaces and faces, their prior experiences with area, and the key attributes of 3D solids they had identified.

Connections between representations. Using the software, students explored the relationship between their printed-out, folded cubes and the cubes’ nets. Students rotated their cubes using the software to explore all six faces. Then they hypothesized how they might calculate surface area.

Rotating and coloring. We asked students to review their definitions of surface area and how it is calculated. Students used the software to rotate and color different faces of their cube and explore the relationship between the colored faces on the cube and its net. After calculating the surface area of their cubes, students constructed other types of rectangular prisms and calculated the surface area of these solids.

Developing strategies and skills

The digital fabrication unit gave students the chance to develop effective strategies for learning about surface area, such as recognizing the qualities of 3D figures they can’t see in a 2D representation and carrying out multi-step processes.

When students look at a 2D representation of a rectangular prism, they are able to see only the top face, front face and a side face. To find the surface area, they need to understand that each visible face has a corresponding face that is not visible. Throughout the digital fabrication unit, students had multiple experiences with nonvisible faces. The software allowed them to rotate 2D representations of prisms to make the nonvisible faces visible, and students could see both the 2D representation of a solid and its corresponding net, which displayed all of the faces. Students were also able to explore physical models of their solids.

They also developed strategies to keep track of their work, such as listing the areas of the faces, labeling each face with letters to account for all six faces, and annotating faces with calculated areas. Again, students used the software to help facilitate these strategies. Coloring each face and exploring the relationship between opposite faces gave students opportunities to keep track as they calculated face areas. When working with their physical prisms, they used their fingers as calipers to count faces in pairs. They would also label faces with a mark, letter or the area.

Take a look at Isaac’s calculations, below, after completing the digital fabrication unit. His work shows his ability to both consider what was not shown and keep track of his calculations.


Digital fabrication addresses the ISTE Standards

The ISTE Standard for Students addressing Creativity and Innovationrecommends that students “apply existing knowledge to generate new ideas, products or processes” and “use models and simulations to explore complex systems and issues,” both of which happened in our digital fabrication unit. One ISTE Standard for Teachers, Design and Develop Digital Age Learning Experiences and Assessments, asks teachers to “design, develop and evaluate authentic learning experiences and assessments incorporating contemporary tools and resources to maximize content learning in context.” And digital fabrication promotes the ISTE Standards for Administrators’ Digital Age Learning Culture, which encourages school and district leaders to “model and promote the frequent and effective use of technology for learning” and “provide learner-centered environments equipped with technology and learning resources to meet the individual, diverse needs of all learners.”

Digital fabrication also supports the National Council of Teachers of Mathematics’ Principles and Standards for School MathematicsTechnology Principle: “Students’ engagement with, and ownership of, abstract mathematical ideas can be fostered through technology. Technology enriches the range and quality of investigations by providing a means of viewing mathematical ideas from multiple perspectives.”

Digital fabrication technology played an integral role in helping our students really understand surface area. Using physical and virtual manipulatives helped them develop conceptual understandings of surface area and facilitated the development of two problem-solving strategies they will be able to apply to other content areas.

Kimberly Corum is a Ph.D. student in mathematics education at the University of Virginia. A former high school math and science teacher, her research interests include students’ understanding of 3D measurement and the use of manipulatives in the classroom to support student learning.

Joe Garofalo is co-director of the Center for Technology and Teacher Education at the University of Virginia. His primary interests are in mathematical problem solving, use of technology to facilitate mathematics learning and mathematics teacher education.