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About Ethan Demme

Ethan Demme is the Chairman and CEO of Demme Learning and is passionate about building lifelong learners. Ethan is an elected member of the board of supervisors in East Lampeter Township, PA. He has never backed down from a challenge, especially if it's outdoors, and is currently into climbing big mountains and other endurance sports. An active member of his local community, Ethan is a well-socialized homeschool graduate who holds a B.A. in Communication Arts from Bryan College.

Free Homeschool Activities to Do During a Quarantine

We have been helping parents teach their children at home for over 30 years and have some free resources that will help if you're in a quarantine situation.

Across the United States and around the world schools are closing their doors to slow the spread of the Coronavirus. CNN writes how ‘regular school’ parents can homeschool their kids, saying, “The most important caveat about temporary homeschooling is that it simply isn’t school.” Forbes Magazine calls it “The World’s Homeschooling Moment” and shares this insight:

Most of the 300 million schoolchildren currently quarantined in their homes will return to school once the epidemic fades. But some parents may discover that learning outside of schooling benefited their children and strengthened their family. They may start to wonder if homeschooling or other schooling alternatives could be a longer-term option. They may realize that education without schooling is not a crisis but an opportunity.

Free Homeschool Activities to Do During a Quarantine

At Demme Learning we have been helping parents teach their children at home for over 30 years and we have some free resources and activities to help you navigate homeschooling.

Visit our Guild page where we have math activities, spelling activities, webinars, and a Homeschooling 101 eBook to get you started, all for free.

We are here to help; feel free to contact us with questions. We can help each other and together we will get through this.

We have been helping parents teach their children at home for over 30 years and have some free resources that will help if you're in a quarantine situation.

Senator Sasse on Building Community [Book Review]

Senator Ben Sasse thinks there is one thing that plagues our nation more than anything else: loneliness.

Senator Ben Sasse thinks there is one thing that plagues our nation more than anything else: loneliness. In his latest book, Them: Why We Hate Each Other – And How To Heal – he explains:

Our world is nudging us toward rootlessness, when only a recovery of rootedness can heal us. What’s wrong with America, then, starts with one uncomfortable word. Loneliness.

There are a variety of causes for this loneliness, and those causes have paradoxically brought us both good and bad. Sasse explains that “the massive economic disruption that we entered a couple of decades ago and will be navigating for decades to come is depriving us psychologically and spiritually at the same time that it’s enriching us materially.”

He further writes that:

the same technology that has liberated us from so much inconvenience and drudgery has also unmoored us from the things that anchor our identities. The revolution that has given tens of millions of Americans the opportunity to live like historic royalty has also outpaced our ability to figure out what community, friendships, and relationships should look like in the modern world.

Connecting Deeply

Senator Sasse believes that we will only flourish as American citizens if we embrace the messy work of connecting deeply with the people all around us, in our homes, neighborhoods, workplaces, churches, etc. He notes that “social scientists have identified four primary drivers of human happiness, which we can put into in the form of four questions:

1. Do you have family you love, and who love you?
2. Do you have friends you trust and confide in?
3. Do you have work that matters – callings that benefit your neighbors?
4. Do you have a worldview that can make sense of suffering and death?”

Sasse explains that “a central component to a contented life is ‘rootedness’ – having a sense of community.” But he warns that our tech habits are often directly at odds with this sense of rootedness. He writes, “we undermine these roots every day by scrolling through feeds [on social media], trying to throw ourselves into a different time and place, because maybe things are better there.” As I’m sure you’ve experienced firsthand if you’ve spent a lot of time online, “the end result is aching loneliness.”

Sasse writes:

I suspect historians will look back at the early twenty-first century and wonder why humans were so willing, often to the point of desperation, to confide our deepest secrets, desires, and fears to machines and inanimate objects, while retreating from actual flesh-and-blood people.

Sasse bluntly states that “when we prioritize ‘news’ from afar, we’re saying that our distant-but-shallow communities are more important than our small-but-deep flesh-and-blood ones.” He asks us to reorder our priorities by remembering that “we live richer, more fulfilled lives when we’re directing ourselves to the right people.”

Of course, the problem with community is that it is really hard work to live alongside other people who are different from us and who think differently than we do. And yet the beauty of building relationships with our neighbors (literally and metaphorically) is that we can experience conversations that change us. Sasse writes that “neighbors see today’s conversation not as the last discussion we’ll ever have, but as a precursor to tomorrow’s. We can and will visit again. We can continue talking, and listening. We can be open to future persuasion – and to being persuaded. We need not win everything by force, and we need not win everything right now.” There is unpredictability inherent to this vision of community and maybe even some degree of risk, but Sasse writes, “I like the idea of investing in a future that isn’t guaranteed.”

Sasse’s preference for persuasive conversation over political coercion is rooted in his understanding of the difference between the political and the civic. He explains that “one of the core problems with our public life together is that we’re constantly failing to distinguish between politics and civics. Politics is about the use of power – how it is acquired and who wields it. Civics is about who we are as a people.” Sasse goes on to stress that while “obviously, politics matters… civics matters more.” He elaborates that “politics is simply the bare-bones instrument we use to protect the freedom to live lives of purpose, service and love. But if we collapse civics and politics together…then we ensure that politics squeezes out community. We give priority to compulsion over friendship, and coerced uniformity over genuine diversity.” Sasse writes that “the only way to preserve sufficient space for true community and for meaningful, beautiful human relationships is to have a political philosophy that emphasizes constraint,” and in practice means learning to recognize our own biases, and seeking to grow through the conversations we have with the people around us.

At the end of the day, Senator Sasse seeks to remind us that “we find lives of meaning and purposes at and near home.” This means that “the District of Columbia is not the center of American life: it exists to maintain a framework of ordered liberty – so that your city or town, the place where you live, can be the center of the world.” Ultimately, all the ingredients for the good life are found in the particular relationships that make us who we are: our family, our friends, our neighbors, and our colleagues.

Further Reading

Parental engagement is a main component of our work here at Demme Learning. In an earlier blog series, we looked at how parents can help their kids grow into responsible citizens who are engaged in civic life.

Sasse writes at length about the challenges that tech and social media pose for us as citizens. But it isn’t just us adults who struggle to figure out the proper role of tech in our lives: our kids are struggling too. One tremendous resource for families is Andy Crouch’s book The Tech-Wise Family, which I recently reviewed.

Senator Ben Sasse thinks there is one thing that plagues our nation more than anything else: loneliness.

The Importance of Improving Math Culture

A recent initiative in math education seeks to situate math culture in a deeper, more embedded context.

As human beings, our social selves are shaped by human culture. The language we speak, the monetary currency we use, the movies we watch; the examples of human culture are endless. Education is also a cultural phenomenon, and that includes math education.

But often the tradition of mathematics is taught as though it exists outside of human history and human culture. A recent initiative in math education (Culturally Responsive Mathematics Teaching) seeks to situate math culture in a deeper, more embedded context. In his whitepaper Knowing and Valuing Every Learner: Culturally Responsive Mathematics Teaching, Mark Ellis, Ph.D. writes:

The idea of culturally responsive mathematics teaching (CRMT) is premised on creating a learning environment focused on mathematical sense making in which each student feels valued for who they are, for their ways of engaging in mathematical reasoning, and for their contributions to the collective success of those within the classroom community.

Moreover, CRMT endeavors to create “a community of mathematics learners who value collaboration and see mathematics as a way of reasoning with and about quantities.” Ellis explains that it is about “inviting all students into mathematics as competent participants whose ways of thinking and reasoning are worth sharing, discussing, and refining.” Finally, this approach is about “ensuring each and every learner not only has success with mathematics but also comes to see mathematics as part of their identity and as a tool for examining their world through a quantitative lens.”

This portion of the white-paper reminded me of an education dialogue with the philosopher Alasdair MacIntyre. In that dialogue, MacIntyre writes:

The teacher should think of her or himself as a mathematician, a reader of poetry, an historian or whatever, engaged in communicating craft and knowledge to apprentices.

For MacIntyre, math culture is about inviting students to see themselves as already participating in the “real thing.” He explains that in a quality education setting:

Students already become practitioners of arts, sciences and games, participants in such activities as reading novels and poetry with both discrimination and intensity, devising new experiments in which their mathematical skills can be put to use, drawing and painting and making music to some purpose.

Seeing our students as budding mathematicians who, even at their earliest ages, are participants in the grand tradition of mathematics requires recognizing that our students need to be active in their own learning. Ellis explains in the whitepaper that “students bring with them ways of thinking and reasoning with and about quantities and quantitative contexts—as well as ways of interacting and communicating—that serve as a foundation upon which to build their understanding of mathematics.” We agree wholeheartedly, and believe that a turn to classical sources in mathematicians (especially Euclid and Plato) help to keep this truth front and center.

To help us as educators to recognize and cultivate the preexisting mathematical reasoning in our students, Ellis gives us some helpful questions to ask ourselves:

• What are the “big ideas” of mathematics my students will learn this unit or year?
• What prior mathematical knowledge will my students need to make sense of these big ideas?
• Am I giving students sufficient time and resources to develop their own mathematical thinking?
• How will I communicate that their thinking is what matters, not just their answers? What language will I use?
• How am I ensuring students are making sense of the concepts and learning with coherence?
• Am I using Instructional routines that encourage both individual think time and partner/whole class discourse?

Nevertheless, though it is important to recognize the mathematical reasoning our students bring to the table, Ellis notes:

Equally important is that teachers allow students to engage with multiple representations of mathematical concepts and relationships, comparing and contrasting these as a means to deepen understanding.

As an example of what this looks like practice, Ellis highlights a teacher who “required students to explore four representations of decimals (verbal, numerical, base-ten blocks, and coins), looking for and explaining connections among them.”

[The recognition that students need to learn multiple modes of representation informs our pedagogical approach and is why we encourage students to “Build, Write, Say, and Teach.”]

Ellis also stresses that:

While understanding mathematical concepts and relationships well enough to demonstrate proficiency with state-mandated assessments is important, it is even more critical for students to learn to use mathematics as a tool for investigating and critiquing issues within their communities.

Similarly to that earlier quote from MacIntyre, Ellis writes that “too often ‘school mathematics’ becomes compartmentalized and seen as something relevant only for an assessment or grade.” This reminds me of an insight from Joseph Aoun, the president of Northwestern University, from his book Robot-Proof: Higher Education in The Age of Intelligence. Aoun warns that without an integrated and conceptual education, students “may find themselves overly dependent on familiar contexts and inflexible to new applications. They also make lack a deep understanding of their domain, knowing the what but not the why. This blinds them from seeing how their knowledge could be utilized in a different setting.” Thus, for Aoun, the goal is to teach conceptual mastery, the “why” of education, not just the how, because doing so will allow students to “take the components they have integrated and apply them to complex, living contexts.”

Ultimately, Ellis’ paper serves to remind us that our goal as educators is to empower our students to “see mathematics as a tool for analyzing the world in which they live.” And he notes that teaching math in alignment with this goal will also “strengthen students’ interest and engagement in mathematics.” In the final portion of the interview with MacIntyre, he similarly reflects: “the test of the curriculum is what our children become, not only in the workplace but in being able to think about themselves and their society imaginatively and constructively, able to use the resources provided by the past in order to envisage and implement new possibilities.”

A recent initiative in math education seeks to situate math culture in a deeper, more embedded context.

Abacus and Algorithm: A History of Math

A scholar introduces us to Gerbert, who is responsible for introducing algebra and the place-value system to medieval Europe by way of the abacus.

We often teach mathematics as if it were ahistorical, as if all mathematical insights – from the concept of zero to theories of infinite sets – were known always and everywhere. Studying the history of math allows us to see mathematics as an ongoing conversation that has its roots in the earliest of human civilizations. In her book The Abacus and The Cross, scholar Nancy Marie Brown introduces us to a medieval mathematician named Gerbert (later Pope Sylvester II), the man responsible for introducing algebra to medieval Europe and for giving us the place-value system that we still use all these centuries later.

The Quadrivium

Brown notes that during his days in a French monastery, Gerbert was instructed in the quadrivium, the four-fold foundation of knowledge in arithmetic, geometry, astronomy, and music that had been formalized by Alcuin of York during Charlemagne’s reign. In these studies, Gerbert was heavily influenced by the thinking of the early philosopher Boethius who taught on the musica mundana: the Music of the Spheres. For Boethius, the turning of the astronomical spheres produced a heavenly music that held the stars together. This music was believed to parallel the musica humana which held together the body and soul. This concept extends forward beyond Gerbert and finds itself most famously recorded in Dante’s Paradiso wherein he writes of “the Love that moves the sun and the other stars.” In short, for Gerbert, the qadrium offered an integrated view of the cosmos and of man’s place in that cosmos.

The Abacus

Gerbert used visual aids to teach the quadrivium, including a wooden sphere to show the motion of the planets, an elaborate abacus, and, the one-stringed instrument known as a monochord. An abacus was an elaborate counting machine, first pioneered by the Romans (see below), and the rudimentary basic for both the calculator and the computer.

Ancient Roman Abacus

Gerbert used a shield-maker to assemble a light, counting board version of abacus that he used to teach his students. Notably, because Gerbert had studied in Muslim Spain, he used Arabic numerals. What are Arabic numerals? They’re the notation you use in everyday math: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Prior to Gerbert, Roman numerals were used, or letter notation.

Introducing Arabic numerals was not the only lasting contribution Gerbert made. As Brown explains:

Gerbert’s abacus board introduced the place-value method of calculating that we still use today.

Specifically, each column represented a power of ten. “The ‘ones’ column was placed farthest to the right, and the numbers increased by a multiple of ten.” Incredibly, “with twenty-seven columns, Gerbert could add, subtract, multiply, or divide an octillion (1027).” Gerbert called these columns “intervals,” which was likely an allusion to Boethius’ use of the word in music theory. He also called any number put in the first column a “digit” which is Latin for finger. Gerbert gives detailed instructions for how to count on your fingers, but those instructions will likely strike you as strange: for example, “when you say one, bend the left little finger and touch the middle line of the palm with it. When you say two, bend the third finger to the same place.”

As another fun but important aside, one of Gerbert’s near contemporaries, Ralph of Laon, helps to explain why then (as now), counting begins with the “ones column.” Trying to explain the concept of zero, Ralph of Laon writes: “even though it signifies no number, it has its uses.” It was only much later that zero as a concept came to signify a number. In Gerbert’s mathematics, zero is mostly a placeholder, whereas for us zero signifies an actual number. Aside from this shift in thought, Brown writes, “For a thousand years, we have added, subtracted, multiplied, and divided essentially the same way Gerbert taught his students at the cathedral school in Reims.”

Adelard’s Algorithm

The main transition between Gerbert’s math and our own was the introduction of Adelard’s algorithm in the in the 12th century. Brown writes:

The step from Gerbert’s abacus to Adelard’s algorithm was very small. The main difference between the two ways of calculating was that the pre-drawn column lines of the abacus board disappeared; they were no longer needed once the place-value system was fully understood. In the algorithm, placement on the page alone distinguished a one from a ten or a hundred, and the use of zero to fill the empty space became standard.

Ralph of Laon was right: zero really does have its uses. And in case the word algorithm caught your attention, note that it was this development in mathematics (first initiated by the Muslim scholar al-Khwarizmi, from whose name the word algorithm is derived) that eventually paved the way for binary coding, and thus for the algorithms that power our computers and smartphones.

I hope this brief overview of Gerbert’s mathematics has piqued your interest in the history of mathematical thought. If you want to introduce your student to more history of math, check out The Story of Mathematics: this excellent online resource traces the story of math in ancient Sumerian and Babylonian culture all the way through to important developments in the 20th century.

A scholar introduces us to Gerbert, who is responsible for introducing algebra and the place-value system to medieval Europe by way of the abacus.

Euclid on Playing with Math Manipulatives

"Things which equal the same thing also equal one another." - Euclid

Imagine the following scene:

A three year old child (let’s call her Susie) is playing with blocks of varying lengths on the floor. She carefully places one block on top of another, trying to find blocks that match in length.

If you ask Susie what’s she doing, she’ll say, “I’m just playing.” But actually, she’s learning mathematics! And I think if you asked Euclid (one of the world’s first mathematicians and the father of geometry) he’d say she’s learning geometry the right way.

Consider this basic principle from Book 1 of Euclid’s groundbreaking text Elements:

Things which equal the same thing also equal one another.

It will likely be many years before Susie will be able to read Euclid’s book and understand what he meant in that statement (listed as “Common Notion 1” in most translations, including this online one.) And yet, when Susie takes block A and places it on top of block B, sees that they match, and then takes block B and places it on block C, she is building a concrete and experiential knowledge of the very principle Euclid is stating here. In high school, knowing this principle as an abstract concept will prove useful in balancing ratios and solving algebraic equations, but the concrete exploration that Susie is engaged in with the blocks on the floor will shape her intuition and make learning algebra much easier.

In Plato’s Meno dialogue, Socrates draws geometric figures on the ground in order to demonstrate to an unnamed slave boy how to find twice the area of a square. It is important that Socrates is teaching an uneducated slave: Plato is showing his audience that math is universal and accessible, and thus that mathematical knowledge is not limited to the formally-educated elite. When Socrates draws geometry in the dirt, the slave boy exclaims “I see!” – because math is fundamentally about seeing the world around us and coming to know that world as exciting and intelligible.

Abstract math rooted in symbolic language (numbers, letters, division signs, etc.) is important. For example, without the discipline of calculus, we wouldn’t be able to do the kind of advanced physics needed to send rockets into space. And yet none of us can begin with calculus: we all start our math education by feeling our fingers and playing with blocks – slowly intuiting geospatial concepts like “longer and shorter” and “more and less” well before we have learned the language needed to turn those intuitions into structured thought.

The Math-U-See approach takes seriously the insight of Euclid and Plato. We recognize that building conceptual mastery requires intuition and formal language, concrete visualization and abstraction. This is why at every beginner and intermediate level of Math-U-See, from Primer through Algebra 1, we use manipulatives to teach everything from 1 + 1 to how to factor a polynomial or solve an algebraic equation.

Alasdair MacIntyre on Improving Math Education

Math education is often seen as preparation for “real life,” where “real life” is seen as an important thing and math education just seems…boring.

Alasdair MacIntyre is one of the most celebrated contemporary philosophers. His 1981 book After Virtue is widely seen as having launched a renewed attention to virtue-ethics in moral philosophy. Throughout his career, MacIntyre has also written a lot about education. I recently discovered a dialogue between MacIntyre and Joseph Dunne on education, and wanted to share some highlights that I think mesh well with our curriculum.

One main idea that is found in all of MacIntyre’s writings is the link between action, practice, tradition, and narrative. Consider the pianist: every day she sits down to work on her scales (action) as part of her participation in the art of piano-playing (a specific practice) in order to learn to play the works of Chopin (the tie to tradition.)

For MacIntyre, action, practice, and tradition are found in the context of community: the pianist, her teacher, her audience, the musicians like Bach who have authored the pieces she is learning to play. Narrative, the story we tell about ourselves, is what makes the context of our actions intelligible and meaningful to us. When the pianist says, “I am a piano player,” she is situating herself within a narrative in which Teacher and Fellow Musicians and Audience Members all have a significant role to play in the formation of her own identity.

In the dialogue, MacIntyre writes:

the teacher should think of her or himself as a mathematician, a reader of poetry, a historian or whatever, engaged in communicating craft and knowledge to apprentices.

In the context of Math-U-See, we want both our parents and students to see themselves as budding mathematicians who are tied to a tradition of mathematics and who are honorary members of a community of mathematicians made up of both those living and those like Euclid, Descartes, and Cantor, who have paved the road we travel on. The exciting truth is that you don’t need to be a math expert to teach math to your kids with our curriculum.

As Gretchen Roe writes:

The good news is that age is not a factor in learning the basic facts; and Math-U-See, with its integer blocks as the central point of instruction, gives any learner an ability to commit those facts to memory.

We don’t believe that people are either a math-person or not a math-person: instead, we cite the research of Stanford University professor and researcher Carol Dweck on growth mindset, and we recognize that just like we become virtuous by practicing virtue, we can all grow in our mathematical knowledge by practicing mathematics.

Most of us have at one time or another said, “Why do I have to learn algebra anyway?!

One reason that this frustration can arise is because math education is often seen as something we are doing as preparation for “real life,” where “real life” is seen as the weighty and important thing and math education just seems…boring.

MacIntyre helps us see why picturing math education in this way is harmful and demoralizing. He writes:

our conception of the school is impoverished if we understand it as merely a preparatory institution, within which the students are contained until they are ready to participate in ‘the real thing.’

Instead, math education is about inviting students to see themselves as already participating in the “real thing.” MacIntyre explains that “…in good schools students already become practitioners of arts, sciences and games, participants in such activities as reading novels and poetry with both discrimination and intensity, devising new experiments in which their mathematical skills can be put to use, drawing and painting and making music to some purpose.”

As exciting as it is to be budding mathematicians, let’s also be honest: learning basic math facts like the multiplication table can sometimes be a dry and wearisome task. Our curriculum development team works hard to spice it up with fun activities, but we recognize that sometimes, especially at the early stages, it just takes some grit to get through it.

In the dialogue, MacIntyre writes:

…small children are able to learn and to exercise some skills while participating in enjoyable and purposeful activities, such as singing or playing some music instrument. But even in those activities there are levels of achievement that require what you call inescapably laborious drills.

But crucially, this labor needs to engage our students and part of that engagement comes through having a multisensory approach. MacIntyre writes: “Children also need to learn to see, to hear, to touch, to taste and to smell; that is, to discriminate the object of the senses and remember what has been thus identified. Consider the varieties of visual object and what has to be learned in order to see what is actually there…”

We agree wholeheartedly.

As Chloe Chen writes: “Well-designed, hands-on activities in the classroom foster connections to real-world situations and increase learner engagement.”

We also want to stress that math manipulatives aren’t just for “little kids”. Our curriculum uses manipulatives to teach everything from 1 + 1 to demonstrating how to factor a polynomial or solve an algebraic equation.

In the final portion of the interview, MacIntyre reflects:

The test of the curriculum is what our children become, not only in the workplace but in being able to think about themselves and their society imaginatively and constructively, able to use the resources provided by the past in order to envisage and implement new possibilities.

This vision of education is strikingly congruent with that of Joseph Aoun, president of Northeastern University. In his book on education, Aoun writes that we need to: “…move beyond the canard that students must choose between an economically rewarding career and a fulfilling, elevated inner life. More than ever before, the capacities that equip people to succeed professionally are the same as the virtues espoused by Cardinal Newman in his paeans to ‘liberal knowledge’ – namely an agile mind, refinement of thought, and facility of expression.”

Math education is often seen as preparation for “real life,” where “real life” is seen as an important thing and math education just seems…boring.

The Lost Tools of Math: Recovering Math in a Classical Education [Article]

When it comes to classical education, mathematics is often the Cinderella sister.

When it comes to classical education, mathematics is often the Cinderella sister: she lives in the home along with her sisters Literature, Language and History, but she’s not really considered part of the family. There are at least two main reasons for math’s odd position in classical education.

First, for us moderns it can be hard to think classically about math as our worldview is often decidedly different from the ancients. Moreover, most of us do not approach math education like Abraham Lincoln who worked his way through Euclid’s The Elements as a self-taught unschooler. And even if we did approach math education in this way, we recognize that innovation in mathematics didn’t end with Euclid and the Greeks, and there is a need to combine study in Euclidean geometry with study in other fields of mathematics, including algebra and calculus.

The second reason math doesn’t fit neatly within classical education is because the classical education movement, while being rightfully skeptical about 20th century math education, has not been critical enough about the lack of emphasis on mastery in the 20th century models.

As will be explored in more depth in this article, a trend in one particular strain within modern American Christian classical education, inspired by Dorothy Sayers’ essay The Lost Tools of Learning, is to divide math into grade levels that are thought to correspond with both the classical trivium and child development theory. In this model, elementary kids are grouped in the grammar stage and drilled in math, middle schoolers are put in the logic stage and taught the abstract symbolism in which mathematics is expressed, and high schoolers are placed in the rhetoric stage and taught the conceptual “whys” of math. One problem with this approach is that it isn’t faithful to the classical trivium model. Perhaps more importantly, the assumed child development theory it is based on is flawed.

In this article, I will survey the history of math education, demonstrate why the Sayers approach to math is in need of modification, and explore an alternative model for classical math education that draws from the insight of deeper sources than Sayers.

History: How Has It Been Applied?

Classical education has its origins in the interplay of hellenistic and Christian thought, culminating in the formal establishment of the trivium and quadrivium in the university model that dominated the Middle Ages.

Many of the ancient Greeks promoted the idea of four major subjects (later identified as the quadrivium), though recall that Plato infamously advocated for the banishment of poetry (which would have included Homer’s The Odyssey, by the way). It wasn’t until the Carolingian Renaissance in the eighth century that the trivium was formally established as the foundation for learning upon which the quadrivium should build.

It was during this time that the concept of a “liberal arts education” composed of seven subjects came to fruition under the influence of Charlemagne’s advisor Alcuin of York (Source). Alcuin’s program makes obvious sense. The texts used in the more specialized technical fields (quadrivium) were all written in classical Latin and Greek, so fluency in Latin and Greek (“grammar”) was a first requirement, followed closely by proficiency in rational argumentation as the preferred method of learning in the university classrooms (“logic” & “rhetoric”).

The modern classical education movement has largely been inspired and influenced by the 20th century essayist and educator Dorothy Sayers and her essay “The Lost Tools of Learning.” In the 20th century, math education in the West shifted its emphasis from geometry to algebra. This shift reflected the modern philosophy of educators such as John Dewey. Whereas in England, the norm had been for students to study Latin and Greek, the latter language preparing them for Euclid and thus geometry, the new education model concerned itself exclusively with practicality and usefulness. Another aspect of this new shift in education was that 19th-century and early 20th-century educators often mistakenly conflated rote memorization with deep understanding. Sayers’ essay is largely a response to both this increasing fragmentation of education and the corresponding lack of attention paid to history and the wisdom found in classical texts.

Current Times: How It Is Being Misapplied

Dorothy Sayers provides a helpful summary of the ordering of the trivium. The student first learned “the structure of a language” by which she meant learning “what it was, how it was put together, and how it worked.”

Secondly, the student learned “how to use language” which included “how to define his terms and make accurate statements; how to construct an argument and how to detect fallacies in argument.” Finally the student learned “to express himself in language—how to say what he had to say elegantly and persuasively.”

The problem comes when Sayers attempts to tie the trivium in her signature “rough-and-ready fashion” to a theory of child development that she has invented based on anecdotal observation and her memory of being a child. She calls her three stages the Poll-Parrot, the Pert, and the Poetic: in the parrot stage, the child memorizes and repeats back, much like a parrot, while in the Pert stage, the (preteen) starts to talk back (arguing as a form of learning), and in the Poetic, the (misunderstood) teen learns to harness creativity leading to self-expression. This description is very much not a caricature of Sayers’ writing, but if you don’t believe me, just read her essay.

If it sounds like I’m being unduly critical of Sayers, it’s worth noting that she herself admitted her limitations, writing that “my views about child psychology are, I admit, neither orthodox nor enlightened.” And as Martin Cothran, an editor for Memoria Press, writes:

Nor did Sayers herself ever explicitly identify her states of development as classical education—in her speech or anywhere else. In fact, the term ‘classical education’ does not even appear in her essay.

This isn’t to say that Sayers’ essay is not valuable, but rather to say we can and should be critical of it in those places where it serves as an impediment to actual learning.

Unfortunately, Sayers’ error of putting K-12 students into these three boxes has been repeated by many in the modern classical education world. For example, in her book on classical education, Leigh Bortins, founder and CEO of Classical Conversations, writes of “grammar school students” preparing to move into “dialectic, abstract math” and advocates for an emphasis on drilling and memorization for those grammar students. This error suggests that students can be grouped into trivium-based stages, such that elementary kids are drilled in “grammar” (math facts) but not given the opportunity to benefit from the dialectical and rhetorical aspects which would allow them to fully master math, both conceptually and practically.

In practice, this means that children’s’ first experience of math is sheer memorization that lacks any meaningful context and that is divorced from conceptual development. Teaching elementary students to memorize math facts without the underlying context is like having them memorize Latin words without learning what those words mean. To be clear, drilling and repetition is important for learning math because it can increase fluency and simplify a student’s ability to reach solutions as math concepts become more complex, but it is not sufficient as a standalone method regardless of the student’s age or development level.

In many ways, curriculum corresponding to the Sayers’ Parrot/Grammar stage, such as Saxon Math, is what has replaced Euclid in modern math education, in public school as much as in many private and homeschool settings. Whereas Euclid’s model in his book The Elements is to begin with axioms and build conceptual knowledge in the form of proofs following from those axioms, the Saxon model has students learn procedures in order to manipulate equations with the goal of producing answers. Neither model is fully sufficient. The Euclid model, at least as it is often practiced in Great Books programs, is akin to a chemistry student who reads a textbook on the periodic table but never participates in a lab experiment. The Saxon model is akin to a student learning how to replicate a chemical reaction in a lab without ever learning about those chemicals and why their combination produces the reaction it does.

Summary of the Proper Approach

While Bortins repeats Sayers’ error, she also recommends a model that, inasmuch it modifies Sayers’ approach, is much more effective. In this model, students emulate all three aspects of the trivium through the course of working through a single math problem. She writes:

Each problem provides a micro-example for practicing the skills of learning. The students demonstrate that they have mastered the math terms used (grammar) and that they understand the rules and strategy of the problem so they can solve the problem (dialectic.) Finally, they explain how they solved the problem rhetorically, demonstrating that they understand the algorithm.

This is remarkably congruent with the model we employ at Math-U-See which encourages students to “build it, write it, say it, and teach it back” as the method for both achieving and demonstrating mastery of each new concept.

Grammar (the language of math), logic (the why of math), and rhetoric (the application of math) are three distinct but complementary and intermingling disciplines, and each of these disciplines is worth studying. If we are to teach math classically, we ought not attempt to split the trivium into fragments, but to tap into its power as a cohesive and unified foundation of knowledge. The best way to model math education on the trivium is thus to utilize each discipline (grammar, logic, rhetoric) for each concept being taught.

The Enduring Value of Classical Education

In the context of this article, I see two main benefits to classical education:

1) Understanding historical ideas in their historical context.

In his beloved essay “On The Reading of Old Books” C.S. Lewis points out that while ancient writers likely had many mistaken beliefs, it is much less likely that they share the same mistaken beliefs that we have. In other words, since writers of our own age are likely to share the same blind spots, reading old books can help us see and correct errors in our own thinking.

Math-U-See doesn’t really offer this benefit, but neither does any homeschool math curriculum so far as I know. As an example of what I have in mind, there are curricula that teaches science through history including Susan Wise Bauer’s The Story of Western Science and Jay Wiles’ new elementary science curriculum.

Of course, for those educators or students who are audacious enough to try it, Euclid’s The Elements is readily available both in print and online, and often is published with illustrations making it more accessible. You can also supplement your math with intriguing books on history such as The Abacus and The Cross which profiles Gerbert of Aurillac, a medieval Pope and one of the leading mathematicians of his day.

2) Correcting a 20th century blind spot by restoring emphasis on mastery and promoting a more unified system of knowledge.

For the medieval university student, mathematics was part of an assumed cosmology, which is to say that every discipline was understood as contributing meaningfully to a unified view of the world and of humanity’s place within that world.

The modern approach to education, which Sayers and classical educators rightfully critique, emphasizes only the practical usefulness of a discipline. In addition, through over-specialization, modern education leads to a body of disconnected ideas and knowledge rather than offering unity. The model I am proposing for teaching math classically does not disconnect the trivium or have students specialize in one aspect of it as part of putting them in a box or stage. Instead, my model says no to the modern impulse to compartmentalize, specialize, and fragment by emphasizing the unity of the trivium in each moment of math education, from the micro level of solving a specific math problem to the broader level of learning a new concept to the macro level of seeing how mastering geometry provides a foundation for trigonometry.

Aristotle took it as axiomatic that all of us, by dint of our very human nature, have a desire to know: his teacher Plato believed that all true knowledge has its origin in wonder. I believe that a classical approach to mathematics that builds conceptually, moves at the pace of the student, and is oriented around an emphasis on mastery is the best model for tapping into our students’ built-in desire to know in a way that connects them to the experience of wonder, and cultivates in them a love for learning for its own sake.

This was originally published in Digressio.

When it comes to classical education, mathematics is often the Cinderella sister.

Mathematicians and Historians Need Each Other

The connection between models in the humanities and mathematical models is deeper than it may appear.

“It isn’t easy to be a citizen in 2018.” So begins Ted Underwood’s essay on machine learning. Underwood continues: “we are told to watch out for bots and biased search engines, but skepticism about new media can also make us easy prey for old-fashioned propaganda.”

For Underwood, a central aim for education needs to be preparing “students for a world where information is filtered by computers” and he argues that to do so, “we will need a stronger alliance between the humanities and math.” The rest of his essay explores how “this alliance has two reciprocal parts: cultural criticism of the mathematical models shaping our world, and mathematical inquiry about culture.”

Humanities and STEM?

The question of the relationship between the humanities and STEM is nothing new. Indeed, undergrads still regularly read texts like CP Snow’s classic 1959 essay “The Two Cultures,” which argued that the humanities and what we now call STEM are fundamentally divided and unable to speak to each other. Underwood subverts Snow’s thesis arguing that the only viable path forward for us is one that integrates the disciplines. We might intuit that humanities can help, well, humanize, STEM, but we might ask, “why does human culture need math? Underwood explains that “The challenges that confront 21st-century citizens are not always arguments that come one by one to be evaluated. Information is more likely to come in cascades, guided both by networks of friends and by statistical models that anticipate our preferences.” As a result of this cascade of information, Underwood argues that “Evaluating sources one by one won’t necessarily tell us whether these computational and social systems are giving us a biased picture. Instead, we need to think about samples and models—in other words, about math.”

The Connection Between Models

The connection between models in the humanities (i.e., a novel as a model of the human condition) and mathematical models is deeper than it may appear. Underwood illustrates this point with algorithms. He explains that in 21st century computing…

Instead of manually writing algorithms that directly govern a computer’s decisions, we often ask computers to write their own instructions by modeling the problem to be solved.

To illustrate this, he explains how email filters spam:

Undesirable email comes in many different shapes, and it would be hard to write an algorithm that could catch them all. A more flexible approach begins by collecting examples of messages that human readers have rejected, along with messages they approved. Then we ask the computer to write its own instructions, by observing differences between the two groups.

In other words, algorithms are models that continually update themselves to function better. This form of computing is known as machine learning, and as Underwood explains, “machine learning increasingly shapes human culture: the votes we cast, the shows we watch, the words we type on Facebook all become food for models of human behavior, which in turn shape what we see online.” He then notes that “since this cycle can amplify existing biases, any critique of contemporary culture needs to include a critique of machine learning.” That latter point means that just as we have a tradition of literary criticism that digs into the stories we tell in order to critique aspects that our problematic in service to helping us become better moral agents or citizens or lovers, so also we need to critique the algorithms that, for example, reinforce unhealthy and addictive behavior on social media. And since the English department has been a place where this criticism has flourished, programmers might learn a lot from taking some English classes, and on the other hand, English majors might benefit from taking a programming course and learning to think in code as another language.

Integrating Humanities and STEM

Integrating the humanities and STEM is not just helpful for ethics. The integration can also help us better understand and interpret the world: put differently, the integration can help us move closer to truth. Underwood writes that “to be appropriately wary, without succumbing to paranoia, students need to understand both the limits and the valid applications of technology.” And he suggests that “humanists can contribute to both halves of this educational project, because we’re already familiar with one central application of machine learning—the task of modeling fuzzy, changeable patterns implicit in human behavior.”

Now, the models of the world produced by the novelist may strike as “slippery and unscientific,” but Underwood asks us to recall that “machine learning can also be slippery and unscientific.” He writes: “Remember that we resorted to machine learning because we couldn’t invent a simple, universal definition of spam. Instead, we had to draw on the tacit knowledge of human readers who had rejected email for a range of reasons. A model based on this sort of evidence will never be stable. It will have to be updated every few years, as old scams die out and new ones emerge.” In other words, while total objectivity is impossible, some models are better than others at getting us closer to truth, and learning to discern between models is an essential part of a well-rounded education.

The integration that Underwood recommends is also essential for dealing with untrustworthy politicians and political pundits. Underwood explains that “tech leaders who argue that machine learning is more objective than other knowledge cannot be trusted. But we should just as fiercely distrust political leaders who use the perspectival complexity of the internet to imply that real knowledge is impossible, everything is fake, and we can only fall back on affinity and prejudice.” He stresses that “it is possible to build real knowledge by comparing perspectives from different social contexts. Historians have long known how.” The good historian sifts through all sorts of past culture – religious tomes and tax records, political treatises and recovered coinage – to figure out which accounts of history are most plausible. That skill-set parallels the work needed to sift through scientific studies, for example, to determine what is most plausible.

Underwood concludes his argument by reminding us that in order to understand ourselves and our world “we will need numbers as well as words.”

Further Reading

Underwood’s writing reminds me of two books that have shaped my thinking on this theme.

Actual Minds, Possible Worlds explores the power of literature as a pathway to this important ability to imagine new possibilities.

Robot-Proof: Higher Education in The Age of Intelligence blends theoretical and practical.

The connection between models in the humanities and mathematical models is deeper than it may appear.

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