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:
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.
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.”
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.
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.
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.”
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.
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.
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.
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.”
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.
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.