It is something of a pastime for political pundits to unfavorably compare America to Sweden. Whether it’s healthcare or parental leave or early childhood education, America is often made out to be backward-minded and inept. Certainly, there are things we could learn from Sweden, but it turns out that freedom-loving America might have something to teach Sweden too.
Gabriela Maya is a Texan transplant to Sweden. In her op-ed in the Houston Chronicle, Maya praises Sweden’s generous social policies but she argues that Sweden should follow Texas’ lead in learning to trust parents to make good decisions for their families.
In the midst of the Covid-19 pandemic, many people have looked to Sweden for ideas on how to respond as a society. Unlike in America, Sweden did not implement compulsory stay at home orders. Maya writes that
with no lockdown restrictions in place and only a few and relatively mild guidelines from the authorities, Swedes are free to soak in the sun at outdoor cafes and parks, or to fill shopping malls, get haircuts and go to the gym, as long as they don’t gather in groups larger than 50.
The Swedish government trusts its citizens to use common-sense to navigate risks and protect each other. It is this approach that leads Maya to observe that “Swedes are suddenly a model for freedom-loving Texans.”
Homeschooling in Sweden?
So far so good. “But there’s a catch”, writes Maya. “Unlike in Texas, authorities have kept elementary and middle schools open, and worried parents are now facing legal repercussions if they insist on keeping children home.” Homeschooling is not legal in Sweden, and even in response to a highly contagious and deadly virus, the Swedish government refuses to trust parents. Maya underscores the irony: “We can choose if we want our eyebrows plucked, but we can’t choose to protect our families from an all-pervasive danger by keeping our children home.”
Maya writes that “Texas gave me the freedom to choose whether my kids learned at school or at home.“ And she adds that she feels “a great longing for the freedom and responsibility I was granted as a parent in Texas.” I like this pairing of freedom and responsibility, and I think that societies flourish when both are emphasized in equal measure. In my own life, I have advocated for policies in both the private and public sectors that are rooted in both freedom and responsibility.
In 2012, I traveled to Berlin to participate in the first International Congress on Home Education in Germany. As a participant, I signed the historic Berlin Declaration calling for all international governments to recognize “the essential, irreplaceable and fundamental role of parents and the family in the education and upbringing of children as a natural right that must be respected and protected by all governments.” I believe the words of the Declaration so deeply because I watched my own father Steve Demme advocate for homeschooling here in the United States in the early 80s, and also because I know how much homeschooling has benefited my brothers and me, as well as countless other students.
It shouldn’t surprise you that as much as I’m a fan of homeschooling, I am more of a fan of trusting parents to make the best education choices for their children. This trust of parents animates our vision here at Demme Learning. We are in the business of building stronger families and we sell excellent educational products to achieve that goal.
To learn more about our vision for empowering parents in education, check out these videos from our past #TrustParents conferences:
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.
We are here to help; feel free to contact us with questions. We can help each other and together we will get through this.
Kids ARE Writing…
Most of us want our kids to have basic writing skills, yet we often place too much emphasis on what medium our kids should be proficient in. We should focus more on what they are actually writing.
In today’s 21st century world, ‘yours truly’ has given way to TTYL (talk to you later). Our kids don’t write letters; they send texts. They don’t keep pen-and-paper journals; they blog. It’s not that our kids aren’t writing; in fact, if you count Facebook status updates (many of which are quite creative), social media messaging, and texting, our kids are writing all the time.
A brief look at history shows that the trend is for the physical act of writing to become easier and easier while the cost becomes decreasingly expensive. Consider how much it cost, both in time and money, to write even one book on a scroll. Only that which was important was written on scrolls, and the vast majority of people couldn’t even afford to buy scrolls. Writing belonged to the elite as did the possession of scrolls, which is why scholars were the only educated people at the time.
With the invention of the printing press, books became readily available and the cost of making a book was exponentially lessened. As a result, the trivial was published right alongside the important. Living in the digital age, one can write almost effortlessly and for only the price of a laptop/tablet/smartphone and internet connection. The ratio of “good” writing to fluff now seems to have completely reversed from the days of clay tablets and scrolls.
Most of us are familiar with the phenomenon of “mommy bloggers.” Many women have found blogging to be a creative outlet and will blog at night while her kids sleep. Many teens blog too, and with new online publishing mediums appearing all the time, there are plenty of creative outlets online.
If we consider how many Facebook messages our kids are sending, how many texts they send in a month, and how many Facebook status updates they write in a single week, it becomes apparent that getting them to write is not the issue – teaching them to write well in different mediums is the tricky part.
We need to teach our children the universals of writing; principles that transcend the limitations and structures of each medium. If we teach our kids how to write clearly and concisely, they can successfully learn to write in any medium. In fact, the greatest thing we could ever do is teach our kids that it’s a valuable skill worth developing. If we do that, our kids will be more likely to grow in their ability.
We as parents need to encourage our children to write in whatever medium they naturally gravitate towards. Once they have developed a love of writing, we can challenge them to experiment in different mediums and genres.
Writing to Reflect
Some people start a diary as casually as they start golf, stamps, or a new digestive cure. Whereas to start a diary ought to be a solemn and notable act, done with a due appreciation of the difficulties thereby initiated. The very essence of a diary is truth—a diary of untruth would be pointless—and to attain truth is the hardest thing on earth. To attain partial truth is not a bit easy, and even to avoid falsehood is decidedly a feat. – Arnold Bennett
Whether you prefer to use a diary, journal, or other medium, writing to reflect is an important discipline to develop.
As someone who had tried to keep a journal knows, the key word here is discipline, because it takes a LOT of self-discipline to main this habit. Reflective writing is about taking time to regularly process and record your life experiences, to find meaning, and to preserve your memories.
An exciting and relatively new field in the world of counseling is writing therapy. Research has found that “writing about traumatic, stressful or emotional events has been found to result in improvements in both physical and psychological health.”
Whether it’s writing stories, recording memories in a journal, or expressing emotions through poetry, expressive writing has the ability to help us heal, process emotions and thoughts, and relieve stress and anxiety. In addition to the health benefits, physical and emotional, journaling is also a fun recreational activity!
Besides the benefits of keeping an individual journal and encouraging our kids to do the same, keeping a family journal is also a great way to preserve memories and reflect on experiences together as a family. A family journal can be as simple as recording birthdays and noting important activities (such as a family vacation) or as detailed as writing about every single day. Family journals can also provide a written records for future generations to learn from and enjoy. We have a tradition in my family which we call a “thanks book.” Our goal is to look back at a previous week or month and record things that we are thankful for. My parents started this tradition, and it is still going a generation later.
3 Journaling Tips
1) Develop a Routine
Figure out what time of day works best for you and then develop a routine that lets you fit journaling into your busy schedule on a regular basis.
2) Don’t Stress
When you write, try to turn off the internal ‘editor’ and don’t stress out about writing “amazing” prose or making sure every word is spelled right and every law of grammar is obeyed. Instead, just think of writing as you would talking: You’re communicating, and the focus is on what is being communicated, not how it is being communicated (substance, not style).
3) It Will Feel Like a Chore
Remember that it is quite natural for journaling to feel like a chore. Most runners will tell you that it is often hard to get out of bed in the mornings and that they often loathe having to go for their morning run, but once they start running, they fall in love with it all over again and by the end of the run, they’re happy they resisted the impulse to skip it. The same is true of the discipline of journaling.
Writing to reflect is a habit that can benefit all of us, and, while it is often hard to get started, it’s well worth the effort. Make a conscious decision to start journaling and encourage your kids to do the same.
This classic essay, “The Diary Habit”, by Arnold Bennett, is a passionate defense of journaling, a comical essay that gets you laughing, and a thought-provoking discourse on the power of the written word.
Do Your Kids See You Write?
My wife, who is a real writer, unlike me, journals every day. She likes to sit at the kitchen table in the morning with a warm cup of coffee and write. Nicholas and I will often notice her writing or typing away on her laptop as we go about our daily tasks, and we’ve grown accustomed to seeing her write. We often talk about what she is writing about with her and offer our own suggestions and ideas.
It is important for our kids to see us write. If we make it a point to write where they can see, they will learn that writing is a valuable skill and valuable use of time. In addition, writing in front of our kids opens those doors for discussion like I mentioned above. Through discussion, our kids can take ownership of these values that we are communicating and can internalize those values.
Life is often very busy and it can be hard to make time to write. The first step in teaching our kids to value writing is to make the conscious decision to write and then to do it. It starts with an act of our will; by making the resolve to write, we’re making a conscious effort to think through how we can fit writing into our schedule.
Simply resolving to write or to just write when you think about it is not enough. We need to actively work to fit writing into our busy schedules — and the best way to do that is to develop a routine. Of course, creating a routine requires experimenting to see what will work best for you: maybe writing in the mornings as everyone is getting ready to face the day won’t work for you and your family — perhaps doing it in the evenings after supper might work better. The important thing is to develop a routine … and then see how it works. Success isn’t the activity, it is the communication of value.
Writing is important, and what better way to communicate that truth to your children than to model writing for them? You don’t have to be a great writer — you don’t even have to share your writing with your children — but by taking the time to write, you’re showing your kids that writing is a valuable skill. We have busy, hectic lives, but making time to write is a crucial activity that we cannot afford to skip — both for our children’s sake and for our own.
Research Says Writing is Important
Writing is one of those skills that is applicable nearly everywhere. Communication is at the center of all relationships, whether it’s emailing colleagues, texting family members, or writing letters to a friend. It’s not surprising then that research says it is an important skill to develop. What may be surprising, though, is the extent to which writing matters. For example, research shows that writing is key in creating opportunities for success in the workplace.
According to a report entitled Writing: A Ticket to Work…Or a Ticket Out, writing has become a needed skill to navigate the workplace. To quote an analysis of the report:
Writing is both a ‘marker’ of high-skill, high-wage, professional work and a ‘gatekeeper’ . . . People unable to express themselves clearly in writing limit their opportunities for professional, salaried employment.
Whether it’s writing an application, reporting problems, sending an email, or writing manufacturing documentation, writing is often an inescapable part of the workforce, especially for white collar jobs. Being able to write clearly and effectively is an important skill for getting good jobs, advancing in those jobs, and creating opportunities for career transitions.
The workplace is not only the place where writing is important. Writing is also important in the classroom. According to research shared in The Wall Street Journal:
Writing by hand is more than just a way to communicate. The practice helps with learning letters and shapes, can improve idea composition and expression, and may aid fine motor-skill development.” The article also shared that adults “may benefit similarly when learning a new graphically different language, such as Mandarin, or symbol systems for mathematics, music and chemistry.
There are also health benefits. In the world of counseling, a newer field has emerged called “writing therapy”. Research has found that expressive writing, such as reflecting in a journal, has therapeutic value and can help us cope with emotional trauma or even just the normal upheavals and stressors of life.
When people are given the opportunity to write about emotional upheavals, they often experienced improved health. They go to the doctor less. They have changes in immune function. If they are first-year college students, their grades tend to go up.
Benefits of Expressive Writing
• Fewer stress-related visits to the doctor
• Improved immune system functioning
• Reduced blood pressure
• Improved lung function
• Improved liver function
Writing is a crucial skill to develop and as the research shows, there are many benefits to becoming a good writer. Whether it’s reducing stress and boosting physical health, gaining a competitive edge in the workplace, or excelling in the classroom, it is important for our kids to learn how to write effectively.
A popular phrase says that “readers are leaders” and the same could be said of writers; equipping our kids with the tools they need to be good writers and communicating the value of writing by doing it in front of our children and talking with them about the importance will help prepare them to be the leaders of tomorrow.
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.
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.