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The Dance of Number
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‘The Dance of Number’ Is the Way Forward for Teaching Mathematics in the Classical Christian Tradition

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I’ve been pouring over James Nickel’s work for the past two months and began vetting his new math curriculum three weeks ago. Since the first four books of his “Dance of Number” arrived I’ve been exploring its structure and going through the lessons. It wasn’t long before deciding it will be the centerpiece of the math education for our boys. I’ll explain why and also present an argument for the title of this article: Nickel’s curriculum is the way forward for teaching mathematics in the classical Christian tradition.

Not Your Father’s Math Textbooks, Please!

My relationship with math textbooks has been no love affair. Whether mathematicians can’t write or clarity is anathema to the profits of their publishers, the words in my books were as clear as mud. That mud trained me to skip right to the examples in the solutions manual (often written by someone else and purchased separately.) Whoever wrote the solutions manual couldn’t play games; they had to list the steps of the derivations. Whatever concepts I managed to grasp were incidental to the derivation steps in the solution manuals.

My “learning process” was devoid of historical context and practical application. Some of the science — made possible by the math — seeped into physics class. However, the only thing beautiful in the whole experience was a GPA that made it possible to get a job.

The Beauty of Math, Revealed

In contrast, Nickel teaches math thoroughly and takes pains to reveal the logic behind the concepts. Math is presented alongside the science, history, theology, and practical applications related to the lesson; and the integration is seamless.

The Dance of Number

None of the leading contenders for the precalculus stages of mathematics even attempt to do what Nickel has done.

I want our boys to have that keen sense of number you sometimes see in carpenters and engineers. The best way to do that is to provide context, application, and meaning to each building block. James Nickel has done this in beautiful sequence. I’m as excited to teach my sons as to relearn mathematics, myself!

 

The Way Forward

With the perspective imparted this summer by a slow read of David Hick’s “Norms & Nobility”1 it seems no risk at all to support Nickel’s curriculum as “The Way Forward.” I also thank Andrew Kern for his distillations of the many terms surrounding the pursuit of Classical Education.2

norms and nobility

Hicks defines a classical education as “a spirit of inquiry and a form of instruction concerned with the development of style through language and of conscience through myth.”1 These are penetrating words but require the context of Hick’s book to unpack and grasp fully.

On the other hand, Kern’s definition is a one-stop-shop:

“A Christian Classical Education is the cultivation of wisdom and virtue by nourishing the soul on truth, goodness, and beauty by means of the seven liberal arts and the four sciences so that, in Christ, the student is enabled to know, glorify, and enjoy God.”2

The Argument for “The Dance…”

On the jacket cover of “The Dance of Number,” Nickel makes lofty promises. He claims his curriculum:

  1. “Teaches mastery of number sense and algebraic syntax.”
    • It does. The student also learns how to use an abacus and an improved version of Stoddard’s speed math (having already mastered the abstracts of number.)
  2. “Integrates math themes with history, science, and personalities.”
    • It plainly does.
  3. “Coordinates beauty, truth, and goodness with rigor and heuristics.”
    • In a math curriculum? Yep.
  4. “Structures mathematics as an interconnected framework, and explores the dynamic interrelatedness of Arithmetic, Algebra, Geometry, Trigonometry, and Science.”
    • This is one of Nickel’s passions. He laments of so many students beginning Algebra before mastering Arithmetic. The error is compounded by attempting geometry and trig before learning Algebra. Problem solved in “The Dance …”.
  5. “Brings to light the multiplicities of the perichoretic nature of creation and mathematics.”
    • Perichoretic refers to the mutual indwelling nature of the Trinity and is the word that inspired the curriculum’s title. In “Mathematics: Is God Silent?” Nickel traces how impasses in mathematics were overcome by the Christian revelation of the Trinitarian nature of reality. The distinction between Creator and Created — when widely accepted — broke the Platonic spell and paved the way for the technological achievements of the middle-ages (which were anything but “Dark.”)

Mathematics, Is God Silent?In short, Nickel’s “Dance” does what no other on the market does; and the classical integration (or interpenetration as James might say) is seamless.

Flipping the Argument

Has any math curriculum you know of even attempted to do what Nickel’s has done?

How do the accomplishments, listed above, compare with the math curriculum deployed at your classical school?

A Whole New World

Do western students know how to use an Abacus? Why not? And what’s the harm in teaching Stoddard’s “Speed Mathematics” (which Nickel further streamlines with Vedic methods) as long as the student has a firm grasp of the fundamentals? These and a seemingly endless stream of surprises are in store for the student. Nickel draws from his 40-year teaching experience and 1400 volume library to show the student the context of the mathematical insights that shape our lives and unify our impression of the Divine Creator.

Nickel’s approach to teaching mathematics can impart that intuitive sense of number you sometimes see in carpenters and engineers. That’s not to say that mathematics is any less abstract than it always has been. But Nickel explicitly reveals its poetry and the stunning natural beauty upholding “The Dance.”

Those with a gift for math will be lit up at the beginning of their study like never before possible. Those less gifted can learn at rest, knowing that a logical and inspiring presentation is in store.

Thoughts on Implementation

This is not an assign-and-forget curriculum; it’ a “hands-on” journey for Teacher and Student to embark on, jointly.

The curriculum is recommended to start at age 12 (through 16). Therefore, a bridge is needed for younger students. Our 11-year-old is ready though we’ll be going through each lesson in tandem.

Nickel recommends the student read through each lesson with the teacher joining in when the student begins the exercises. The lessons are quite accessible but also what one might expect from a curriculum integrating mathematics with history, theology, science, and the beauty of practical applications: Deep with a capital ‘D’.

Until our youngest is ready for “The Dance of Number” we’ll be using Math U See. However, since dad is going through “The Dance …”, in advance, I’ll be able to verbally fill in gaps per Nickel’s framework.

The Dance of Number

Conclusion

Nickel seems to have used Kern’s definition of a Classical Christian Education as a specification. More surprisingly, his curriculum delivers on that specification. What do you do with something like this and whose first paragraph defines the word “Elohim”?

You’re reading my answer: adopt it as the centerpiece of the mathematics curriculum for your school and tell everyone you know about it!

James Nickel has given us the way forward in Mathematics! For imparting the subject in the classical tradition, there’s not even a close second out there to this monumental achievement.

  1. David Hicks, Norms & Nobility, a Treatise on Education, University Press of America, 1999  ↩
  2. Andrew Kern, Circe Institute’s “Definition of Terms.” ↩
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Why Mathematics Work
Creativity

Mathematician James Nickel on ‘The Unrelenting Issue of Intelligibility’

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James Nickel explains why mathematics work. Or, as scientists put it:  “The Unrelenting Issue of Intelligibility.”

He also describes why most mathematical breakthroughs (and mathematicians) are driven by the pursuit of beauty rather than utility.

How could it be that mankind is able to predict behaviors in the universe based only on abstract mathematical principles “invented” in his mind?

Could it be that mathematics is the language of God’s creation?

Nickel expands on this theme and topics in his excellent book, Mathematics: Is God Silent?

Even better, he’s finally fulfilled his life-long ambition to create a math curriculum that inspires the student by tying math with wonder, meaning, applications, & philosophy. He calls it “The Dance of Number.” Perhaps the myth of mathematics having no applicability to life and daily inspirition are finally over!

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leonardo da vinci
Creativity

8 Things You Don’t Know about Leonardo

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This year marks the 500th anniversary of Leonardo da Vinci’s death. Widely considered one of the greatest polymaths in human history, Leonardo was an inventor, artist, musician, architect, engineer, anatomist, botanist, geologist, historian, and cartographer.

Though his artistic output was small, Leonardo’s impact was great, reflecting his deep knowledge of the body, his extensive studies of light and the human face, and his sfumato (Italian for “smoky”) technique, which allowed for incredibly lifelike images. Leonardo regarded artists as divine apprentices, writing “We, by our arts, may be called the grandsons of God.”

Twenty-first-century scholars at MIT ranked him the sixth most influential person who ever lived. Like Rembrandt and Michelangelo, he is so renowned that he is known by only his first name. Yet despite his fame, there are things about Leonardo that many people today find surprising.

1. Shady Parentage

Leonardo was born out of wedlock on April 15, 1452. His father, Piero, was a wealthy notary, and his mother, Caterina, was a local peasant girl. Although the circumstances of his birth would place Leonardo at a disadvantage in terms of education and inheritance, biographer Walter Isaacson regards it as a terrific stroke of luck. Rather than being expected to become a notary like his father, Leonardo was instead free to develop the full range of his genius. People surmise that it also imbued him with a special sense of urgency to establish his own identity and prove himself.

2. Physical Beauty

Leonardo created some of the world’s most beautiful works of art, including the “Last Supper” and the “Mona Lisa.” In his own day, he was known as an exceptionally attractive person. One of Leonardo’s biographers describes him as a person of “outstanding physical beauty who displayed infinite grace in everything he did.” A contemporary described him as a “well proportioned, graceful, and good-looking man” who “wore a rose-pink tunic” and had “beautiful curling hair, carefully styled, which came down to the middle of his chest.” Leonardo is thought to have entered into long-term and possibly sexual relationships with two of his pupils, both artists in their own right.

3. From Scraps to Notebooks

One of his best-known notebook drawings is the ‘Vitruvian Man.’ Leonardo da Vinci/Wikimedia Commons

The paintings generally attributed to Leonardo number fewer than 20, while his notebooks contain over 7,000 pages. They’re the best source of knowledge about Leonardo, housed today in locations such as Windsor Castle, the Louvre and the Spanish National Library in Madrid. Their diverse content ranges across drawings—most famously, Vitruvian Man—notes of things he wanted to investigate, scientific and technical diagrams and shopping lists. They comprise perhaps the most remarkable monument to human curiosity and creativity ever produced by a single person. Yet when Leonardo penned them, they were just loose pieces of paper of different types and sizes. His friends bound them into “notebooks” only after his death.

4. Outsider’s Education

As a result of his illegitimacy, Leonardo received a rather rudimentary formal education consisting primarily of business arithmetic. He never attended university and sometimes referred to himself as an “unlettered man.” Yet his lack of formal schooling also freed him from the constraints of tradition, helping to instill in him a determination to question authority and place greater reliance on his own experience than opinions expressed in books. As a result, he became a firsthand observer and experimenter, uninterested in serving as a mouthpiece for the classics.

5. Prolific Procrastinator

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Apprenticeship

Preparing Learners for the Uncertain Future of Work

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By Katherine Prince

Continuous learning, cultural awareness, change expertise, adaptable and effective communication and the ability to learn from failure. These are just some of the capabilities that participants in KnowledgeWorks’ convenings on the future of work identified as being important for graduates. Finding resources to solve problems, time and project management, reflective leadership and a sense of responsibility to the broader community also promised to help all young people thrive no matter what future of work emerges.

That question – what future of work will emerge – is unanswerable, making it critical to help young people, along with other education and employment stakeholders, plan for multiple possible futures. From today’s vantage point, we can identify two critical drivers of change shaping the future of readiness for further learning, work and life: the rise of smart machines and the decline of full-time employment. But we cannot yet know what extent of technological unemployment we will face or how much support individuals will have in navigating the changing employment landscape.

A New Foundation for Readiness

In the face of such uncertainties, stakeholders need to help people develop our uniquely human attributes along with developing flexible skills that we can apply across settings. Putting social-emotional skill development at the center of learning promises to help individuals develop the foundation necessary to navigate uncertainty throughout their lives. The new foundation for readiness shown below illustrates how redefining readiness from the inside out – focusing on human development rather than attempting to prepare learners for any particular future of work – can provide a platform for future success.

This new foundation for readiness is grounded in the human qualities that are most central to our relationships with one another and which are most difficult to code. Social-emotional skill development will need to be supported in integrated ways alongside the mastery of content and the application of skills and knowledge to specific contexts. Education institutions will need to balance supporting learners in preparing for their first-careers while also helping them develop the adaptability and resilience needed to navigate the changing economy and the ways of thinking necessary to address complex problems.

Flipping Education’s Focus

Establishing a new focus on feeling and relating will help education institutions and systems align with a future of readiness in which foundational skills and practices will be more important and enduring than specific content or job- and task-related skills.

For K-12 education, flipping the focus of learning to whole-person development could mean that:

  • Curriculum needs to be inverted, with core social-emotional competencies shaping the design of inquiry projects and the school and classroom rituals that anchor the learning climate and culture.
  • Students need to be grouped in new ways to follow flexible learning pathways.
  • Classrooms need to become more fluid and open, enabling new ways of structuring learning.
  • School schedules need to be transformed to allow for more interdisciplinary collaboration, deep reflection, and personalized learning.
  • Educators’ roles need to be reconfigured to focus less on content or grade specialization and more on foundational skills and practices, as well as on interdisciplinary, phenomenological or challenge-based learning.
  • Community partners need to become key assets for introducing new kinds of learning experiences that stretch students’ comfort zones and expand their aspirations.
  • K-12 schools and districts need to explore where and when it may be more appropriate for them to serve as brokers, rather than direct providers, of learning experiences.

At the postsecondary level, institutions might need to:

  • Focus more on supporting deep personal development as well as context- and discipline-specific skills and knowledge.
  • Diversify offerings and business models, with a multitude of formats and structures engaging learners and increasing access.
  • Contribute to student-driven and student-designed ecosystems of support that evolve over time and reflect students’ strengths, weaknesses, and needs.
  • Help students plan for both their careers and their lives and respond to changing conditions.
  • Enable learners to weave in and out of learning experiences as their career development needs dictate.
  • Collaborate more extensively with workplace partners.
  • Shift the focus of faculty professional development toward supporting students’ development of foundational skills and practices and attaining ongoing learning related to relevant workplace skills.

Strategies for Redefining Readiness

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Apprenticeship

Patent Education Buys Time and A Foundation for Financial Rewards

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by Clare Donovan

Meaningful innovation education exposes kids to this vital element.

I love watching the rise of makerspaces in schools. More and more students have access to 3D printers, kid-friendly coding programs, and other great technical tools. This investment in STEM project-based learning means that students with all kinds of learning styles are directly engaged in issues critical to their future.

But there’s a key ingredient missing. When you give students access to awesome tools and the freedom to design, what happens when they hit on something great? Most often, that’s where the project stalls. In the business world, that’s when a patent comes in.

Teaching patenting concepts at a young age is not only feasible, but I believe it’s a way to help women and other underrepresented groups play a bigger role in tech.

Tinker While You Learn

At the Future of Education Technology Conference in Orlando, Fla., I saw some amazing curricula and technology demos [Ed. note: Clare spoke there]. Schools have a huge selection of tools to teach engineering to young students.

Engineering, however, is only one component of the innovation process. Creativity, business and the law also play a critical role. Patents motivate people to invest in the arduous process of inventing by giving them economic ownership for continued development. For a startup, a patent buys you time to do the hard work of bringing your idea to life. Patent education is an important ingredient in helping kids become real-world problem solvers.

Patent education also lays a foundation for financial rewards in the workplace. The U.S. Patent and Trademark Office finds that wages in patent-based industries are 74% higher than other verticals, and this premium is growing. This is according to the USPTO’s latest update to “Intellectual Property and the U.S. Economy.”

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Creepy Monkey
Creativity

10 Un-creepy Ways to Use Google Search

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Google is a crystal ball in reverse: while you’re asking it questions about the future, it’s probing you for the answers. Indeed, when millions of people participate, Google can use end-user data, queries, and clicks to “predict the future.” Of course, the real power remains with the people using the platform and the choices they make. One of the choices I’ve made was to switch to DuckDuckGo when Google started directing the future instead of just predicting it.

But Google and I are still friends. I’m happy to use them for lots of drudgery work around the office.

1. Timer & Stopwatch

Just type “timer” into the search box for an instant timer (or stopwatch) at your disposal.

2. Calculator, Formula Calculations, Equation Graphs

Type “calculator,” “area of a circle,” or “2x +1”, or “y=x^2+2x” and get a desktop calculator, a plug-in formula, or the corresponding graph.

3. Conversions

“Celsius to Fahrenheit,” “ounces to liters,” “dollars to pounds,” and just about anything stumping you at the moment.

4. Color picker

“color picker” enables me to get the RGB numbers of a hex color, or vice versa, in a jiff. Same goes for CMYK, HSV, HSL.

5. Time in “Place”

“time in Nepal,” “time in Wyoming,” “time in Cleveland”; they all give me location specific times.

Bonus: “time 5:00 p.m. Nepal” shows what time it’ll be in your location when it’s 5 p.m. in Nepal.

6. Weather “place.”

Same as for Time but returns weather in location.

7. Speed Test

Is your internet connection slow or is there something else going on? Type in “Speed Test” and find out.

8. Flight Status or All Flights Available

“New York to Paris” brings up all available flights from … you guessed it … and “United 4885” shows the status for that flight.

9. Site Specific Search

They’re tracking everything so why not have Google show you everything a site has on my term of interest? Searching a specific site this way is usually better than navigating to the site and using their search box.

The syntax is “SearchTerm site:URL”

Example:
ducks site:nationalgeographic.com

10. IP Address

“IP Address” returns your current location’s IP address.

Bonus: Forget Me Yes

“Hey Siri, tell Google to forget about me.”

This only works if Siri takes you to the correct web address to request that Google delete all the previous web search data they’ve saved about you. Take a look at everything they’re keeping, and perhaps you’ll agree that asking them to delete it is the un-creepy move.

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A Closed Mind
Business

The #1 Reason Innovation Efforts Fail

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by Kumar Mehta

Despite the countless books and articles written about innovation, we don’t have a good understanding about what drives innovation within organizations. Instead, we have multiple, often conflicting, theories about what makes innovation happen. We follow the one that resonates best with us or is in vogue. As a result, innovation efforts at many companies suffocate as they stumble along, thinking they are on the right path but not really knowing where they are headed.

The way to build an environment where innovation happens consistently is to make certain that the foundational building blocks that drive innovation are present. Few companies have built such an environment, what I call an innovation biome . The companies that have built this environment are the ones we admire as they are the ones that bring the most innovative offerings to the world.

In the rest of the companies, though, the most common reason that innovations fail is this: others reject an idea because they don’t have the right tools to evaluate the idea.

Overcoming disbelievers and naysayers

In most companies, when someone has an idea it has to climb up several layers of management, requiring a yes at every level for it to keep climbing.

A single no going up the management staircase can kill an idea. Since no one is ever penalized for saying no (you only get hurt for saying yes to the wrong thing), companies often develop cultures that are conservative and not conducive to innovation.

Just about every major innovation in history was rejected by the experts of the time.

Whether it was the earliest people who believed the earth is round, or Darwin’s theory of evolution, or Pasteur’s theory of germs, disbelievers and naysayers have always shown up in full force. This has never stopped. Experts thought the Personal Computer would not be successful, nor the automobile, nor the telephone (presumably the telephone was never supposed to catch on because there was no shortage of messenger boys).

This still happens every day in companies around the world.

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Dieter Rams
Creativity

Dieter Rams’s 10 Principles of Design, illustrated by his ingenious products

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by Mark Wilson

Every designer alive has heard of Dieter Rams’s 10 Principles of Design–the legendary designer’s quick-reference rules for making products, developed in his early days at Braun. But never has the list been presented with such a strong visual thesis as it is in the documentary Rams, by Gary Hustwit. The documentary was released in select theaters in 2018 (read my story on it here) but Hustwit recently shared this intriguing new clip online. It’s a fascinating, four-minute thesis about how Rams articulated his design philosophy through consumer products.

In the documentary, the 10 Principles of Design sequence feels quite different than what comes before or after. While Hustwit films Rams himself with a locked-down tripod, creating crisp, impeccably balanced frames, the list breaks out of this stoicism, embracing multi-panel animations and a bit of whimsy.

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Mentorships
Apprenticeship

Mentorships and Kids

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by Joanne Foster, EdD

Synopsis

Here’s an overview of why mentorships are increasingly popular, including benefits, structuring guidelines, and lots of helpful information for parents, teachers, and kids.

“The term ‘mentor’ comes from Greek mythology: Odysseus’ son Telemachus was entrusted to the care of Mentor, a wise advisor. History and literature from classical times to the present abound with examples of mentorships in politics, business, science, the arts, and education. Aristotle benefited from his mentorship under Plato, as Mickey Mouse benefitted from his in Fantasia’s ‘The Sorcerer’s Apprentice.’”

~ Excerpt from p. 160 of Being Smart about Gifted Education 

WHAT, EXACTLY, IS A MENTORSHIP?

A mentorship is a supportive relationship established between a learner and someone who is more experienced in a particular domain. (For example, sciences, business, creative arts, technology, and so on.) The mentor offers guidance, knowledge, and understanding. Mentoring requires an investment of time and patience, and a willingness to support and encourage the learner. Typically, the “mentee” is deemed to be the learner, but in truth all strong mentorships are mutually rewarding experiences wherein both parties interact meaningfully and respectfully with one another, learn, and derive benefits.

WHAT KINDS OF BENEFITS? 

Here’s a list of ways mentorships can strengthen a child’s or teen’s learning:

  • Enriched perspectives relating to an area of interest, including useful information, skill sets, creative and critical thinking opportunities, and practical applications
  • Transmission of values and attitudes
  • Enjoyment
  • Enhanced and authentic connections to important domains of competence, and to others within the “real world” (This includes exposure to fields of interest—leading to greater career path awareness, preparation for taking on roles, and appreciation of accomplishment in the chosen area.)Emotional support
  • Discovery of resources beyond the classroom
  • Intellectual challenge and increased competence, including perhaps the creation of a possible portfolio of acquired learning achievements
  • Encouragement and guidance for self-directed learning
  • Expansion of understandings of diversity and possibility (For example, non-traditional minority professionals can challenge gender and cultural stereotypes, and mentorships can be particularly beneficial for students from culturally diverse or economically disadvantaged backgrounds.)
  • Respect for expertise
  • Relationship-building experiences
  • Positive role models, including helping kids better understand pathways to high achievement
  • Potential for academic credit

Here’s a list of benefits for mentors: 

  • Ongoing learning
  • Rejuvenation of spirit
  • Sense of fulfillment
  • Sense of respect and of being valued
  • Fresh perspectives—seeing things anew from the point of view of mentees
  • Involvement and enjoyment
  • Contribution to the skills and expertise of young people interested in possibly entering the field of interest
  • Vicarious satisfaction through accomplishment of the protégé
  • Connections to the educational system
  • Inter-generational friendship
  • Community engagement

HOW TO STRUCTURE A MENTORSHIP?

In any mentorship arrangement, it’s important to clarify expectations. These should be agreed upon by both the mentor and the mentee, with parents and teachers overseeing the process, and with their approval. It’s a good idea to draw up a written agreement outlining intents and responsibilities. This includes the right of withdrawal from an arrangement if it does not seem to be working out well. Periodic review of this “contract” will help to ensure that everyone’s expectations are being met.

A mentorship can be a one-on-one program between two people, or it may take the form of more a complex arrangement with others involved. Either way, it should be part of an individual student’s overall educational plan, and it should also be valued as an integral component of it.

Heads up—any program involving kids requires careful supervision and consistent monitoring by adults. With that caveat uppermost, here are a few possible “models” for mentorships.

 Co-creation of an individualized program by the mentor and the student, always under the guidance of parents or teachers

School visits by vetted community experts who can help to increase the depth of programming that classroom teachers are able to provide

Job-shadowing programs, whereby students prepare for the mentorship phase in school, and then spend time in the pre-approved career setting of their mentor

Online or virtual mentorship programs—especially good for children and teens who want to investigate a field or learn about something that is not otherwise readily accessible to them (Note: Online options demand attentive supervision.)

Creative approaches, whereby a mix of the above might be contemplated, or an innovative mentorship format is designed for specific purposes. For example, mentorships are a frequently recommended practice in gifted education. Unique or differentiated learning experiences can provide gifted learners with targeted and enriching educational opportunities and challenges in areas of heightened advancement. (Click here for an article with additional information about fostering giftedness.)

WHERE TO FIND A MENTOR?

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