A Math Garden: Mathematics Everywhere and All Around
John F.C. Cheong (Updated: May 19, 2015).
I. Popular Theses about Mathematics
Q: Could there be a curricular design for a one-year course offered at an advanced 8th grade level that sensibly captures all of the four major facets of mathematics today? In clearly seeing mathematics this well-balanced way sooner, tomorrow's innovators would be much better prepared — than many today or in the past — to appreciate what math is; understand what it can (or cannot) do, as well as know when and how to use math. Given the early start, the next generation of innovators would have ample time to hone a more nuanced sense of mathematical intuition; avoiding many of the bad habits that might otherwise befall human judgment in many realms of life, e.g., in finance or economics where human follies abound that the right mathematical thinking would have prevented. At least that's the plan — or hope — for the future.
A: For an advanced 8th grader undertaking directed independent study (upon completion of a typical sequence of algebra and geometry), the core of the curriculum can be designed based on the content areas of Discrete Math (e.g., number theory and combinatorics) and Statistical Thinking (i.e., practical aspects of applying statistics). The curricular coverage of discrete math should be comparable to how the subject of discrete math is taught academically in Singapore or in the Eastern European tradition of math circles. To augment learning, interesting projects from the real world can be researched for their mathematical content, from which a collection of seven projects can then be selected, i.e., expected to progress at the rate of about one project per month over 9 months. We can think of it as "project-enhanced learning." These projects are organized in a thematically coherent manner that reflect their ubiquitous relevance and recurrence in the real world, e.g., "Mathematics Everywhere and All Around." Most critically, a trio of well-chosen modern mathematical tool, environment, and language (e.g., Wolfram Alpha, MATLAB, and Julia) equips the student to independently explore the mathematical landscape of the chosen problem domains. A final presentation plus Q&A with interactive demonstration to an audience marks the completion of each project.
Most of the course materials can be readily assembled from self-contained chapters in four different math textbooks of the well-regarded "Art of Problem Solving" series, plus suitable contents excerpted from a particular statistics book that emphasizes concepts over formulas. We outline a proposal for an integrated 8th grade "Mathematics + Technology" curriculum (aka "Project-Enhanced Discrete Math + Statistical Thinking") — to be further developed — for an advanced 8th grader conducting supervised independent study at a small private school in the San Francisco Bay Area. The rest of this article is sprinkled with our further thoughts and musings on educational aspects of computational mathematics — the role that it could play in furthering the math education of tomorrow's innovators. We are especially interested in exploring how the right computational tools in the hands of mathematically inclined students could enable them to not only "solve real-world problems using math," but also learn to "do interesting math research."
II. A "Big History" View of Computational Aspects of Math
Greek mathematicians like Euclid introduced proofs, i.e., an abstraction device embedded in rhetoric, in the absence of computational tools. The Greeks as philosophers are very good at logic, and proofs naturally arose from their attempt to extend common sense by logic. The Chinese are practical problem solvers who invented a positional numeral system for use with the counting rods, and later codified this knowledge into the abacus to facilitate commerce and trade. Thanks to the Hindu-Arabic notation that Leonardo of Pisa (aka Fibonacci) imported to the West in the 13th century to replace the legacy of unwieldy Roman number system, mathematicians in Europe not only had a transcription system but also learned to compute very well on numbers using just ink and paper, with expeditious help from logarithmic tables prepared by John Napier. European mathematicians such as Fermat, Euler, Gauss, and Riemann were very proficient at discovering new theorems by a process of trial-and-error experimentation, i.e., exploratory calculations with numbers or symbols based on guesses and hypotheses. Driven by the needs of science and engineering, mathematical progress in that era was rapid. Mathematics by now has developed into a "partial script" that is the universal language of science and engineering, a precursor to mathematical computation at the dawn of computers. A good college-prep math curriculum traverses many of these well-worn paths, often presenting the polished results of mathematical research while taking care to avoid certain pitfalls in curriculum design.
III. Math Education for Tomorrow's Innovators
Pedagogy notwithstanding, the subject of mathematics as taught at the pre-collegiate level has remained largely unchanged for close to a hundred years. In the intervening decades, many of the important advances in computational methods have found their way into a range of practical mathematical tools that are now accessible to a wider range of younger students. This new breed of modern mathematical tools: MATLAB, Wolfram Alpha, and Julia, promises to not only make tedious calculations simpler, but are excellent exploratory tools for young students, too. The potential impact of computational methods on mathematical education is expected to be no less than that of the scientific method in the teaching of science.
Throughout much of history, the activity of mathematical research (which is often messy) is often confused with its final product (which is often polished and well-crafted). Early mastery of these mathematical tools, like doing research using Google and Wikipedia, not only helps students better navigate the mathematical landscape, but also enables them to explore and discover interesting new worlds by themselves, or in their peer groups. More importantly, it will change the social dynamics of mathematical education in a big way. Students can now do math and learn math at the same time, just as they could conduct scientific experiments to learn science. All else being equal, these tools provide better alternative use of time for the mathematically inclined that might otherwise be spent playing computer games.
Innovators in the 21st century, according to one educational scenario, "will need to have, above all else, a good conceptual (in a functional sense) understanding of mathematics, its power, its scope, when and how it can be applied, and its limitations. They will also have to have a solid mastery of basic mathematical skills, but it does not have to be stellar. A far more important requirement is that they can work well in teams, often cross-disciplinary teams, they can see things in new ways, they can quickly come up to speed on a new technique that seems to be required, and they are very good at adapting old methods to new situations."
"To produce the twenty-first century innovative mathematical thinker," today's middle schools and high schools will need to provide a classroom and campus environment that is conducive to "project-based, group learning in which teams of students are presented with realistic problems that will require mathematical and other kinds of thinking for their solution." Gauss, the Prince of Mathematics, would have agreed, for he had already understood that: "it is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment."
IV. An Integrated 8th-Grade "Math + Technology" Curriculum
1. Curricular Goals for "Project-Enhanced Discrete Math + Statistical Thinking"
- Develop mathematical intuition for understanding the physical world and society around us.
- Able to detect and identify patterns in numbers or other mathematical objects and structures.
- Able to sketch "mathematical blueprints" for computational experiments, modeling, or simulation.
- Proficient with using math and statistical vocabulary to describe and narrate ideas/concepts/theories.
2. A Directed Independent Study Tailored for an Advanced 8th Grader (Easy as "ABC&D"?)
(A) Mathematics: Foundational concepts for analysis and computational modeling
- number theory and modular arithmetic, sequences and series, piecewise defined functions.
- sets and logic, recursion, conditional probability, events with states, generating functions, graph theory.
- vectors and matrices, cross products and determinants, analytic geometry, combinatorics.
- constructive expectation, sample size estimation, confidence interval, significance test, correlation and simple regression.
(B) Technology: Tools, environment, and languages for computational mathematics
- MATLAB/Simulink: easy-to-use modeling and simulation tools with good library support.
- Wolfram Alpha: computable knowledge from around the world updated constantly.
- Julia/IJulia: programming language and interactive math notebook interface that is evolving rapidly.
(C) Projects: Select topics organized according to thematic coherence
Theme: "Mathematics Everywhere and All Around"
- Picture Proofs: Assemble a gallery of creative visual proofs (aka "Behold!") that will make Harvey speechless. Can all proofs be rendered into pictures? Why or why not? Are pictures no more than beautiful illustrations that sometimes deceive? What is a proof, really?
- Testing for Primes: Mathematical research using computers. Collect in a math notebook computational studies of patterns in prime numbers and illustrate their statistical properties. What do you see? Why has prime testing captured so much attention? Can numerical evidence ever replace proof in mathematics? How does one do research in mathematics?
- Words and Secret Messages: The history of secrets. Take the Enigma "cipher challenge" to test your code breaking skills and win the war on the side of the Allies! Is there a "master key" that can unlock any secret messages using cryptanalysis via statistical methods?
- Numbers Everywhere: Serendipity in mathematics. Empirical data study of Benford's Law (aka first-digit law) from chosen domains (e.g., surface areas of 335 rivers, the sizes of 3259 US populations, 104 physical constants, 1800 molecular weights, statistical physics, etc.). How can one guess when Benfor's Law applies, and when it doesn't? For example, does Benford's Law apply to fast-growing bacteria population? Why?
- Interactions in the Wild: Cycles and resonance. Predator-prey models of population growth, e.g., moose and wolves, or cicadas and birds. Interaction matrix. How might changing various parameters to the model affect population dynamics? Does the chosen model fit the empirical dataset under study? Do cicadas survive better by having a prime lifespan (i.e., 13 years or 17 years)?
- Movements Around Us: Big effects of small rules. Modeling and simulation of flocking behavior and swarms (e.g., fish, bats, fireflies, dolphins, spiders, people, etc.). Produce your favorite movie special effects!
- Music Around Us: What sets music apart from noise? Experiment with mathematical harmonics and synthesized music. Assemble your very own duo, trio or quartet of instruments to play ensemble music!
(D) Discourse: Discussions, board work, show-and-tell, interactive demos, presentation plus Q&A
Communications of mathematical ideas is paramount; one's idea is never more effective than one's ability to make others comprehend them. A wide range of activities in different communication formats, e.g., regular conferences with the supervising teacher, informal small-group discussions and board work at the beginning of each project, followed by show-and-tell in front of the class as the project nears completion, help generate conversational flow. A final presentation plus Q&A with interactive demonstration to a wider audience marks the completion of each project.
Over the span of the 8th grade 2016-2017 academic year lasting 9 months, a total of 7 projects are to be covered; each project lasting about 4 weeks on average (but could also be as short as 3 weeks or as long as 6 weeks). Of course, with good time management more than one projects can be worked on concurrently for higher productivity and greater efficiency, just as in the real world. What's more, the frequent practice of "spaced recall" in a concurrent projects setting improves retention — for both the student and his audience.
For the student of mathematics undertaking directed independent study, having an audience for sounding board and feedback is a critical aspect of learning from doing projects and effective mathematical communications. One could easily imagine that the rest of the class would be equally curious as to what the more advanced student is learning or doing for his projects. This creates a natural classroom setting for conversations, information exchange, and Q&A. A wide range of communication practices can thus be carried out in a supportive classroom setting on a regular and frequent basis to prepare the mathematics student in a well-rounded manner, as advised 400 years ago by the great philosopher Sir Francis Bacon: "Reading maketh a full man, conference a ready man, and writing an exact man." After all, we are preparing tomorrow's innovators to be ready today!
V. SPECIAL Features of Curricular Design
- Geared towards actually doing mathematics in a real-world context. Why? Better motivation for the interactive learner: easier to understand when put into context, a richer environment that is more relevant, concrete and less abstract for the development stage of an 8th grader. Continue to build and strengthen linkages between mathematics and the surrounding world. Lay a strong foundation for continuing STEAM education where math plays an important role. Doing mathematics in a real-world context actually drives the learning process (i.e., demand-pulled rather than supply-pushed) and is particularly suited for students who enjoy hands-on projects.
- Introduce useful mathematical tools at the earliest opportunity. Why? Better retention of new things learned if it can be practiced repeatedly in visual, auditory and computational setting. More fun if can be used for projects, show-and-tell, etc. A different way to explore the mathematical landscape and conduct supervised independent research. "Practice makes perfect" only works if it is fun and enjoyable and repeatable without becoming tedious busy work or over-simplified drills. Middle school students are increasingly tech-savvy users of tools, so let's give some of them useful mathematical tools for actually doing fun math projects!
- Build a working set of mathematical vocabulary to facilitate communications. Why is it important to know what each word mean and to use it precisely in conversations? Because math reuses many common English words, but gives them different meanings. If all these words can be cognitively organized into a working set through frequent conversations, then not only will the mathematical explorer be able to read, write, research and label mathematical objects more accurately and precisely, greater fluency in mathematical discourse can also be readily achieved. Unambiguous communications and clear thinking goes hand in hand.
- Wide range of communication formats practiced in a supportive school setting. Why? Math is inherently an interactive social process for certain mathematicians (e.g., the prolific Paul Erdős, who had written or co-authored 1,475 papers with 485 other mathematicians across 25 countries in his lifetime). So math can be made more enjoyable for the sociable type in the classroom or around the school when its every step along the journey can be shared with classmates, teachers, and friends. For those who love to talk about their projects, there are now many different formats for sharing at various stages of progress: classroom discussions, board work, show-and-tell, interactive demo, presentation plus Q&A, and the school website.
- Balancing breadth of curricular coverage with depth of understanding. How? By first decoupling 23 lesson units from 7 practical projects; we can think of them as two orthogonal dimensions that may proceed at different pace of student learning, i.e., solving problems or doing projects. Since the projects are well-chosen to utilize the concepts and techniques learned from lesson units in a recurrent fashion, the initially learned materials can then be recombined together in the context of multiple projects down the road to further enhance learning. We can think of this matrix organization akin to growing "mathematical roots". The "spaced recall" aspect of project-enhanced learning will greatly help in the retention of learned materials, especially for the quick learner (who sometimes may also be quick to forget!).
VI. Mathematical Growth
Organizing the mathematical landscape for current retention as well as for future exploration is always a good learning strategy. Math is interconnected — often in serendipitous ways. Different sub-fields sometimes have deep links between them, usually below ground. When taught as separate courses, students see each tree up close but not the forest. They also have no idea how the trees look like together when assembled into a forest. Just as we learn language through frequency of contextual use in different social situations, students learn mathematics by traversing the mathematical terrain in different ways. Get organized early — 8th grade is about right for Sean — before the mathematical forest gets too big and each tree grows too tall, all becoming dark and intimidating causing one to easily become lost in the woods. Most importantly, interconnection helps retention and deepens overall understanding. Building a mathematical garden that is both broad and deep, based on shared mathematical roots and organizing projects around a coherent theme, could address the "a mile wide and an inch deep" syndrome (as well as the "mile-wide and mile-deep" chasm) that was said to afflict U.S. middle school and high school math (and science) education.
Here is a set of related metaphors that might be helpful for charting the course ahead. A 5th grader's view of the mathematical terrain is like "watching bonsai in a terrarium"; an 8th grader's experience is more like "a walk in the garden"; a 12th grader's is closer to "hiking in the woods without getting lost"; whereas a college graduate should be able to "trek in the dark forest and survive a few days of camping." A professional mathematician, of course, could easily navigate uncharted mathematical universe with ease. I think we are here trying to organize and layout a customized mathematical garden for "that particular 8th grader undertaking directed independent study" (aka Sean) to walk around in, and while there, for him to learn to do some gardening along the way — water the plants, pull a few weeds, trim the hedges, and perhaps plant a few acorns that will grow and bear fruits in the years ahead.
So what are some of the little acorns that were recently planted or just starting to put down roots in Sean's mathematical garden? We look towards the textbooks, i.e., the flower pots that hold the saplings, for clues. We specifically seek out textbooks that are written in a style that is precise, simple, and concise, as well as math books written using a coherent narrative. One shouldn't have to spend too much time parsing words when re-reading the text, or expend unnecessary cognitive energies when the narrative is inconsistent or disjointed. We prefer illustrated contents along with free-flowing contextual narrative in a modular book design. As a rule of thumb, there should be no more than two (or maybe three) authors on a math textbook, we think, before its narrative tends towards consensus-driven blandness as the textbook grows heavier with each new edition. Textbooks authored by a committee of authors in arbitrary reflection of dogmas or ideologies, instead of informed opinion, suffered the worst. The following is a list of math books and video lectures that Sean enjoys, and forms a part of Sean's evolving garden plan going out to 8th grade (and maybe a little beyond):
- Gelfand, Israel M. and Shen, Alexander (2002). Algebra. Birkhäuser.
- Rusczyk, Richard. Algebra Videos (Ch. 1~13). Art of Problem Solving.
- Rusczyk, Richard (2008). Intermediate Algebra. Art of Problem Solving.
- Kiselev, A.P. (2006). Kiselev's Geometry. Book I: Planimetry. Sumizdat.
- Crawford, Matthew (2008). Introduction to Number Theory. Art of Problem Solving.
- Rusczyk, Richard. Counting and Probability Videos (Ch. 1~14). Art of Problem Solving.
- Rusczyk, Richard (2012). Intermediate Counting and Probability. Art of Problem Solving.
- Lehoczky, Sandor and Rusczyk, Richard (2006). The Art of Problem Solving, Vol. 1: The Basics. Art of Problem Solving.
- Lehoczky, Sandor and Rusczyk, Richard (2006). The Art of Problem Solving, Vol. 2: And Beyond. Art of Problem Solving.
- Rusczyk, Richard. Mathcounts Minis (2009~2014). Art of Problem Solving.
- Large, Tori (2007). The Usborne Illustrated Dictionary of Math: Internet Referenced. Usborne Pub Ltd.
- Elwes, Richard (2013). Math In 100 Key Breakthroughs. Quercus.
- Walonick, David (2013). Survival Statistics. StatPac.
- Stankova, Zvezdelina (2008). A Decade of Berkeley Math Circle: The American Experience. American Mathematical Society.
NB: Select chapters gathered from a collection of 4 "Art of Problem-Solving" (AoPS) textbooks — as highlighted in the list above — can easily be the "Discrete Math" anchor of Sean's 8th grade curriculum. The AoPS collection of books presents the right level of challenge for Sean; all have a clear, consistent style of pedagogy that Sean is already familiar with. The one additional statistics book (highlighted above) focuses on concepts, not formulas, and has just the essentials of what one needs to know to conduct a research project: (a) what statistic to use for a given situation, (b) how to interpret a statistic; and (c) how to make a statement with a statistic. A short reading list of another 4 (or perhaps 5) supplemental papers should round up the remaining of the "Statistical Thinking" anchor of the curriculum.
A good part of Sean's 7th grade would be to continue watering the plants already germinating in the garden, e.g., becoming proficient in writing proofs and to properly learn geometry, while also preparing for a new round of middle school math competitions. In addition, to pave the way for his upcoming adventure into actually doing mathematics in a real-world context for the academic year 2016-2017, a well-aerated garden bed will first need to be laid during the 2015-2016 academic year, such as to: improve problem-solving skills, learn proof techniques, and acquire programming skills. Only then can this particular mathematical garden spring to life at the start of the 8th grade, i.e., embodied in a pair of math notebooks — paper and electronic — which is portable and can be readily shared with others in various modes of communications, or simply be exhibited on the Web like an artist's portfolio. In summary, a healthy set of well-chosen mathematical roots not only provides good nourishment, but also serves as strong anchors while little acorns grow into mighty oak trees.
VII. Reminiscence of Art and Mathematics
Sean becomes animated naturally when he talks about projects he's working on — models of airplanes, rockets or other flying contraptions that he wants to build. He read voraciously and always has ideas for what to do in his free time. He has been adept since a young age at learning about and working with a wide range of tools and materials, procured from a great many sources — online or from around the neighborhood. One time, when Sean was little, he meticulously drew an airplane with the greatest precision that a four-year-old could muster — crayons on paper — all on his own. Took him a good half of one Saturday afternoon. All the parts were colored in nicely like never before, and the final drawing crisply folded for bringing to Sunday art school in the next day to show his sensei.
Just when his parents were overjoyed that their little boy finally took to drawing and coloring — expected of boys his age — for no higher purpose than simply one's own enjoyment, Sean proclaimed with great anticipation that he had just discovered a new way to "precisely tell" his Japanese art teacher what he needed help building despite his somewhat limited vocabulary at the time, and how all different parts should be painted to his color specifications once his wooden "delta-winged" airplane was finally glued together. The art school had apparently just started woodworking projects for the older students, and Sean saw potential in the seemingly limitless supply of fine wood at school.
It soon became clear to his parents that — before the "year of moratorium on toys" was over during which it was reiterated that no further family funds will be expended on new toys to add to the growing clutter of his childhood possessions — their resourceful boy had already figured out how to recruit his Sunday art school teachers to help build new toys which he carefully designed on paper. All wood, all good. So, at the tender age of four, our son had figured out art, a colorful visual rendition of what is to be built. A blueprint for toys, no less! Art is a beautiful medium of communication, we have to admit. But above all, art is purposeful because it leads to the construction of beautiful objects.
For Sean, mathematics is just like art. It has a purpose beyond its own very existence that only his mind's eye can glimpse at, but for his hands to bring into existence so all the rest could see, hear, touch — and for him (and his brothers) to play with. But isn't that the whole entire goal from the start? What makes the object of creation possible is the subject of mathematics.
VIII. Back to Counting?
To understand the world around us, we build up conceptual maps over time as its explorers. These cognitive maps enable us to reason about the world intuitively. However, human brains succumb too readily to such cognitive traps as: availability bias, hindsight bias, problem of induction, fallacy of conjunction, confirmation bias, contamination effects, affect heuristic, scope neglect, overconfidence in calibration, etc. To overcome our all too human shortcomings, we resort to using mathematics as an extension to common sense and a refinement of our intuition. In particular, we make use of mathematical thinking based upon our built-in cognitive senses, for example: arithmetical thinking (based upon our number sense), algebraic thinking (based upon our logical sense), geometrical thinking (based on our spatial sense), and statistical thinking (based upon our conceptual sense). A "Big History of Math"-type understanding of the logical progression of mathematical thinking fits our long-standing human tradition of telling original stories; and provides a coherent narrative that underlies all of the various mathematical ideas:
A well-rounded mathematical education for tomorrow's innovators, we believe, should cover all of these aspects using the progression of mathematical thinking as an organizing principle; so they can better understand the past, explain the present, and imagine the future. Statistical thinking, which is a relatively recent development compared to other cornerstones of mathematics because statistics is heavily reliant on computing power, should therefore be given greater emphasis for its practical utility in engineering, and across both the natural and social sciences. After all, it is only by seeing mathematical thinking from a rich historical perspective that one can arrive at a deeper appreciation of mathematics. For example, it clarifies why it took humankind more than two millennium of progress in both mathematics and computational tools to get to the current state-of-the-art in computational mathematics that we have today, which despite its complexity is surprisingly accessible to everyone with a computer plus an Internet connection. Most importantly, a mathematical education that teaches mathematical thinking can better illuminate the possibilities that guide the path to our future.