In an article for The New Republic, editor Ryu Spaeth ruminates on his impromptu homeschooling due to Covid-19. In particular, he reflects on the experience of teaching his five year old daughter how to read. Spaeth writes self-deprecatingly that “teaching a child to read is hard” and “I have no idea what I’m doing.”
I loved reading Spaeth’s account, and I think it demonstrates just how important parents are in the learning process. He may not feel like he knows what he’s doing, but I think he’s doing a lot right. Regardless of what educational option parents choose for their children, parental engagement is essential. A large part of our mission here at Demme Learning is to encourage parents to trust themselves; you are doing a good job, and your investment into your child’s learning really matters.
There are three key features of quality at-home learning that stand out to me in my reading of Spaeth’s article.
1) The Best Learning Experiences are Embodied, and Rooted in Relationship
Often when people are first experimenting with home education, they are tempted to recreate the school at home. But the beauty of learning at home, whether full homeschooling curriculum or summer learning, is how deeply personal it is enabled to be given the familial context. Your home does not have to look like school to be a good learning environment.
Spaeth writes, “when my daughter and I first began this regimen, curling up on the couch with a book, it was an inversion of our usual routine: She would be reading the book to me.” There is obvious warmth in the texture of this experience, a texture made possible by the level of intimate connection in the parent-child bond. This kind of bond can be of particular help in inspiring perseverance in the learning process. Spaeth observes that “when facing a particularly daunting block of text, she will wriggle out from the nest I have made of my arm and lie stretched out on the couch, eyes closed tight.” And he subsequently reflects that “teaching her to read is less about helping her with the more difficult words and explaining how particular arrangements of letters are pronounced, than about urging her to sit up, to concentrate, to put her shoulder to the wheel.”
There are plenty of quantifiable academic metrics through which we could analyze the learning taking place in these interactions. Yet I think most of us can agree that the qualitative benefits here are of at least equal value as the academic gains. And the beauty is that the two kinds of benefits intersect.
2) The Learning Process Unfolds Slowly, and So Good Education is Rooted in Patience
Every child develops and learns at an individualized pace. (This is why our programs are not based on grade levels: we want the learning experience to be fully customizable for the needs of each student.) We also believe that the visible effects of education are like the tip of an iceberg with layers of deeper learning lying underneath the surface. Those inspiring moments where something finally clicks are possible because of that much slower, less visible process of building understanding.
At school, there are schedules to keep, and efficient uses of time and resources to worry about. But at home, there is much more space and time for qualitative learning, rooted in an educational approach that isn’t in a hurry to get where it is going. Spaeth writes about how his “normally loquacious child was having trouble with her words. They came out slowly, haltingly, sometimes not at all. The finger tracing the page would hover for a ponderous moment over what was, apparently, a senseless jumble of letters. Sometimes, frustrated, she would have to reset her brain to proceed: “OK,” she’d sigh, then vigorously shake her head clean of its confusion like an Etch A Sketch.”
In this description, Spaeth draws our attention to that slow process of building understanding that happens under the surface. Whether it’s developing word sense or number sense, this process can feel tedious – for both instructor and student – but it is an important part of authentic learning. In the case of English, Spaeth notes that it isn’t surprising that his daughter struggles to learn to read the language. He explains that “much of English is basically ideographic, requiring readers to simply memorize the sound of an otherwise incomprehensible collection of dots and loops and strokes.” As an example, he considers the word ocean: “so lovely to look at, so strangely evocative of the thing it represents—is a phonetic catastrophe, the c suddenly switching to the “sh” sound, the ea inexplicably approximating a u.” Learning to read (or later to spell) in the English language really requires spending a lot of time fumbling, making mistakes, and slowly creating a web of connected learning, stored in long-term memory. (blog post here) As Spaeth explains, “mastering the ability to read, like so much else in life, requires practice and repetition.”
Your student is learning. Trust the process! You will see the fruits soon enough And when you let your student know that its okay to take her time, you are building the kind of confidence and resilience that will let your student grow as a lifelong learner..
3. Students are not blank slates, and learning is optimized when it is tailored to each child.
3) Students Are Not Blank Slates, and Learning is Optimized When It Is Tailored to Each Child
Every student is unique. We say this so often that it might sound cliche. But it truly is at the heart of our philosophy of education here at Demme Learning. Kids are not blank slates, and learning is not an automated assembly-line. Instead, every child is a dynamic learner with particular skills and weaknesses, whose particular learning preferences and individualized pace are all important factors to consider. The great news is that as a parent, you are more of an expert regarding your kids’ learning than anyone else in the world! Of course the expertise of others can be really helpful – for example speech therapists who are trained to help your student overcome a stutter – but no one has spent as much time as you have in observing how your kid learns. Trust the expertise that you have built up over the years.
I like how Spaeth incorporates family history and shared culture into the learning process. He observes that he and his daughter “have better luck with classic children’s books, the ones I used to read to her in her infancy: The Runaway Bunny, Green Eggs and Ham, Where the Wild Things Are.” For his daughter, these are not just any random books. Instead, these are beloved books with rich memories attached. All of these particularities enhance the learning experience, both the qualitative and quantitative outcomes. Of course, Spaeth is also right to note that “there is a rhythm to good writing that eases the passage from page to tongue, that makes reading more intuitive and natural.” Many of us fall in love with Dr. Seuss books as kids, for example, because they are truly exemplary books that help aid in our acquisition of language. But I also think that the specific books matter less than the fact that they are attached to important memories of shared delight.
You do not need an education degree to help your kids learn. All you need is a willingness to engage, to put in the time, to invest in your child with all your love and support. I hope that Ryu Spaeth’s article is a source of encouragement for dads (and moms) to lean in, and partake in the joys of the learning process. There are so many great memories just waiting to be made!
This back-to-school season may be unlike past years in terms of limits on what we are able to do, but there are still many ways to have fun. Playing strategy games is a great way to fill those long evenings while keeping your student’s (and your own) mind sharp.
Here is list of 10 strategy games that you can enjoy as a family:
The classic game of dueling ships recommended for ages 7 and up. At the onset of this two-player game, players hide a series of battleships on the grid on their side but are unable to see where their opponent’s ships are hiding. On your turn, you call out a location on the grid (eg., “A10)”, and your opponent says “miss” or “hit” until the ships are sunk.
Strategy: Battleship invites players to think strategically about both where to position their own ships, and how best to locate and sink their opponent’s ships. After you’ve played this game a lot, it gets way easier to figure out where ships are located, but especially for young players, this game is a great introduction to the world of strategy.
Sharpen your spatial reasoning skills with this geometric puzzle/board conquest game for 2-4 players (ideal is 4 players). Each player has 21 different shaped tile pieces, and the goal is to be able to place as many of your tiles on the board as possible by the end of the game. On your turn, you must place a piece connecting to another of your piece’s on the board, but only corner-to-corner touching for the same color is allowed. If you are not able to place a piece on your turn, you forfeit your turn. The game ends once no player is able to place a piece, and points are then counted. This game is recommended for ages 7 and up.
Strategy: Unlike some of the games included on this list, luck has no role in this game. It all boils down to strategic placement. Can you ensure a path for your own pieces while blocking your opponent? I love this game because the mechanics are simple but learning how to play optimally takes time. As players grow more experienced, the competition grows fiercer.
3) The Settlers of Catan
In this award-winning family board game for 4-6 players, dice rolls lead to collecting resource cards which let you build settlements and cities that are worth victory points. Players can also initiate trades for resources they need, as everyone competes to be the first to get 10 victory points. On the Catan website, you can find the version of the game that best fits your family in terms of age, amount of time per game, and number of players.
Strategy: While there is quite a bit of luck in Catan based on dice roll, the game also invites a lot of strategic thinking about where to build your roads, settlements, and cities. You can also use resource cards to buy development cards which give you advantages that you can use to gain more victory points. Some players like to diversify their resources, while others double down on a couple of resources and use that monopoly to leverage trades with other players.
4) Cat Crimes
In this single-player game for kids ages 8+, you must use your powers of logical deduction to determine which cat broke the flower pot, spilled the coffee, etc. Each game scenario includes a card with pertinent information, various tokens, and a game board to help you visualize the information and solve the logic puzzle.
Strategy: Logic puzzles like these (written as abstract word problems) are a core part of the LSAT test taken to get into law school. Many people falsely believe that kids are incapable of logical reasoning, but the truth is that if they are supported through the kinds of visual aids that the game provides, they can begin to build those reasoning skills at a much younger age than we typically expect.
Compete as the head of an Italian city-state using a mix of strategy and bluffing in this card game built for 2-6 players (best for age 10+.) At the beginning of the game, every player is randomly dealt two character cards – with each character having a specific power (possible action) unique to him/her. On your turn, you can choose to gain currency, launch an attack on other player’s character cards, etc.. The game continues until there is only one player with at least one character card remaining.
Strategy: Because different characters have different abilities, each round of the game will require you to develop a strategy for how to win with which character/powers you have (unless you opt to switch cards later in the game.) Successful players will need to figure out how many resources to devote to offense versus defense. As another fun layer, you can choose to act as though you are holding a character card that you are not, and as long as no one challenges your action, your bluff will succeed. But it’s always a risk, because the penalties for being caught bluffing can be steep.
6) Exploding Kittens
This is one of my favorite card games. The premise is simple: each turn, you play a card from your hand, and then draw a card from the deck. If you draw an exploding kitten card, you are eliminated from the game unless you happen to have a diffuse card. Each player begins the game with a diffuse card, and there are four exploding kitten cards: and then there are an assortment of other fun cards like Attack, Skip, See the Future, Shuffle The Deck, etc..
Strategy: This game is strategic because in choosing which cards to use and when, you can play aggressively (trying to eliminate other players), defensively (gathering as many cards as possible to protect yourself), or a mix of both. Thinking about probability is also relevant to this game: the further into the game, and the fewer cards left, the greater the chance of drawing an exploding kitten card.
7) No Stress Chess
Chess is a complex game and it can feel daunting to learn. No Stress Chess provides an easy introduction (for kids as young as seven) that combines a standard chessboard with illustrated cards of each piece that shows you how that piece can move. On your turn, draw a card which tells you which piece you can move on your turn. As you grow familiar with chess, draw 3 cards (or even 5) and decide which card you will play on your turn. After a while, you’ll feel confident playing chess without needing the cards at all.
Strategy: There are typically two ways to become really good at chess. The first way is to study moves: memorize openings and endings, learn optimized move sequences, and drill on classic blunders so that you can avoid those mistakes while taking advantage of situations where your opponent makes them. Professional players who use this approach analyze the board logically and make predictions for several moves in advance, with contingency plans for when the game doesn’t go as predicted, The second way, which is my preference, is to develop intuition by playing a lot, and then to rely on that intuition as you seek to maintain competitive advantage in position and movement.
The original game of world domination, this game mixes the strategy of where you place your troops and when you choose to attack your opponents with the sheer luck of the dice roll. A game of Risk can last for hours, which means it’s a great way to spend a rainy Saturday.
Strategy: In addition to planning out your attacks, and deciding which locations are worth fortifying and trying to hold, this is also a game that also rewards shrewd alliance-making. Also, as the name suggests, this is a game all about evaluating risk: because attackers use an extra dice when rolling, playing too defensively can be costly. But even the most sure-fire attack can sometimes backfire, leaving you vulnerable to counter-attacks.
9) Tiny Epic Galaxies
This is one of the coolest games I’ve played. The game is similar to Catan inasmuch as it revolves around spending resources to build cities, but while the game mechanics are not more complex than Catan, the strategy aspect far exceeds it. On your turn, you roll dice with various symbols on them: the symbols designate actions which you can use for your turn. The order in which you act matters, and as another fun twist, other players with the right amount of particular resources can “follow” your move (basically copying it) out of turn. This game is recommended for ages 14+ and can be played by as many as 5 players or as a solo game.
Strategy: to win this game, you really have to plan out your actions, steward your resources well, and anticipate what other people will try to do. This is a game that will likely frustrate younger players, but older players and adults will enjoy the layers of strategy.
10) Zeus on the Loose
Zeus on the Loose is a card-based strategy game that reinforces addition and subtraction skills. The goal of the game is to have possession of the Zeus figurine when the deck equals 100. Each turn, players add a card to the deck and do the math (eg., the pile is at 15, I added a 3 card, the pile is now 18). Whenever the deck equals a multiple of ten, the player steals Zeus. Whoever has Zeus when the pile reaches 100 wins. But there’s a catch: interspersed with the numerical playing cards are various Greek deities who can affect the game by subtracting, skipping, etc..
Strategy: I’ve played this game with students ranging from early to late elementary and they love it. The ability to strategize based on anticipating moves, and saving special cards for later in the game allows students to develop their own theories about how to win. And all the while, they’re sharpening their math skills and not even realizing it!
As always, we encourage you to be actively engaged in the entertainment choices for your family, so make sure you use your best judgment in deciding what games are a good fit for your family.
What strategy games do you enjoy playing? Did we miss one of your favorites? Tell us in the comments!
Related Blog Post
(The following post is Part 4 in a four-part series on studying math through the lens of other disciplines. We believe that students thrive when they can form meaningful connections across different areas of study. Previous installments in the series have included History, Art, and Philosophy.)
A pizza is divided into half, and then half again, and then half again. Sally eats one slice of pizza. What fraction of the pizza is left?
For many of us, word problems like this feel like torture. But did you know that your brain is doing similar calculations when it listens to music? The mathematician Gottfried Leibniz famously observed that “music is the pleasure the human mind experiences from counting without being aware that it is counting.”
If your musically-inclined student is struggling to enjoy math, exploring the connections between the two disciplines can help to reinspire your student in both studies. Music theory is the formal study of music, with a focus on the interplay of number and sound. And this post will feature some light introductions to music theory. But music is also intuited and felt through the body, and this post will also try to provide opportunities for you and your student to feel the math in music.
Simple Addition, Ratios, and Fractions in Music
Let’s begin with musical notation, the sheet music that tells musicians what and how to play. In notation, we have whole notes, half notes, quarter notes, etc., which all designate how long a note should be held in proportion to other notes. We also have time signatures, which specifies how many beats are contained in each measure, which are expressed as ratios such as 4/4. The top number indicating there are four beats per measure, and the bottom number indicating that one quarter note equals one beat. So for example, if my time signature is 4/4, I know I can fit four quarter notes (♩) into that same measure. To see this in action, write 1234, 1234, 1234, 1234 (that’s the four beats of the measure, in four measures); then have your student count those numbers while clapping for each number. Next, try out the half note which in 4/4 signature, takes up two beats. Write out 1234 (four times) again, but this time underline the 1s and 3s and tell your student to only clap when they say those numbers. (You can also circle 1 and 2, showing that half note takes up two parts.) Now try whole notes which are notes that last for a full measure. Underline the 1s and only clap when you say them.
Now it’s time to experiment with how ratios can overlap. Try this exercise with your student: have your student clap the quarter notes while you clap the half notes. If there are other people available, have them clap the whole notes. Try to keep tempo together as best you can. As a bonus, see if your student can demonstrate how to continue the pattern by halving the quarter notes while keeping the time signature the same. (Hint: there are two eighth notes in every quarter note.) In solving this problem, your student is doing the exact same work as in the pizza word problem. A whole pizza is like a whole note, half a pizza like a half note, and then quarter, eighth, and etc..
Okay, now let’s practice counting out a famous excerpt of music from Beethoven’s Ninth Symphony. While watching the video below (it’s 50 seconds, so watch it several times for these exercises), have your student start by counting aloud 1234 over and over, practicing the 4/4 time signature and quarter notes. Your student can then experiment with clapping the whole notes (1, 1, 1), the quarter notes (1234, 1234), or half-notes (1, 3, 1, 3, 1,3).
Your student can also experiment with adding in the other notes. Identify each type of note and add it to your clapping. You will quickly notice there are a few extra notes types (dotted quarter notes are ⅜ of a beat). Or you can follow along to the bottom line of music which is only half notes and whole notes. If your student is looking for an extra challenge, see if they can figure out how to count out the eighth notes.
I studied piano for several years. And I also took tap dance lessons and participated in musical theater. Dance is another discipline that lets us feel the math in music, and also the math in our movement. Tap dance most obviously corresponds to counting because the tapping sound actually helps us to keep the count. But all dancers have to keep count in their head to be successful. In this famous tap dance routine from the beloved classic film White Christmas, notice how the dancers use the tapping of their feet along with clapping and even snapping to help keep track of the intricate rhythms. What you’re seeing and hearing in this routine is ratios brought to life. As you and your student watch, try clapping or tapping along with the beat and don’t feel bad if you lose track of it from time to time, just listen and try to regain it again. See if you and your student can count it out at various points in the song.
You probably found this a much harder exercise than Beethoven’s Ode to Joy. The main reason for this is that the dance number uses a musical technique called syncopation which is all about switching which beats get stressed. When you and your student count and clap 1,2,3,4 in Ode to Joy, notice that 1 and 3 sound naturally stronger as beats, while 2 and 4 are softer. But if you reverse this, so that 2 and 4 are the strong beats, and 1 and 3 are the soft beats, it changes the sound and feel of the count dramatically. Oh, and in case you were wondering, I definitely can’t tap dance this routine; in fact, I can barely keep track of the count!
Properties of Multiplication in Math
In thinking about this relationship between math and music, one of my inspirations is Dr. Eugenia Cheng, a professional mathematician who is also a concert pianist. Dr. Cheng is passionate about the intersection of music and math, and she has created videos that blend instruction in both music and math.
Most of us do not have the level of knowledge in abstract mathematics needed to understand the kind of advanced research that Dr. Cheng is involved with. But the mathematics in music are often far more accessible to us on the level of intuition. Recall Leibniz’s insight quoted at the beginning of this article about how we make sense of music through counting though we typically don’t realize that’s what we are doing.
Here’s a fun example that Dr. Cheng highlights. The principle of commutativity in multiplication tells us that we can multiply numbers in any order without changing the product. So for example, if we know that 2 x 3 = 6, then we also know that 3 x 2 = 6. As it turns out, it’s this principle that allows us to have varied meter in music in a way that still feels meaningfully patterned. By varied meter, I mean when a song starts out with one time signature (say 4/4 rhythm) but slides into another one (for example 3/4 meter which is often heard in waltzes.) In the short video below, Dr. Cheng uses several pieces of music including a familiar Broadway song from West Side Story to illustrate how this principle sounds in music. As Dr. Cheng states: “the commutativity of multiplication sounds like an obscure rule in mathematics but it turns out to be something that we can feel in music. Feeling things in music is a kind of obvious thing to do; feeling things in math is less obvious but we can feel things in math as well.”
If you and your student enjoyed this video from Dr. Eugenia Cheng, be sure to check out this website that features ten more videos where she explains cool aspects of math in music.
Hopefully, these exercises have helped demonstrate that disciplines like math and music are deeply interconnected. Once we realize this, we are empowered to approach these disciplines in less conventional ways. Whether your student is currently taking piano lessons, inventing their own songs on a guitar, or listening to their favorite album on repeat, they are actually engaging with mathematical ideas and processes, though it is mostly subconscious. And of course, the more they engage with music theory, the more they can see those mathematical ideas consciously as well. The goal, in the end, is for your student to interact with all kinds of music from various vantage points – listening, studying, making, and playing. All of these modes of engagement provide your student with opportunities to feel the math in music, and perhaps later, to hear the music in math.
(The following post is Part 3 in a four-part series on studying math through the lens of other disciplines. We believe that students thrive when they can form meaningful connections across different areas of study. And we also know that many students who are disillusioned with math can find fresh inspiration in seeing how math interacts with other subject areas that they care about. With this in mind, in this series we want to provide ideas for supplementing math education to reinvigorate your student’s learning. Previous installments in the series have included History and Art. Our final entry in the series will include studying math through music.)
If your student is like me, textbook math is often…less than engaging. Order of operations, differential equations, and long division…yawn! We need to learn all of these things, but sometimes it’s more fun to get lost down a rabbit trail of ideas.
Math is About Big Ideas
Math really clicked for me as a junior in college when I took a course entitled Mathematics in The Western Tradition which doubled as a philosophy and history of math course – while also counting for my quantitative reasoning requirement. In this course, we worked our way through the development of mathematics, from the mystery cults of the Pythagoreans through Cantor’s radical realization that there can be greater and lesser sets of infinite numbers (I still only somewhat understand this one.) For the first time in my life, I realized that math is about Big Ideas: beyond dry word problems and rote memorization, the field of mathematics opens up into an ongoing conversation centered on exciting philosophical questions like, “why is there something rather than nothing?”
Your student may find the philosophy of mathematics to be an engaging way to supplement their mathematical education, and a source of inspiration for those times when the discipline feels tedious or dry. Sometimes it’s worth putting the textbook aside (it’ll be there waiting when you return), and putting on your philosopher cap.
Related related blog post: How and When to Take a “Math Break”
You and your student can start by just asking questions: What is a number, and does it exist? How do you know?, and letting the conversation unfold from there. I have some sample “Discussion Prompts” at the end of this post, but before I list those, here’s some background on the philosophy of mathematics.
Ontology: What Is a Number, and Does It Exist?
Numbers are profoundly strange. We have names for them, we talk about them, and we use them in our daily activities. But we don’t see numbers the way we can see butterflies, or feel numbers the way we can feel the wind on our skin. And while music and math are very obviously linked, particularly in terms of rhythm, we don’t hear 2 singing out good morning. So in what way can we say that a number is “something” that exists?
The subfield of philosophy that these questions fit into is ontology, the study of being, essence, and existence. Various philosophers have addressed these very questions in differing ways. The ancient Greek philosopher Plato believed that numbers exist as Forms – immaterial and invisible realities that shape the material world. Medieval theologians like Augustine often taught that numbers exist as ideas in the mind of God. The 20th century mathematician Bertrand Russell believed that all math is fundamentally a self-contained system of logic. And many postmodern thinkers maintain that math is basically a made-up language that helps us achieve practical goals but isn’t true in any absolute sense.
I have my own beliefs about what a number is, but that isn’t the point here. The great thing about philosophy is that it is something we can all participate in these debates, at the level that we are able. When a group of first graders wonder why the sky is blue, they are just as engaged in philosophy as the tenured Harvard professor.
Epistemology: How Do We Know?
Okay, so we might have strongly held beliefs on what numbers are, but how do we go about justifying those beliefs to ourselves and others? How do we know that our beliefs about numbers are true?
Epistemology is the subfield of philosophy that deals with questions about how we know what we think we know. The goal in epistemological investigations is to produce what is called a “justified true belief,” a fancy way of saying that we want to demonstrate both the right belief and the right reasons for that belief. Suppose, for example, that I was locked in a room without windows in the month of December and I said, “I believe it is snowing outside, because it is winter.” Now, it could very well be snowing outside which would make my belief true, but it doesn’t snow every day in winter, and so there’s no adequate justification for my belief. But suppose my room did have a window, and I observed it to be snowing. Most of us would readily accept that I have adequate justification for my belief. Though of course it could conceivably be the case that I’m trapped in a virtual reality simulation and am being tricked into seeing snow where there is none, the far more likely scenario is that it is indeed snowing.
When it comes to the epistemology of math, we can produce proofs (leading to justified true belief) for all sorts of things, such as demonstrating that 1 + 1 = 2 or that “a squared plus b squared equals c squared.” But a belief in the validity of something like the Law of Noncontradiction – which says for example that a shape cannot be both a square and a triangle – is something that is not provable but must be held as a self-evident first principle. Of course, the question of what is a first principle and what is a contingent (and provable) idea gets tricky fast, and that’s yet another dimension of the ongoing conversation that is mathematics.
For most of us, our daily experience of mathematical reasoning is tied to the practical – like doubling a recipe while cooking. But I hope this post opened up possibilities for engaging with mathematics on a more theoretical level. For another take on engaging with math in unconventional ways, check out this post on exploring math through its often incredibly exciting history.
– For the youngest ages, I would simply start by asking, “what is a number,” and seeing what your student comes up with. I’m often shocked by how perceptive even the youngest learners are, and how far their curiosity and wonder naturally take them.
– For middle school students, consider also asking about other immaterial things we speak of as existing, like love, and how numbers are or are not similar to these other entities.
– For high school students, consider supplementing your discussion by reading this article on Plato’s famous Meno dialogue, which will be sure to provide all sorts of interesting questions to ponder and discuss. If your student feels particularly inspired, they can even read the full Meno dialogue.
– What does mathematics have to do with virtue formation? According to Francis Su, former president of the Mathematical Association of America, studying and practicing mathematics can help cultivate virtues like perseverance, integrity, and love.
– One core belief that we have at Demme Learning is that students need to be active participants in their own learning, and part of that means being invited to see themselves as already being mathematicians participating in the tradition of mathematics.
– The understanding that mathematics is inherently philosophical fits well within a classical education framework.
(The following post is Part 2 in a four-part series on studying math through the lens of other disciplines. We believe that students thrive when they can form meaningful connections across different areas of study. The first post in the series looks at exploring math through history. The final two installments in the series will include studying math through philosophy and music.)
Consider the following word problem.
Andre and his nine friends are at a birthday party. Andre’s friend Taylor is the fourth person in line to receive cake. Three additional friends receive their cake after Taylor, but before Andre. How many friends receive cake after Andre?
Are you already groaning?
I’ve always hated abstract math and logic puzzles like this. For many students, abstract learning that is disconnected from a concrete, sensory experience can feel alienating and frustrating. But using simple art, I have helped a classroom full of early elementary students work through math puzzles like this. Initially many students feel overwhelmed with the number of details and trying to solve this problem abstractly through logic is almost impossible for this age group. But there’s a simple way for young students to solve a problem like this. They can draw Andre and his nine friends (stick figures are sufficient), label them, and then count. Inevitably, the more artsy students spend time sketching elaborate hair or pronounced facial expressions. But regardless of how detailed their drawings are, visual representation gives them the key to solve the problem.
Redefining the Lines Between Disciplines
It is often popular in contemporary education to draw strict lines between disciplines. We often talk about STEM fields versus the Humanities or say things like, “I’m a creative person” or “I’m more of a math person.” We even use the categories of “left-brained” or “right-brained” – though most research suggests that the whole brain is generally engaged in most activities (Source). While these categories of thought can be helpful sometimes, they can also force us into boxes or prevent us from seeing how one area of study can positively influence another.
Math Isn’t All Abstract
Mathematics certainly involves abstract thought, but it also involves sensory perception, intuition, number sense, and spatial awareness. Perception and intuition can actually serve as scaffolding to support more abstract engagement and to sustain the feeling that math is tangible, knowable, and connected to our experiences of the material world. Many artistically-minded students become easily frustrated with sheer mathematical abstraction but thrive when visualization makes that abstract math click for them. One of the goals of our approach to math education with Math-You-See is to foreground visualization. We appeal to sensory perception as a valid and indispensable building block for conceptual understanding. That’s why we encourage students to use manipulatives to represent everything from simple addition to dividing fractions to solving algebraic equations.
Visualization in Math and Art
Art is all about visualization and art and math are often connected. We tend to recognize this reality with early learners, for example when we have students count the number of apples on the page in a picture-book. As students age, it’s easy to think that this connection no longer matters. But your highschooler studying geometry can also benefit from looking at art and connecting it to mathematics. Ask your student to think about and discuss how lines in a drawing affect their perspective, suggesting three-dimensional space or creating visual layers of sky and land. Have your student consider how principles of proportionality are used in portrait paintings. The closer you and your student look, the more you can see that art is math in action. And I think the inverse can also be true, that the more your student can see the math in art, the more likely they will be able to see the art in math.
Traci Jackson, a math teacher in San Diego, provides another great example of how to draw on the connections between math and art to build conceptual understanding. Recently, Jackson has been using sidewalk chalk to create engaging math problems for local residents to encounter as they go about their day. One of Jackson’s most creative techniques is to use shapes like circles or squares to stand in for X in solving equations. This is a playful way to represent abstract math in a modality that’s far more fun than a math workbook.
Here are some examples of Jackson’s sidewalk art:
Today's #mathwalk problem is from @mashupmath! I couldn't draw candy, so substituted shapes. It is so interesting that people are more willing to engage in algebra if the variables aren't letters. #sidewalkmath #iteachmath #MTBoS pic.twitter.com/wdhMWDiZJO
— Traci Jackson (@traciteacher) May 4, 2020
— Thy Dinh (@Dinhclass) May 6, 2020
In her reflections on the success of this project, Jackson talks about how art can change our perception of math. “The [common] perception of math is a set of sterile problems but in reality it describes all the patterns of our world. … [Sidewalk math] opens the conversation to what math is.”
If your student loves to create art, challenge them to consciously use the math they are learning in their artwork, and then to explain their process to you. Consider also giving your students opportunities to take a handful of word problems and render them in elaborate artistic fashion as part of their solving process. You can even combine art, math, and history by letting your student pick some famous paintings to explore through the lens of mathematics, and then having them attempt to copy those paintings for themselves.
Ultimately, mathematics is a creative discipline and engaging with art is a fantastic way for your student to lean into that creativity. Have fun and let your imagination soar. And we would love for you to share your fun math art projects with us on Instagram with the hashtag #MathUSee.
(The following post is Part 1 in a four-part series on studying math through the lens of other disciplines. We believe that students thrive when they can form meaningful connections across different areas of study. And we also know that many students who are disillusioned with math can find fresh inspiration in seeing how math interacts with other subject areas that they care about. With this in mind, in this series we want to provide ideas for supplementing math education to reinvigorate your student’s learning. Future installments in the series will include studying math through art, philosophy, and music.)
For a given triangle labeled ABC, side A is 5 centimeters while side B is…wow, I’m yawning already! So let’s take a math break. Instead of calculating angle measurements using the Pythagorean theorem, I want to tell you a story about a mathematician whose “irrational” thoughts led to his probable demise at the hands of Pythagoras and his buddies.
A long time ago, in a far-away land, a secretive group of mathematicians were united by one core belief: all is number. These Greek philosophers were disciples of Pythagoras, and they lived an ascetic life. They avoided eating meat or collecting wealth, and they venerated beans as a sacred object because, well, we don’t really know why. These mathematicians loved rational numbers, those numbers that can be expressed as a whole number or ratio. In fact, they believed that all numbers were rational and that this reflected the logical divine order of the universe.
Then one fateful day, one of the Pythagoreans named Hippasus discovered a devastating secret: not all numbers are rational. In an attempt to find the square root of 2, Hippasus realized that it was simply not possible to represent that number as a ratio. And he was able to demonstrate to his peers that this was the case through a technique called proof by contradiction. Grieved by this paradigm-shifting discovery, the Pythagoreans vowed never to tell anyone what they had learned. But as the legend goes, Hippasus spilled the beans (wink, wink), and so he was taken out in a boat and drowned at sea.
Knowing the story of the Pythagoreans may not be strictly necessary for your student to solve geometry problems using the Pythagorean theorem, but there is great value in learning the history of mathematics. Textbooks alone can often feel cold and lifeless, as though math is not tangibly connected to what makes us feel animated, like our desires or hopes. But exploring the history of mathematics shows us that math is very deeply connected to core questions and ideas that have shaped the way we think about and interact with our world.
We all desire meaning, structure and agency, and we can see these desires driving mathematical discovery at every point in world history. Behind every key discovery, there lies either a How question or a Why question, some kind of puzzle that keeps mathematicians lying awake at night until that beautiful moment, Eureka! Studying the history of math is thus a great way for your student to feel inspired by these questions and puzzles, and to feel connected in their efforts to find answers for themselves.
3 (Additional) Reasons Why Studying the History of Mathematics is Valuable
1) Math History Provides Us with Meaningful Context
This allows us to connect facts together to form a meaningful web of learning. Our brains thrive on patterns, and the more we can connect one idea with another, the easier it is for us to remember everything that we are learning. This is all the more true when we engage with history through narrative, which can embed pertinent information in a memorable plot. Remembering the difference between rational and irrational numbers becomes easier when it’s linked to a frightening tale of murder on the high seas!
2) Math History Deepens Our Respect for Human Cultures and Collaboration Across Time
The story of math spans continents and centuries. For example, the insights of algebra and the numeral system we use today originated with the medieval Spanish Arabs. Initially, much of medieval Christian Europe was hostile to this new approach to abstract math because it came from an outside, foreign culture. Thankfully the medieval mathematician Gerbert (who later became Pope Sylvester II) understood that Europe did not have a monopoly on truth, and under his influence, European mathematicians began to integrate these concepts and systems in their own mathematics.
3) Math History Presents Us with Role Models
These role models are people who practiced mathematical reasoning with integrity and courage. One of the reasons that I love the story of Hippasus is that it reveals how destabilizing new discoveries can be to our view of the world. This is a theme seen throughout the history of mathematics and science, and often the men and women responsible for these discoveries were alienated or worse by the academic community around them. But part of being a good mathematician or scientist is being willing to follow the truth wherever it takes you, no matter how many cherished beliefs might need adjusting along the way. Think also about the integrity of Gerbert to accept the mathematical concepts that originated from outside his own circles, and the courage to advocate for them no matter how unpopular it may have been with his peers. By exploring these and other stories, we gain examples of how to conduct ourselves in our pursuit of truth. (For more reflection on how studying math can cultivate virtue, check out this post [insert link].)
Math workbooks are valuable, and the discipline of mathematics often requires methodological and rigorous learning that utilizes those workbooks. But sometimes it is worth putting the workbook aside and exploring math from a different vantage point. Digging into the history of math can be a fun break that still assists in math education.
To explore more key moments in the history of math, check out the online resource The Story of Mathematics. This cool website traces the history of math from the account-keeping of the ancient Sumerians and Egyptians all the way through modern developments like Gödel’s incompleteness theorems.
If you wanna dig into the specifics of why √2 is an irrational number and how to prove it, check out this insightful article. You can also read more about rational and irrational numbers in Isaac Demme’s entertaining article.
Why Study Math?
There are a few answers to this question, some more obvious than others. We study and practice mathematics because it is useful for everyday activities like calculating the tip based on the percentage of the bill, or measuring the length of wood before cutting it. But there is also a deeper reason why we study mathematics, and that is because it is an activity inherently worth doing, just like playing the violin or drawing in a sketchbook.
Francis Su, the former president of the Mathematical Association of America, wants to convince us that studying and practicing mathematics is a path to building virtues, which lead to a flourishing life. In a series of speeches and publications, Su draws from ancient philosophers like Aristotle who understood that “people flourish when they exercise virtue.” Su invites us to see how mathematics, as a core component of a liberal arts education, can foster growth in virtue, which is a benefit to ourselves and a benefit to everyone around us.
Su always begins his reflections by introducing his audience to his friend Christopher, an inmate at a federal prison serving a 30 year sentence. The two became pen pals, and over the years, Christopher has sought Su’s guidance as he studies increasingly advanced mathematics in prison. For Christopher, math is not simply a distraction or something to fill up the time. Instead, math provides him with a source of profound meaning, allows him to achieve goals, and helps him develop as a person. In other words, math is helping Christopher to become a better person.
How Can Studying Math Build Virtue?
Francis Su’s friendship with Christopher has led Su to reflect concretely on how studying math builds virtue. The process of struggling with a problem, while trusting that there is a knowable answer, leads us to exercise hope that truth is attainable and strengthens the perseverance and grit required to solve the problem. Because mathematics is external and (mostly) objective, it requires us to submit to the discipline itself in order to progress, and that fosters both honesty and humility. As we advance in our understanding of mathematics, we can begin to see all kinds of unexpected patterns. This attentiveness can lead us to make all sorts of connections to the world around us, for example, seeing the Fibonacci sequence in flowers and seashells. In making these connections, the beauty of math can foster gratitude for the intelligibility of the world.
Math and Human Desires
Su agrees with classical thinkers like Plato and Augustine that both genuine learning and virtue are powered by desire. We act according to what we love, and what we love in turn shapes the kind of person we become. Su highlights five core human desires that mathematics can tap into: the desire to play, to perceive beauty, to understand truth, to render justice, and to express love. The best education helps to cultivate these desires while channeling them away from ignoble ends (like feeding the ego by winning competitions) and toward noble purposes such as using our knowledge to serve others. Christopher exemplifies this by using his math knowledge to tutor his fellow inmates and help them earn their GEDs.
Mathematics is useful for all sorts of practical goals. Su helps us remember that beyond its usefulness, the study of mathematics invites us into that deeper, sacred work, known as contemplation. Plato famously said that all philosophy is born in wonder, and mathematics shows us that the world truly is a wondrous place.
Here’s a 5-minute speech from Francis Su that outlines his thinking on math and virtue.
Su has also written an excellent book on this subject, titled Mathematics for Human Flourishing. Each chapter considers a different virtue, and Su interweaves his reflection with insights from personal experience as well as insights from thinkers like Simone Weil. At the end of each chapter, Su provides an optional mathematical puzzle that loosely connects to the theme of the chapter. Some of these puzzles are harder than others, and if you get stuck, there are hints as well as the solutions in the back of the book.
Related Blog Posts
I recently read a sobering cover story in The Atlantic on the rise of anxiety in children. The article’s author Kate Julian begins by looking at the grim statistics from PEW Research on the growing percentage of kids who are diagnosed with anxiety or other related disorders. There has been a steady uptick since the early 2000s, and there is also preliminary research that suggests each generational cohort is more at risk than the preceding one.
After this survey of research, Julian seeks to understand why anxiety has increased, and how to deal with it. One sentence from the article really jumped out at me. Julian writes:
Anxiety disorders are well worth preventing, but anxiety itself is not something to be warded off. It is a universal and necessary response to stress and uncertainty.
In fact, she goes on to note that trying to protect kids from any feeling of anxiety or shielding them from any risk of suffering can undermine the resilience we all need to deal with the curveballs of life.
Can Reading Aloud Help Children with Anxiety?
I’m not an expert, and the question of how to fully deal with the rise of anxiety in today’s kids is above my paygrade, but I do believe one source of help is found in parents reading stories aloud to their kids. I have increasingly grown to realize how important this was in my own life, in helping me to cope with early childhood struggles, and in giving me language to understand and express my own anxieties. My experience of reading aloud fits with the empirical data. In an article for PBS, Deborah Farmer Kris writes about the immense power of reading aloud. She cites several studies that have found clear cognitive and emotional benefits associated with reading aloud to kids, especially when it is the parent who is doing so.
In one such recent study cited in the PBS article, the researchers write that when parents read aloud, their children “learn to use words to describe feelings that are otherwise difficult and this enables them to better control their behavior when they have challenging feelings like anger or sadness.” This makes intuitive sense: stories provide us with language and imagery to make sense of our own experiences, desires, and anxieties. And when parents and children direct attention to the same story, the language and imagery can be a shared basis for dialogue about what children are experiencing.
As it turns out, even the very act of reading can help us (adults and children) to destress. The Telegraph reports on a British study from 2009 that indicates that even six minutes of reading a day can lower stress levels by 68%. Reading is a calming activity, both physiologically and mentally. And I think even just devoting the mental space and time is a helpful way of lowering anxiety. In my opinion, the content of our stories matters too. I believe stories that feature children acting bravely and with integrity and purpose can be an especially profound source of empowerment for kids. And indeed, such stories are able to delve deeply into heavy themes like death in ways children can respond to, which is far more helpful for their development than shielding them from these topics altogether.
In my own life, stories like Where The Wild Things Are, Charlotte’s Web, and To Kill A Mockingbird were all formative. These stories, and so many others, provided me with protagonists I could identify with, who wrestled with emotions I wrestled with, and who learned to face their world with courage and responsibility. And these stories provided my whole family with a shared reference point for our own family culture, helping us all to better communicate with each other.
In the end, reading aloud to your kids will not fully eradicate all anxiety, and to Julian’s point, banishing all anxiety probably isn’t best anyway. But stories are powerful; they inspire us, give us tools to name our fears and process our emotions, and provide a common foundation for dialogue with the people in our life who stand ready to love and support us, come what may.
Harvard University’s Center on The Developing Child has an excellent introduction to the science of resilience, that includes practical advice for how to deal with particularly stressful times.
Related Blog Posts
As a 90s kid, my introduction to the world of high stakes negotiation was trying to convince my parents to let me play the latest violent game. While I rarely prevailed in these conflicts, (yes to Star Wars, a resounding no to Grand Theft Auto), my parents were hardly the tyrants I felt them to be at the time. And in a surprising turn of events, some of the games I remember most fondly now were the games my parents were most enthused about: educational computer games that were more about creative thinking and problem solving than the shooting and slashing games my friends wanted me to play.
In this post, I’ll share some of the games that I played, and that I recommend because they are both genuinely fun and actually educational. Of course, we at Demme Learning always encourage you to research any game for yourself before deciding to let your child play it. And we also encourage you to decide with or for your child how best to balance screen time with other activities including outdoor play. Having a conversation with your child about not only the amount of time spent playing computer games, but which games can or cannot be played is a great opportunity to explore your family values together.
History: Oregon Trail
In this award-winning classic for kids of all ages, your student will choose a character who will venture out in a wagon with American pioneers to Oregon. Along the way, your student will need to keep track of how much money they have to spend on needed supplies, while making key decisions like how best to cross a river, and dealing with unexpected catastrophes like a venomous snakebite.
You can play the retro version of the game online for free.
Suggestions for Older Students
Consider games like Civilization, CivCity: Rome, and Crusader Kings II which are all turn-based strategy games that teach a wealth of history. Please note game ratings, and decide what is appropriate for your family.
Geography: Where In The World is Carmen Sandiego
The notorious art thief Carmen Sandiego has just stolen a valuable jewel, and the whole world is looking for her. Your student will need to sharpen their geography skills to track Carmen as she galivants around the globe. Your student will be so caught up in the adventure, they won’t even realize how much geography and history they are learning!
Science: Zoo Tycoon II
It’s time for your student to put on their small business owner cap, and design and run their own zoo. This game is a blast to play, and the learning opportunities are rich. To successfully run the zoo, your student will need to do in-game research about what each of their animals need in terms of living environments and diet, while making sure those tigers can’t escape their cage! This game also functions as a crash course in business, letting your student intuit principles about supply and demand, pricing, and leveraging savings to slowly expand infrastructure.
Civics: Democracy 3 (suggested for older students)
In this turn-based strategy game, your student is the newly elected leader of the nation of their choice. Your student will have a budget to maintain, and will need to use a slider bar of policy outcomes to keep enough voter bases happy to prevent being assassinated and to ensure reelection. In the midst of governing, your student will have to deal with random events like natural disasters and make tough split-second decisions about everything from labor disputes to new trade agreements.
Parental Note: Policy decisions can include controversial social issues. This can provide you with great opportunities for discussion with your older students.
Creative Play: Minecraft (suggested for older students)
Let your student build a world of their own imagination, using blocks as the primary resource. Minecraft is a completely open-ended game that allows for maximal creativity.
Here’s a short trailer that gives you a feel for Minecraft’s unique visuals.
Strategic Thinking: Chess.com
Whether your student is a chess novice or a grandmaster, chess.com will provide plenty of opportunities to grow. You can choose to play against the computer or a live human being (including your friends!). You can also watch tutorials or read articles to help you improve your game.
Chess.com is free, but you do need to sign up using an email address or by linking to a Facebook or Google account, etc. You can also pay for more extensive features, like a comprehensive analysis of your games to pinpoint strengths and weaknesses.
Parental note: Because chess.com lets you play against other humans online, you’ll need to decide whether or not you want your student to disable the optional chat window (our recommendation).
(Disclaimer: Demme learning does not own, benefit, or profit from the links provided in this post. Recommendations are the opinion of the author, and do not necessarily reflect the views of Demme Learning. Linked site owners are responsible for information and content on their webpages. We hope to provide resources to parents but encourage parents to do their own research and have family conversations about online safety as well as finding an appropriate balance with screen time and other activities.)
(If your student is struggling to feel motivated studying algebra or geometry, have them read the following about the exciting ways mathematicians aid healthcare professionals to keep us healthy.)
“Ugh! Not more lines on a graph! How will I ever use all this math I am learning?”
Friend, I totally get it! Throughout high school, I would constantly wonder why I needed to study abstract ideas that seemed so far removed from daily life.
Learning math can sometimes feel impractical and unrewarding. But did you know that public health scientists are at this very moment using math to figure out how to protect us during this coronavirus epidemic?
In fact, there’s a whole field in public health called mathematical epidemiology in which researchers use the kinds of graphs you’re learning about to read to answer questions like, “How long before the number of people infected doubles, and how long before that number doubles again?” Answering questions like these can be a matter of life and death, and by providing this data, researchers are able to help everyone from local governments to hospitals prepare for and respond to challenging situations.
What Does It Look Like to Be a Mathematical Epidemiologist?
In this short video, Dr. Lauren Meyers talks about her job, and how her nerdy childhood science camps and college education prepared her to tackle the complex challenges of public health.
One major job of a mathematical epidemiologist is creating mathematical models to provide hypothetical scenarios that can help us make key decisions. In the short video, the
Society for Industrial and Applied Mathematics, high school students talk about how they’re using math models to study disease, decide which recycling program is most cost effective, and even figure out what rollercoaster is the most thrilling!
So, Are You Excited About Math Yet?
Can you picture yourself as a mathematical epidemiologist, fighting the spread of disease? Try developing a math model for yourself – start by figuring out what question you want to answer, then decide what the variables will be for your model, and then determine what mathematical concepts you’ll need to find the solutions.
For more on developing your own math models, watch these short videos:
Note for Parents and Teachers
The Society for Industrial and Applied Mathematics has helpful resources on teaching and evaluating mathematical models.
The CDC has put together some online resources for teaching kids about epidemiology.