Cake
• Hi All! My name is Dan Finkel. I'm the founder of Math for Love. It's a pleasure to be on this panel with all you fine people!

While I always enjoyed math, my moment of real math conversion came when I attended the Hampshire College Summer Studies in Mathematics program when I was fifteen. That was my first taste of "higher math," and the beginning of the passion for mathematics that still guides my work. Since then I went on to teach math to high, middle and elementary school students, get a PhD in algebraic geometry, found Math for Love, and design games, puzzles and lessons that I hope share the same excitement that I found in those first moments of genuine mathematical thinking.

There are, for me two reasons to share the ideas from higher math with students at a young age. The first is that mathematics is much, much bigger than arithmetic. It's bigger than all the math that students see in school. Consider that calculus, the endpoint of the tradition high school math sequence, is about 350 years old. Ignoring more modern results would be like never teaching anything in English more current that Shakespeare. (Are his plays great? Sure. But a LOT has happened since then!)

But there's another reason too. The central point for me is not necessarily the content you share (though there's an argument for sharing beautiful ideas with kids at any age, as long as they are ready for it); it's about sharing the process of mathematical thinking. Accelerating kids into higher grade material isn't worth doing if the process of genuine mathematical thinking isn't part of the deal. On the other hand, when you ask students to consider interesting patterns, ask questions, make conjectures, and try to break them with counterexamples—another "advanced" idea that very young kids can play with—the students may take you into territory you weren't expecting.

Many years ago I was working with a fantastically gifted second grader. One of his great gifts was the ability to ask natural, fascinating questions. He once came in and asked whether you can always cut a collection of identical squares into a single square. For any n, that is, can you dissect n identical squares into one single square? For four squares you don't need to cut them: just arrange them in an array. For two squares it's relatively simple. Five is trickier, but still doable. Three seemed impossible.

I didn't know anything about this problem at the time. It came from the student. But I ended up finding it discussed in an old Martin Gardner book, with the three square case solved, gorgeously, by a ninth century Persian astronomer. The problem, in fact, was part of the inspiration for one of Hilbert's 23 problems (the third). Did we go looking for "higher math?" No. But by letting the act of asking questions about novel ideas and situations play a part in our process, we played the game of mathematics that gives rise to higher math.

There are so many beautiful areas of mathematics, from games, to knots, to topology, to modular arithmetic, that are accessible to young kids. But the point is not just to convey the knowledge. It's to participate in the thinking.

Even arithmetic comes alive when you attack it this way. A student I knew once asked his mom if 3^(-2) had any meaning. When she started to answer, he stopped her, saying, "Don't tell me. I want to figure it out."

• Thanks for pulling this together. I actually have a degree in chemistry, but I’ve always loved math and share that passion with my children. I homeschool 4 of them, from a 6th grader working through Art of Problem Solving’s Introduction to Algebra down to a pre-k girlie who still often skips the number “firteen” when she gets past twelve.

One of the best things I’ve done is to make math a totally natural part of everyday life. When we are putting the silverware away, my 4 year old and I can count by twos or fives for fun, and fractions, negative numbers and decimals come up in conversation before they come up in a curriculum.

My kids and I recently saw a special when you buy 3 medium pizzas, but they were sold alongside personal pizzas and large pizzas, and I asked my kids what was the most pizza they could get for a certain amount of money. That was the first time my 6 year old really got interested in pi. He’s a big fan of pizza and it blew his mind that you could get more than twice as much pizza for the same amount of money if you just do the math.

Introducing topics as they come up in real life eliminates any preconception that math is dry or incomprehensible. Instead, it’s useful, interesting, and amazingly beautiful at times.

I like to explicitly help develop number sense with my young kids because I think it gives them a framework to build upon. My preschoolers can count, but it’s like magic when they realize what “sixty-five” actually means and how it relates to other numbers, rather than simply coming after sixty-four.

These super simple handmade cards are fun to help them connect the dots.

• A collection of projects I put together in 2014 is probabily a nice peek at some of the ideas from advanced math that we explored when the kids were in elementary school. At the end of 2014 my older son was in 5th grade and my younger son was in 3rd grade.

Looking back on that list now, there are certainly a ton of fond memories for me. The Terry Tao "Cosmic Distance Ladder" for example, was an incredible source for ideas to share with kids.

Another favorite moment was talking through through the famous / infamouns Numberphile video about the strange "equality" 1 + 2 + 3 + 4 + . . . . = -1/12. If you start the video below around 6:50 and watch for about 2 min you'll see how much it bothered my younger son, but also why it was so fun to talk through with kids:

Another great memory for me is talking through the so-called Chaos game wth them. Watch for about 2 min here starting around 2:15 to see the joy that advanced ideas math can bring to kids:

• The fun part of knowing the trajectory is being able to see the progress kids are making, but I don't think parents need to know the difference between kids who do and don't have things like one to one correspondence. Parents will naturally demonstrate counting objects by assigning one number in the counting sequence to each object they are trying to count. And it will be obvious to adults that the answer to a "how many" question is the last number that they say (that's cardinality). Kids learn these things over time but they certainly don't need to be told "don't forget to use one to one correspondence when counting!" Anyone can make a rekenrek, the ones on my math play mats are pipe cleaners and beads. https://tinacardone.com/math-play-mat/ But any object that kids can move around is great for counting. A fun challenge is to count a set of objects by lining them up, then see if the number stays the same when you put them in an array. Which way is easier to count? Do you want to count the rows or the columns? They can learn the commutative property of multiplication this way!

• Wonderful to have your participation in this weekend’s panel.  I thoroughly enjoyed your insights shared during last year’s event.

Cake’s co-founder @Chris is doing TedX this month.  As someone who’s TED talk video has received over half a million views on YouTube, what were some of the opportunities that occurred for you from that experience?  Would Prime Climb have been created without TED?

***********
From your exploration, Why We Love Counterexamples

It is possible to play Counterexamples with kids as young as kindergarteners as a kind of reverse “I Spy” (“I claim are no squares in this classroom. Who can find a counterexample?”).

I love this because you’re taking a higher math idea and turning it into a game that children that age are familiar with.  It’s also a game that parents and grandparents can play with the child. And it instills in children at a young age that math play is enjoyable.

Can you talk about other math explorations of Higher Maths that parents can do?  Or that an elementary school teacher, who doesn’t consider herself a mathematician, can try?

• Note to audience: Enjoying this conversation? Why not join Cake?

***************

Happy to be chatting with you again about math, @amacbean16 . I received a question from Twitter on the cover image used for this panel.

Can you talk a bit about the gifted curriculum you use and how you decide what’s a good choice versus what to avoid? What other resources do you draw upon as a homeschooling educator?

And do your kids really like math?

😉

• Math understanding came very early to my oldest child. She seemed to just intuit concepts, and plowed through the solid mastery-based curriculum I had chosen to teach arithmetic (Singapore or Primary Mathematics).

She was on track to start algebra in 4th grade, but I read an opinion piece from a mathematician who urged lingering and applying tools in more creative ways, rather than being in a hurry to introduce more tools. So I shifted gears and found Art of Problem Solving’s brand-new elementary curriculum. It’s called Beast Academy, and I was a little skeptical at first about the comic book approach to the textbook. But it’s highly respectful of kids’ intelligence, curiosity and creativity and it’s lots of fun besides.

My younger kids use a combination of Singapore and Beast Academy. It was a perfect fit for my oldest, but her younger siblings don’t want to miss out so they rise to the challenge. That picture was from Beast Academy’s 3A Practice book.

I’ve also pulled together a small homeschool team for the Math Olympiad contests, but it’s difficult to find other homeschool moms enthusiastic about math, unfortunately. Most feel that is a weak subject for them, and they often pass that insecurity along to their children.

• I am also a huge fan of the Art of Problem Solving books and have used (and still use!) them with my kids.

• We also keep tangrams, pattern blocks, and logic puzzle books around, but even something as simple as stocking graph paper for doodling has yielded fruit in understanding geometry.

• I’m so inspired by what I’ve seen on your site! As I’ve been busier with her younger siblings, I think my oldest has been more independent with math, and she and I both miss the companionship as she explores new concepts. I want to get back in the arena with her.

• A fun challenge is to count a set of objects by lining them up, then see if the number stays the same when you put them in an array. Which way is easier to count? Do you want to count the rows or the columns? They can learn the commutative property of multiplication this way!

This is brilliant! It’s amazing the curiosity of young children and this is an activity that can fascinate them as much as non-math oriented discovery activities.

You wrote Nix the Tricks in part, I believe, to end practices that inhibit a child’s natural curiosity for math exploration.

Can you give an example of common short cuts that get in the way of teaching Higher Maths to young children?

• Everyone teaching tricks to kids means well, they're trying to get them through the assignment, the lesson, the test etc. But tricks jump past the concepts that form a foundation for later. We can start by counting objects more efficiently by organizing them in arrays, then see that as multiplication and then further simplify by using an area model to represent partial products (instead of drawing out twelve boxes per row, use a large box to represent ten and two smaller boxes to represent the ones). This approach can be extended in all sorts of fun ways, including using variables. But if students only learn to multiply with something like the turtle head method (Google image search if you must, I don't want to send them any more web traffic than they have already), there's no opportunity for connections.

• Fun that @Chris will be doing a TEDx Talk! I believe we had already invented Prime Climb by the time I had my talk, but the color scheme presented itself as a perfect example to show, which would lure people in to thinking about math :-)

For a super accessible entry into higher math ideas, I like 1-2 Nim. This is an ancient 2-player game, and super simple: you take turns taking 1 or 2 counters from a small pile. Whoever takes the last counter wins. (The "poison" variation, where whoever takes the last counter loses, is also popular.)

You can start playing this game right away, even with young children (say, 5 and up). At first it seems like there's not much strategy until you get down to a low number. But it's actually possible to concoct a perfect strategy, and always win, no matter how large the starting pile is. (As long as you get to choose whether you go first or not.)

Though 1-2 Nim is completely solvable, it quickly relates to Nim variations like this one I posed as a TED-Ed riddle. From there it just gets deeper, like the classic game of Nim, which gives rise to what John Conway (probably my personal favorite living mathematician) called Nimbers, a subset of surreal numbers. There's a crazy rabbit hole you can go down. But the game is fun and engaging, and perfect for elementary school teachers or families to start playing.

The point is not that we're getting to advanced math earlier. It's that most students never get to see the study of games as part of mathematics. Sharing higher math isn't about speeding up, it's about slowing down and expanding our sense of what's possible with mathematics.

• There's something really powerful in this idea that really understanding 65 is actually totally nontrivial. The base 10 system is a ridiculously powerful tool that took centuries to develop.

How do you draw out the depth when you work with your kids? Is it just in making the numbers and playing with them, and noticing where these things come up naturally? Or are there specific questions or activities you like to introduce?

• AoPS and Beast Academy specifically are pretty great.

• I love the emphasis on connections and understanding over tricks to get the right answer. A world of difference.

Do you find that parents are open to making that change? Shifting that emphasis?

• I do get to work with children that comprehend and are able to reason above standard expectations. I don’t like the word gifted, but I will use it to maintain context. (It is a label that is often misunderstood as being pre-wired with knowledge, or the ability to grasp concepts quickly.) Gifted brains are not that simple. It is researched and proven that the brain of gifted individuals work differently than normal brains. The child may exhibit more emotion, another a complete disconnect from social awareness. One may show high levels of excitement, while another appears almost stoic. They may exhibit great command over one, two, or many subjects, but still need to hone skills such as spelling, or eye-hand coordination. Information fires across neurons faster and memory whether short or long term is more efficient.

The experience I think that will surprise people most about doing math with gifted children is that there is not often an epiphany or ah ha moment. Their thought process, conjectures, and actions are often very deliberate, or very abstract and vague. They rarely come to the same answer from the same direction, which is less telling of their ability, and more evident of the flexibility of mathematics.

I know the problem well that Dan Finkel and I explored deeper. We found some interesting patterns. I think we left off wondering does this sequence work all the time, and what if the pattern changed for example to 1,7,13,19, is it still true. It’s a beautiful sequence but the questions about patterns, and variation is where the real investigation starts. So the set problem in that pattern will always work. Dan moved it to conjecture using other sequences, and patterns. Those are questions for others to explore. We had a lot of fun with that. I do believe we posted some of our findings on Twitter (the problem is pinned @lilmathgirl).

Doing problems that are not typically taught in school or those problems not at grade level is where the fun begins. It’s where we get to play with math. And I mean play with math. I’m not one for dressing a wolf in sheep’s clothing. No masking of the subject. Math is a wolf. It is fierce. But if you get up close you’ll see just how beautiful it really is, and it becomes less scary.
The best environment is relaxed and familiar. The math usually surrounds social interactions: cooking which requires accuracy, building which is spatial development, drawing which uses imagination, reasoning, coordination and memory, I also love twisting a standard game or methodology. A few examples: Scrabble only spelling math terms or using words that relate to math. Telling time in fraction, decimal, or ratio form, dividing fractions without using the multiplicative inverse. Games that require finding and making patterns, such as Set, DaVinci’s Challenge, Othello, Checkers, Chess, all of these require reasoning and thinking ahead. Creating a path and strategy to get to the solution. There’s also a need for conjecture, if I do this the outcome will be ____.

Much of what gifted children know comes from letting them lead their curiosity. Let them puzzle it out. I answer their questions with a question, sending them Into deeper learning. Often times we don’t speak about math at all, because the more we understand one another as a person, the easier it becomes to communicate mathematically. I want everyone to think about that for a moment. Think how you talk with others. How you adjust you speech or train of thought based on to whom you’re speaking.

🤣 Everyone always asks how I come up with my ideas. So first a bit of a backstory. I was born 3 months and 2 weeks early. All of 2 pounds, 3 ounces, with pneumonia. I wasn’t suppose to survive the week. If I did survive, I most certainly wasn’t suppose to have full brain function. Well I did, and I do. I was clearly talking at eight months, and reading at two years old. At five years old I was working at a fifth grade level. I spent most of these years in the hospital. The doctors, the same ones that thought I would not live, were my teachers. They folloeed my mother’s wishes, and taught me everything. Never limiting me by any factors. Now the question stands, am I gifted or was I in the right environment, surrounded by those intent on nurturing my curiosity? I don’t know.
I know my brain doesn’t function normally, I know I thrive in the muck of what if, and curiosity is my driving force. Amongst everything that I am, I’m a master flutist, I speak or understand 6 languages fluently, and I truly don’t understand if one can see and draw a triangle, why can one not draw a face?

A mind like this is often a hindrance in traditional school settings because institutions are set up for balance and leveling education. Students not meeting requirements, and students excessively above requirements cause displacement of structure. Education systems fail to embrace this real life dynamic, and instead they push and pull on students, trying to fit them into one box.

Now back to the original question. How do I come up with creative ideas. I approach math from every direction. Not just a problem on paper. Math has to come out of the textbook, and become relatable to the world we live in. It’s when we take this abstract thing called math, and correlate it to reality, we give it tangibility and substance. Math becomes real and not just symbols. I explore all of math. The pedagogical approach, and the avante garde. ( That includes not shying away from “tricks”). For instance multiplying by 9 and knowing : 0 and 9, 1 and 8, 2 and 7, 3 and 6,.. the digit sums add to 9 and all the other patterns of 9. However it does not ensure comprehension of repeated addition. And that’s where we have to be mindful with clarity of information. I often take a problem or topic and explore it from several different angles. Word play, reverse the equation, change the sign, think through history, how does it relate to science or current events, can it be modeled and how? Some of the simplest ideas turn out to be the most complicated.

Don’t underestimate children. Don’t put a limit on their comprehension. Don’t expect the same reasoning. Math is limitless, and so are the minds of children.

This puzzle is not easy ( the blocks are from the game Visual Eyes). The switch up asks you to pick 1 or more blocks that represent a math word, phrase, problem, Etc. For example 2B and 4A = even plane. Children were very creative with this. When I did this on Twitter the responses were very good. When I did this with a group of educators they struggled. 🤔

• Mostly we look for patterns at first, what all the numbers in a row or column have in common. Then we might try to find where a certain number belongs, looking for clues in all directions around it. I also use it as a more tactile 100 board so you can see patterns for finding multiples, primes, etc. But my preschooler isn’t there yet. 😉

• I’m a fan of art puzzles, and one other thing that I enjoy having students do is a math scrapbook (I call them lookbooks) as a group or individually, (or in this case Imparting their thoughts, and ideas into my book.)

• I’m happy to see amacbean16 talk about looking at the pattern of numbers in all directions. Place value is tricky. To the left of the decimal the place values are from least to greatest starting from right to left. Counter the direction we read. Yet we write numbers in increasing value from left to right. Our brains have to learn to put so many actions into place to comprehend numbers above 10. Math is complicated and it takes daily practice, and exposure. There are no shortcuts.

• The beginning of adding 2, 2 digit numbers, and Carry Over. Carry over fails to give a clear understanding of what occurs when the ones column has a value of 10 or more. The carry over of 1 is a misrepresentation of the value moving up by 10.

• The Hamkins perspective is crazy cool!
I love repetitive shapes, and creating Tessellations from nature.