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?
Love that list, Mike!
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.
I am going to throw this question out to all of the panelists.
I think a lot of our audience on Cake are not educators. They may have done well in math during their undergraduate or graduate studies. They may even be recreational math enthusiasts for problems like these from @1to9puzzle .
But that doesn’t mean they know how to teach it to young minds. Let alone higher maths.
I know @amacbean16 is an accomplished homeschool educator, but assume for the moment that she was looking for specific ideas to introduce higher maths to both her 6 year old and her 6th grader.
What are your suggestions?
I think parents want what's best for their kids, but if they only know the way they learned so of course that way seems best. Hilary wrote a book called Adding Parents to the Equation that helps address this. The more teachers can engage parents in conversation and explaining why we do these baffling things the better!
Playing with James Tanton’s explodingdots using clementines at snack time to model.
We started with 9. Since were doing binary every two fruits must kapow! And one of the two fruits moves to the space at the left. Now we have1 left in the first space and 4 in the second. 4 gives us 2 sets to kapow! Now there are 2 in the third space. Since there are 2 they must kapow! Once more leaving 1 in the last space. There is one in the 1st and the last space. Every orange is a 1 and an empty space is zero. So 9 in binary is 1001. Checkout the link above. This is just the beginning of what can be done with xdots.
I invite you to step away from the workbooks and structured online programs that mimic school learning. Enrichment can look so different from a traditional classroom. Ask questions. Share what you're thinking when you make a comparison or divide a strategy or wonder something. Don't be afraid of not knowing the answers to kids questions, instead explore together by testing theories and researching. I love the blog and book What If by Randall Munroe. https://what-if.xkcd.com/ While your kids won't know all the physics he does, they can certainly engage in a similar thought process.
Patience. We live in a world that values math as a marker for intelligence. Don’t chase an invisible finish line.
Most parents understand elementary math being constant rote memory and just do the steps of math. This does not allow for interpretation, reasoning, expression, and explanation. The experiences needed to understand what is being done through the use of math. Learn to implement mathemization. Use math language as often as possible. Introduce new concepts. Discuss relationships. Ask questions, prompt answers, and pose new problems. Have plenty of manipulatives, tools, work space, and supplies around.
Good advice. One of my favorite books is Mitchel Resnick’s LifelongKindergarten, Cultivating Creativity Through, Projects, Passions, Peers, and Play. It has very little to do with math. Instead it focuses on project base thinking, collaboration, creating passion, learning through play (repetition with purpose), and sharing with peers.
For younger kids I think it is a super fun challenge to find ways to share advanced math ideas in ways that are accessible to them. Our paper folding "FamilyMath1" was already shared above, but here is FamilyMath2 - my younger son was 5 in these videos (and I just did FamilyMath985 with him today!):
Another fun thing I did with both kids when they were learning arithmetic was making binary adding machines out of duplo blocks - this is quite similar to the already mentioned "exploding dots" exercise from James Tanton:
It was a surprise to me how far you could go just with blocks -> here's a fraction division example:
And here's a "proof with blocks" that a negative times a negative is a positive:
For the 6th grader - by luck I'd previously written a post about 15 fun projects for a 6th grade math camp. I'd be happy to go into detail on any of these projects if there are any specific quesitons:
Another thing that I like to do even today is find projects for my kids that sneak in review of material we've covered before like arithmetic. This is a really neat project based on an unsolved problem in number theory - but beyond seeing a neat problem, I think a lot of the value for k-12 kids is just getting in some sneaky arithmetic practice:
Even the project that I did on the morning this conversation was launched was an advanced idea designed to hide a bit of sneaky review on material he's already covered. On project like this one it is so fun to see how kids apply their existing knowledge to "new to them" areas of math:
Hi all, glad to join you here. (This discussion already has so much meat in it, it might take me a while to digest it all!)
I teach math at a community college in California. I've also worked with homeschoolers, and run some math circles, and math "salons" at my home. I put together a book showcasing some of the amazing work being done by math enthusiasts with kids of all ages, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. I have a blog at Math Mama Writes, but I haven't been writing as often there lately. (I'll definitely link to this fascinating conversation!) Basically, I love playing with math, I want to share that love, and I look for ways to bring that playful spirit into my college classrooms.
I'm not sure what the term "higher math" means. To me, all of math can be approached playfully, and I'm not sure what makes something 'higher'. If we simply mean the topics that aren't arithmetic, there is so very much!
I hope I'm not being too much of a curmudgeon by questioning each word (I think that's a math skill, perhaps...), but I also know that 'teaching' can be problematic. I want to offer delights to young children, and then follow their lead. Does this game intrigue them? No? How about this puzzle? Or maybe this art project? So much math can be achieved this way.
My department asked us to wear something math-related on Thursday. I was grumbling to myself for a bit, and then I found a t-shirt I love in the back of my t-shirt drawer. It has the dragon curve on the front, and on the back are images that show the stages you'd get when folding a strip of paper (this site might be a good way to see how it works). So I pulled it on and went to work, where I chopped up some paper, and started showing people the coolness of the pattern you get when you keep folding and opening it back up the same way. (We looked at the pattern of "valley folds" and "mountain folds".) It only took a few minutes of classtime, and maybe, just maybe, one or two students will be intrigued enough to keep playing.
I agree that teaching people tricks that having nothing to do with the underlying math is a big turnoff. I think there are some things that fall into the trick category that we don't even notice sometimes.
While working on my book, I discovered a blog post (now a chapter in the book) talking about Montessori math , in which the author vented about a dad who turned multiplication by ten (profound for a 5-year-old) into just a trick ("you just add a zero at the end"), and ruined the kid's discovery. Wow. It had never occurred to me that "adding a zero at the end" is a trick! I love realizing something basic like that.
Welcome to Cake and thank you for joining our 2nd annual maths panel!
@amacbean16 shared that
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.
What suggestions do you have for homeschooling parents who want to move past their own insecurities with math?
Play with it. Relax and have fun. It would depend what age their kids were. I'd try to find activities both the parent and kid would enjoy. Katamino is a great game using pentominoes. It doesn't seem like math to most math-averse people. To me, it's one part of math. But maybe this family would prefer to do math scavenger hunts. Or maybe the parent will relearn their math along with their kid. Beast Academy has been mentioned in this conversation, and it seems like a great way for parents for start fresh.
If they were interested in reading Playing With Math, I'd want to know which chapters they engaged with most. That would help me get an idea of what works for them. The goal would be to be comfortable enough not to pass along their fears and displeasure with math. And that's definitely doable. But those parents might still find competition-level math too intimidating.
