Cake
• 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.

• 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.

• If anyone would like to see what a project with Dragon Curves and paper looks like - we tried it out for FamilyMath8 a while back (my kids were 5 and 8 in this video)

• 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.

• ... so on the topic of familymath1... in case you were wondering, my 11 year old only got 8 folds in for an entire Costco roll of 2-ply. 😁 She was hoping to beat your effort from the video. It certainly was more effort (141ft+ long!), but only a tie score. We had a great time.

• Math is a wolf.

That's got to be one of the best sentences I've read about math in a while. What an incredible math scrapbook too!

• This is fantastic. I could do a lot with this, I think.

• ... so on the topic of familymath1... in case you were wondering, my 11 year old only got 8 folds in for an entire Costco roll of 2-ply. 😁 She was hoping to beat your effort from the video. It certainly was more effort (141ft+ long!), but only a tie score. We had a great time.

I love that exploration! It’s about experimentation and having fun while doing it.

I am beyond thrilled with this annual math exploration that was started a year ago, when four math educators had a DM exchange on Twitter on a Tuesday night and then pulled off “How do you Make Maths Fun?” a few days later.

The preparation this time took well over a month, and the weekend panel event was expanded with a Guest Blogger Week leading up to it as well as interviews from educators in the MTBoS community (thank you @PonderingDan , @jlaib , @henrip , and Heidi Allum!)

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We also had a Saturday night conversation on the multiplicative inverse that was sparked by the maths panel, moved to Twitter and then branched off to Cake when 288 characters wasn’t enough for conversation.

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Thank you, thank you, thank you

to all of our panelists who shared their ideas, passions, suggestions and insights on How to teach Higher Maths to young children. This was amazing and I could not imagine even how to try to summarize the key points because every post was chock full of them.

Before you go.

How can we best keep up with you?

Final thoughts?

🙏

• I'm most active on Twitter so that's the best place to continue following up on these ideas. I'm @TinaCardone. Thanks for inviting me to join this conversation and I hope everyone leaves inspired to continue engaging with math from a variety of perspectives!

• What a fun and insightful panel. Thanks everyone!

If you'd like to follow my work, you can find me at mathforlove.com, or on twitter at @mathforlove. I strongly recommend signing up for my mailing list at Math for Love, since I send out a lot of materials and info there.

You can check out my games, Prime Climb and Tiny Polka Dot, at the links. They're also on Amazon, and in other spots.

If you happen to be in Seattle on Pi Day, 2020, I'll be hosting a Julia Robinson Math Festival at the University of Washington. Those festivals are a great way to share beautiful ideas in math with children. If you'd like to host one, or just check out the material they provide, go check out jrmf.org.

Thanks!

• I'm pretty active on math twitter - you can follow me @mikeandallie. There I spend a lot of time sharing the ideas I'm exploring with my kids and also learning from all kinds of people interested in math and math education.

The blog where I share the projects with my kids is mikesmathpage.wordpress.com. The boys are in 8th and 10th grade this year and are studying from Art of Problem Solving's Precalculus book and Strang's Linear Agebra, though the bulk of our projects this year have come from Mosteller's 50 Challenging Problems in Probability.

Thanks to everyone for the weekend conversation and especially to Stephen for inviting me to participate. This was the first panel discussion - virtual or otherwise! - that I've ever participated in.

• Wow. First thank you Stephen for putting together this great panel. Thank you fellow panelists for bringing so many different views to light. It truly shows the flexibility of mathematics, and approach.

Final thoughts:

Young minds are open to learning. Numbers and counting is part of most children’s lives before they can speak. It has been argued that infants think of their fingers and toes as the numbers 1 through 10, because of how often they are referenced in this manner.
But children’s brains are wired to discard information they don’t need. A brain that hears, “math is stupid”, “we don’t need math”, “math is too hard”, etc...is a brain that will avoid and discard math. There are two quotes that I love from Daniel J. Siegel, Professor of Psychiatry at UCLA, and author of The Whole-Brain Child

1.) “...Rather than trying to shelter...children from life’s inevitable difficulties, we can help them integrate those experiences into their understanding of the world and learn from them.”

Teach children how to persevere. Not everything is easy. With patience and time they can succeed.

2.)”....integration ...it coordinates and balances the separate regions of the brain that it links together. It’s easy to see when...kids aren’t integrated—they become overwhelmed by their emotions, confused and chaotic. They can’t respond calmly and capably to the situation at hand. Tantrums, meltdowns, aggression,...a loss of integration, also known as dis-integration.“

A loss of integration, disintegration, or disintegrate. To break down. No matter their age or level, we need to be mindful of disintegration. It is often read as disruptive, or unnecessary behavior, when at it’s core it is often the inability or unwillingness to communicate confusion.

Note: As adults we tend to look away when we know what’s going on, and bring our attention back when we need more information. Children are very much the opposite. They give attention to what they know, and turn away from what confounds their thinking. They need time to process information. Figure out how it fits. They do this during play, imitating, mimicking, imagination, and quiet times.

Teach, and nurture appreciation. Allow space for processing. The how and why will happen. Math is not training for the standardized test, not a race to the end of the textbook, not made for collecting A’s on a report card.
Math is a way to understand and express our world symbolically, through comparisons, and relationships. Math is ever evolving, because we are ever changing.
Math is a journey and an adventure. There’s plenty to discover, and lots of ways we use it everyday.
Learn it, share it, add to it. It belongs to all of us.

If you’re looking for me:

I’m Sophia Stier, creator LilMathGirl™️
Follow on Twitter and Instagram @LilMathGirl Contact by DM or at weplaymath@lilmathgirl.com Vist the website at LilMathGirl (Still a work in progress, but you can meet Lil’s Friends and see what they’re working on)

Upcoming speaking engagements:
CMC north, Asilomar CA, Dec. 2019
NCTM Chicago, April 2020

LilMathGirl Podcast launches mid January with guests such as Dr. Christopher Brownell (Math Recess), Dr. James Tanton (The Global Math Project, and Exploding Dots), Patricia Dickenson (Teaching Outside The Box -Technology Infused Math Instruction)