# Sharing Brownies

The other night at dinner as we all debriefed our days, I mentioned that I’d given my college students a problem about sharing brownies. “I should give the problem to you and see how you would solve it,” I mused to my kids, and my husband immediately said, “Let’s make brownies and solve the problem in real life!”

So we did. We threw together a pan of brownies from a mix we had sitting in our cupboard, and when they had cooled I cut and plated three square brownies for each kid. Then I handed them each a plate and a dinner knife and told them to figure out how to give every member of our 4-person family the same amount of brownie.

We like to ask sharing problems about food (“If there are 8 pancakes, how many can everyone have?”), because our kids, like all kids, are highly motivated by food and by fairness. But this is a challenging problem, without a clear, immediate solution.

Still, my kindergartener dove right in, deftly cutting all three brownies in half. She then paused for a moment with the two extra pieces before cutting them in half too and stacking a quarter brownie on top of each half-brownie portion. “Everyone gets a half and a piece,” she said when I asked her about her solution.

(Surprisingly, very few of my college-age students come up with this particular method initially, although once they have seen it they tend to prefer it. This semester one student commented on how surprised he was that a kindergartener would come up with this method right away when he, a math education major, didn’t think of it on his own.)

My 7-year-old stared and stared and stared at the brownies and I could see the gears turning in his head. Finally, several minutes after my daughter had confidently offered up her half and a piece, he said, “Okay, I think this will work,” and embarked upon a complicated cutting exercise that I would call “split the brownies into smaller and smaller pieces and hope it will all work out eventually.” First he cut one brownie into thirds, cut another brownie into fourths, cut a fourth in half, and put the half-quarter together with a full quarter to make another “third”. He then cut the last brownie into fourths, and proceeded to cut and re-cut any odd pieces out until he felt confident that he had a workable number of pieces.

He explained that everyone got a third and a quarter and a half quarter and a “small quarter” (half of half of a quarter). But when he actually distributed the pieces onto each of our plates, there were a few extra bits left behind. “Hmm, I don’t know if that really worked,” he said, and then shrugged and popped the extra pieces into his own mouth.

It was obviously interesting and fun to watch how my kids approached this problem on the very same day I watched my college students approach the same problem. But it was also interesting to listen to the informal language they used to talk about their solutions. They both already had some language for talking about fractions, and they both ran up against limits. The kindergartener could talk about halves, but once she got to quarters they became “pieces”. Nevertheless, with the motivation of actually getting to eat the brownies at the end, she was remarkably accurate at splitting the brownies into equal parts. The second grader could talk about halves and thirds and quarters, but then began talking about half-quarters and “smaller quarters” when he got down to eighths and sixteenths. And he made an interesting approximation of 1/3 by combining 1/8 and 1/4. This was not precisely equal to 1/3, but it was certainly close enough to feel fair.

My college students like to ask me questions like: “When would you start teaching this to students?” My answer is often: “Much earlier than you’d think!” When can kids start understanding fractions? Much earlier than you’d think! When can kids begin making sense of probability? Much earlier than you’d think! When can you give kids multiplication problems? Much earlier than you’d think! When can kids understand that a square is a type of rectangle? Much earlier than you’d think!

This particular problem involves both division and fractions, and while I had no idea what my 5- and 7-year-old children would do with it, I knew they would be able to do something. Kids have great ideas, and they have great ideas much earlier than you’d think!

I knew I loved to write long before I knew I loved mathematics. As a child I was a voracious reader, and dreamed of creating the same magic with my own pen.

Instead I became a math teacher.

I kept up a personal blog on the side, and my writing background served me well when I decided to earn a masters degree in math education, and then doctorate. Then followed a new career, and marriage, and a baby, and in the last few years my writing has mostly fallen to the wayside. Still, in the back of my mind I had a feeling that writing could still fit in my life, that there was some way I could merge my professional passions and my personal life in a way that would be useful and interesting to a wider audience than just my friends and family.

One night, when our son was still very young, I shared some of my thoughts about mathematics and parenting with my husband, Brian. I felt it was important for our son to get positive messages about mathematics from both of his parents, and for both of us to engage with him in mathematical thinking.

Brian, a trained opera singer who hadn’t taken a formal math class since his first semester of college, agreed that this was important, but expressed some concern. “What would talking mathematics with Monkey even look like?” he asked me. “I don’t think I would know where to begin.” Honestly, I didn’t have a good answer. Talking math felt intuitive to me, but that was because I was totally immersed in the world of mathematics learning. My husband, like so many other parents, is not.

Not long after that conversation I had a chance to give a 20-minute presentation to a group of mothers of young children. I focused my presentation on mathematical literacy for toddler-aged children, and when I rehearsed the Power Point with my husband he said, “That. That’s exactly what I was looking for.” The presentation was very well received. Several mothers told me that they very much wanted their children to have a good experience with mathematics, maybe a better one than they themselves had, but they knew very little about what they could do as parents, and that my presentation had given them a way to think about it. They felt excited to begin talking math with their kids.

That was the moment this blog was born.

There are many, many early literacy resources for parents. We know to talk to our children from the moment they are born, and to read to them. In fact, giving your children a strong foundation in literacy is one of the best things you can do to provide them with a strong foundation for mathematics. But some very literate adults can nevertheless remember struggling with mathematics, and I have found that many parents are looking for more. They want to actively help their children to feel excited about mathematics, and able, and empowered. These parents also want to feel empowered themselves when it comes to helping their children learn, understand, and enjoy mathematics. I created The Kids’ Quadrant to be a resource for these parents. For you!

The Kids’ Quadrant lies exactly at the intersection of my personal and professional life. I have a PhD in math education and a familiarity with and access to research on teaching mathematics to children. I teach, coordinate, and write curriculum for mathematics classes for future elementary teachers, which means I spend my career working to impress hearts and minds with an understanding of how children learn, perceive, and sometimes struggle with mathematics. And with my one-year-old son now at the cusp of language, I am deeply, personally invested in creating a natural and positive environment for him to explore the world of numbers and shapes and patterns.

I have much to share and much to learn, and this blog is my journey as well.