Over the last year and a half or so, I’ve taken a deep dive into the world of mathematical art. My current explorations include geometry and beads.


What’s equally interesting about these hobbies is that I’ve learned a great deal about they way we learn new things. And specifically, it drives me to think extensively about the way we teach and learn mathematics.

Something important I’ve discovered: I like to start a project with a procedure. Although I will avoid turning to a YouTube video if I can (probably because of residual trauma from Khan Academy), I like to be given clear directions, steps and models from beginning to end. It gives me comfort to know what I’m getting into but also know what the finished product should look like. But in order to be successful, I need to then work to understand the procedures conceptually.

There is an elusive stitch that beaders struggle with called the Cubic Right Angle Weave. I was obviously attracted to it because of its mathematical name but also because it’s used for some beautiful designs (and it’s difficult). Yesterday, I loaded up a YouTube video and dove into the beads. This 20 minute video took me 7 hours. SEVEN HOURS. But again, I needed to conceptually understand what was going on with this stitch. I needed to visualize where the next set of beads would go to make these connected cubes.


I see such a parallel to the purpose of school mathematics and the way it’s taught. I learned the cubic right angle weave so that I could use the stitch in a bracelet I’m making for my mom to for her birthday. I’m going to use it as well as a number of other stitches that will work together to create what I’ve envisioned. I most certainly did not learn this complicated stitch so that I could continue to practice strings of different types and sizes of beads – glass beads, stone beads, crystal beads, beads with fractions, and beads with a coefficient on x squared. My goal was instead to have a usable end product and practice the stitch in the process. When we isolate skills in mathematics, we are teaching kids over and over that their knowledge of a process is only useful in problems with a similar structure. Knowing how to solve a quadratic equation is a lot less useful if all we are ever doing is solving increasingly complex quadratic equations. Just ask an adult how many times since high school they have needed to rely on their knowledge of the quadratic formula to solve a problem.

I’m grateful to be learning so much about my learning with my extra free time.  I hope to research more about these learning parallels through my doctorate program.  However, my next steps are to complete my mom’s bracelet.  I can’t wait to give it to her.  It’s going to be beautiful.

Holy Hundreds Chart!

One of the best things about working with first graders is that they have internalized very few procedures and algorithms.  So when they are posed with a problem or scenario, they are eager to develop their own problem solving method and most of the time, they are enthusiastic to share their thinking.  In contrast, high school students, when presented with an unfamiliar problem, often try to scroll l back through their mental library of mathematical procedures and then attempt to apply one that seems to fit the scenario.

Because of spring break and other conflicts, it has been about a month since I’ve worked with the first graders so I decided to bring something a little different.  Rather than pose an open ended problem with multiple solution paths (but ultimately one correct answer), I brought something more exploratory.  And I started with the 100s chart.


The hundreds chart:  a capstone to the foundation of a solid elementary math program.  One of the first tools students use when working with bigger numbers.  And there is so much beauty and structure to it that kids can see when working with it slightly under its surface.

Thanks to this week’s featured  primary level problems on Nrich, I decided to let the children explore some relationships with numbers on the hundreds chart.  I had them choose any two “next-door” numbers and then add them together.  (Surprising to me was that at least one person in each group chose 99 and 100).


First graders have not had much exposure with adding two two-digit numbers, but based on my experience with these kids, I knew they could give it a go.  For the most part, we were able to break the sums down into friendly numbers.  This was exciting stuff to see them work through.

Now the fun part:  I listed all of their sums and asked them what they noticed.  And when you give kids a set of 4 things, and they are familiar with the Which One Doesn’t Belong routine, they can’t help but tell you which one doesn’t belong.  In summary, we were convinced, based on the variety of examples we had, that when we added the “next-door” numbers, the result would be odd.


Another amazing characteristic of first graders is that once you explore one avenue of a problem, their minds open up to an infinite number of other possibilities for exploration.

  • What do you think will happen if we add numbers that are on top of each other?
  • What will we get if we add diagonal numbers?
  • What about if we add three numbers in a row instead of just two?

And the music to any educator’s ears at the end of a lesson:  Can I take this home so I can figure more stuff out?


Growing the Mathematical Mind through Reading

Kickstarter has a magnetism that I am powerless to resist. I’m not sure what draws me in most: the creative novelty of the projects themselves or the inspiring hustle of the imagineers turning their dreams into reality. So when Math-with-Kids enthusiast Christopher Danielson threw out the bait, I was caught, hook-line-sinker.

My track record on reading with my daughter is not great, (if I have the choice of reading or mathing with Maria, I’m going to choose math 110 times out of 100) but the colorful cover and intriguing story drew us in immediately. We faithfully read a chapter every night and were quickly drawn in wondering what happened next. The book serves as a story to introduce functions to young children, but the plot is so captivating that the mathematics runs so naturally underneath.

When I ask high school 11th and 12th graders to recall what a function is, they usually respond with some jargon about inputs and outputs or recite something relating to the vertical line test. They are versed on the process of determining if something is a function but not very proficient in why functions are important with respect to the study of algebra. But what if their understanding of functions began in elementary grades like this:

When I visit my daughter’s 1st grade classroom, the kids get most excited when I say, “this is math I do with the high school kids.” The foundational work they do with “what’s my rule” machines, data collection, and graphs are also important pillars of a strong high school math program. Reading about the use of functions in such a creative way as in Funville Adventures, helps kids make sense of them in a relatable context. The functions become as alive as the characters in this book. Each Funvillian has a unique power, and we learn why some powers cannot be undone while others can.

I cannot recommend this book highly enough. As an avid non-reader, I was drawn in from the start and so was my strong-willed child (who seeks to avoid anything suggested by mom). This is a book that needs to be on your bookshelf, at school or home. I will be ordering many more of Dr. Fradkin and Dr. Bishop’s books, and I’m very grateful that we were introduced to this one as it served as a fabulous platform for some mother-daughter bonding.

Fraction Frenzy

“Mommy, will you play school with me?” Those words usually send me into an anxious panic given that I spend all of my working hours “playing school.” But now that I am on a leave of absence from my regular teaching gig, I’ve been able to play school with my daughter with calm humility.

Sidebar: If you want to know what really goes on at your child’s school, ask them to play school with you. It’s fascinating.

Anyway, so a few nights ago, Maria lays this gem on me: “Mommy, do you want to play school? I want to learn about fractions.” I have never dropped what I was doing so quickly to go play school. Fractions. YES!

Oh, wait. No. I don’t know how to teach fractions. Especially foundational work on fractions. I mean, yes, I know how to compare fractions, I know where they are on the number line, I know the algorithms for fraction operations and I know how they work for the most part. But helping my 7-yr old develop a conceptual understanding of a fraction. Nope. One thing I was certain about: I needed to bust out the pattern blocks.

I’ve heard many a secondary teacher complain about how the kids don’t understand fractions and don’t remember the rules for operations with fractions from 3-5th grades. Having recently taken a plunge into the world of discovering how an understanding of fractions is developed, two realizations emerged: 1. Teaching fractions from a conceptual framework with a classroom full of students is really REALLY difficult, complex work. 2. It isn’t surprising that kids don’t understand them very well given the constraints we have as math teachers (time, etc) to help develop that conceptual understanding.

Luckily, I have a lot of friends who have. “Start with asking her how she would share two cakes equally with four people.” I gave her two yellow hexagons (cakes). Because of her (always helpful) assistance in the kitchen, she knows about half in the sense that it divides a whole into 2 parts. So she quickly grabbed two of the red trapazoids and determined that we could divide the cakes in half with those.


She did the same with the blue diamonds and green triangles for 3rds and 6ths.

Fast forward to last night. I’ve been reading Extending Children’s Mathematics and the first chapter outlines how kids begin their initial understanding of fractions based on equal sharing problems in 1st or 2nd grade. Knowing that Maria is always happy to do math if it means putting off bed time, I tried this problem with her:

Four children want to share 10 brownies so that everyone gets exactly the same amount. How much brownie can each child have?

She went through a number of strategies first determining that two per child was too small because there would be brownie leftover. Then she figured out that 3 per child would be too much because there wouldn’t be enough brownie.

M: It’s not possible to share 10 brownies fairly with 4 friends. Can we have 5 friends?

Me: Nope. These are BROWNIES. And we aren’t sharing this chocolate goodness with any more people.

M: Well, if each friend got two brownies, then there would be two leftover. (lots of thinking) Then we could split those two in half and each friend would get one of those halfs.

Me: Excellent strategy. So how much altogether would each friend get?

M: (It took my suggesting that she use her visual model to determine this) Two and a half brownies.

Me: Good. Now go brush your teeth and get ready for bed.

What I learned:

  • Kids have an intuitive understanding of fractions that builds from their experiences with ‘fair sharing’ problems.
  • The shift from working with whole numbers to working with fractions is a big one because of the variety of ways we use fractions (beyond part-whole).
  • Helping kids develop a conceptual understanding of fractions is really hard work, and it’s really important for secondary teachers to learn more about the complexity of this work.

First Grade Tales from a Former High School Teacher

It’s Thursday, which means at 2:00, I’m off to Mrs. Quick’s class to play math with the first graders. Hooboy, they were wound up today. This was going to be fun.

While scanning through the Math Forum’s problems of the week, I came across this one about puppies. And who doesn’t like puppies?! Puppies and math! Double win.


Being new at this whole elementary classroom thing, I naively thought that maybe this problem would be too easy. But trusting my pedagogical prowess (and the expertise of the Math Forum’s problem-writing team) , I was confident I could extend the problem if necessary.

Again, we were in rotations with groups of 4 – 5 students each. I had a student read the problem out loud (note to self: In the future, have copies for the other children to follow along.) The next thing ABSOLUTELY BLEW MY MIND, again, because of my initial thought that the problem would be too easy. Out of the 20 students I worked with, exactly 2 of them got the right answer the first time they approached the problem. And the other 18 all did the same thing: 4 + 2 = 6. I was so delighted with this, I might have actually let out an audible squeal. Their explanations all revolved around the same main theme: There are 4 of something, 2 of something, and the questions says ‘altogether.’ Therefore, 4 + 2 = 6.

I asked them what we knew about the problem. We knew we had 4 crates and 2 puppies in each crate. I then prompted them for how we could model the 4 crates. We love drawing on whiteboards. So we drew 4 crates. Then the wheels really started churning. Puppies, in crates. We got this.

Then we had to make sure we were convinced that the answer was truly 8 and not 6. A few groups had time to tackle the “extra” question: Can there be 9 puppies? Why or why not? One thing was clear: we were absolutely certain there could NOT be 9 puppies altogether.

My favorite “why nots”:

  • There are two puppies in each crate and the answer has to be an even number.
  • The answer is 8. There can’t be 9 because we just convinced ourselves that the answer was 8.
  • If we had three crates with 3 puppies in each crate then there could be 9.

What I love about these 1st graders is all of their thinking and reasoning relies on their own sense-making. Very little of their explaining involves a reliance on algorithms and memorized processes.

I’m marking today’s experience solidly in the win column. All of us learned today. I learned more about how to facilitate problem solving with first graders, and each and every student shared their mathematical thinking today on a non-routine problem. Moreover, we were convinced that there are 8 puppies and not 6. And we all know that 8 puppies are much better than 6.

Elementary Interlude

Imagine you are in charge of making the cake for your grandmother’s 90th birthday. If you know anything about me, you would know that this task is quite anxiety inducing, and I’d pay good money to a bakery to complete this task for me. But Grandma really wants a homemade red velvet cake baked with love by her favorite grandchild: you.

You realize that you are out of vanilla so you have to go to store. I mean, you could use the imitation vanilla that your spouse bought once because it was cheaper but even at 90, grandma is going to pick that out from a mile away. Of course, it’s snowing buckets outside, but no matter. Grandma needs a cake, and for this occasion, you have been appointed Cake Boss. So you fight through the snow and the (honk honk) people driving like they’ve never seen snow in their lives (This is Minnesota, people! Why do we all get snow-driving amnesia over the summer?!).

You somehow arrive back home in one piece when you realize that your precious daughter mistook your cookbook as a coloring book. And she, no doubt, didn’t care about staying inside the lines because what used to say “preheat the oven to 350°” now is a covered in crayon streams of pink and purple. Luckily, you’ve made this cake enough times that you don’t really need the recipe, and you figure baking powder and baking soda probably are interchangeable. You have a 1 in 2 shot at picking the right one anyway, and after all of this, you are feeling pretty lucky, right?

And as you continue your kitchen adventure, you forgot to shut the pantry door and realize that your dog is neck deep in the open box of Coco Puffs. By now, you are pondering things like Maybe grandma wants a dog for her birthday instead and maybe next year someone else should be in charge of the cake. And naturally, your dear daughter wants to help with grandma’s cake and instead of a cup of sugar, you end up with 3/4 cup in the mixing bowl and 1/4 cup on the counter. (Luckily you see this as a nice teachable moment to talk about fractions). Instead of red velvet cake, your daughter insists on purple velvet and since the color of the cake is not the hill you wish to die on today, you give in to purple velvet. Maybe grandma won’t notice.

When it’s party time, you present grandma with a lovely cake made with your very own blood, sweat, and (lots of) tears. It isn’t exactly a culinary masterpiece, but given the obstacles, it’s a work of art. And regardless of how she truly feels, grandma tells you how much she loves it. And the icing on this metaphorical cake is that your junior chef wants to make it on her own next time, just like mommy.

This, my friends, is what teaching 1st grade is like. But instead of one helping hand in the kitchen, there’s 20, sometimes 25, even 30. A few will take the flour and the oil and create something totally unexpected while others will constantly stick their fingers in the batter to make sure it tastes right. You’ll have some that will try to grab it out of the oven before putting on an oven mitt, and a few that will fight over what color the frosting should be. But in the end, the cake is perfect and Grandma loves it.

I’ve always had a deep appreciation for elementary school teachers, but after spending a few days in my daughter’s 1st grade classroom, I have an even more profound respect for the brave individuals who teach our young, most vulnerable children. They are charged with teaching the foundational skills upon which mathematics is built, and they do it with a room full of kids who are still developing control of their bodies and emotions. And on top of that, they have to listen to middle school teachers complain about how their kids don’t know their “math facts” and high school teachers complain that they are bad at fractions.

There was a lot to love about my experience in this first grade class, and I’m sure my weekly visits will provide me with much more insight into the mathematical development of 7 year olds.

My favorite routine so far is rotations. Basically 10-15 minutes at 4-5 stations, 4-5 kids per station. I was delighted that “We are going to tackle a challenging problem today” was met with cheers from 1st graders while high school kids usually respond with groans. (I have a hunch that grades and GPA’s have something to do with that, but that’s a topic for another time). My station was a Math Forum problem of the week.

Credit: The Math Forum

I wish I had pictures of the awesome strategies the kids used to solve this problem because watching them think through a non-routine math problem was nothing short of joyful.  From the ways they explored the problem to ways they excitedly shared their solutions. I’m looking forward to the weeks to come and the opportunity to witness the development of their approaches to non-routine problems.

Learning is not linear and teaching is anything but routine.  Although a first grade teacher is dealing with eight kids suddenly all needing band-aids while I’m dealing with cellphone distractions, there are common threads that tie us all together as educators.  We are all baking cake for grandma.  I’m grateful for this chance to see first hand the workings of an elementary classroom.  It confirms my belief that when kids show up to my classroom in high school with gaps in their understanding of fractions, the solution lies in first acknowledging the delicate and difficult job it is to facilitate learning at any level and appreciating the work that middle and elementary school teachers do to help prepare students as best they can.

Calling for a Compass Comeback

Have you ever handed a compass to a child? After they satisfy their initial curiosity that one of the ends is pointy, have you ever then seen what a child does with that compass? I recently attempted this experiment with my 7 year old daughter. Her immediate discovery: She could make a flower by strategically constructing 7 circles.

I wanted to have a conversation with her about the 7-circle arrangement and see what geometric understanding we could build on from what she noticed and wondered.

She identified shapes and “non-shapes,” talked about shapes that could fit inside other shapes, and developed methods of determining whether or not certain shapes were congruent. She was most interested in cutting out figures and seeing what the result was when she folded them in different ways. My favorite observation of hers was with respect to the image on the left. She described it as “the top of a rocket ship being built on a really strong stick.” What’s important here is she made sense of what she observed and then justified it based on what she knows to be true: The rocket needs to balance on a strong, vertical stick in order to be completed. She is developing her own understanding of geometric relationships. It is much less important that she can produce an output that matches one of the 1st grade Minnesota Academic Standards in Mathematics. We can build up to understanding specific standards, but only after we have a foundation based on what she has already made sense of herself. In this case, the sense-making began with my handing her a compass and then having a conversation with her as she explored with it.

If you walk into a typical secondary math department, often the compasses can be found collecting dust and developing rusty hinges in the back of the supply closet. And these ancient tools have been traded for more updated technology such as animations of classic geometric theorems such as the perpendicular bisector theorem. Here is an example from a popular geometry textbook:

(Source: Envision Geometry, Pearson 2017)

You can draw your own conclusions about the usefulness of this example and how it can help students develop an understanding of perpendicular bisectors. But what I notice, besides that the perpendicular bisector theorem has been dwindled down to a series of steps to understand, is that the reasoning behind the compass arcs is unclear and somewhat ambiguous. Additionally, the students are not given an opportunity to explore their own curiosities and play with the mathematics. They are left with a set of steps to follow without ever having connected this idea to a circle or a compass.

What would happen if instead we gave students a copy of this image (or better yet, let them draw it for themselves):

(Source: Don Steward)

What questions would they ask? What properties would they explore? What would they notice and wonder? And if they didn’t hypothesize about perpendicular bisectors (or regular hexagons or symmetry), how could we as secondary math teachers lead them down the path of making sense of this theorem?

Here is another image from Don Steward’s blog which could ignite similar curiosities:

(Source: Don Steward)

He begins with an initial question, but what else might your students ask about this particular design?

Geometry class is ripe and ready for these kinds of interesting, beautiful explorations, yet our textbooks insist that students need to be shown steps and procedures to understand these theorems well enough for a standardized test.

Sometimes we are fearful of using manipulatives in math class because we are afraid that kids will “play around” with them. But in order for them to do sense-making, to develop a lifelong appreciation for, or even love of mathematics, isn’t that exactly what we have to let them do? They need a chance to play and wonder, conjecture and experiment. We need to believe that children (and yes, high schoolers) can develop powerful mathematical ideas, and if these ideas do not perfectly align with our mathematics standards, we need to develop strategies to build from student sense-making.