# Today We Are Going to Act like First Graders

I teach a class called Mathematics Content and Pedagogy; it’s a course for Elementary pre-service teachers. Most days are pretty fun. Today was amazing.

We began with introducing numbers in base 5.  Why on earth would we use base 5 with elementary teachers?  I wondered this myself the first time I encountered numbers in different bases in the course syllabus. It immediately occurred to me as the students were working…now I can get them to think like first graders.

In this course, we often ask the students to think like a student and imagine how their students would approach the task. The problem with that is…once you know the standard algorithm for the various arithmetic operations, it’s very tricky to un-know it. It’s difficult to think like a person who doesn’t know the standard algorithm.

Think like a student who doesn’t know the standard algorithm for addition.  Until I had a child of my own who has built from counting all to counting on to derived fact, I didn’t really understand how young children built their understanding of arithmetic (See Children’s Mathematics, Carpenter et al., 2015 for more information).

But if I put them all into base 5 (or any other base), now they can think like 1st graders. First graders who do not yet know how to add 4 + 3 (in base 5).

We started with representing the numbers with unifix cubes. At first they were resistant to the cubes, but then they realized they were first graders. They NEEDED the cubes.

Then, we agreed on some common language for hundreds, tens, and ones (flats, longs, ones).

Here’s the beauty:  They needed to model the problem. Our brains are so wired to think in base 10, that base 5 required a physical model, whether that was the number line or the cubes or a picture. They were truly thinking like a first grader. And it was beautiful.

Here are some of the models they came up with for subtraction:

Can you see the borrowing in the picture?  Can you FEEL the borrowing?

I’m grateful for teaching moments that make me want to stand up and cheer. Today was one of those days.

# Math is Beautiful (and other lies)

When I was growing up, I really wanted to be able to do art.  My mom was incredibly crafty, and my younger brother seemed to follow in her footsteps.  My brother filled canvases and sketchbooks, painted on the walls and even designed some of his own tattoos. He studied art in college and went on to pursue a career that relied on his artistic eye.  My brother was an artist.

I wanted so much to be artistic.  I’ve always been creative, no doubt.  But I wanted to be one of those people that could carry around a sketchbook and draw when the mood struck.  I wanted to be able to decorate my walls with my own creations.  As I got older, I seemed to accept that my brother got the “art” gene and I, well, did not.  Not that I wasn’t good at things.  But art was a real, tangible skill that seemed to extend beyond formal schooling.  Being good at math just doesn’t seem to have the same inviting ring to it.  “Hey!  I still know how to factor a polynomial.  Wanna see!?”  Nope, not a party trick that gets much of an audience.

Math teachers will tell their students that Math is Beautiful.

Yes, math is beautiful and if you do the whole worksheet, you’ll get the answer to the riddle at the bottom!

Geometry is beautiful, here’s a bunch of two-column proofs to prove it.

To me, math wasn’t particularly beautiful.  Art was beautiful.  I thought math was cool, and I liked teaching it.  But “math is beautiful” was just another lie that teachers use to get them to buy into the quadratic formula ever being useful beyond the 11th grade.  The covers of textbooks perpetuate this lie.  That pretty peacock blowing rainbow bubbles on the front seems to say “Open me!  I’m full of color and beauty and wonder!”  Then sadly, the pages beneath just reinforce the ongoing myth that math is about rules, procedures, and memorization.

Somewhere in the midst of being a high school math teacher, wishing I could do “art,” and not being particularly convinced that math was beautiful, someone handed me a compass and a straight edge.  Yes, a compass – one of those rusty metal contraptions with a point on one end and a pencil on the other.  I drew a circle.  And then another.  And another.  Then I drew strategic lines connecting parts of those circles.  And I colored in pieces that I wanted to highlight.  And I kept drawing circles and lines.  And kept coloring.  I dove head-first down the internet rabbit hole of geometric art.  Soon, I was beginning to believe two important things:  1.  Math was actually quite beautiful.  2.  I was, in fact, artistic.

At the age of 33, I went to the art store and confidently bought myself a sketchbook.  I filled it from cover to cover with math-inspired art.  I bought drawing pencils, artist markers, and one of those shading tools that looks like, well, nevermind.  I bought art stuff.  Lots of art stuff.  I bought big, giant water color paper.  And I put my art in a frame and hung it on the wall.

Earlier in life, it wasn’t that my brother was the artistic one and I was the analytic one.  It was that at a young age, we were inspired by different things.  My definition of “art” was freehand drawing, something my brother enjoyed and excelled at.  Because my narrow definition didn’t include mathematical design, I believed that I just wasn’t an artist.

The same is true of mathematics.   In school, we are led to believe in this narrow definition of math based on state standards and aptitude tests.  We are taught a math hierarchy that starts with counting, continues with multiplication facts, leads into algebra and ends with math’s high priestess:  Calculus.  In reality, there are many complex, high-level areas of mathematics that don’t have anything to do with calculus.  We are also led to believe that some people can do math and others simply cannot.  Our experience in K-12 mathematics cements these beliefs and we’ll go through life believing we simply are incapable of being good at this thing that we wish we could do better.

My daughter is entering the 2nd grade this school year.  Over the last 8 years, I’ve worked very hard at widening her definition of math, fearful that a future classroom experience is going to snap that door shut.  I’m very grateful that this year I will be working with future elementary teachers so that I can help not only expand their definition of mathematics, but also help them believe that they are brilliant mathematicians.  It’s been a while since I’ve been in front of a classroom, but I’m inspired, and excited and I’m more ready than ever.  It’s time again, to change the world.

# Today, I Finally Set Down the Chalk

Today was the last day of school. The bell rang at 2:25, and like every year, the students cheered, the teachers breathed a collective sigh of relief, and the school emptied to close out the year. But this time, I wasn’t there to experience the abrupt transition into summer as I have the previous thirteen Junes. Yet I couldn’t get this particular last day of school off my mind today.

I have been on a leave of absence since November, and halfway through my time out of the classroom, I resigned as a high school mathematics teacher. My computer has been turned in for months. My personal items have been recovered from my classroom for ages. My keys were given back weeks ago. And recently, I deleted my school google account from my phone. I even let myself have a necessary, healthy cry while going through endless boxes of documents and binders acquired from 13 years of teaching math, deciding what could be recycled and the things I couldn’t let go of.

But today, on the last day of school, it finally felt, well…final. That is complex chapter in my life, filled with more love than heartache and more hope than regret, has ended. And although the story didn’t end in happily ever after, my new venture allows me to dig down deeper to my educator core as I pursue my PhD in math education starting in September.

St Francis high school was my first and only teaching job, and I had a tremendously positive experience serving as a mathematics instructor and as a leader in the district.  I have been very fortunate to have done such meaningful work while employed there and to have worked with such committed, passionate people. My work there has propelled me to become involved in the greater math education community and engage with teachers across the country in order to keep our profession moving forward for children.
When I reflect on what I’m going to miss the most, there are a lot of things, but my mind keeps going back to the dog wall. (Wow, unexpected teariness). Each picture on that wall isn’t just a dog. It’s a student who handed me that picture and told me why that dog was important to them, whether it was their dog, their friend’s dog, or a dog they found on the internet. Every picture has a student, every student had a story, and every story had a piece of being a teacher that I cherished so much. Thankfully, my spouse captured some photos of the dog wall, and a collage of it hangs prominently in my new office. I’ll apologize in advance to the teacher who takes over that classroom next. But I hope they can appreciate for a moment before they take the pictures down what’s really joyous about being a teacher: the students, the unique stories they bring into our classrooms, and the value of their perspectives.

Here we go, Megan. Onward.

# The Compass and the Straight Edge: a Complicated Love Story

“Hi. My name is Megan Schmidt. I’m a former high school math teacher and future PhD candidate. This presentation has nothing to do with either of those things.”

“Full disclosure: This session will not directly connect to something you can take back to your classroom. My goal is for you to learn some simple geometric constructions and my hope is that it inspires you to try more. If, in turn, that gives you some ideas on how to teach standard 9.3.3.2 (transversals and angle relationships), I think that’s fantastic. But in an hour, I’m going to walk you through some of my beginning explorations.”

Here’s what I learned:

1. Anticipating supply needs is hard. My session started with more people than I had compasses. Some MCTM angels found round paper cups, but the best tool was totally a compass, and I only had about 25. I appreciate those who stuck with me, sans compass.

2. Teaching anything in an hour to a room of 60 people is hard. This was more or less a crash course. I was hoping people would be inspired to investigate and play on their own. But there’s a learning curve to working with a compass and straight edge that probably can’t be overcome in 60 minutes. I’m thankful that the session we are doing at Twitter Math Camp will be done over 3 mornings.

3. It’s ok if it doesn’t connect to the math standards. Even though my description didn’t mention a curricular alignment, plus my explicit statement from the beginning, I still received some (anonymous, angry) feedback that my session did not include activities to relate this artwork to “the curriculum.” But the tangible and intangible takeaways from a math conference do not need to be directly transferable to your math classroom. Much of what we teach (in secondary math especially) isn’t transferable anywhere else but to the next secondary math classroom.

4. I really needed a document camera. I was able to adapt my laptop camera for this purpose, but I really needed a document camera. I was facing away from the group the whole time and it wasn’t as good as it could have been for that reason. I did the best I could. But a doc cam would have made it much better.

5. I still really love being deeply entrenched in math education. Being around other passionate math educators is inspiring, empowering, and enriching for me. I’m changing the direction of my career a little, and the last two weeks at NCTM and MCTM reminded me that I still really love mathematics, math education, and the unique conversations that happen amongst math teachers.

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’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?

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.