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.

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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.

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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.

Spiraling the Hundred Chart and Beyond

Now that I have a little more time on my hands, I decided to write up some of my spiral explorations so that others can more easily play along or use them in the classroom.

Spiral Instructions

Templates:

Spiral Templates – 100 1 sheet

Spiral Templates – 100-4sheet

Spiral Template 1 to 196

Spiral Template 1 to 1089

Google Document Files:  Instructions   Templates

Suggestions:

  • Because my school is not 1:1, I printed these out and had the kids use markers or colored pencils.
  • I have them start with the spiraled hundred chart (printed 4 to a page) to have them play around with patterns at first. This is also a good opportunity for them to test conjectures.
  • Connecting the numbers in the pattern adds another element of visual interest.
  • Students looking for an extra challenge can try a variety of quadratic patterns.  I usually have them put a function into Desmos or a graphing calculator and use the table.

Here are the pentagonal numbers done on the large spiral:Pentagonal.jpg

#NCTMRegionals Chicago Presentation

As promised, here are the slides and resources from our NCTM Regional session, “Stats Trumps Hate.”

I want to thank Carl Oliver for coming all the way from New York for the day and leading this presentation with me.  Carl, you are such an incredible human being, and you have so much awesome to offer the math educator community.  I am truly humbled that I have gotten to work with you so many times on this topic that is close to my heart.

https://docs.google.com/presentation/d/117234ae5AWL_2P3UODDqJGUO1CvOvs_VuoECkMoj3fQ/edit?usp=sharing

…But the End is Beautiful

Every term, a lot teachers (myself included) are disappointed that students didn’t learn exactly what we intended them to. Furthermore, they didn’t learn as much as we wanted or thought they could. I’ve been there before many times, and that result is the main reason I changed the structure of my class to focus on student discourse and making sense of problems.

The last question on the final today was “Tell me something you learned about yourself this trimester. (It does not need to be math related).” Of course, it’s a math test so most of them tell me something math related anyway which doesn’t bother me one bit.

I originally was going to just share some highlights, but reading these answers brought me so much joy, I am just going to share the whole list.

I learned:

  • I understand math better when I’m given problems to try rather than just talking notes.

  • How to help myself learn on my own by asking more questions to deepen my understanding.

  • I can learn math with a group of people I don’t know at all and we can have fun as well.

  • I can develop my own solutions to problems and make sense of ideas myself.

  • I could be independent and do my work with less and less help.

  • It’s ok to ask questions but I learned how to believe that I can learn math myself and figure things out.

  • I don’t give myself enough credit for what I am able to do.

  • I learn things a lot better when working with others.

  • I am capable of being independent but still learn what I need to in order to be successful.

  • I am able to figure things out myself even if it’s not explained to me first.

  • Making sense of WHY I am getting an answer is much more important than the answer itself.

  • Understanding why something happens is much more useful than understanding just how to do it.

  • If I work through problems and bounce ideas off my teammates I CAN figure things out instead of being shown first. I enjoyed this, which surprised me.

  • Applying yourself to learn the material instead of memorizing it is much easier.

  • I CAN in fact understand the abstract concepts of math.

  • If I can’t explain WHY something works, then I don’t fully understand the concept.

  • I am a very hard worker and determined to reach my goals. I’m unstoppable when starting a problem.

  • I am able to graph functions without a calculator because I understand how they work.

  • The graphs of equations are really satisfying once you understand why they work.

  • I enjoy being able to figure things out myself first.

  • I really like to work with people to figure things out and I didn’t think I would.

  • I am able to make sense of my answers and not just get the answer, which is much more important.

  • Once the connections between concepts were more clear, math becomes much easier to understand.

  • While I’m not the best at understanding things while I’m being taught, when I teach other people, I retain the information.

  • How to explain myself better by teaching math to others.

  • When I need to explain something to someone else, I understand it better.

  • If someone doesn’t understand something, I am able to explain it in a different way that they can understand.

  • The more I am engaged with the math, the better I do.

  • Ask when you get stuck. The teacher will help you.

  • I’m glad I took another year of math even though I don’t like it.

  • I am able to teach other students when they don’t understand.

  • How to work through things better with a group of people.

  • The struggle is part of the process and you have to go through that barrier to learn new things.

  • I can handle more stress than I thought I could.

  • I am a visual, hands-on learner rather than just being shown how to do it.

  • I can be a mean person and that can turn people off from friendships.

  • I can better understand if I apply my mind.

  • Applying what I learn to everyday activities, I understand them better.

  • I should cherish my time with my friends and family before it’s too late.

  • I need to stop slacking on homework because it only gets harder from here.

  • There’s only so much in my control and not everyone will care about some things like I do.

  • If I actually take time to study and make sure I understand, I will do better.

  • I should probably study more in college.

  • It’s ok to not understand and ask questions.

  • When left to my own devices, I don’t do anything productive. I need to plan.

  • I do better in a classroom where there aren’t strict rules and things are more free flowing.

  • Don’t judge a person because you don’t really know what they are going through.

  • Not to overwhelm myself with school work and if I’m tired, I should go to bed.

  • If I study for my college algebra tests, I usually do better on them.

  • I love my job and my coworkers.

  • I don’t need to stress so much about the future.

  • If I want to get good grades in a college class, I need to study more.

    I am better at math when I have a therapy dog.