I Noticed a Pattern in My Pattern

Introduction

“What is a pattern?” – “It is when something is repeating like ABAB.” I found at the beginning of grade 3 that my students could build repeating patterns. We’ve been working throughout the year on identifying rules in increasing and decreasing numerical and visual patterns, and I wanted to find a way for all this scattered practice to come together.

The credit for the idea goes, yet again, to Simon Gregg  who shared how he used Desmos with his students to graph the visual patterns they created. He shared his blog post with the details of the lesson and students’ work. I wanted to adapt it for my classroom and I hoped that my students would notice the connections between the model, the symbolic numerical representation and the diagram/graph.

Questions and Thoughts: Before

I was anxious about the logistics of the lesson. Thank you, Sarah, for moral support!

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Will everyone be able to build an increasing pattern?

Will all students identify which step is which when translating the pattern into the table?

If we work in math journals, will someone spend an hour creating a table to fill out?

Will the students be able to handle the technology part independently?

Preparing for the Lesson: Getting Ready to Make a Pattern

I used one of the three-act math prompts from Graham Fletcher’s website to get students thinking about the growing patterns. After watching the video, I asked my students to build steps 4 and 5 of the pattern, label the steps and the number of tiles in each step and write down the pattern rule. Then students had to figure out how many tiles would be in step 10 without building or drawing all intermediate steps. This part has gone smoothly enough to proceed to the next lesson.

Preparing for the Lesson: Getting Ready to Make a Table and a Graph

Many of my students used the table to organize the numerical information about the pattern to find the number of tiles for step 10. We created and extended the table together with the class. I decided to use teacher.desmos.com to utilize the dashboard feature that would allow me to keep track of the students work and to bring it on the interactive whiteboard for future discussions. Here is the activity screen I used to for my lesson. We graphed the pattern together and discussed the behaviour of the graph.

The Creation

The lesson went much smoother than it did in my nightmares. I made a table template to simplify the logistics. Most of the students were successful in creating an increasing pattern that follows a specific rule. Some students created patterns that were increasing, but they needed support to identify that their rule was not consistent.

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There was often a “break” in the rule between step 1 and the rest of the pattern.
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Some patterns had obvious visual rules but were tricky to represent in a table.

While most of the students worked with addition, some students tried to triple or double the number of cubes they use in the pattern.

Pattern 1: A Pyramid

Pattern 2: A Fire

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Pattern 3: Plus Four

Pattern 4: “It’s a bit up and down but still increasing.”

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Students had some great notices and wonders.

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The really exciting things started happening when students crowded around the interactive whiteboard and started analyzing their own and each others’ graphs. Someone noticed that if there is an unusual change in behaviour in a graph, there might be a mistake somewhere in the calculations. One student said, “If the number of cubes is the same on every step, I think the graph will be just the straight line, not tilted.” Someone else asked, “How can you make a circle if there will be more than one Y for one X?”

I had a moment of insight.

Making Predictions

During my lunch break I threw together my template for the afternoon reflection which included three questions and three spaces for the sketches.

-Without numbers, sketch how the graph of your increasing pattern looked like.

-Sketch how you think a graph of a decreasing pattern might look like.

-Sketch how you think a graph of a repeating pattern might look like.

I asked students to try and explain their intuition.

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There were a few interesting ideas considering the behaviour of the graph of a decreasing pattern.

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Many students tried to look at the graph differently rather than switching the direction of the slope. As if they re-located their point of view rather than changing the orientation of the graph to represent the decrease. I wonder why and I currently have no explanation.

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This reminded me of a graph of an absolute value function, and it seems to me that the student’s logic was also describing something similar.

As expected, repeating patterns had the largest variety of predictions. Most of the students went with the equally spaced bumps. A few sketched a graph of a constant.

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There was an attempt to express a repeating pattern as a periodic event.

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And I need to ask some questions about this graph: How far do you think these lines can go up and down?

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Questions and Thoughts: After

My students need more Desmos in their lives. The interface appeared to be 8 year old friendly. I did not need to explain anything about the coordinate grid. I think, if we try graphing decreasing patterns, there will be more clarity about the structure. I loved the teacher dashboard with all the functionality of tracking my students’ work and the ability to pause the lesson and to control the slides. I would love to have the function to print students’ work from the dashboard implemented because printing 50 clipped screenshots was somewhat time consuming.

I wonder if showing all three steps of the process before engaging with it pushed many of my students to work numerically rather than visually. Many of the students were just building the “rods” with the cubes instead of going for more creative shapes. Would the outcome have been different if I asked them to build the patterns first, took the pictures, and then introduced the next two steps? If anyone tries, please let me know how it goes.

I also wonder what my next step(s) should be. I am considering giving kids their graphs and pictures of their patterns and finishing their notices and wonders first. Some of the students wanted to test their predictions for decreasing and repeating patterns. Some started experimenting with creating pictures with their lines. I am also thinking about reversing the process; giving my students a graph and asking to create a table and to build a pattern. We will start with wonders and then I will make my decision.

 

Shoelace That Makes Life Harder

Introduction

When I started teaching grade 3 last year, I printed out the Program of Study and went over it with three different colors highlighters. One for the concepts introduced or explored deeper in grade 3, one for the numeracy skills my students need to master (e.g. subtraction of three-digit numbers), and one for applications (measurement, data analysis and representation). Oftentimes, I intentionally integrate my “applied math” lessons with other subjects, sometimes the need to measure something arises in a context. I try to be sensitive to these contexts to catch the teachable moment and to turn it into a lesson. Last week, we ended up learning about measuring length.

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Tuesday: Counting

About a week ago, Christopher Danielson wondered about one of his prompts for his future book How Many? How many shoelaces will the students see, two or four?

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Image from talkingmathwithkids.com

On Tuesday, I gave the prompt to my class; we noticed, wondered, and counted everything from shoelaces to letters on the box, to dots inside the shoes. It was wonderful. One girl said, “I have these kinds of shoes at home, I’ll bring them tomorrow and we can check how many shoelaces are inside them.” I said, “Sure”.

After school, we had another Mount Holyoke College session, this time with Dan Meyer. The big question from the session: Can you involve your students in the co-development of the activity, rather than just assigning it? I didn’t have an idea yet, but it put my mind into the “search” mode.

Wednesday: Estimating and Measuring

The girl handed me her shoes first thing in the morning. We were about to take the shoelaces out, when someone asked, “How long are the shoelaces?” This was perfect!

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Here is the Outcome from Alberta Program of Study Mathematics:

Students will demonstrate an understanding of measuring length (cm, m) by selecting and justifying referents for the units cm and m, modelling and describing the relationship between the units cm and m, estimating length, using referents, measuring and recording length, width and height.

First, we recorded what else we might be able to measure: shoe size, width, weight, how tall the boots are. We discussed what units we need to use to measure the shoelaces. The moments like this I appreciate that I teach in Canada and I don’t need to figure out Imperial units. I brought in the meter stick to use as a referent point.

Then, we gave our too high, too low and just right estimates.

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After students used post-it notes to give their just right estimates, they tried to figure out the length of one shoelace by measuring anything they need to measure without taking the shoelaces out. Someone asked if they could use the rulers. “Yes, if you think it’s helpful.” The math notebooks were used to document and explain students’ thinking and to write down their final estimate. Some students tried to get the answer out of me. “I have no clue, these aren’t even my boots.”

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The reveal was truly grand with lots of cheering and jumping. The shoelaces had the total length of 125 cm! We put some estimates on the number line to see who managed to get the closest estimate.

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We discussed the possible sources of error.

-”Shoelaces go inside the holes.”

-”Centimeters are different on different rulers.”

-”Everyone was touching the shoes, pulling and stretching the laces.”

-”Some people would start a little bit ahead, not at zero.”

We had to pause for a discussion of “centimeters are different” to compare personal referents to standard units of measurement.

Then I had another moment of insight. “Let’s have a challenge. We can measure the shoelaces in my boots on Friday.”

Friday: Challenge

Here is why it was a “challenge”. What is your estimate?

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When my students saw my boots they looked at me like I am a magician who just pulled a rabbit out of a top hat. We went through the same routine, but this time we had a new referent point. The knowledge of the length of our Wednesday shoelaces. I honestly had no clue again how long the shoelaces might be, and my own estimate was quite a bit off.  I loved the range of the strategies that my students employed to support their problem-solving.

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Saturday: Thoughts and Questions

I do estimation challenges with my students somewhat regularly. They know the routine which is helpful when your audience is eight years old and the logistics can interfere easily with the flow of the lesson. Dan asked us to think about the verbs in our lessons. With my students, we noticed, wondered, selected tools, compared, modeled, estimated, discussed, measured, and some of us jumped and made not classroom appropriate sounds when our estimation was close. Most of us came up with reasonable answers, and all of us tried. Some students estimated that the shoelaces in my tall boots will be shorter than in student’s boots. I had to explicitly point at it for them to notice the disconnect. I wonder what I could have done differently to encourage them to make sense independently. I think that during our next estimation, we should have spent more time comparing our strategies and reflecting on our work.

I have previously done one week estimation challenges and I am considering bringing this format back . I would introduce the challenge on Monday and do the original estimation,(usually some kind of physical prompt), then leave it around for a week to allow students to get their ideas together. Finally, on Friday we would do measuring, calculating, summarizing, checking and reflecting.

I wonder what we can measure next?

 

Mix Ideas Together and Make a New Idea

Discussions and arguments about definitions were defining features of week two of our hexagons explorations. I found it was a challenging week for me to manage because my students’ wonders yet again went places. Simon Gregg and Christopher Danielson were supporting me along the way, and I feel like they were part of the conversations that happened in my classroom last week. #MTBoS at its finest.

Which One Does Not Belong?

After realizing that there are, indeed, more hexagons than you think, I wanted to prompt my students to start focusing on particular properties. Not surprisingly, Simon already had a shapes WODB ready for the cause with four not obvious hexagons. I could anticipate some of the responses (like identifying a regular hexagon), but I was not sure what my students will make of the red and yellow shapes. I was underestimating my kids.

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Late idea

At some point, students started comparing the shapes to the real objects and getting really sidetracked with it. I asked them to get back to focusing on properties. Things I should’ve asked instead: What makes it look like a hammer? What makes other shapes not look like hammers? Those observations that students made were not irrelevant, but I failed to connect their relevance to the goals of my lesson.

I’ve noticed that students had a lot of observations around composing and decomposing shapes, and after discussing it with Simon, I’ve decided to continue with tangrams hexagons challenges.

Tangrams the Hexagon Game

The Rules: Build a hexagon with 2 tangrams. Call a teacher to take a picture. How do you know it is a hexagon? Proceed to 3, 4, 5, 6, and finally 7 tangrams. Is it possible to build a regular hexagon with tangrams?

I also decided that it was the right moment to introduce some formal vocabulary like concave and convex to define “tucked in” vertices more conventionally.

Here are some of our hexagons.

And then there were some different ones.

Students kept building these funky shapes, and I had to admit that I am not certain how to count the edges and vertices on them. My students weren’t certain either. In our table groups, these shapes caused a lot of discussions. I took a few pictures and asked  for now to stick to the polygons that don’t look like their edges self-intersect. Then I called Christopher and Simon for help.

How do mathematicians agree on definitions?

After looking through a chapter in Christopher’s teacher’s guide to his WODB shapes book, I decided to revisit the conversation about the definition of polygons. I brought back the pictures of shapes in question. We still did not have the agreement on what these shapes are. I told that we all had different ideas, and mathematicians have different ideas too. But the definition that many mathematicians agree on is that polygons are not self-intersecting.

Then one of my students asked this question: “How do mathematicians agree on definitions?” – “What do you think?”

I did not expect that almost every student had an opinion on that.

“They have a box, they write their arguments, and they try to convince each other. Then they agree on some idea. But the ideas might change later. Or maybe some parts of the idea.”

“They vote, but they have a debate first. People give proof and evidence to support ideas.”

“They write their theories and talk about them.”

“Someone comes up with the name and has an idea what this name means, then others discuss it and agree or disagree.”

“They have an argument and they prove to each other why their theory works.”

“There are lots of Greek words in math, maybe people in Ancient Greece invented them and then we borrowed them in English. So we borrowed the words and we borrowed what they mean.”

“In olden times in Ancient Greece people would come up with an idea, and then they would invite other people to discuss it and try to convince others and give proof.”

“They mix ideas together and make a new idea.”

I promised to interview some professional mathematicians and to share their responses with my students next week. Daniel Finkel and David Butler agreed to help and to share their thoughts on the matter, and on Tuesday I plan to bring their ideas to our classroom conversation. In the meantime, Simon Gregg made two more hexagons WODBs that can also be put to a discussion ( WODB#1, WODB#2). Also, I think I am ready to face “square is not a rectangle and if you turn it around it’s a diamond” challenge.

Final Thoughts

I struggle with giving definitions. There are always a lot of questions. There is always some disagreement on the properties that are deemed important. Where does the classification system come from and how do I justify it? My students offered me a solution. “After some discussions, many mathematicians agreed to call the “closed” 2D shapes with straight not self-intersecting edges polygons.” Then we can reason why. Sounds much more reasonable to me than “Polygons are…”

And thank you again to everyone who helped me and my students to make some new ideas.

Like Someone Walking Through Time Without Even Noticing He Is

Sometimes there are unpredictable diversions in my lessons that are really hard to resist. They don’t really fit into any Specific Outcomes in our curriculum. I have an excuse for those. The Alberta Program of Studies front matter has a couple of goals that I like to quote. “Students will gain understanding and appreciation of the contributions of mathematics as a science, philosophy and art” and “Students will exhibit curiosity”. And eight year olds are curious about all sort of things when it comes to mathematics.

This lesson happened some time ago. The pictures and notes from it has been sitting in my folder for a while. Sometimes I come back to look at them because they amaze me. I’ve been sharing bit and pieces with my colleagues and on twitter, but I wanted it all in one place, and this post will be the place.

Can Number Line Be a Number Circle?

We have just had a few clothesline lessons on integers which extended our number line to the left of zero. Students were gathering on the math carpet, and I overheard a few boys arguing about infinity on the opposite sides of the number line. Will it be the same, or will the other one be negative? If it is the same, can’t you just connect infinity with infinity, and then make a number line a number circle? More students were joining the conversation. I grabbed an anchor chart and started putting down some questions that were happening.

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Based on the idea that numbers are creatures that live on the number line, we tried to figure out if infinity is a number.

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Someone said, “I saw a video about an infinite hotel, but I don’t know how it works, can we do this puzzle?” My original lesson plan would have to wait until tomorrow.

Infinite Hotel

Here is the link to the paradox in question.

So, I told my students a story about a hotel manager, and a lonely traveler arriving at night, and asked what room number will he get if the hotel is full but infinite. At first, everyone jumped with “I get it!”.

S: “There is infinity of rooms, so he will just go to the next room!”

Me: “But what number is that next room?”

Ss: “Infinity.” “Infinity is not a number!” “Maybe it is!”

Me: “And what room number is right before infinity room? How will the guest find it?”

Ss: “Is it an actual normal number?”

Me: “It is a normal number that you can count, so you can give the guest a key.”

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Students got together with their table groups to discuss their ideas.

“There are rooms with fraction numbers. So he can go into 1 ½ room.”

“He can quickly build another infinity hotel”.

“He can give the guest his own room.”

“There was a secret room 403 that was left empty.”

Me: “Nope, it was really full”.

S: “People can share a room.”

Me: “Nope, hotel’s motto is We Have a Room Just For YOU.”

Infinity And Us

By the afternoon, I was curious to know how my students would describe infinity. I tried to come up with an art project, but nothing really came to mind. So the proper art lesson did not happen. Instead, we read the beautiful book Infinity And Me by Kate Hosford, and then I asked students to write how they imagine infinity. Then draw it on a square of black paper. Not a full blown art lesson, but it worked with the writing.

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“Infinity is like a cloudy number line.”
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“I imagine infinity like a never ending wall you need to climb.”

“I imagine infinity as millions of dust in the air.”

“I think infinity is like a rainbow because it disappears and comes back.”

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“Infinity is living forever to me.”

“Infinity means to me is freedom.”

“I imagine infinity as my friendship and love for my family and friends.”

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“Infinity is a race across the sea.”

“I imagine infinity as a bunch of rocks cause you can always break rocks smaller and smaller and smaller and smaller and… a lot of rocks.”

“Infinity mean to me a color wheel with many colors and the wheel spins around and around forever.”

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“I imagine infinity is like a black hole because a black hole goes on forever”

“When I think of infinity I thing of earth because it is a circle and circles are endless,”

“I imagine infinity would look like a ball rolling around a cup for forever.”

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“I think infinity is a clock because it keeps going.”

“I picture infinity as a dragon going around and around  looking for something.”

“I think of infinity as the space because if it ends what is next?”

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“I think infinity is like a cat chasing its own tail around a stop sign or a tree.”

“I imagine infinity like someone walking through time without even noticing he is.”

It will be interesting to look at a different kind of infinity when we get to fractions. I wonder what images will come to my students’ minds when infinity will appear between the familiar numbers, when it will be close, not at a far distance, not vast, but unimaginably small. Sometimes I wonder if diversions like this are justified from the point of view of learning intentions, assessment and outcomes.  I think, sometimes it’s worth to stop and look at mathematical landscape and just admire the beauty.

There Are More Hexagons Than You Think

Geometry is always fun to teach in elementary. Students feel liberated by not having scary three digit numbers flying around. Shapes you can bend, stretch, fold, connect and transform; not everyone is comfortable with bending, stretching, folding, connecting and transforming numbers yet. You can prove claims by cutting things and moving pieces around. Students have a lot of first hand experiences with the shapes world, they have a lot of ideas and opinions about it. They are not afraid to step into this world, they feel at home in it.

Background

After my teaching partner discussed the attributes that separate the polygons from other weird flatland creatures, we decided to focus on reviewing our polygons vocabulary and refining our understanding of the attributes of different polygons.

We started with the lesson shamelessly stolen from Graham Fletcher Where is Poly? An Exploration in Geo-Dotting. Here is the picture of the prompt that we used.

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Students engaged in a dots scavenges hunt, shared their ideas on the SMART board, everything was going pretty similar to the original lesson. Many students noticed that if you connect some of the dots, you can make a hexagon in the middle. When one student started connecting the dots, the running commentary was as follows: “It is a hexagon because it has six sides, six vertices, it looks like a circle, and all sides are equal.” This last part caught my attention.

I drew a triangle with obviously not equal sides. We all agreed that it is a triangle. I drew a wobbly quadrilateral. While the students have just recently learned this word, they identified that it is “quad-something because it has 4 sides and vertices”. I drew a bowtie looking hexagon. I have never seen my class agreeing so unanimously with anything before; this was NOT a hexagon.

I had a moment of weakness when I was about to turn around and announce that it is, and then explain how it works. Instead I told my students that we will investigate it later.

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I took a picture and on my recess got to twitter. It’s not the worst way to spend recess. This was the idea I was looking for.

Instead of just taking a photograph I decided to make some sorting cards for my kids and it was a good excuse to spend my evening playing with the compass and pretending I’m working. I narrowed the set down to fifteen trying to find the hexagons that would cause the most confusion. I anticipated the responses for each one of them. I was close.

Hexagon Scavenger Hunt

Here is the picture of my hexagons and my students’ descriptions of some of the shapes. See if you can match them.

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  • “A hexagon”
  • “It will be a hexagon if you push the sides out”
  • “It would be a hexagon but one of the sides is off balance”
  • “A triangle with a hole in the middle or an Illuminati”
  • “It’s a rectangle, that’s missing a piece”
  • “It’s a rectangle and a square together”
  • “This one is a triangle but a little piece missing”
  • “A pacman”
  • “A heart”
  • “A heart #2”
  • “Z”
  • “A part of a star”
  • “A star that is also a triangle with bent sides”
  • “Looks kind of like a trapezoid”

Lesson

I gave students the cards and asked them to find hexagons, put them into a “hexagons” pile and try to describe what makes them hexagons. After about 10 minutes of logistical hiccups (“I need a pencil sharpener!”) and discussion, we put all the attributes on the board.

“It is a hexagon because it has six sides and six vertices and I think a hexagon only needs to have six sides and six vertices to be a hexagon.”

“This is a hexagon because hexagons usually look like honeycombs.”

“Hexagons always have even sides”

“If you cut it in half it will be the same on both sides.” (I think we are talking about symmetry here).

“If you cut a hexagon in half, it will have one, two three, four sides, it will be a trapezoid.” (Did not see this one coming, and kids really picked this one up, one of the last attributes they dropped.”

“The sides should not go into the middle, they should be either straight or go out.” (No tucked in sides)

“All angles are the same”

“All angles are 120 degrees”

“Inside sides don’t count” (I think this one was about the illuminati triangle)

“All sides should be parallel”

“It shouldn’t have sharp angles”

I find it fascinating how this conversation jumped all around K-6 geometry curriculum, and students made some observations that they don’t need to officially approach until grades 5-6. I think it takes care of some concerns that more student-centered lessons might not be the most efficient way to cover the curriculum. Actually, they are.

We looked through all the attributes, and found that there is only one hexagon that fits them all, the honeycomb one. Then I told the students that there are more than one hexagon. They went off back to their groups to decide which attribute they might consider dropping to allow more shapes to join the hexagons ranks.

After the second round we had 3, the regular one and the slightly crooked “honeycombs”. The attributed of equal sides was dropped first.

Here is when the real fun started. A few shapes were so resembling triangles and rectangles that reasoning through attributes was coming into conflict with the shape intuition. This triangle is just missing a little piece, if I put it back it will be a perfectly fine triangle again, even with six edges and vertices it is way closer to regular triangle than to the regular hexagon.

These conversations made me wonder: If our geometric vocabulary is usually develops through the context of regular shapes, do we subconsciously allow the attribute of “regularity” to take over everything else?

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We made a few more rounds of dropping attributes and adding more and more hexagons to the family.

“I have an idea! I think they are all hexagons!” This idea was appearing around the table groups and spreading around the classroom. One student came to me, “I think the pentagons might be like hexagons, all different. And other shapes too.”

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Thoughts and Observations

My goal was to have a conversation about the properties of the polygons without me being on the spotlight. I did not anticipate hexagon being a topic for such a passionate debate for 8 year olds that couple of groups required my intervention to direct students from speaking at each other to speaking to each other. While everyone had some aha moments, I am not sure everyone is equally convinced, but I think I can work with it. I also loved how students started extending their ideas to other polygons, and I plan to use their ideas to shape my next lesson.

Moving Forward

I think I want to make a WODB with 3 pentagons (2 “weird ones”) and one funky quadrilateral. Or maybe bring in a hexagon friend. This will be another fun filled evening because I don’t remember how to construct a regular pentagon with the compass.

Special Thank You to

Graham Fletcher for Geodotting lesson, it was engaging, interactive and brought up a lot of vocabulary

Christopher Danielson for giving me an idea when I didn’t know where to take my kids confusion

Simon Gregg for continuing the conversation and helping to organize my ideas into lessons

Zak Champagne for a great session Talk Less Listen More in Effective Practices for Advancing the Teaching and Learning of Mathematics online course in Mount Holyoke College. You are responsible for my “shut up don’t say it” moment that lead to this lesson.

 

Everything Is a Shape But I Don’t Know What Shape Is.

It all started with the tweet from Sarah Caban a few days ago.

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I decided to have a chat about it with the Kindergarten class, ask my grade 3 students, and one of our grade 5 teachers kindly agreed to ask the same question to her grade 5s. Here are the answers and some context from three different grades. At first, I planned to throw it all into a google doc and share with Sarah. Kindergarten kids’ ideas got me excited. When I started reading the other grades’ responses, I realized it’s worth sharing with anyone who might be interested.

Kindergarten

Students in kindergarten have already learned some shapes vocabulary and identified different shapes in their environment. I started with pointing at their shapes wall, “I see you have been learning about shapes, we are learning about shapes in grade 3 too. Can you tell me what does it mean – “a shape?” A forest of hands.

“It’s a heart!”

“It’s a diamond!”

“It can be a circle.”

“A small triangle can be a shape.”

Students were pointing at shapes around them.

After recording all their “shapes” ideas, I asked, “What do all these things have in common? Why do we call them all shapes?” I was surprised how quickly kids considered the question and were all ready to discuss their ideas.

“Shapes are everywhere”.

“Everything is a shape but I don’t know what shape is.”

“We are made out of shapes”

Then someone mentioned that some letters are shapes. But not all. Students did not seem to have a consensus on that point, and the discussion continued around particular letters.

“C is not a shape, but you can make it a shape.”

Our kindergarten teacher who kindly invited me to talk math with her kids today, ask the student to explain what he means.
“You need to draw a straight line to make it a shape.” He drew a line connecting two “ends” of the letter C together. “Now it is a shape.”

Grade 3

Grade 3 students have just started talking about polygons and sorted their polygons and non-polygons into two groups. Their thoughts about shapes were likely affected by this recent activity. Students answered the question on the post-it notes and we did not have time to discuss it yet.

“A shape is something that has vertices and edges.”

“A shape is random line but with curves and edges.”

“A shape is a structure that most things are made out of. It is something that makes up everything.”

“A shape is something that you can outline.”

“Shape is something that you can use for a pattern.”

“A shape is something that connects. A shape means a thing that has no opening.”

“Shape is an object.”

“A shape is something to represent slots, key slots and other items.”

“Shape is 2D which means it’s flat.”

“A shape is something that is 2D or 3D and it needs a face.”

“A shape is something with vertices and everything is a shape.”

“A shape is something that has to have vertices/faces/edges and more than 2 edges or vertices.”

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And the one that really made me pause and contemplate.

A shape is a 2D or 3D object that is drawn or held by humans.” – I thought bringing in the point that shape is something manipulated by people was really interesting.

I am also curious about this.

“Shapes are everything you see in life. You can find anything and it falls under 1D, 2D, 3D and 4D shapes.”

Grade 5

Grade 5 students worked on describing 3D and 2D objects and sorting quadrilaterals in the beginning of the year. They explored points, lines and did a lot of “hands on” geometry work.

“A shape is a thing that makes everything.”

“A shape can be a polygon and it can be irregular.”

“A shape is some sort of structure.”

“A shape is a structure of an object with all connected lines. Sometimes shapes have lines with holes.”

“A shape is a form of geometry that represents objects in their form. They are the appearance of an irregular or regular object.”

“A shape is something that can have sides, vertices, straight edges or none, so they can be anything like a soccer ball.”

“A shape is something like a symbol of math.”

“A shape is geometric combination of lines and curves making a closed region.”

“A shape is a closed 2D object with nothing inside it.”

Thoughts

I found it interesting to observe how the intuitions unfold with age, to see how students would use new and more sophisticated vocabulary to grasp the same concept. How they start looking at new mathematical concepts and to compare them with their idea of “shapes”. How they try and test new learned attributes to define a shape. When I started organizing these notes, I realized how much this simple question can actually tell me about my students’ mathematical thinking.

Questions

When other teachers look at all this students’ thinking, what do they notice? Just like these kids use different structures to define “a shape”, how are our “teaching structures” that we use to interpret students’ thinking are the same or different. Not sure it makes sense. “Everything is a shape but I don’t know what shape is.”

Proof and Evidence: Reasoning About Subtraction Part 2

Introduction

Last week I shared how my grade 3 students were preparing to prove their conjecture about subtraction: If you add +1 to both numbers in subtraction, the difference will not change. I really appreciate all the feedback and conversations that happened around the topic on twitter. It was also interesting to read blog posts from Jenna Laib , Simon Gregg and Sarah Caban about their experience working with proofs and representations.

My class finally finally got to our proof lesson last Monday, it was an extra long lesson, students came up with many great models and their teacher ended up with many questions. All this made for an extra long post.

Towards Lesson 4

We ended last week with building and sharing different ways to represent subtraction. I promised that on Monday we will work on Proving the Claim. I admit, I was nervous regarding the level of frustration and confusion the task could cause, and I was not certain how to prepare my students for it. So I did the only thing that came to mind; I scared them.

Lesson 4: Proof and Evidence

I started with warning my students that this is going to be the hardest part of the whole process. They came to the point where they need to prove their conjecture for all positive integers. The only time my students attempted proving was in the beginning of the year when we used snap cubes to reason about sums of odd and negative numbers (the lesson inspired by this post in TCM blog).

I reminded my students about this experience, pointed at the selection of tools and manipulatives and sent them off to work. I sensed an air getting heavy with confusion just as I anticipated. Before it ascended to panic, I made a suggestion to rebuild the models that students used last time and then see if they could tweak them to prove our conjecture. I tried really hard to not impose my thinking on my students, so I was mainly wondering around asking students how their models work, recording their explanations and asking for clarifications. The sense of direction was all that was needed, and I will leave the rest of this post to my student’s work. I wish I could post the videos, but my school board is very particular about not sharing students’ voices or pictures online so I will have to stick to the transcripts.

Students’ Proofs

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“Music is actually a lot like math”.

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“How do I know that it always works? Because it never ends if we just keep adding by 2s, even, even, even”.

The next one got me really excited.

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“All of the bottom is white. I added this way as a handle or to add more of these squares. Green is the difference and the red is the subtraction amount. Red and green are the main number. The black adds one more to the total number and subtracts one more.” -“Why do you add one cube? In our conjecture you add +1 to the first number and +1 to the second, and you just add 1?” – “Because this one cube adds 1 to the main number and the other number, like the minuend AND the subtrahend.”

There was also a trio of “story models” that showed that if you subtract 1, the conjecture will also work. I really enjoyed the narrative part.

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“This is 10 – 6. The white is 6. I made it like this because white is like extinguished water and that’s why I also have blue which is the remaining water. And that equals 4. The original number is 10”.
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“I made this like cheese. This is like an entire square of cheese. Five of these got stolen by mice cause they like cheese and now there is only 4 bits of cheese left. 4+5 equals 9, 9 is the original number”.
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“This is the leaf, burning. This is an entire leaf of 8. This one works like 8-4=4”.

Students’ Reflections:

The lesson took longer than I expected, so I asked for a more open reflection with our general prompts: questions, challenges, observations, discoveries.

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I knew this one was coming!

My Notices and Wonders:

I wish I could say that all of my students experienced the same degree of success with these lessons, developed multiple models for proof and expanded their range of subtraction strategies and their understanding of the operation. I am afraid it is not the case. When I was analyzing my lesson, I got back to thinking about my goals. I wanted my students to

  • explore different representations and contexts for subtraction
  • gain deeper understanding of subtraction and its properties
  • attempt and experience proving
  • construct effective arguments
  • explain our ideas in words, models, diagrams and mathematical symbols

Even students who were not able to come up with a working model on their own, had a chance to discuss and review other models. They had a chance to reason, to try and to model. They might not have gotten the result but it doesn’t mean that they didn’t get the experience. I wonder if they count it as their success.

The Best Intentions..

For the “We All Fall Down” edition of the blogging initiative! I skipped the last two prompts with the valid excuse of report cards writing, but I couldn’t skip this one.

The Fail

I finally get my students to calm down from all the excitement and ask, “So… do you see the problem? How can we solve it?” I look around and to my horror realize that no, they don’t, because there is no problem.

The Premise

I really bought into Dan Meyer’s “If math is the aspirin then how do you create a headache?” idea.  I always felt that teaching is close to storytelling, to creating an engaging narrative. Then after watching some of Dan’s videos and reading some blog posts, articles and discussions around, it appeared very intuitive. Of course, I won’t be interested in solving problems that I don’t have. I won’t spend my mental energy or waste my memory capacity on that. If I don’t care, I won’t learn.

All this seemed so obvious and easy that the real challenge got obscured: How do you plant and grow the problem? What seed do you throw in? What notices and wonders do you nurture?

The Lesson Plan

In early September, I planned a lesson to introduce standard units of measurement. The idea was to let kids experience how standard units allow us to have common language to communicate our measurements. Students were asked to measure different items in the classroom with… whatever they felt like. Then we were to discuss the length of their desks and to realize that we can’t come to a consensus because we all used different units.

The Lesson

Students were very enthusiastic and measured a lot of different items around the classroom, shared their measurements and observations, recorded them. Then the moment came to discuss our problem and to see how we all need to use same units… “What problem?” When I looked around I saw twenty-five happy eight year olds who have been having time of their life measuring the classroom with everything from pencils to their heads. They didn’t have a problem, they had a blast. They did not care that we all had different measurements for our desk. It was exhilarating to find out how many different ways we could measure our desk! I wanted my kids to need metric system. Well, after my lesson they didn’t want anything to do with metric system.

The time for the lesson was almost up. No way to salvage the lesson materialized in the last 10 minutes of the class. I spent this time mumbling about standard units of measurement who nobody cared about.

I started typing in how I fixed the lesson next day building on my failed one, and then decided I should leave it out. I’ll share if someone is interested. The most important impact of this fail was that it made me more sensitive to my assumptions, made me question the questions I ask my kids and the prompts I give to them. It made me experience that I can’t force a problem on my students, it has to come from them.

The Question

What would you have done to save the lesson?

Proof and Evidence: Reasoning About Subtraction Part 1

Before

It’s the third year that I spend some intentional time in my classroom working on a “making friendly numbers” subtraction strategy: 234 – 19 = 235 – 20. I was not happy about my previous two attempts. Some kids I think just bought into it, some kept confusing it with addition (take from one number and add to another); in any case, I felt like students were not able to figure out why this works, and they either took my word for it or tossed it away. In year one, I gave it to them. In year two, I made a lesson with Cuisenaire rods. I offered my students a great model they didn’t really care for at that moment.

A few students brought this strategy up this year during our number talks. “Why does it work?” – “I don’t know, my teacher told me it always does.” It was time to try again for the third time.

Original plan was a bit blurry: get the number string on, get students to notice the rule. Then model it with the snap cubes in two towers; let students work with partners to change the number of cubes in each tower without changing the difference.

Intermission 1

The new idea came after Mike Flynn’s session about exploring multiple representations, a first session in Effective Practices for Advancing the Teaching and Learning of Mathematics course in Mount Holyoke College. I tossed my original plans and decided to try and follow the example that was shared in the session; let students notice the pattern, come up with the claim and then prove the claim using their own representations.

Lesson 1: Developing the Claim

First lesson was a brief number talk. 3-1, 4-2, 5-3. Students noticed that if they add “1” to both subtraction numbers, it seems like the difference stays always the same. I also started recording the steps of mathematical thinking that we will follow. My worries during the lesson were that a few students did not seem to notice a pattern. I hoped that the Testing lesson will help.

Lesson 2: Testing the Claim

The plan was to ask students to spend 15 minutes testing the claim with as many examples as possible. Then things went sideways. “Can I try negative numbers?” We don’t teach integers operations until grade seven. I teach grade three. I panicked then said, “Go for it.” Then there were groups of kids engaging in some interesting discussions.

  • It doesn’t work! (-5) – 3 is not the same as (-4) – 2!
  • You are not subtracting -3, you are subtracting positive 3.
  • If (-5 )– 3 = (-8), then what is -5 – (-3)?

Not sure what I was thinking when I briefly introduced integers earlier this year. You can’t give eight year old kids a new math toy and expect that they won’t play with it. I’m bracing myself for multiplication and division of integers in the spring.

One student discovered that if you subtract a number from itself, you will always get zero. From that moment on, he forgot about original claim and was testing his new discovery. I think we will be proving it next. It was all fun if not quite what I anticipated. At the end of 15 minutes we all agreed that the rule seems to always work.

Intermission 2

The plan was to now ask kids to use any models to prove this claim. And that’s when I realized that I didn’t allow my students to play with different tools and to develop their own subtraction models. We built numbers, we did not build “subtraction.” I felt we are not ready. I decided to let kids to explore different representations for subtraction first.

Lesson 3

“Today we will be testing the tools that we will later use to prove our claim.” Students worked in 4 groups to build a model, draw a diagram and write a story for 9-5=4. I had a selection of different items ready for the students to pick; from ten frames and balances to dominoes and snap cubes. First ten minutes were terrifying. Students used manipulatives to build symbols 9,5 and 4. I told that with their manipulatives they are trying to explain it to someone who might not anything about subtraction or symbols for numbers, but can count.

After all groups seemed to have come up with some working representations, students interviewed each other to find a representation that was different from their own and reflected on it in their math journals.

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Closing Thoughts and Plans

These few lessons made me realize that when I ask kids to explain their thinking, I often give them my tools and they end up modeling my thinking rather than developing and explaining their own. They need to find their own tools and develop their own reasoning and I need to give them more opportunities to do it. On Monday we will move on to our proving.

Here is the template I made to plan my future Mathematical Reasoning Lessons.

Questions:

How do you lead students from drawing the picture to drawing the diagram? From building numerical symbols to building a model?

My Favorite: From Sharing to Discussion

My grade 3 students love sharing: stories, artifacts, ideas. They are much less interested in listening – to what others have to share. Which kind of defeats the purpose. Math time is not an exception.

I believe that discussions play important role in elementary mathematics classroom and all students need to become comfortable with constructing arguments, defending their reasoning, questioning and proving. I try to design my lessons and routines so that they would include these skills as their natural component. And yet sharing doesn’t turn into discussing as often as I hope it would.

The background:

There were some things that worked in the past for me: Circle counting (I learned about this routine from Sadie Estrella here ) encouraged students to listen to each other and WODB (from Christopher Danielson and Mary Bourassa) seemed like a scavenger hunt to my students who were interested in “who found what else and how it can help me to find something else”.

But I am always on a lookout for more ways to engage more students into productive and attentive mathematical discussion, and my favorite “aha” this week was this “one new way”.

About a month ago in December, I joined a Global Math Department webinar lead by Andrew Stadel (@mr_stadel) , Clothesline Math: The Master Number Sense Maker.

The lesson:

So as January came around, I decided to start my first day of school in grade 3 classroom with Clothesline lesson: looking at numbers, addition and subtraction equations and arranging them on the number line; first in the right order and then in the right spot relative to each other.

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Students worked in groups first to put the numbers in order, then each group moved one number on the large clothesline at the front. Everyone agreed on the order but more precise locations were already causing a lot of disagreements. Usually, I would ask students to come up one by one to move one number and justify their reasoning and then have some discussion routines to roll. The same kids usually want to come up. The same kids usually want to have a discussion. The same kids want to argue about making the location more perfect. And all these kids do not constitute a majority of my class.

Using some webinar ideas, this time I asked a student to come up, move one number and remain silent. Other students had to try and “read her mind” to figure out the reasoning. Whatever the reason, this shift worked. Kids who are usually not comfortable presenting did not hesitate to come up, to move the numbers, and then to listen to others try and figure out their reasoning. “Not quite”, “On the right track”, “That was what I was trying to do”, “There is more to it” – the list of responses developed on the go. At the end, they found it easier to summarize their explanation after listening to all these other people trying to “read their minds”.

The “favorite” stuff:

The best conversations happened around some solutions that most of the class quickly identified as incorrect. They still had to try and figure out what was the reasoning behind it instead of jumping straight to disagreeing.

Student 1 comes up and puts the card 5+7 between 0 and 10-5.

Me: “What do you think is the reasoning?”

Student 2: “Maybe 5+7 is close to 0 because 5 and 7 together are smaller than 10 and 5?”

Student 3: “Maybe it’s smaller because 7-5 is 2 and it’s very close to 0. But one is minusing and another is adding, so it might be important.”

Student 1: “Can I revise my thinking?” Moves 5+7 past 10-5 and 20-10. I did not think that I need to add.”

What worked well:

-New students joined the discussions.

-More conversations happened without my interference.

-Students listened to each and “revised their thinking” based on each others feedback.

-Students who find it challenging to explain their reasoning felt more confident when they had the “help” of others working not to challenge their ideas but to understand them.

My Questions now:

  • How can I adapt similar routine to other activities?
  • What other strategies and routines other teachers use to encourage authentic discussion?

I appreciate your ideas and suggestions!