Space for All Learners

In my ongoing quest to find great math resources on Twitter (not hard), I started following Steve Wyborney (@SteveWyborney). Steve is a math teacher and coach in Oregon. He has been posting high quality math activities immediately usable by elementary teachers. I will share a few of them here.

Teachers often struggle with math tasks that feel just right for some students but not others. Steve’s activities are accessible by a wide range of learners in the elementary grades. They provide access points for many but also opportunities for challenge within the realm of grade-appropriate number sense, place value understanding, and additive or multiplicative reasoning.

Provide Massive Space to Notice

Steve has created this subitizing/number patterns video that teachers can show to students, pausing at a certain spot to allow thinking and noticing. It couldn’t be much easier. He provides a printable page to give to students. Simple to try, powerful results.

The Maze Hundreds Chart

Watch Steve’s video of ideas for how to use this resource. Then you can get the interactive powerpoint slide he created and try it with students. I can’t get it to work on a Chromebook, so you probably need a computer with Powerpoint installed.

Also try The Green Light Hundreds Chart and The Animated Multiplication Table.

These are simple tools, already frequently used in elementary mathematics, but often without the kind of exploration and reflection encouraged by Steve.

If you explore Steve’s blog, I’m on a Learning Mission, you will find lots more high quality instructional strategies as well as tools that are immediately usable with students.

Fix It

I appreciate Marilyn Burns’ presence on Twitter and her blog. She offers great contributions to the math education community.

Marilyn's recent blog post, entitled Fix It: An Activity for Ordering Fractions, is a well-written reflection of a lesson she taught to fifth graders. Marilyn describes how she used an engaging format and includes details of some masterful scaffolding for a student who needed help. It's worth reading the whole thing.

Fix It Fractions reminds me of the Clothesline Fractions activity Rebekah Thomas did with her summer school class. Before class, she’d hang a set of fractions (using clothespins) on a length of string in her classroom. A different student was in charge of “fixing” them each day (putting the fractions in order). The student presented his or her work to the class along with an explanation.

In Marilyn’s activity, students create their own Fix It Fractions sets for others to try.

There is something inherently compelling about fixing. It is different than doing math work that involves simply performing a calculation or solving a problem. Students look at the work of someone else that may be intentionally or unintentionally incorrect and perform an error analysis. There are lots of formats for doing this, including My Favorite No, or presenting two or more solutions and asking students which is correct and why.

I always appreciate when I see error analysis type tasks in Eureka Math lessons, Smarter Balanced Assessments (both of which we use in our district), and other materials. When I think back to different jobs I have held in my life, I think the majority of my time was spent interpreting, analyzing, adapting, correcting, and reimagining work that had been begun by others.

Let’s think about how to include more of this type of approach for students, and see if it improves their school experience.

If I Had a Hammer

Peter, Paul & Mary

I just read a great blog post by Tracy Zager. Her blog is called Becoming the Math Teacher You Wish You’d Had.

It is about getting students out of the normal (sometimes boring) routines they’ve become accustomed to in math class.

Tracy describes being in a workshop with Brian Hopkins and doing a bunch of math problems in groups. Her group solved a problem and then there was discussion and learning about the mathematics that best fit that problem. When Brian posed a seemingly similar problem, Tracy assumed they would be applying that same mathematical construct to the new problem. That was not the case, which surprised Tracy.

“...Brian disrupted the predictable, pitter-pat routine of math class...

What I see in schools is we cue kids to know what tool to use. If we’re two weeks into a unit on fractions and we give them a story problem, the kids figure fractions are involved. If the name of the chapter is “Multiplying Two-Digit Numbers” and it’s written on the bottom of the worksheet, the kids are going to assume they should multiply some 2-digit numbers. If we’ve written an objective about linear equations on the board, kids figure the answer is going to involve linear equations. If my new tool is the hammer that divides fractions, I’m going to use that hammer until my teacher tells me it’s time to switch hammers.”

As educators, we are often frustrated by our students’ lack of ability to make sense of and solve problems (the first Common Core math practice standard). Yet, are we giving students experiences in math class that help or hinder their ability to solve problems?

Geodesic Dome Building, Part I

Ada Leaphart, Integrated Arts Academy Art Teacher Extraordinaire, and I have embarked on an adventure. We’ve got all the fourth graders at IAA building geodesic domes out of newspaper.

I really don’t know how this is going to end up, so that is why I have named this Part I. I hope to report back with more news as the project progresses.

Ada and I decided to keep things loose. We didn’t want to figure everything out for the students by creating our own dome first, learning all the lessons, and then presenting a tidy scenario.

Instead, we have never attempted to create a geodesic dome out of newspaper or anything else, for that matter. Sometimes STEM+Art (I, unlike others, am not wanting to call that STEAM) should be messy and students should have the fun of making mistakes and doing all the figuring.

Before the students got started building, Ada showed a photo of a geodesic dome and asked students what they noticed. We got some math+art conversation going from that, as students noticed many geometric shapes, like triangles, pentagons, hexagons, and trapezoids. Then we talked about old Bucky Fuller (every class asked if he was still living) and how he really wanted to make the world a better place for everyone by using an efficient structure like a dome for shelter.

Kids couldn’t wait to get started. Here are the instructions we are using from PBS.

It turns out, you can roll newspaper in a loose, floppy, weak way or you can roll it in a very tight, very strong roll. Students shared successful and unsuccessful techniques. Among the successful techniques invented by students are 1) Asking someone else to help you tape the roll, 2) asking someone else for help, period, 3) twisting the roll when it is finished to make it even tighter, and 4) using a pencil to act as the center of the roll, then shaking it loose once the roll is finished.

We ended up with enough rolls to make one or two domes. 65 usable rolls are needed. Next time we will need to establish a Quality Control group to assess, select, and count the rolls we’ve got. Ada and I aren’t sure how the whole thing is going to go. At the end of it all I would like to set the dome on fire in the playground. I am not sure we will get permission for that. Oh well, we’re going to roll with it.

Vermont Fest 2014 and a Snake

The sign of a good conference is when you come away from it questioning everything about your work.

I like this snake drawing from the New York Public Library more than the three terrible photos I took of people in banquet rooms at the Killington Grand. I would never consider publishing a blog post without a picture.

At Vermont Fest this year, I found myself wanting to make a list of a bunch of things all students should get the chance to do in school.

Skype/hangout/video interact with people in other parts of the world

Type an equal sign into a spreadsheet

Make a personalized, LED-lit badge to explore how light works

Have maker/genius/whatever hour/day/week to explore something interesting to them

Publish their content (writing/photos/videos) on a blog and comment on the work of others

Learn about how computers work by writing some code

Be asked the questions How are you feeling? What do you think? and What do you need?

Use Google Research Tools to find and cite things.

Sit in chairs listening to a teacher talk a lot less.

Some great presenters I encountered include Chris Lehmann (@chrislehmann), Bonnie Birdsall (@bonniebird), Dan French (@danfrench), Adam Provost (@batman44), Lucie deLaBruere (@techsavvygirl), Charlie Macfadyen (@csmacf), Tricia Hinkle (@VTScienceFair), Matt Dunne (@@mattdunnevt). Check them out.

Here are a couple of links.

Using Google Sheets in the Classroom

Tricia Hinkle’s site

Bonnie Birdsall’s writing tech site

The Vermont Fest site with all kinds of other links to presenters

Fibonacci and Fractals Links

I had the pleasure of being the co-presenter of a workshop called “Fibonacci, Fractals, and the Common Core Math Standards” at the VCTM conference at Saint Michael’s College on Friday. My partners were Professor Tim Whiteford of Saint Michael’s College and Laura Sommariva, a math teacher at Colchester High School.

The three of us shared what we’d done with students during a fifth grade field trip to Tim’s famous Penny Arcade. Here is Tim’s original write-up about that.

As promised, for the participants in our workshop and others, here are the resources we used to teach about fractals and Fibonacci numbers.

The Fractal Foundation Fractivities. How to draw Sierpinski triangles, do fractal cutouts and more.

Worksheetworks. Create your own graph paper and triangle paper.

On Being a Plant, Part 1, the 6 minute Vi Hart video we watched. She explains Fibonacci numbers, and demonstrates how to count the spirals on pinecones, draw realistic pinecones, and use graph paper to make a golden spiral.

Laura teaches summer math classes for elementary students in Switzerland. She has photos of her students here doing wonderful, related activities like building giant tetrahedrons out of mini marshmallows.

Have fun!

Shelburne Farms Mini Maker Faire

I went to the Maker Faire at Shelburne Farms today for the first time.

This is what I did.

I made a magic wand with an LED light that turns on and off by touching a wire to a battery on the wand. I first had to figure out how to connect the LED to some wires, then run them to the battery correctly. After that part was working, I added sparkly silver ribbon to the stem of the wand and encased the LED light in crumply clear plastic tape for light refraction purposes. Voila! I am ready to put spells on people. Joanna Elliott, Flynn Elementary parent and teacher, was the wizard behind this project. See her fabulous art blog.

I made a puzzle book, a square flexagon (a previously unknown-to-me relative of the hexaflexagon) for comic-book type story-telling, and a mini book that could contain anything from math facts to the secrets of the universe. A matchbook size mini book can be made and then kept in an actual matchbox. Book Arts Guild of Vermont people helped me do this. Students might want to make these after reading the Red Clover Book entitled The Matchbox Diary by Paul Fleischman. This is an activity for any budget.

I spoke with Richa, who is going to assist with a course called Intro to Relational Databases at Girl Develop It Burlington. There are classes and meet-ups. I want to go.

Champlain College Emergent Media Center folks explained what they are working on. Their new Maker Lab that had its grand opening party last night.

I saw a robot-building challenge and presentation by Joe Chase and his team of students from Essex High School. Joe is my neighbor and it was great to see him up there advocating for more design and engineering work in schools. My daughter took his robotics class a few years ago and loved it.

I saw and did many other cool things, including experimenting with magnets with Frank White from CreateItLab and speaking with the effervescent Michael Metz of Generator, Lucy deLaBruere, Courtney Asaro, and Graham Clarke, both of Flynn Elementary School in Burlington.

What a great day! Takeaways included an Arduino Robot Kit made by YourDuino.com and the knowledge that so many people are working on creating engaging opportunities for people of all ages in the Burlington area.

Hit the Target

I was fortunate that my boss found out about a STEM (Science + Technology + Engineering + Math) Coaches course taught by Dr. John Tapper and registered me and our district science coach. We spent four whole days at the Hampton Inn, which is a strange venue for a STEM course but it worked fine. I gained about 5 pounds because the hotel served muffins every morning and cookies every afternoon and I ate them.

I already knew John Tapper from afar because of his excellent book on math instruction called Solving for Why. This course wasn’t about math menus and differentiation; instead John got us thinking about teaching STEM and instructional coaching. There was lots of discussion, not lots of lecturing, and plenty of hands-on investigations involving buoyancy, force, and motion. John teaches college level math and education courses, but has experience as an elementary classroom teacher, administrator, instructional coach and researcher. I found his perspective refreshing and some of his ideas about STEM new to me.

I’d been wondering how to best integrate math with other content areas. In the STEM Coaches course, John pushed us to use math in a meaningful way rather than as an add-on. He asked, “What about challenging students to create a mathematical model that they will then use to make a prediction?”

It turns out there are fun, engaging ways to do just that. John gave us Straw Rocket Launchers (go here if you want plans to make your own). Two teams experimented with these, adjusting the launch angle and launch force to see how to get the straw to travel different distances. When each team felt they had enough data, John placed a hotel napkin on the sidewalk. “It must hit this target!” Teams measured the distance to the target, then decided how to set the angle and force of the launcher. No more practice runs - each team had one chance to get it right. See photos below of classmates crunching the numbers, then trying to hit the square.

There were a few different targets. Both teams were able to hit the napkin, but the small soup cup John put out was too tough.

It helped to experience these challenges as a learner. Everyone was engaged in the task and we had to really use the math and rely on it to be successful. A key point: hitting a target at a given (initially unknown) distance is very different than simply sending a rocket as far as it can go. The math ended up being the centerpiece of the activity, and the whole thing was really FUN.

We also tried to determine the minimum surface area for an effective parachute, create our own timer (pendulums, a hole in a water bottle), and decide how to predict if a given object would sink or float.

I have a new sense of how to use all four letters in STEM in a meaningful way, and how to give students greater ownership of their learning.

We should all try to hit the target!

Organizing Information

New York Public Library

Max Ray’s talk at NCSM was about a math lesson he taught twice. The first time things didn’t go as well as he’d hoped. Max had the opportunity to reflect with his collaborators afterward. Luckily, they were going to teach the same lesson again to another group of students the same day. They decided what they wanted to do differently based on the results from the first group.

Here is the excellent write-up of that lesson by Max. The second go at the lesson is one in which the teachers respond to students who need support by modeling an organizational strategy. They actively avoid giving students hints about how they might mathematize the problem situation they’ve been given. It is no surprise that Max has done a nice job organizing the student-teacher-student volley into tables in his article. The outcomes in the second lesson are superior.

The ability to organize information is an important skill. When I observe students engaging in math problem-solving, often their success hinges on if and how they organize their work. Organizing information is also the key to great writing, troubleshooting, designing websites and databases, waiting tables, using spreadsheets, teaching, talking, and many other things. Here is Rob Lamb’s Collaborative Infographics for Science Literacy site, which also relates.

If you are a teacher, test it. See how students respond if you honor their ideas and teach only organizational strategies.

The Importance of Curiosity

One student was curious about tornadoes. Photo from NPR.

I look forward to getting the Marshall Memo via email. This week I found myself reading an Edutopia article entitled “Curious Homework: An Inquiry Project for Students and Parents” by Suzie Boss. It emphasizes the importance of curiosity in life and education, and laments the fact that students don’t always find a way to express their callings and passions in the context of school.

In the article, we learn about a teacher named Scot Hoffman who started his work on curiosity by bringing in folks with interesting careers or passions - a mountaineer, a doctor - and interviewing them in front of the students. Then it was the students’ turn to explore their own sense of curiosity. It would be fun to try this in the Burlington community, let students choose and interview folks, and partner with community members in addition to parents in order to pursue their own interests.

Vermont Math Leadership Council

the view from my porch

Vermont has a new education organization - Vermont Math Leadership Council - thanks to Julie Conrad, Tracy Watterson and others for getting it off the ground. Thanks to me we have a blog with the easy URL vermontmathleadership.org. Thanks to Tim Whiteford, there is a bunch of great posts on the blog! Check it out. We’d like to get more material from more voices on this blog and increase our readership. Anyone can join the organization. Our next meeting is January 29, 2014.

I was about to post something here on my blog about the new Institute of Education Sciences Educator’s Practice Guide called Teaching Math to Young Children. But then I saw that Tim beat me to it by posting about the very same thing on the VMLC blog. So, read his post here.

What I like about this publication is that it helps both rookies and veterans understand how best to work with young children and math. There are clear, specific recommendations that are easy to follow and a selection of easily usable games and ideas that illustrate the concept.

Whereas we often encounter activities or lessons and need to figure out what math we’re really going for, this resource communicates the math goals clearly and first, then hands us the activity to use.

Letting Go

By Mica Angela Hendricks and daughter

My dear friend, Rachel Wynne, kindly emailed me the link I am about to share. It’s a blog post called “Collaborating with a Four-Year Old”. I just love it. It’s about a mom and her young daughter creating artwork together. The pictures are priceless, and the ideas about sharing and collaboration are profound.

“Those things you hold so dear cannot change and grow and expand unless you loosen your grip on them a little.” - Mica Angela Hendricks, The Busy Mockingbird

Shrub

Edible Wasatch 2013, photo by David Vogel

My brother, David Vogel, and his partner, Rachel Hodson, are the owners and editors of a magazine called Edible Wasatch. It’s about local food in Utah, but I can’t wait for it to arrive in my mailbox in Vermont each season. It’s always full of gorgeous photos, great articles, and interesting recipes.

Shrub is the name of a syrup made from fruit, sugar and vinegar. I found it in the summer 2013 issue of Edible Wasatch I just received. You can add a small amount of shrub to seltzer water to make tasty, homemade soda kids will love.

Teachers and parents: Try making shrub with your kids. With this activity, you can combine wellness, local food appreciation, culinary skills, and math. What could be better? Then reward yourself by using some of the kids’ leftover shrub to mix up a delicious summer cocktail.

Strawberries have been so perfect around here lately, I think I would make shrub with those. However, you can use any fresh, ripe berries or other fruits. I have raspberries growing at my house now, so I might combine those with some local strawberries.

Here’s the recipe, straight from Edible Wasatch:

Basic Shrub

One part fruit or berries

One part sugar

One part vinegar

Wash and cut up the fruit and put into a bowl or a jar large enough to contain all of your ingredients. Add the sugar and stir or mash it with the fruit until it’s evenly distributed. Cover with cheese cloth or a kitchen towel to protect it from fruit flies and leave it to macerate - at room temperature for a few hours or overnight - until the fruit has broken down and become juicy.

Add the vinegar to the fruit and sugar mixture - apple cider vinegar or any kind of wine vinegar, not distilled vinegar. Stir or shake until the sugar is completely dissolved. Strain off the liquid and discard the solids. Store your shrub in a bottle in the refrigerator and enjoy for up to a year.

See their original article and recipe here.

Ideas for teachers and parents

Let kids do all the work, including deciding which fruits or berries are in season and would work well, picking or buying those, making the shrub, and making the soda. They could even sell it in the neighborhood instead of lemonade and hand out the recipe to customers.

Here’s some math. Ask kids things like...

• What does “part” mean in this recipe?
• How do we know how much of an ingredient to use when “one part” is called for?
• How is this measurement different than other kinds of measurements we’ve seen in recipes (teaspoons, cups, etc.)?
• How should we decide how much shrub to make?
• How much fruit, sugar, and vinegar would we need if we wanted to double the amount next time we make it?
• What would we do if we wanted to try making shrub that was less sweet?
• Have you ever seen another recipe that called for “parts”? How could you figure out how much you need for “parts” when you know the total amount of the recipe you desire?

Other possible questions

• If we use a few teaspoons of shrub to make homemade soda, about how much sugar will one glass of soda contain?
• How does our homemade soda compare to a commercial soda? Amount of sugar? Type of sugar? Other ingredients? Taste? Color? Cost? Environmental and economic impact?
• What does “macerate” mean?
• What other ways could we use shrub besides making soda?
• Why can shrub stay in the fridge for so long without spoiling?
• What questions do you have?