People joke with me and say, “You must do calculations in your head for fun.” My instinct is to deny this accusation, but it probably is true. Math educators run through numbers all the time. We play numbers games in our head like would you rather, without prompt or encouragement to do so. I see a fraction that could be written in different way, so I try. I see a complex shape that can be broken into parts, so I do. These instincts were fueled by years or even decades of me talking numbers in my head.
I should never assume that this is normal, even though it is one of my life’s missions to make it so for the general population. The CCSS Standards for Mathematical Practice says to make sense of problems and persevere in solving them (among other things). This idea of having a sense for numbers has been discussed by some awesome educators like Fawn Nguyen, and Sadie Estrella, and it’s been the hot topic of journalists and bloggers alike. I’m in the camp that says, “Feeling comfortable with manipulation of numbers directly supports comfort with manipulating the abstract.”
So today we were talking circles, and proportions of circles. Instead of just giving a formula…
…we developed our own.
It started with a spreadsheet.
then a geogebra applet
then into paper practice.
Students shouldn’t be given a formula, and then be expected to make sense of the abstract values without having developed a sense for the concrete values.
Math educators need to talk numbers with students daily. It just makes sense.
In #slowchated a while ago the topic was on Change. Inspiring change, cultivating change, and the purpose of change. I remember going through a course on leadership and psychology that was focused on the text Change: Principles of Problem Formation and Problem Resolution. The discussions often overlapped with the concept of thinking outside the box in order for change to be possible.
Example from parenting
(not from experience, no kids yet):
Son keeps locking himself in room to try and avoid interacting with family. Parents want their son to interact more and stop being so evasive.
Remove the locks.
The answer is simple, but it breaks a rule that wasn’t even a rule. There’s a lock on the door. It was already there, so it must be a requirement.
I feel like this is how educators and learners get stuck. That’s how it was, so that’s how it should be, and that’s how it will continue to be. The educator I started out being is only some arbitrary portion of the one I am today. How did that happen? I’m pretty sure I didn’t just do what everybody else was doing. I also did do the same thing that I did the day before. Often times I had an inclination that a lesson could have gone a different way, an activity or assessment could have been more authentic, or there may be an alternative to what I had tried in the class.
What are we so afraid of?
Why don’t we try new things or different things in the class? Maybe it’s a matter of effort and exhaustion. Maybe it’s discomfort with what others may see as failure. Maybe we would rather be safe than sorry because the development of the youth in the class is at stake. I don’t know what it is, but I’m more afraid of thinking that this is how things should be for the rest of my life and I might as well get comfortable with it.
Too Much Change?
Is it possible that change actually turns into chaos? I would say yes. In fact, this year has probably felt more like chaos than progress. That is, until I start to reflect on what has been accomplished through interactions with others. Blogging, tweeting, GHOs, and meet-ups have been huge stabilizers for me. Sharing experiences with others and learning from others experiences supports taking these leaps of change.
Where do I start?
Write it down. Talk about it. Be social about it. Journal. Locking yourself up in a room for 7+ hours a day won’t get you to change. Break the locks and start looking for others that are trying to break out as well. Start small or jump in, but however it may be:
My brother was in a band called “Sounds Familiar” back in high school. They played up the idea that no one should forget their name.
My dad posed the question, “Isn’t it possible that we can run out of songs. Don’t they eventually sound familiar?”
VSauce breaks down the countable differences in music with a video:
(in which he mentions a website to compare familiar sounds)
And TED.com recently highlighted a talk discussing the culture shifts in sharing and sampling sound from one another.
These reminded me of a question on counting and probability. If you haven’t already, let yourself get distracted with Incredibox. You won’t be disappointed. How many unique songs can one make with it? I think this would be a great project for a math class discussing counting principles. Let the students determine parameters for uniqueness. Maybe there can be more than one level of uniqueness (same beat+different melody <different beat+different melody). I’m laying this down, #MTBoS. Who’s up for the challenge to break it down?
The are only a few givens in geometry, building on those we derive many other patterns theorems. One theorem I often prefer to focus on visually is the sum of the interior angles of a convex polygon. There’s different ways to approach the process, and most of them refer to creating a fan of triangles inside the polygon:
Early on in most geometry studies, we learn that the sum of the interior angles of a triangle is equal to 180, building off of this we can use the above animation to calculate the sum of the convex hexagon: (180)(4)=720.
There’s often the student who wants to divide the hexagon up differently. The student wants to draw lines criss crossing all over the shape.
Too often I would have totally passed up this opportunity and said, “That’s not how we do it, so it can’t work.”
Taking The Long Way isn’t an Error:
Thankfully, this year we saw something different. There were still triangles inside, and with a few more lines added, the picture had only triangles inside.
Once we have all triangles, we can just count those and multiply by 180.
(16)(180)=2880. That can’t be right, can it?
From here we get a better definition for what is an interior angle of a hexagon. It’s an angle inside the hexagon, but still attached to the edge of the shape. Somehow we need to get rid of the angles that are inside the hexagon but not touching the edge of the shape, like these red circles:
There are 6 of these circle sets of angles, each of them having a value of 360 degrees. We just need to discount those (aka subtract).
This kind of process also builds a more flexible understanding of how to decompose a geometric shape in multiple ways. I encourage all math teachers out there to try this next time a student chooses to slice it up however he/she wants.
It’s Thursday. Yesterday we talked about circles, chords, and kites. Today we asked a very similar question:
Here’s the applet in action: (click the animation to open the applet)
This was similar to Wednesday. We asked the same type of questions. We saw similar relationships. But it stood out enough for a unique post because a student enlightened me with an observation I didn’t articulate. When I repeated yesterday’s question, “What shapes do you see inside the triangle?” one student almost immediately replied, “Is that a kite?” I had to look at it myself. “Yeah, wow, that makes this conversation easy.” My original plan was:
- Focus on the Right angles/right triangles
- Question if certain segments were congruent
- Look at the reflection or congruence theorem that helps confirm the congruence
- then finish off with some color coding.
Instead this students recognizes the 3 kites, then refers to her knowledge of the symmetry in kites. Congruence, simplified.
To help students in transcribing the diagram onto paper to start doing some hand calculations we took a tip from a student in the first class:
- Lay Chromebook on its back
- Increase brightness of screen, turn off some/all light in class
This was another good introduction and discussion with segments in circles. We of course spent the last 1/3 or so of class practicing and becoming fluent with the skill with problems like:
At the end of the week we took a short quiz. 2 questions, same as these. Today/tomorrow I’m going to try something new with how I grade them (thanks to some inspiration from Michael Fenton, Michael Pershan, and Ashli Black). More on that later this weekend.
In the meantime, let’s keep rounding out this circle thing and see what other shenanigans we can come up with.
In geometry we had some fun with kite like shapes for the last couple of days. On Wednesday we asked:
See it in Action (click image to open)
I asked the students to play with the applet. I prompted them to ask for more information. Some noticed the sample questions below the applet and asked those to start the discussion. The key question had to do with decomposing the shape into other shapes that we had more familiar tools to work with. The students were quick to see the right triangles. Once we were able to identify that, the next question was, “How does that help me?” Getting students to connect and then apply relationships they know into a seemingly new context is a constant challenge, but they are getting better. I reminded them to recognize what are we trying to find. Once they narrowed in on the task of finding a length, and the length was a part of a right triangle, some started to see it, “Pythag Thyrem.” (this seems to be a tongue twister for the general high school math population). “Okay, how do you mean?” was my reply. Some didn’t see the given values for the hypotenuse that was also a radius.
Circles seem to be a pain for many geometry teachers, and I feel that it’s because so many people approach circles as a never ending list of formulas. We need to find ways to simplify the overall question and give students an opportunity to fill in the structure(s) needed to respond to the question. I know there are some awesome activities out there dealing with volume and area with round objects. Here I’m trying to put together a series of interactive questions the see the overlaps and relationships with circles, segments and angles.
This is Anthony (aka best bud):
We were playing this:
And we were talking about these:
We focused on SMP 1 (making sense and persevering) as well as 7 (find /use structure). Mostly 7. Anthony and I are from the Sega and Nintendo generation. Beyond a tech ninja, volleyball all-star, and just an awesome guy, Anthony is a video game connoisseur in its truest form. There have been many a late night where we would stay up around the context of a video game, but it’s not just because we have to get that next achievement, or finish the level. Instead it’s the conversation.
I always enjoy our discussions that break down the elements of engagement and strategy of a game. As he and I were reminiscing on the design of a game as old as DuckTales, his wife Maira simply commented, “I don’t see it.” I immediately pictured the student in the classroom with the blank stare that says, “I don’t get it.” Learning is about structure. Those that can master the game identify and then manipulate this structure. We noted how a game often has predefined mechanics, and it is up to the user(s) to learn and then apply said mechanics. If you are in the world of education you should be seeing the correlation at this point. I’m not saying that video games = learning. What I am saying is the design structures built into gaming could teach us a lot about engagement and learning. One could find plenty of discussion on this following the twitter feed #gblchat or #gamefication.
Anthony and I discussed how clever design of an interactive experience allows the user to identify relationships of objects, and then progressively learn more about such relationships in order to use them toward advancement. One of the games that is entirely dependent on this learning process is Portal. A player has to identify how to use a simple set of tools in varied combinations to accomplish tasks. Questions like,
- Does it matter the order in which I travel through these rooms?
- Is there another way I can use this tool?
- What will happen if I ______?
- Why is that platform there when clearly I can’t get to it right now?
- I can see an item through the window, but can’t reach it yet. When will I come back to it?
- Why do I need to get that item?
- Can I use the items in combination such that an entirely new outcome is possible?
These are questions that a user intuitively asks and rarely articulates. Such analyses happen so fast that most hardly bother to even out words to the thoughts. The amateur simply play point and shoot while looking at the flashy colors and listening to the cool sounds. In similar fashion the amateur student just comes with the simple tools and plays the game of school on Easy. He/she might maintain low scores along the way but it doesn’t matter because you can often reset the level or maybe even purchase a power up for $.99.
The gamer/student progressing in his/her skill and performance takes advantage of the clues hidden within the game and exploits them. The learning experience appears more like a puzzle worth solving, and the experience becomes self guided at times. Those that excel in school aren’t better at memorizing formulas and passages. The strong student is the one that can identify structure and use it.
For some teachers and learners though, this game of school is out of date, too simple to play, and tedious like no other. We as educators need to take a bigger interest in the design of our game. Not just the standards and outcomes, but the structure that guides the path of the user. So where do we start? I suggest following some tips from Dan Meyer.
Really, you should. GO now and read it.
— Joseph Williams (@jswilliams) May 1, 2014
I enjoy math, thoroughly. I also enjoy design and technology. Recently I’ve had a wildfire like experience in processing and learning material through Geogebra and Desmos. I’ve been learning and experimenting with these much faster than I could possibly archive organize the material. I look forward to the upcoming extended break at summer to truly polish this material. Currently I’m feeling more like
This little presentation I had tonight was a breath of fresh air. It was a mixed crowd and we had a great time. About half mathies in the room, and half techies. It wasn’t a large crowd, so we had a casual, yet productive time. At around 2/3 of the way through the time allotted, one of those in the room inquired about the coding behind some of the applets. Others seconded the question, so then we transitioned from math to tech.
Technology geeks, myself included, often dive into the code and lose some of the social part of the experience. Working on the backend of a program experience too often is a lonely one. Talking about this experience in a real life, social platform was great for myself and them. I was able to reflect and process on my wildfire experience of learning, and from what it seems they were about to start their own versions of something similar.
On the other end of the screen, it all looks so easy.
Geogebra and Desmos both use clean user interfaces that allow for wide audience. Knowing some of the coding and design on behind the scenes still has it’s place though. Recently education has seen growth in the art of coding. It has become more accessible with drag and drops like scratch, tutorials and screencasts shared freely online, and organized movements from large institutions like Khan Academy.
Now I would like to throw another idea into the mix of developing a coding mindset. For those of you that know me, you could probably guess that I’m thinking of Geogebra (and Desmos as well). Tonight we talked about the object oriented code experience it offers, and it’s simplicity with design and interaction. To toggle a picture or make an object move, all you need is a slider control and some checkboxes. I’m not sure where this can go from here, but I like it.
If you haven’t already you should check out these online interactive tools. And when you do, look at them as tools for tech and coding, not just math.
End of Line
PS: bonus points if you get the geeky references.
The Battle Begins
So I tried something, and some of it worked. We did an entire math unit, without paper.
So how do you do math, without paper?
It wasn’t easy. First we needed the technology platform to work with. My district has gone 1:1 with Chromebooks in a program we call #ScholarPlus. Using the Chromebooks in the classroom has been awesome, especially when it comes to visualization and assessment. We can show graphs, diagrams, geometrical shapes, 3D perspectives, real time effects of changing numbers. It can and does get really fun. There are some major downsides as well so let’s break down the basics.
The following are some excerpts of common comments I heard from students throughout the last 3-4 weeks.
Paper beats Computer:
My battery’s dead. I forgot my chromebook. The network crashed. My computer is acting slow. Where is the document again? Where do I click? How is this math? I miss paper, I learn better that way. My computer , is going, sooooooo, s l o w.
Computer beats paper:
That was easy. I can see it. Can I add this to it? So you mean I don’t have to turn it in? What do you mean I can’t lose my copy…it’s always there when I need it? Can my parents see this? How will I be graded? I’m not a good artist, does that matter?
The biggest advantage to using the technology, discussion opened up. I had high level comments from students on congruence, shape transformation, making sense of values with number talks and problem solving methods I hadn’t even come up with. The content being pre loaded and interactive allowed us to focus on this discussion time.
Start with the interactive text. We took a unit created by another school district and turned it into a Google document. This required a lot of screenshots for pictures, copy and paste for text, and tables tables tables to keep things aligned and organized. Here is a link to the actual online document we used.
Repeat the success
- Google Drawings: These worked awesome in getting the students to accurately draw lines, decompose figures, and label appropriately.
- Doctopus and Goobric: Probably the best document delivery, collection, assessment, and overall management system to help organize the workflow of digital content.
- Geogebra Applets (examples: 1 , 2, 3): These weren’t embedded into document. That needs to change, but incorporating them into our discussion was powerful. Also allowed students to use as instant feedback device. If the applet disagreed with me I need to check my work.
- Desmos: for graphing data from investigations (look at rectangles with fixed perimeter section).
- Lot’s of coordinate plane examples. This needs to happen more in Geometry type of units.
- Practice using an equation editor. Preparing for tech like questions on SBAC and other CCSS computer based assessments.
- Use headings for Table of Contents in Google Document
- Have students reflect with written responses.
Learn from the struggles
- Shorten and Simplify. Don’t rely on long explicit directions. If document is getting too long (too many pages), split into smaller chunks
- Practice Paper for fluency. Digital was great as a curated portfolio like experience. Pencil and paper is still preferred for practice and repetition.
- Build in quick assessment opportunities into document for student to explore and get own feedback.
- Find something else to do besides staring at a screen for more than 20-30 minutes. More discussion and activity outside of seat/desk whenever possible.
- Start every day with starting up device and opening up document page (hardware and network sometimes need a minute to get functional).
- If your going to do constructions, like a perpendicular line, just use Geogebra with shortcut commands. The conceptual understanding does not require compass and straight edge style of work.
- Incorporate Parallel and Perpendicular relationships of lines earlier and more frequently.
- Discussions are great, archiving them somehow (reflections with feedback built in) would be even better.
This experience has taught me a lot, but I think the main ideas that hit me square face every day of the unit are balance, preparation, and assessment. In terms of balance, digital tech needs to be near 50/50 with non-tech experiences. It get’s overwhelming and students get disconnected. Preparation, like for any solid learning experience, is much higher when technology is involved. We need to get the kinks out first. For assessment I’m now focusing on more than just mathematical understandings. I’m grading for using tools appropriately, precision and fluency, and articulation. My best brainstorm so far is to have a rubric for digital notes with categories: Precision, Process, and Problem Solving.
This isn’t my last thougths on the paper(less) debate, and I look forward to hearing other perspectives too.
Now it’s your turn:
- Do you teach math? If not, what do you teach?
- Have you gone paperless? What did you learn in the process?
- What do you wonder about going paperless?
- How would you assess the experience?