Friday, November 28, 2014

Video Error Analysis (Anti-Khan style)

Something I tweeted this week:
Crystal (colleague) and Lynda (fellow) wanted to know more about this. So here's the story:

The previous week, I met with one of my high school fellows who teaches Algebra to freshman. As with all my fellows, it's been an extreme pleasure to work with her because she's hungry for ideas and will take suggestions and run with them. It was so cool to walk into her class this past week and see her running with an idea, again.

She had already taught her students ways to solve linear systems; graphically, substitution, elimination, etc. On this day, she prepared six short videos of her solving linear systems and linear inequalities using Educreations on her iPad. Students were to watch the videos and do error analysis, reporting the following on their handout:
  1. Identify the mistake(s) for each question.
  2. Explain what should have been done.
  3. Fix the mistake and complete the question correctly.
Each video was between 60 and 90 seconds in length. We both discussed what we thought would be most effective for her students and short videos was a must. Have you ever noticed how the majority of Khan videos can be extremely lengthy? Sal Khan usually talks (and repeats himself) while writing things on his digital blackboard. To me, that's a waste of someone's time. It's like you watching me type this blog post while I reread every sentence two or three times, stalling so I can finish typing. Another thing I can't stomach in Khan videos is when he fumbles around searching for colors to write with. Lastly, I find it unfortunate that the videos rarely suggest the viewer to pause and consider what's happening. Here's an example. Sorry, here's a 9+ minute example:

I suggested my fellow pause the recordings often and write the equations "offscreen" when not recording. Then, press record again when she's ready to talk and/or write something important on her screen. She also took advantage of this offscreen time to select different colors in order to emphasize different equations, steps, lines, or shading (linear inequalities).
*See the video structure below with suggested notes and style points.

It took my fellow one prep period on a minimum day to create six videos, a supplemental handout, upload the videos to Educreations, and create hyperlinks on her Haiku page for students to access all the videos. That's super impressive. Talk about an activity with meaningful and HUGE return from an efficient investment in her prep time.

When debriefing with my fellow after class, she was completely ecstatic.
I asked her, "What elements made this awesome?"
She replied:
  • it was video and new
  • they liked figuring out someone else's mistake
  • the videos were short
  • students could pause, rewind, and start the video over
  • using Desmos to show a graph of the original equations at the end (comparison)
  • gave students the idea to use Desmos to check their work/answer
  • self-pacing
  • very little hand-raising or students drowning
  • the videos were easy to make
  • she passed out the handout and said "go" instead of modeling
  • the handout had a simple structure 
  • the students did most of work, not the teacher
I love this last element the most. The two of us talked about this specific element the previous week. Now she experienced it first-hand and it's an amazing feeling. As an observer, it was awesome to see the students working hard on a meaningful task and helping each other out so it allows her, the teacher, opportunities to calmly circulate and provide support where necessary.

Student engagement and interest were high. Discussions were plenty and authentic. Students were thriving using thinking skills in the "Analyzing" category of Bloom's Taxonomy or Strategic Thinking category of Webb's Depth of Knowledge. Here's a tip I suggested when I noticed some kids plowing through a video and hadn't caught the mistake: pause and make predictions. The video structure will explain pausing and predicting more.

Video Structure:
Part 1: She takes about 8 seconds to explain her plan
*All of this was written on the screen prior to her pressing record. Style points.

Part 2: Multiply the top equation by (-5) in order to eliminate the x-terms

*Here's where we need to ask students to pause and predict what the top equation will look like after being multiplied by (-5).
  • Model this for students.
  • Build "pause and predict" prompts into the video. 
  • Circulate the room and ask students to pause and predict.
SO valuable. Don't skip "pause and predict".

Part 3: Write the new equations "offscreen". Don't record yourself writing these equations.
*Notice the new equation is written in red inkStyle points!
**Pause and predict what it will look like when combining the equations
***Catch the mistake?

Part 4: Combine the two equations.
*Another great use of "offscreen" writing.

Part 5: Find the value of y.

Part 6: Substitute the value of y into one of the original equations.
*Yet, another use of "offscreen" writing here.

Part 7: Solve for x this time.
*Ask your students to check for reasonableness.
**Find an alternate way to validate (or invalidate) their conclusion.

Part 8: Insert a screenshot of the system graphed in Desmos.
*Mind grenade: the graph doesn't match the algebraic procedure.
**HUGE style points by inserting a visual representation of the correct answer.

For those of you who don't have 1:1 devices in your schools, no sweat. I still recommend you make a video of some sort. Borrow an iPad from someone. Create an Educreations video for error analysis. Use the tips and techniques mentioned here. Your videos should be less than 90 seconds. Play it to your class. Pause the video to have students make predictions and/or discuss possible errors. I guarantee you, good things will happen.

Style points,

Tuesday, November 11, 2014

#PuzzleMath ideas

Tonight, my son wanted me to work with him on his new puzzle.

I don't know your strategy for doing puzzles, but I find all the corners first and then start putting the border together before I start working on the inside. Look at that box again. Would you be able to determine the dimensions (in puzzle pieces) of this puzzle by knowing the total number of pieces?

That was my first question:
A) If you know the total number of puzzle pieces, could you think of the all the possible dimensions (in puzzle pieces) of the puzzle?

This puzzle will either be a 1x35 or 5x7.

Then came the next question:
B) Estimate the actual dimensions (in puzzle pieces) given the picture on the box?

I'm going to go with 5x7 because five and seven are the only factors of 35 that would reasonably make the rectangular picture on the front of the box. The puzzle should be 5 pieces high and 7 pieces across from left to right.

With a box of 35 pieces, these questions aren't too ridiculously challenging. However, I know there are crazier puzzles out there in the world; puzzle with 500, 750, 1000 pieces, etc. That's where I called on Twitter to help out. Like a champ, the #MTBoS came through and hopefully will continue to come through with #puzzlemath.

Below are some of the tweets I received followed by additional math questions I'm curious about.

Additional Questions:
C) If you think you know the dimensions, could you determine:
  • The number of corner pieces
  • The number of border (non-corner) pieces
  • The number of inside pieces
If so, could you write a rule for any of these?

D) Knowing these quantities, say you randomly choose a puzzle piece, what are the chances it's:
  • A corner piece
  • A border (non-corner piece)
  • An inside piece
  • The exact center (if possible) piece
Please add to the collection of puzzles and questions by tweeting with the hashtag #puzzlemath.


Roofs Are Expensive!

Today is Veteran's Day. If you personally know a veteran, say "thank you" to them somehow: call, email, text, or in person. If you don't know a veteran, then ask a friend if they know a veteran and if they do, ask them to say "thank you" on your behalf.

Because today is Veteran's day, my district isn't in service today. Therefore, I was able to sleep in a few minutes longer today. I woke up to the sound of this pitter-pattering on the roof of our house. Not five minutes later, my 4.5 year-old son walks into the room and says he hears something that woke him up. I wonder what? Ha.

He climbs in bed with me and I ask him if he knows what the sound is. I proceed to tell him the sounds he hears are birds walking around on the roof hitting the roof with their beaks. When we moved into our house, there was a fake owl tied to our roof. Supposedly it keeps birds away. Not too sure it's all that effective, but we keep it up to have an occasional laugh.

My son thinks that the birds might start to ruin the roof and that we'll have to replace it.
Son: That will probably cost more than $100 to fix.
Me: How much more?
Son: Probably like one-hundred, one thousand dollars.
If you read my Pumpkin Seeds post, you know that his vocabulary includes a thousand now which is the biggest number he currently "knows". I use "knows" very loosely.
Me: Oh, I see. Well, which number is bigger? One hundred or one thousand?
Son: One thousand!
Me: Right. So we say the bigger number first, like this, "One thousand, one hundred."
Son (whispering to himself): One thousand. One hundred.
Me: That's a lot of money.
Son: Maybe one hundred twenty?
Me: Oh, so that's smaller than one thousand, one hundred.
Me: How do you know?
Son: Well, we have a big roof. We would need a lot of wood. We'd have to go to Home Depot and get all the things. 
Me: So we can buy all the things from Home Depot?
Son: Yes.
Me: And then would we put the roof on by ourselves or have some worker-men do it for us.
Son: You and I can do it Dad.
It dawns on me. Geez, I hope it's not $10,000. That's a lot of money. At the same time, I'm glad my son doesn't know the concept/magnitude of "A million" yet. Regardless, I have no idea what a new roof might cost, let alone how to put a new roof on. But maybe I'll save a lot of money on the labor because I have a 4.5 year-old son extremely willing to climb on top of our house and help install a new roof.


P.S. I'm enjoying these conversations and posting about them.
Today felt like Mathematical Practice 4 with my son.

Monday, November 10, 2014

An Amusing Context for Zero

From the mouths of babes:

Tonight, my 4.5 year old son was getting ready for bed. Getting ready for bed means he needs to shower. And as any parent can relate, there's the usual banter of "get your jammies out" to "let's brush your teeth." Tonight went in this direction:
Me: C'mon. Let's get ready for your shower. I have three clothes on still.
He comes around the corner.
Son: Dad, I have zero clothes on still. 
Yes. Yes, he did. He was, indeed, ready for shower. Bahahahaha.

Like I said, from the mouths of babes: an amusing context for zero.


Saturday, November 1, 2014

Pumpkin Seeds

For Halloween, we gutted the larger pumpkin from Day 197 for two reasons:
1) To carve it and
2) To roast the seeds
*Little did I know this pumpkin would produce many number sense opportunities beyond Day 197's weight estimate.

My 4.5 year-old son helped gut the pumpkin with me. We gathered as many seeds as possible and set them aside for the roasting process. The recipe I use can be found here. Before mixing the seeds in olive oil and seasonings, I lowered the pan down to my son and asked how many seeds he thought there were. [How many do you think there are?]
Me: How many seeds do you think there are?
Son: Well, ...[thinking for a few seconds]... there's definitely more than a hundred.
Me: How do you know it's more than a hundred?
Son: It just looks like more than a hundred.
Me: Make me a pile of seeds that would be about 100. 
He takes his hands and makes a pile toward the bottom of the pan. I can't contain my excitement to see what he has done.

His pile actually makes sense to me and definitely looks close to 100, give or take. Now that we've made a pile of his "one hundred" seeds. I ask:
Me: How many piles of 100 seeds do you think we can get?
I see he's a little confused by my question, so I ask it differently.
Me: How many piles do you think we could make where each pile has 100 seeds?
He starts to think, but by this time, my almost two year-old daughter sees the pan of seeds is now at her level, so she comes running over and wants in on the action of moving seeds. I thought our conversation was derailed as I brought the pan back up to counter height. But my son keeps it going. He didn't necessarily attempt to answer my question (which is fine by me). He took it in a different, pleasantly unexpected direction:
Son: What if we could get 10 piles?
Me: Do you mean ten piles of one hundred?
Son: Yes!
Me: Are you asking me how many seeds we would have if we had ten piles of 100?
Son: Yes.
This blew me away. Where'd this come from?
Me: Well, ten piles of one hundred seeds would mean we have a thousand seeds.
Son: A thousand??!!
Me: Yes.
Son: What if we had twenty?
Me: Twenty piles of 100 seeds?
Son: Yes.
Me: Well, what two numbers make twenty? Ten and...
Son: Ten!
Me: Right. So if ten piles make a thousand and another ten piles make a thousand, we would now have two th...
Son: ...ousand! 
Me: Right.
Son: What's 100 piles? 
Me: You mean, what's one hundred piles of one hundred? 
Son: Yea.
Me: That would be ten thousand.
My wife chimes in.
Wife: No. 
Me: One hundred piles of one hundred? Ten thousand.
Wife: Wait. You're right.
And our kids are now off in the other room playing and getting ready for going outside and looking at Halloween decorations before trick or treating. So many cool things happened:
  1. I love how we pursued my son's questions and not mine. 
  2. I love that he was fascinated by the word "thousand" even if it didn't make sense.
    1. Because he thinks a "hundred" is big.
  3. I love how we started with actual seeds (concrete) and went way more abstract.
  4. I love how my wife had the best estimate in the house [375 seeds].
  5. I love how the "pumpkin seeds" estimation challenge is just large enough so you can't accurately eyeball the amount and it's just small enough where it wouldn't take me an absurd amount of time to accurately count.
My wife and son watched the video answer this morning. My wife's response:
This is perfect for my [1st grade] students!

We paused the video when we saw ten groups of ten and paused it again when I made a larger pile of one hundred seeds. He struggled with making sense of final amount. 
Me: How would you say that number?
Son: Forty-eight eight? 
Me: How many piles of one hundred are there? 
Son: Four.
Me: So we say "four hundred" and let's count the piles of ten together. Ten, twenty,..
Together: thirty, forty, fifty, sixty, seventy, eighty,  
Me: And. Wait. There are only eight in this last group. So we say eighty-eight. Four hundred eighty eight. Say that.
Son: Four hundred eighty eight.
Me: Great. Let's go make breakfast.
Here's the recipe again and some pictures of the process.
Ready for the oven.
Roasted and ready to eat!

P.S. Thanks to Christopher Danielson and all he does at #tmwyk!