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2016 Blogging Initiative

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I am participating in the 2016 MTBoS Blogging Initiative.  I am doing this in part in to open my classroom up and share my thoughts with other teachers. I hope to accomplish this goal by participating in the January Blogging Initiation hosted by Explore MTBoS

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I’ve just dusted off my “About Me” page to include the schools at which I’ve taught, and I’m excited for the next month.  You, too, could join in on this exciting adventure. All you have to do is dust off your blog and get ready for the first prompt to arrive January 10th!

Lies and Collaboration

I actually enjoy the puzzle-like aspect of exponent rules, and simplifying radicals.  For me, there is something satisfying about learning ways to manipulate numbers and letters – probably why I love algebra so much.  But I am tuned-in enough to my students to know that many of them don’t find the same satisfaction from doing this work just for the sake of intellectual exercise.   And since calculators came into vogue, it’s been harder to justify the need for rationalizing the denominator or expressing the square root of 50 as 5 times the square root of 2.  But we are tasked to follow standards that often include these kinds of skills and it has been helpful for me to turn this into exploration or game learning as much as possible.

IMG_4135I did some mining of the MTBoS for ideas to teach rules of exponent arithmetic and came across this post, which includes a nice exploratory worksheet from Andrew Stadel.  He describes a similar issue with contextualizing exponent rules for middle schoolers – one of the really great things about our online community are these moments where we are reminded that we are not alone.  He asks his students to find the mistakes in the equations, to explain where the author went wrong, and to find the correct solutions.  He used a bunch of the common misconceptions found on mathmistakes.org to help students to catch themselves in the common errors.  Very nicely done.  This would have been a good lesson as is.

IMG_4147Then I remembered the Bucket O Lies protocol from Nora Oswald at Simplify With Me.  Nora manages to gamify math like no one else that I’ve seen.  She manages to add entertainment even to potentially dry topics like this one.  I combined Andrew’s worksheet with Nora’s idea to make a bucket-o-exponent lies.  I printed the 3 worksheets, cut them out into individual problems, folded them up, and put them into buckets (or baskets).  Voila! Drama and Motivation.  In pairs or threes, learning happened.

IMG_2885Of course, I hammed it up with the students.  There’s nothing like telling teenagers that someone is trying to get one over on them to motivate them.  This has worked well for me in the past, especially when it came from advertisements.  I riled them up by acting outraged that someone had created this whole set of math material, which was full of mistakes!  (Actually, I blamed Andrew :) ) …Lies I tell you… these baskets are FULL OF LIES!  Let’s find the mistakes so we can write a self-righteous set of corrections back to this author who was deliberately spreading bad math.

They quickly saw through my act, but it was enough.  They were already motivated in spite of themselves.  Andrew’s worksheet was just enough for everyone.  I started by coaching the groups who needed help getting started and moved to pairs who were making mistakes with fractional exponents.  For my honors group, I added a few more examples with rational exponents.

Thanks Andrew!  Thanks Nora!  Our generous community is the Best!

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Revealing Learning Targets

DSC06230I am not always sure about how explicit to be about learning targets.  I have seen some convincing research, which seems to indicate that letting students know exactly what is expected of them for each lesson helps them to take ownership of their learning, and to make sure that they are getting what we think they are getting during each class session.  I agree with this practice in general, and I believe that it definitely has a positive impact on some students.  My current school, as well as the previous one have required that we post targets each day, and there are many educators who I respect who advocate for this practice.  But sometimes, I feel like a learning target can put a limit on where we can go as a class, and can feel a bit stifling, especially when we want a problem or exploration to feel open-ended.

Lately, I have adopted a practice of “Hidden Targets.”  I do post the learning target, but I often leave it covered up during class.

IMG_4121As part of our end of class routines, students make conjectures about what they think that today’s learning target was.  We reveal the target, assess how well the lesson matched the target, and whether the learning matched what was expected.  Although I think that I am good at starting class off, and generating enthusiasm, I sometimes am not as good at synthesizing and wrapping up.  Being conscious of synthesis and wrapping up class in a richer way has been one of my goals for this year, and this routine has been a good protocol for me and for my students.  It quickly reminds us about what we learned during class, and how this lesson fits in to the bigger picture.  Students have been highly engaged in figuring out the day’s learning goals; I hear students talking throughout the class period about what they think is under the flap for today – and you know that they remind me if I forget to do the “reveal!”

IMG_4119…And it doesn’t hurt that we have created this sweet Appolonian Gasket on which to showcase the day’s targets.  Who doesn’t want to stare at this and contemplate infinity?!

First Week: Building Culture

The start of the school year is one of the most important moments for my classes.  Setting the right tone and attitude right from the beginning can mean buy-in from students right away – and conversely, a bad start can be really tough to recover from.  I had a pretty good start this year in my Algebra 1 and Algebra 2 classes.  I wanted to share some things that worked for me in case someone else might benefit, and to document the week, as I may repeat much of this work next year.

My students have been working on a pseudo-Appolonian gasket on the whiteboard. It makes a nice frame for our learning targets.

My students have been working on a pseudo-Appolonian gasket on the whiteboard. It makes a nice frame for our learning targets.

I have several goals for how I want my classroom to “be,” and the first week is a chance to work on some of the big picture ways that we will be working together this year.

  • It is important to me that as a group, we celebrate scholarship – and the struggles involved in becoming scholars
  • I want to nurture a love of learning and of curiosity
  • Our classroom has to be a safe place to take chances and to make mistakes
  • We need to be able to work collaboratively – even more than in other subjects, I believe that we really need to see how others think in order to understand math
  • To that end, we need to learn to be comfortable talking (and arguing!) about math
  • We need to work independently as well, and to trust and value our own ideas
  • We need to respect each other, and hopefully to love each other at least a little.  Of course I love all of them.

I used a series of activities (all sourced from the MTBoS of course) to try to help establish this culture.

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DAY 1: What does it take to do math?

Very first thing, I assigned each student a “secret partner” for the week, based on this idea from Origins.  Students are to observe their partner throughout the week, and are responsible to report back an acknowledgement of something positive that they observed at the end of the week.  The payoff for this happens on day 5.  Next, I introduced a version of Jasmine’s Tabletop Twitter.  I set up 5 stations around the room.  Each had chart paper with a question/prompt on it.  Students moved around the room in two minute rotations, and were asked to respond silently to each question.  I followed Jasmine’s lead in asking students to take a marker and write their name with that marker, so we could look back and see who had authored each comment.  My five prompts were:

  1. Why do we learn math?
  2. What will make our math class a good learning environment?
  3. What does it take to be a good math student?
  4. Respond to this quote: “Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.” – W. S. Anglin
  5. Add a song to our class playlists. Write a genre instead of a song if you prefer.

At the last station, I gave out this Capture your thoughts organizer, and asked students to synthesize and summarize the most important points from their station, to add anything they thought was missing, and then report back to the whole group.  We hung our new “posters” in the hall on this year’s “Sweet Wall Of Math,” to help establish that our work will be public this year and we are proud to show our thinking to the world.  I’ll use the ideas they shared to create our learning agreements for the year.  For anyone who would like more detailed plans for day 1, I’ve written them up just for you: Day 1 Plans. :)

Day 2: How can we create the questions?

I am totally convinced of the positive impact that Dan Meyer’s 3-Act format can have an a group of math students, so I was excited to introduce 3-Act math tasks right away on day 2.  Students are so used to arriving in math class, and just imitating the teacher that they often don’t know how to react when they are asked to think of a question themselves, and then asked to figure out what they actually need to do to solve their question.  The first tasks like this can be really tough and even painful, often for some of the “top” students.

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The Super Bear was a nice one for both groups.  The math was easily accessible, which gave us room to learn the structure of how we should approach these kinds of tasks.  I made a new 3 Act handout for students to use, and guided them through the process.  I was strict about keeping silence in the room for the entirety of Act 1, except when they were asked to share their guesses and to establish a high/low range.  I stressed the importance of this “grappling” time, when they get to really think for themselves without the bias of hearing others’ ideas, and promised that they would get to work together for the rest of the task.  This is one of the important routines in my class, and is one of the few rules I impose on the group without their input.  Every group suggested weighing the bears, and several came up with ideas for how to measure volume (displacing in water, melting down the bears…).  Act 3 provided some rich discussion about the discrepancy between their solutions and the revealed answer, and the drama of the reveal of Act 3 can’t be beat!  Even reluctant students can’t look away as the gummy bears are weighed out.

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Day 3: Metaphors for perseverance.

We spent much of our class playing the Game About Squares.  I followed Annie Fetter’s suggestion to try this out with students, and read her post about this several times, so it was fresh in my head. She has a clarity about the importance of these tasks that I wanted to hold on to, emulate, and embody for this lesson.  This is a game that does everything that we want in our math classes.  It meets kids where they are, and little by little gives them slightly more challenging tasks to accomplish.  I (mostly)refused to help them at all, but made it clear that they were expected to figure out what to do.  After some initial discomfort with the whole idea that they were going to be playing an on screen game, and that I wouldn’t help, they dove in.  They grappled, made mistakes, started over, helped each other, groaned, and persevered.   They were competitive and proud when they solved each level.  We used the last 15-20 minutes to debrief the activity, to list the things that helped us succeed, and to respond to a short survey.  We talked about how these skills actually encompass just about everything that they need to be successful math thinkers.  Interesting that the number one thing they did that helped them to be successful in this game was to make mistakes.  I bet that we will be referring back to this often.

Annie Fetter is the best.  Just Saying.

G of Squares Survey results

Survey results after 30 student responses. Notice the top result!

Day 4: Number flexibility: You mean there’s more than one answer?

Day four, we worked on the four 4s.  This has been a favorite of mine since I began working with students.  It allows for multiple approaches and creativity in math thinking.  I’ve written about it further here and here.  This year, I decided to keep it to one class period.  In groups of 3, I challenged Algebra 1 students to create every number from 1-20, and Algebra 2 to shoot for 1-30.  We put their work out on our Sweet Wall, and they may go later to try to fill in any blanks.  Jo Boaler and the Youcubed team put together an excellent week of inspirational math, which began with this activity.  The rest of the inspirational week’s activities were tough to resist.  There are some good ones in there, along with great growth-mindset messages for students.  I may get back to the others later in the year if we have time.  I did play her day one video, and led a short discussion hinting at growth mindset to end Thursday’s class.  I was also especially tempted to jump on the explicit growth mindset work that Julie Reulbach has shared, but we can’t do everything.  I will be following Julie’s reflections closely to see how her implementation goes this year.

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Day 5: Assessing Numeracy + Mathematical Drama.

Trying to do Algebra without a solid understanding of arithmetic is rough.  I’ve seen students suffer through this, and it is not easy for me or for them, and there just isn’t the time during Algebra 2 to work on dividing fractions or operations with negative numbers.  So we’re implementing an after school numeracy workshop this year for grade 8, 9, and 10 students who need more support in this area.  We used this class to assess students’ arithmetic skills, and to identify those who might be most helped by the after school program.

I saved the last 20 minutes of class to follow up on the secret partners activity, and for a read-aloud.  Secret partners takes just a few minutes, but has a nice impact on student attitudes.  They act reluctant to speak nicely about each other, but they are grateful for this opportunity to celebrate each other’s good qualities.  Comments ranged from “I noticed A looking out for the new student at lunch” to “Y worked really hard on the science lab” to “X is really funny and cracked me up in English yesterday.”  I ended the week by reading the introduction to Zero: Biography of a Dangerous Idea.  This is an excellent book, and the introduction is high drama!  And kids just like to be read to.

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Although we didn’t get deep into new content this week, we did some valuable math together.  But equally important is the positive feeling that students left with on Friday afternoon.  With confidence in themselves from their successes, with trust in each other and the knowledge that their peers notice their positive behaviors, and with the assumption that their teacher cares about them, we are set up for the year.  Now we need to hold on to this feeling when the going gets rougher!

 

 

 

Quadratics: Mighty Square! (start by completing the square)

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Many texts I’ve seen ask students to solve quadratics by factoring, then by graphing, then memorising the quadratic formula, and then if there’s time left, they are introduced to the idea of completing the square.  I’ve done it before like this myself, and I have seen students struggle mightily and miserably.  For me, beginning with factoring is problematic.  While factoring trinomials can be satisfying, especially for students who like the puzzle solving parts of math class, this technique generally only works with problems that have been contrived by math teachers or textbook writers.  Most quadratics we come across don’t naturally factor to nice clean numbers like y=(x+4)(x-2).

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I was looking for a better approach, and came across James Tanton’s take on quadratics.  He emphasizes symmetry as the key to studying and understanding quadratics, and right from the start, teaches completing the square (Although he relentlessly resists formal vocabulary, and calls it the “box method”) as the way to solve for x.    His course gives a sequence of problems, each of which adds one level of complexity until students can solve just about any quadratic thrown at them.  As he says, “The box method will never let you down.”

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Developing a solid conceptual understanding of completing the square leads naturally to moving back and forth between standard and vertex form, and to the derivation of the quadratic formula – and students know why it works.  And the best part is, only one of my students from this year’s group uses the quadratic formula as her go-to method.  They all go straight for completing the square because they understand why and how it works, and are totally comfortable with the techniques.  Most of them can recite the formula, but they are worried about making arithmetic mistakes, and are not as confident in their results as they are when they use symmetry and completing the square.

 

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I strayed from Tanton’s approach in some ways – we figured out imaginary solutions with the honors group (Tanton advocates to leave out imaginary solutions when studying quadratics with Algebra 1 or 2 classes) – because they asked, and because they were ready to expand their horizons.  And I did eventually teach factoring as a method of solving quadratics.  Tanton suggests that factoring might be better included when we’re working on a discreet study of polynomials – but honestly, it was easy for me to fit in as the last method I taught.  We took less than one full class period to discuss and practice factoring, and by the time we got there, students understood what the factors meant, and how they related to the roots of the quadratic.  They appreciated the quickness of the solutions when factoring worked, and understood that if it didn’t work easily, that they could fall back on methods they know.

Every one of my Algebra students, even those who really struggle with math have had success moving through this sequence.  Although this is only one year, and there are always other variables, I am convinced that this order makes much better intuitive sense.  I’ll report back next year after I’ve had the chance to try this again.

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I love the way that the white boards look when studying quadratics

BTW, some of my favorite resources to go deeply into studying quadratics include

Some of my favorite Geoboard activities

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I love using math manipulatives.  For some kids, using visual logic, or a hands-on approach can help to remove barriers to understanding, and can take some of the intimidation and fear out of learning about difficult concepts.  Using Algebra tiles to teach polynomial multiplication and factoring quadratics help to reinforce an area model of multiplication, and completing the square – well it does make more sense if you actually complete a square.  (And if you get the lab gear designed by Picciotto, you can complete the cube too!)  I’ll post about my experience with these sometime soon.  For today, here are some ways that we have used Geoboards in my Algebra classes.

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My Algebra students had a blast using geoboards to find triangles and squares.  The Math Forum has put together a set of very rich ideas for using geoboards, and we worked over a few days to discover line segments, triangle area, and practice with the pythagorean theorem.  I asked students to use the geoboards to find solutions, and then to translate their ideas into pictures on dot paper by drawing all of their solutions.

I had some challenges handy for some students who were ready. Here are some extensions to the challenges posed by the Math Forum, which my students grappled with.

  • How many different squares can you find in a 5 pin x 5 pin board?
  • How many different triangles can you find in a 5 pin x 5 pin board?
  • How many total squares can you find in in a 1 x 1? …2 x 2? …3 x 3? How about nn?
  • How many total rectangles?  IMG_1322

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There was some beautiful and deep thinking going on, and this is a highly differentiable activity.  Students who are at the concrete level could physically count squares, while students who were ready to generalize could find the cubic (I guess that’s not too much of hint), which describes the number of squares in an area.  Teachers, feel free to drop me a note or tweet at me for my solutions.  In an effort to keep the problems from being Google-able, I haven’t included them here.

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Geoboards are excellent to physically see how  slope works as well.  For next year, I’ve ordered the 11 x 11 pin versions.  Stay tuned!  How have you used Geoboards in Algebra class?