Reflections of a Techie

Music as Magic Potion to Turn on Math Engagement

 

Neuroscientists can't explain it, but music is a great leveler.  The absence of words, the touch of emotion has been a powerful potion in helping my students find their math mojo.

I've been experimenting with incorporating all sorts of the "arts" in an hour of math class that really struggles.  They've repeatedly said how much they don't like it, don't feel confident about using it and have no connection with it.  So I've started bringing in more of the story behind the math....

You know....

Doing a problem comparing different pricing plans based on car washes.  Not exactly the topic to excite 8th grader.  So I added in music....in fact one of the all-time favorite songs, "Car Wash" from Soul Train days.  Started the music as they entered the room and we had a dance party before we started considering the math.

Hearing the music inspired them and loosened them up....enough so that we could turn our attention to the math.  And they were inspired and fantastic.

Have you heard the love story of Mr. Radans and Mrs. Degree?

A fantastic video by one of my former students on converting one measurement to another. She used stop action video and it's informative and funny.  I have her younger sister in my current math class and she helped with the video production and shared it with me tonight. 

It's wonderful.  Funny, informative and helpful.

So many possibilities with these kinds of math products beyond the worksheet are available these days.

Practicing Habits of Minds--students request a hashtag swag for their efforts

Extending this notion of mathematical discourse...we tackle another set of statements. Normally what I would have done to review the distributive property and its application to expressions and equations is to generate problem sets.  Students would have practiced the procedure and talked about how to do the procedure, but rarely would I have challenged them to think what lies underneath it.

What's under the hood of this math engine?

We've been using the Always, Sometimes and Never strategy for thinking about the "underneath"  And this time it's varied by asking is it true for all, some or no numbers!!!

• Adding 8 to a number gives the same result as multiplying that number by 4
• x + 8 = 4x
• Subtracting 10 from a number gives the same result as adding 10 to that number
• 5n-6 = 3n + 34

And so we started into the discussion.

What's amazing is that my students now know they have to "test" each statement by picking a variety of situations.  So they usually start with zero and see what happens...then they try several positive numbers and several negative numbers.  Finally checking fractions and decimals.

Looking for patterns and having a generalized method of testing each statement has been one of the habits of mind we've developed.  Pretty quickly students were able to tell if something was never true or not.

"There's no way that the statement Subtracting three from a number gives the same result as adding 3 to the same number, one student announced.  I wasn't sure how they would handle this statement since it doesn't use the actual numbers...instead it's written out in words.  Didn't seem to throw them off at all.  Subtracting 5 from a number and then squaring the results gives the result 16 sometimes..."and I found 2 examples of when it works."  But there weren't any other numbers that made it true.  I was so impressed because most students realized that there were two successful solutions.

They really derive a sense of pride from finding these bits of evidence to support their thinking.  Even if a student didn't find the "evidence", listening to the class conversation makes them strong.  "I may not get it all the time, but then during the discussion I figure it out."  My favorite part of the whole lesson happened near the end of the day...I'll admit that I was tired and having been through the same lesson 4 times already I might have lacked a bit of enthusiasm.  But my 6th hour picked me right up...admantly suggesting that we should announce every good answer and thinking processing by saying hashtag swag.  Since I hadn't a clue what this meant, the class explained it was their way of saying "Good Job" with some style.

#swag

I can do that!  So lots of #swag's erupted as different students defended their ideas and announced their evidence.   Did we actually Tweet these out?  No.  We just awarded #swag to the people for their contribution and did a little happy dance.

 Discourse can be fun.  It should be fun.  It should be playfully serious.  I'm learning and getting better with every time I try this strategy.

 

Car picture offered under Creative Commons Attribution license.  http://www.flickr.com/photos/kirkschoiceshots/2733343814http://www.flickr.com/photos/kirkschoiceshots/2733343814

Related articles

Dance P-A-R-T-Y with irrational numbers

Dance P-A-R-T-Y with irrational numbers

Knowing I had to lead 8th graders in a discussion about rational & irrational numbers drew me back to junior high school math.  Ugh.  It made me shiver just to think about it.  So boring.  Remembering that I kept thinking and thinking about what I could do inject something into this activity. 

Then I remember Kagan's Mix-Up strategy....not sure that would work for my 8th graders.  (At least I think it's what it is called...took those workshops so long ago.  I remember the structures but maybe not their official Kagan name.)

But a dance party would!!!!  

So I re-mixed the old musical chairs kindergarten game into Property of Numbers Dance Party. Who knows what it should be called? 

Thank goodness for Pandora because I gave DJing responsibility to one of my most trusted 8th graders who knows Pandora much better than I do.  She selected the Michael Jackson channel.  We were off.

For all the negative Nellies who didn't want to play or thought they were above it all....I simply said, "This is as good as it gets.  It's a math class.  Let down your guard and have fun with it."  I had to tell myself that because, of course, I  looked ridulous dancing.  That's probably part of the fun of the class is laughing at me and my dancing style.  Honestly...don't care a lick because I had a blast and I never really thought I was a good dancer!!!

They initially sat as pairs.  Then we'd call right or left partner and the music would start...and around they'd go...some dancing like crazy people, some sheepishly waving an arm or hand in the arm, some just praying the music would end their misery of having to look they are participating when they wouldn't dance if I paid them a $1,000,000 bucks.  The music would stop and they'd scramble for the nearest chair. 

Partner A would explain their position to Partner B...and then Partner B would ask questions and then share their position on these 8 statements

• The sum of a rational number and an irrational number is irrational.
• The circumference of a circle is irrational
• The diagonal of a square is irrational.
• The sum of two rational numbers is rational.
• The product of a rational number and an irrational number is irrational.
• The sum of two irrational numbers is irrational.
• The product of two rational numbers is irrational.
• The product of two irrational number is irrational.

Now wouldn't the possibility of discussing these statements make you want to jump up and dance???  These are the statements that are part of a fantastic math resource called the Mathematics Assessment Project.

While it all may seem pretty silly, I used the Dance PARTY as a way to interject playfulness into math class.  These are conversations where students must risk taking a stand on something that can be pretty daunting....and you want to convey to them that it's the process of the discussion not if they got it right or wrong.  It's also why they wrote their mini-posters in pencil and were free to change them as a partner helped them see a mistake or counter-example that changed their idea.

It was a brilliant success.  Students had at least 7 or 8 different partners with whom to discuss their ideas....they refine/edited/fixed their positions...and they smiled and laughed. This is the meat of the lesson and I got them to talk more and do a better job discussing than if I'd done this in a serious, be on-task kind of activity.  They danced & were loud AND then they settled to do their 3 or 4 minutes of serious math discussion.  Repeated over and over until everyone was tired and ready to sum things up.   To wrap up the class period, we had an "All Dance" and then finished with a class discussion where we decided what we collectively believed to be true about these statements one by one.

Lots of learning.  Lots of smiling.  A good way to inject some "spicey stuff" into Properties of Numbers.

 

Picture licensed under CC Attribution http://www.flickr.com/photos/jeremywilburn/5428009431/       http://www.flickr.com/photos/commonbond/500397279/

Debating Always, Sometimes or Never ---furthering mathematical discourse

Students debated whether 8 statements about rational and irrational numbers were Always, Sometimes or Never true.   Thanks to an activity posted by the Shell Center, University of Nottingham and UCBerkeley, I had the coolest lesson(s) with students about property of numbers.  Lots of conversation and writing. If you try this kind of activity, be sure that students can "rewrite" their answer....because it's all about understanding. Not being right. They loved it.

I'm going to have each class pick what they think are the most compelling arguments for each of the 8 statements and we'll make big posters for the hallway.  All along the way, I'm trying to incorporate more student voice, more student choice and more student empowerment. 

This is the cumulative body of information students found in 2nd, 3rd, 5th and 6th hour classes.

Two steps for incorporate PBL learning in math class

I've been thinking, experimenting and testing out ideas about how to move math instruction forward in my classroom, using technology and Problem-Based Learning (PBL) as much as I can.  It's a daunting task even if I've done it before in science....very different set of inputs and outputs in a math class.

I've also been wondering what "Teaching in the Age of Siri" really means.  I believe we have students who believe they can Google the answer to anything that is asked.  When they run up against questions that aren't Google-able, it stumps them.

This week, I gave students pretty tough problems about functions that involved square roots.  They really didn't know anything about square roots.  So rather than just telling them....I took a bit to explain that we were going to learn what to do when you can't find the answer on Google or have a friend tell you.  I gave them 10 minutes to find everything they could about a square root that seemed like it might help answer the two-part question I gave them. 

Update from 12.06.12--- I'm inserting a little revision in here to clarify what I first wrote about last week.  I received so many requests for the problem, I thought I'd post it here in addition to leaving it in the comments.  I also think I may need to restate that it is really two questions.  Knowing about square roots help with answering both questions which is why I called it a tw0-part problem in my original writing.

The problem they were solving is really pretty simple

x----> SQRTx    and the second problem was

 x----> SQRT ABS(x).

For these problems, they had to define what inputs, if any, were not valid.

Connecting it to the real world...(educational-speak would be career-ready skills)

I wanted to connect it to the real-world and explained that rarely have I had a boss/supervisor that takes the time to show me how to do my job....most of the time, they expect me to know how already.  That was very puzzling for my students.  Why wouldn't my boss help me, they wondered?  It wasn't until I explained that bosses (or at least the kind I've had) hire me to do something that they I can either already do or that I'm smart enough to teach myself.  It wasn't that they were being mean to me.  In fact it was the opposite, it was that they had confidence in me to solve the problems of my job assignments.  That was a huge flip of a circumstance.....something none of my 8th graders had considered before.  I assured them that I wasn't going to suddenly assume they knew all of algebra and be like my boss, but the experimental learning style we were about to do was a first step towards being that kind of independent learner.

They used laptops, the Algebra references books I have in the room, the textbook, notes that I have given throughout this unit, their phones and some even asked Siri for help.  I really didn't care where or what they brought back to the table.  Then I led them in a class discussion about what they thought was relevant in all that they found to solving the problem.  Once the class decided (and I have to tell you that they learned a ton that they didn't need to know but was just there for the learning) what was most helpful, we started a discussion about what do you do now that you've found info that could help you solve the problem.

Honestly....I don't think many of them have ever thought about that kind of reasoning.

Turning found info into useful knowledge and something that solves the problem

And they really loved it....they argued and wrestled with all the info they collected and then built their conjectures.  After conjectures they had to use evidence from "tests" they could run with their graphing calculators to see the info they found matched up.  My role wasn't to teach...it was to coach them.  I helped them realize that once they had info NOW they had to something with it. 

It's a bit like taking a recipe and a bunch of ingredients...and creating delicious cookies.

And that was thinking...tying all the little discrete factoids they had from prior knowledge and this mini-research into a new problem.  Eventually they correctly solved the problem....along the way we discovered way more than the single answer to the two-part problem. I think they'll probably retain the info better.

I realized most of all....they need some hand-holding and practice as they learn how to take "found" information and "think it" into meaning.  I'm definitely doing this again....try to refine the process and sharpen it.  This isn't all the way to PBL yet.  But I think it's the foundational shift of responsibility for learning onto the student's shoulder that will be a precursor for doing a full-blown PBL or maybe a STEMy kind of lesson.

http://www.morguefile.com/archive/display/541092 ---map picture

http://www.morguefile.com/archive/display/83127--- recipes

Love the way you can read PDF documents on my iPad using the Kindle reader. I don't want to lug my whole computer on an upcoming trip but I have to read a PDF version of a book. Voila! I simply email the PDF to my iPad's Kindle email (who ever knew it had one???) and presto/chango....it appears in the Kindle app. Ready for me to read on the plane. Now that is cool

Let's Count Goats---Number Sense continues in Algebra

Mem Fox starts kiddos thinking about numbers with her well known and loved book, "Let's Count Goats".  Kids love to participate in telling you why something is going to happen the way it does. You'd think that number sense would be well developed by the time my 8th graders get to me, but there's still more to do.

Phase One: Equivalent ways to express answer

This is a pretty simple idea but it makes a big difference.  When we are working with writing equations, for example, I will always play the straight man.  So if someone says they have 1.5x, I will say, "Oh, I don't think that's right because I have 3/2x."  And then wait to see what happens...

It used to be that students would dutifully mark their 1.5x answer wrong!

Now they're smart enough to stop and consider what I've said....at least a little bit.  Once they caught onto that I then had to intermittantly make some conversions that weren't equivalent.  After almost 4 months of this kind of embedded practice, students are getting pretty strong in their number sense and I can hardly even stump them....and now they love to ask the question, "Mrs. Ratzel I don't have 5.2 for that answer, I have 5 and 4/10. "  And they wait to see what I say.  Sometimes we have pretty good conversations about simplifying answers and what makes the most sense in that particular situation.  But the real deal is that I have them sucked into mathematical conversation and dare I say, mathematical thinking.

 

Phase Two---Extending this idea to distributive and additive identity property

Once I saw this was effective, I started looking for other places to embed number sense practice...mostly with the distributive property.  I picked it because it is the property where we are the weakest.  I know you can already predict where I'm going to go with this.

Right now we're just starting functions....and we're using images of function machines that say things like "multiply by 5" and "then add 2".   Simple enough to write the rule 5x + 2.  Then we move onto function machines like "add 2" and then "multiple the result by 5".  So students immediately write (x+2)*5.  So we work a bit on rewriting it as 5(x+2)....and then I start to throw in my distributive property jabber.  "Oh, I got 5x+10 not 5(x+2)". 

Sadly they dutifully marked their 5(x+2) wrong.   But this time, it only took me doing that one time and they were wise to be on the lookout.  And we've had loads of opportunities to practice the distributive property.  Won't it be fun when we are going to be able to go the other way?

The other variation of this is writing answers.  Since we're sort of in the second year of their writing equations experience, they are very apt to write things like y=4x+0 when the y-intercept is 0.  It's taken lots of experience where I tell them that I had the answer as y=4x.  I really waited to start doing this until I was sure they had the idea that +0 was really there, but just didn't need to be written because of the additive identity property.

Maybe not the most thrilling way to work on number sense, but very effective

OK...so this isn't the most thrilling idea or biggest ah-ha moment any one has ever shared with you.  But the real deal for me is that it has been incredibly effective...much more than I think I'd ever get with loads of practice problems that weren't attached to the other work we were doing.  So....if haven't already caught onto this idea, I'd really recommend give it a try.

On Halloween's eve....a witch's brew of YouTube, Geogebra & Questions

Mike's Car Wash is a pretty amazing place to get your car shiny and new.  Maybe not such a good place for a person to go through a car wash though!

 

The problem, for you as the new owner is that there are multiple pricing schemes.  And you don't know which one will give you the best return on the money you've invested.  Your accountant hands you this analysis but has to rush out the door before she can explain what they mean.  So you're stuck....

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Is there anyway to know what it means and what parameters you should consider in picking the optimal price to charge your customers?

What do you think all these lines represent?  Why would the accountant have done this kind of thing instead of writing it out in English?

This will be my setup tomorrow for our next problem set.  In thinking about how to pull them into thinking about linear equations, this video will appeal to their middle school sense of humor and the Geogebra gives them a visual representation of the options. 

I'll need to help them get oriented to the problem.  Making sure that they know what's the problem really want you to do?  what should you be looking to find?  If the graph makes them stuck, I might offer the idea of using a different representation...can they generate a table from the graph's information or do they know enough to create equations? Are there any patterns that you can find on this diagram?

Anything so that they are steered in the direction of being able to solve the problem.  Sure hope this gets them hooked and into developing their thinking and I hear great conversations tomorrow.