My 6th Grade Version of Algebra

This is probably better fitting for a twitter tweet twit, especially considering how delinquent I've recently been but couldn't pass up sharing a few questions/comments from my sixth graders today.

Ok, so maybe a little context to defend my choice of not being on twitter:

EDIT: Ok, maybe more than just a little context.  If this is too much to read, at least skip down to the end and look at the questions/comments.

My sixth graders just started talking about algebra.  I love this because, at least with the way I approach it, it's the first time that I don't have to do much unteaching (fractions, on the other hand...).

I began by asking them to come up with mathematical questions motivated by a 1x1, 1x2, and 1x3 strip of squares made from Construx.

We shared out.  The good: Lots of kids asked questions about counting the pieces and looking for patterns.  The bad: I was going in this direction regardless of the questions they asked.

Then in groups they played around with this Edges, Faces, and Vertices activity for a while, building models and looking for patterns so that they wouldn't have to build huge models.
The next day I asked them what they thought their new unit was on.  Most guessed geometry.  Others guessed graph theory and combinatorics.  No one used those words.  A few said algebra.  I told them all of these are reasonable choices, but we're going to make the choice to explore algebra.  I then talked ranted for a few minutes about some misconceptions of algebra:
1. It's something you do in 8th grade.
2. If you do it before 8th grade, you're ahead.
3. It's something you check off the list and don't do again after 8th grade.
4. It's math with letters.
5. It's something my smart friend did in 4th grade.
*6. It's something people use all the time in their jobs.

*There may be some disagreement with this misconception.  It'll have to wait for another post.

I then gave my algebra schpeel.    Talked a little history.  Talked about how the word comes from the Arabic al-jabr roughly meaning restoration written by a guy named al-Kwarismi (where the word algorithm came from) and how these ideas managed to avoid Europe for close to 1500 years.  Spent the few minutes to share the aside that while the mathematics of algebra didn't make it to Europe, the word did as barbers in the medieval times were called algebristas because in addition to cutting hair they were also bone setters. As an aside to my aside, they were also bloodletters which is where the red in the barber pole comes from.  Well, at least that's what the internet tells me.

Then we talked about three aspects of algebra:
1. Finding the secret number or numbers, ie solving algebraic equations.  Told them that the problem
3+?=7 was really not so different from x^2-5x-6=0 except that the latter was harder to find the secret number(s).
2. Describing and using patterns, ie writing algebraic expressions and using formulas.  For these types of problems, it's not about finding secret number(s). Instead, it's about describing relationships and making generalizations.  Hey, we've already done this when we described the number of edges, faces, and vertices in a 1xN strip.
3. Proving equality or inequality, ie manipulating expressions.  Hey, we've already done this when we showed that one students pattern for vertices of 2*(N+1) was the same as another student's pattern of 2*N+2.

I'm sure there are other great ways to think about the subject of algebra.  Feel free to share your own in the comments.  After making a few examples, I left them with 2 questions:

1. So we've already done some of #2 and #3.  What about #1?  Can you make up a problem in the context of edges, faces, and vertices that would use the first aspect of algebra?
2. Someone mentioned that 3+?=7 has a solution of 4.  Does it have any other solutions?  Can you create an algebraic equation that has more than one solution?

Mixed results with #1.  Some kids were right on it and came up with the expected "What size strip has 68 vertices?" along with some other good questions.  Others were pretty confused by the task.
Great results with #2:
0*N=0
(11-11)*N=0
1*N=N
N/N=1
7*N=E where E is even

There were more, and this second question prompted some fabulous questions/comments, the point of this post.  While I've tweaked some of the language to be more concise, these are all from the 11 year olds.

Can I make it so N has to be a whole number?
Can I make it so N has to be an even number?
Is 2+3=x an algebraic equation? (anyone have thoughts on this?)
In the equation 0*N=X, N can be anything but X has to be 0.
Can I make the operation a variable? (I had a hard time containing my excitement)
If variables are used to talk about a bunch of numbers at the same time, can we use variables to talk about a bunch of algebraic expressions at the same time? (more failed attempts to contain my excitement)

Project Euler

My brother told me about Project Euler today, and I've more than used up my extra hour working on some of the problems.  What is Project Euler?  Well the simplest explanation is "just click on the link and find out."  I can't give you a much more detailed review since I only found out about this today, but here are my initial thoughts.

So far, the problems I've worked on are good.  They are very accesible.  They are problems that a middle schooler could work on (and solve), but at the same time problems that keeps me--a middling recreational mathematician--challenged.  So far there has been no context aside from the context of puzzle solving, so they won't help you determine how the speed of the shadow of a sandbag being dropped from a hot air balloon is changing.  Somehow, they've still kept me engaged.  :)

The problems are posed (and from the solutions I've read, generally solved) with the lens of a computer programmer.  So far I've solved 5 problems without any coding (well, to be fair, I used someone else's code to solve one), but it looks like programming will end up being a requirement for many of the problems.  I've skipped one so far that either required some programming, more thinking time, or someone more clever than I.  Looking down the list this appears to become more common.  Regardless, I find this to be an interesting lens (and maybe a great motivation to bone up on my Java/C/C++/Mathematica/Matlab/Scratch/Perl/Dylan/... skills).

All of the problems (so far, at least) have unique answers.  All of these answers (so far) are numbers.  While I generally frown on such a focus, I admit that it feels good to put your number in and get a green checkmark (instead of a red x).  I also like that answering a problem correctly unlocks a forum where other people have posted solutions (which include both code and explanations).

I also like this response to whether or not you can use google for help.

Making use of the internet to research a problem is to be encouraged as there could be hidden treasures of mathematics to be discovered beneath the surface of many of these problems. However, there is a fine line between researching ideas and using the answer you found on another website. If you photocopy a crossword solution then what have you achieved?

All has made me wonder how this or something like this could be used in or outside of school to promote the activity of doing mathematics (or to allow robots to replace teachers).  Feel free to wonder with me, or just register and try your hand at some of the problems.

Alexander's Fax Machine

I've been meaning to share some activities from this year...both successes and failures not-so-successes.  Here's an activity I did on the second day of school.

 

Alexander Graham Bell invented the telephone, but if he wanted to “fax” something, he would have to describe in detail the item to be replicated to someone else on the other end of the telephone.  You and your face partner will replicate this.
1.     Everyone should build a wall so your image to be copied and work can’t be seen by your partner.
2.     I will give the window partner a manila folder that has parts cut out.
3.     You and your partner will talk to one another (nothing allowed other than talking…definitely no peeking) and the door partner will attempt to replicate the shape.  The door partner also can’t show her window partner her progress.
4.     Once the door partner has a sketch on paper, she will cut out a manila folder.
5.     Once you have checked and rechecked, show each other your result and see how you did.
6.     Reflect on this process:
How’d you do?
What went well?
What did not go well?
What was hard?
What tools would have made this easier?
What would you do differently next time?

Examples (two columns of originals on the left):

Take away:

language is important

precision is important

articulating confusion is important

checking is important

collaborating and listening is important

Overheard during discussions:

How far from, how big, how long/short, tools for measuring: inches, cm, finger width, nametag width, diagonal, vertical, horizontal, landscape, triangle, isosceles, up/down, positive/negative space, hamburger/hot dog, line, left/right, corner, gap, arrow, width, slope, NW, ___ degree angle, curving, squiggly, acute, obtuse, points

Reflection: * I didn't create the original puzzles with any specific shapes or language in mind...maybe something to think about in the future. * Most students found this super challenging, but I didn't see any of the normal math anxiety that can rear its ugly head. The lack of numbers have anything to do with this? A sense of no black and white right & wrong answer but instead a spectrum of close to far?

*When we switched roles and did this a second time every pair felt better about how they did.  

*I chose not to quantify "how well they did. Felt ok about this decision.

I gained a good bit of insight from their subsequent homework, Alien Encounter.

While I considered this activity a success, I'd still love your thoughts and/or feedback.

Show Your Work

I can’t speak for other math teachers, but I sometimes fall into a zombie state where I wander the classroom with outstretched arms continuously repeating my math-zombie mantra braaains show your work. Don’t believe me? The best present I’ve ever received from a student (hope I don’t offend any former students reading this who gave me a Starbucks gift certificate) was an art piece by a fourth grader titled “Show Your Work.”

I love my job. But I digress.

I’ve always felt a little bad about telling my students to show my work, partially because I was resistant to doing this when I was a student, especially when I saw showing my work as a pointless exercise. A colleague of mine recently told me a story of her son who, in his early elementary years, would respond to instructions asking him to show his work by drawing his version of The Thinker next to his answer.
 
I empathize with this stance and sometimes feel that I am in fact going around eating the brains of my students (metaphorically, of course).

Don’t get me wrong or quote this out of context. I think explaining your reasoning is one of the most important parts of mathematics.

This year I have a few particularly studious sixth graders who—off the record—are showing too much work. I wish I had a scanned original, but my version will have to suffice for an example:

So can a student really show too much work?  I think the answer is yes.  First of all, I wouldn’t want students to get turned off by the subject because they felt that they felt forced to do something IF this something isn’t helpful.  Secondly—and this is more subtle—I think that explaining your work can sometimes (I should emphasize sometimes) undermine one of the most powerful aspects of the subject: the ability to use mathematical symbols to tell a story or solve a problem.  The righthand symbolic representation above IS the work.  Here's another example, with the original written by al khwarizmi, an 8th century mathematician everyone should know (although I think it's ok if you don't know how to spell his name):

Date	1200 years ago (translated)	Current
Problem	"If some one say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times."	(10 − x)2 = 81x
Solution	Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts	100 - 20x + x2 = 81x x2 – 101x + 100 = 0 (x – 100)(x – 1) = 0 x = 1, 100

Ignoring how the English language has changed (this was translated in the 18^th century), I think we’ve come a long ways.

I’ve never had more than a surface level discussion with my students of what I mean by “show your work”.  Students are constantly making decisions about what is necessary (what constitutes reasoning) and what is not (explaining why 1+1=2) and, for the most part, the only feedback I’ve ever given them is “not enough”.  To complicated matters, the same amount of work will sometimes be deficient and sometimes be plenty.  This brings me to audience. I haven’t done this yet, but I am thinking about having a conversation about audience with my kids in the near future.  They’ve been asked implicitly to write for each of these audiences, but I’m hoping to make this even more clear.

Our audiences:

• The friendly alien (who is intelligent, has a strong command of language, but no experience with mathematics)
• Classmates who understands the problem, but may have a different solution method (peer review)
• Classmates who do not understand how to solve the problem (peer assistance)
• The student
• The teacher

The audience I want them to write for will depend on my goals of a problem/task/assignment.   Here’s a not-even-close-to-an-exhaustive list of different reasons to show your work:

• The friendly alien
  - For big picture reflection.   Example: Explain to the alien why THIS is mathematics.
• Classmates who understands the problem, but may have a different solution methods
  - For problems that lend to multiple solution methods, which I suppose should be every good problem.
  - To share a solution method
  - To compare solution methods
  - Proof
• Classmates who do not understand how to solve the problem
  - To teach others
  - To reinforce your own understanding (I think Bloom said something about this)
• The student
  - To ensure your work/reasoning is sound
• Me- To exhibit mastery

As always, a work in process...

The Pinnacle of Journalism?

4,100 Massachusetts Students Prove ‘Small Is Better’ Rule Wrong

I think this title sucks, but what about the message?

Claim: Brockton High School has moved from being unsuccessful to successful in spite of its size, “proving” that size doesn’t matter (tee hee).  This contradicts “certain education circles” that believe small schools are a useful reform tool (to be transparent, I fall into this category…I personally find it much easier to build a positive learning community in a place where people know each other).

After reading, I think the message it also pretty lousy.  In a two page article, the following is the evidence given for how the school was previously unsuccessful: “[a decade ago] only a quarter of the students passed statewide exams. One in three dropped out.”

Pretty damning.  Evidence of improvement? This is it.  All of it. “In 2001 testing, more students passed the state tests after failing the year before than at any other school in Massachusetts. The gains continued. This year and last, Brockton outperformed 90 percent of Massachusetts high schools.”

Don’t get me wrong.  This is an improvement worth commending.  Regardless of where you stand on the importance or measure of state assessments, it’s no small feat to make this kind of improvement. But the skeptic in me didn’t linger long in this success.  What about the dropout rates?   Here’s an interesting article from the local newspaper that appears to contradict the “one in three dropped out” fact and claim that, in fact, the drop out rate was lower in 1997-1998 than last year (about 3.5%).  I guess it is possible that both of these statements are “correct” since drop out rates are notorious for statistical meddling. 

Most of the remainder of the article talked about how these reforms were put in place which, again with the transparency thing, some of which made me cringe.  The sole example of a math lesson: “Bob Perkins, the math department chairman, used a writing lesson last week in his Introduction to Algebra II class. He wrote ‘3 + 72 - 6 x 3 – 11’ on the board, then asked students to solve the problem in their workbooks and to explain their reasoning, step by step, in simple sentences.  ‘I did the exponents first and squared the 7,’ wrote Sharon Peterson, a junior. ‘I multiplied 6 x 3. I added 3 + 49, and combined 18 and 11, because they were both negatives. I ended up with 52-29. The final answer was 23.’  Some students had more trouble, and the lesson seemed to drag a bit.”

Really?  Drag a bit?  Please shoot me now.  Oh, keeping with the theme of transparency, I should say that I think this ‘lesson’ is worse than useless from a mathematical or a writing perspective.  Alas.

“It had become dogma that smaller was better, but there was no evidence,” said Mr. Driscoll, who since 2007 has headed the National Assessment Governing Board, which oversees federal testing.

Nice logic buddy.  Test scores now > test scores ten years ago imply large school = good.  Ergo, small school ≠ better.

Ack.  Apology for all the snark.

For an apple and I'll let you paint this fence

So this'll be fun.  If you were to write a 10-minute-ish individual assessment on pattern sniffing for...ah, you pick the age (or tell me it doesn't matter)...what would it look like?

Here's my most recent definition of pattern sniffing:

• On the lookout for patterns
  "Ok.  We’ve been working on this staircase problem and it seems that you can’t write powers of two as a sum of consecutive whole numbers."
• Looking for and creating shortcuts
  "It would be nice if there were a faster way to find the greatest common factor of 2 numbers other than listing all the factors. Think we can find a way?”

Challenge question: How would you grade this little quiz of yours?

EDIT:  I should be clear.  The end goal is for students to be able to apply these habits in unfamiliar problems without a big neon sign saying *Use Pattern Sniffing*.  This is in no way an assessment of that end goal, but the first step in what I suspect will be a relatively long road.  

Confessions of a Math Teacher

I believe that my job as a math teacher is not to make things as easy as possible for my students.  I do not give them the shortest explanation with the most straightforward formula and ask them to solve different versions of the same exercise for homework and tests.  I do not tell kids how to do problems from the next section in the textbook so that their 1-29 odd homework will be as painless as possible.  I will not become impatient and compromise a deep conceptual understanding and let kids get away with just knowing the procedure.

Or at least I try really hard not to do these things.

I always let kids explore an idea and try to build an understanding themselves before I tell them anything.  I will give homework problems that are unlike any problem we've talked about in class.  I do think "play with it...look for patterns...make observations" is a legitimate homework assignment.  I do ask students to work on messy problems, problems that I don't know the answer to, and open problems.  I have let kids leave the classroom with a misconception that I know about and did not address.

Or at least I try really hard to do these things.

I see the need to teach math as a verb.  I teach the activity of doing math as much as I teach the subject of math.  This means teaching kids to enjoy, embrace, and feel confident working with hard problems that they won't solve in thirty seconds, five minutes, one sitting, one week, or maybe ever.  This means caring about beliefs, perceptions, and attitudes as much as I care about fractions.  I want kids to enjoy the process of thinking, playing, trying, failing, and lots of other -ing words.

But this post isn't my call for you to see the wisdom of my ways and throw out your lesson plan for tomorrow.  No, it is a plea for help.  Tonight I don't need help with fractions.  I need help with teaching the activity.

This student has been on my radar after an email from the father about frustration at home and "disliking math for the first time."  A conversation was had, but I now know that the attitude equivalent of a deep conceptual understanding has not been made.  The one positive note is that the student is talking to me about this instead of repressing these feelings that will develop into yet another adult who hates math.

Via email: "  I just want you to know that even though you said [all] the [homework] problems aren't mean't [sic] to be finished sometimes when you hand in unfinished work it can make you feel stupid and like you failed. I thought that that was something you should know."

So any ideas on how to teach "the activity of not finishing problems in five minutes" to a super smart sixth grader who still participates actively in class but is clearly not enjoying math on the home/homework end?  If it helps, here and here are the homework assignments that sparked this frustration.

It also might be helpful to know that my policy is that students should spend no more than 30 minutes on homework.  If they spend any more than 30 minutes it should be because they'd rather be doing this than playing, reading, running around outside, talking with friends, or whatever else they're allowed to do.  For these two assignments, the first 2 problems gave me all the information I needed to know about their understanding of the content at hand.  Aside from the students who tried to compute 2 to the 57th power, everyone made it past #2 but only a handful finished the sheet(s).

Beliefs and Attitudes about Mathematics

What beliefs and attitudes about mathematics do you see in your students, in society, in the media, and elsewhere? Try and think of both positive and negative beliefs and attitudes. These can be beliefs that you agree or disagree with. I'll start with a few, but please add your own in the comments.

• If you're good at math, math problems can be solved in a relatively short amount of time.
• People do not solve math problems for fun; they do it for school, for their job, or to balance their checkbook.
• Every math problem has been solved by someone.
• Math is about numbers.
• Math is a language to describe the world.
• If you are good at math, you are smart.
• If you can do computations accurately and quickly, you are good at math.
• People who are good at math are eccentric and/or not socially adept.
• Boys are good at math.
• Asians are good at math.
• There can only be one correct answer.
• If I don't know how to solve a homework problem, I must be doing something wrong.
• Math topics/classes are sequential; I need to understand A before I can learn B.
• It is socially acceptable to say you're bad at math.
• Math is more analytical than creative.
• Using a procedure correctly to get the right answer is more important than understanding why the procedure works
• With current technology, arithmetic is not important.
• To be an engineer, you need to be computationally strong.
• Math is a gatekeeper.
• The value of math is in its connection to real world applications.
• Math teachers sleep under their desk at school. 

Ok, I guess that was more than "a few".  Hopefully I left some for everyone else.

Fun with Data Fitting

Holy page hit batman.    I will readily admit that I am a curmudgeony luddite when it comes to twitter.  The format perpetuates what I see as a disturbing trend towards shortened attention spans and disinterest in depth or subtlty. …or even knowing how to spell subtlety.  To be honest, though, my feelings may also stem from the question “What has twitter ever done for me?” 

Well, this has been a big week for me in the educyberbloggerinterwebmathsphere.  Firstly, I was really excited to meet three—count ‘em three—bloggers that I follow at an evening professional development get together where I may or may not have actually made the comment “One-half is not equal to two-fourths.”  It’s a long story that may or may not deserve a separate post, but somewhat related to Simpson’s Paradox. I had already put a face to Dan because I was blown away by his presentation last winter at CMC-North and, frankly, he’s sorta’ famous in that “famous within a specific subset of a specific field” sort of way. Nonetheless, it was great shaking his hand (ok, the handshake was actually just ok, but subsequent conversations were great).  I had not had a chance, though, to put a face to Jason or Sophgermain (although M. LeBlanc might be more appropriate).   

It appears that I also owe Dan and, begrudgingly, twitter a big thanks because my recent post about mathematical habits of mind was “picked up” via twitter by The O’Reilly Radar (and I thought they just wrote nerdy computer science books) and kottke.org (which I had never heard of, but it appears that other people have).  Anyway, the table below says it all.

Date	Page hits
Friday, August 20th	4
Friday August 27th	10
Friday, September 3rd	33
Friday, September 10th	6449

It’s growing exponentially!!  And by exponentially, I mean fast.  And by fast, I mean something completely different from exponentially.

So don’t fall for that malapropism.  With a little data fitting, you can instead see that these numbers fit the function:

f(t) = 1063t³ -6368t² + 11670t - 6361 (with a little rounding)

where t is the number of weeks after August 13^th and f(t) is the number of page hits.

What does this mean? The table below says it all.

    

Date	Page hits
Friday, August 20th	4
Friday August 27th	10
Friday, September 3rd	33
Friday, September 10th	6,449
Friday, September 17th	25,634
Friday, September 24th	63,964
December 31st, 2010	6,181,370
December 31st, 2014	12,266,700,000

Since this is greater than the United Nation’s projection of the world population at the end of 2014, this is unequivocal proof of either:

1.     Extraterrestrial intelligent life in the universe that will be reading my blog

2.     A tea party landslide in the 2010 midterm elections leading to a universal ban on contraception

3.     Bad math

I’ll let you decide.

Habits of Mind

This is still a work in progress (and feedback would be greatly appreciated), but I've decided to explicitly teach (and assess...more on that later) 4 "categories" of mathematics this year.

1. Skills (I know how to...)
2. Concepts (I understand and can explain why...)
3. Connections (I see and can explain the relationship between...)
4. Mathematical Habits of Mind (I can use and appreciate the process of...)

I've decided not to use the term "problem solving" because I believe this term is often misused to include be limited to solving problems and because the motivation for problem solving skills seems to be to solely help you get an answer.  While I believe that they can be very helpful in finding answers, I see mathematical habits of mind as also being mathematical in and of themselves.  So...while searching for patterns may help you solve a problem it is also DOING mathematics.

Here's the current version of the mathematical habits of mind I think are important.  I hope to explore (in varying depths) every one of these and have already shared the list with my 6th graders.

This is definitely a work in progress and some of these are based on work by Cuoco, Driscoll, Schoenfeld, and others.

Habits of mind

1.    Pattern Sniff

A.     On the lookout for patterns
“Ok.  We’ve been working on this staircase problem and it seems that you can’t write perfect squares powers of two as a sum of consecutive whole numbers.”

B.     On the lookout for Looking for and creating shortcuts
“It would be nice if there were a faster way to do 57x34 than adding 57 to itself 34 times. Think we can find a way?”

2.    Experiment, Guess and Conjecture

A.     Can begin to work on a problem independently
“I’m not sure how to solve this problem, but I’m confident I can make some progress.”

B.     Estimates
“Without doing any calculations, I’m guessing that it will take him 30 seconds to walk up the down escalator.”

C.     Conjectures
“Based on my work, I think the following is true.”

D.    Healthy skepticism of experimental results
“Boy, it sure seems like this 4, 2, 1 thing always repeats but we don’t have a proof yet.”

E.     Determines lower and upper bounds
“I know it will take the people at least 10 minutes to cross the bridge because the 10 minute soldier has to cross the bridge.  I also found a solution that takes 19 minutes so I know the final answer is somewhere between 10 and 19 minutes.”

F.     Looks at small or large cases to find and test conjectures
“I made a table of the first 5 cases and I think I see a pattern.  I’m going to see if this pattern holds for the 100^th case.”

G.     Is thoughtful and purposeful about which case(s) to explore

H.    Keeps all but one variable fixed

“So I’m exploring the equation y=mx+b and I’m wondering how the graph changes as m and b change.  For now, I’m going to set m to 1 and just look at how the graph changes when I change b.”

I.      Varies parameters in regular and useful ways

(Even/odd example)

J.      Works backwards (guesses at a solution and see if it makes sense)

3.    Organize and Simplify

A.     Records results in a useful way
“I’m going to make a table.”

B.     Process, solutions and answers are detailed and easy to follow

C.     Looks at information about the problem or solution in different ways

D.    Determine whether the problem can be broken up into simpler pieces
“I think I can solve this problem by solving these other 2 simpler problems.”

E.     Considers the form of data (deciding when, for example, 1+2 is more helpful than 3)

“I’m going to leave my fraction as 6/36 because the 6 represents the number of ways you can roll a 7 with 2 standard dice and the 36 represents the total number of rolls.”

F.     Uses parity and other methods to simplify and classify cases
“Next time we play 21 Nim I’m going to keep track of whether the running sum is a multiple of 3, one more than a multiple of 3, or 2 more than a multiple of 3.”

4.    Describe

A.     Verbal/visual articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions

B.     Written articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions

C.     Can explain both how and why

“The algorithm for dividing fractions is simple.  Now I just need to work on making sense why this works.” 

D.    Creates precise problems

E.     Invents notation and language when helpful
“For the sugar weighing problem, I don’t want to have to write out every solution in words so I’m going to let the symbol 3w~3s stand for the act of putting the 3 pound weight on one side of the balance scale, measuring out 3 pounds of sugar on the other side of the scale, and then setting aside the sugar.”

F.     Ensures that this invented notation and language is precise
“I need to be careful that I am differentiating between sugar that I am measuring and sugar I am using as a weight.”

5.     Tinker and Invent
      A.   Creates variations

B.     Looks at simpler examples when necessary (change variables to numbers, change values, reduce or increase the number of conditions, etc)

C.     Looks at more complicated examples when necessary

D.    Creates extensions and generalizations

E.     Creates algorithms for doing things

F.     Looks at statements that are generally false to see when they are true

G.     Creates and alters rules of a game

H.    Creates axioms for a mathematical structure

I.      Invents new mathematical systems that are innovative, but not arbitrary

6.    Visualize

A.     Uses pictures to describe and solve problems

B.     Uses manipulatives to describe and solve problems

C.     Reasons about shapes

“I see how this structure is made.”

D.    Visualizes data

E.     Looks for symmetry

F.     Visualizes relationships (using tools such as Venn diagrams and graphs)

G.     Vizualizes processes (using tools such as graphic organizers)

H.    Visualizes changes

I.      Visualizes calculations (such as doing arithmetic mentally)

7.    Strategize, Reason and Prove

A.     Moves from data driven conjectures to theory based conjectures

B.     Tests conjectures using thoughtful cases

C.     Proves conjectures using reasoning

E.    Looks for mistakes or holes in proofs

F.  Uses indirect reasoning or a counter-example (Park School)

E.  Uses inductive proof

8.    Connect

A.     Articulates how different skills and concepts are related

B.     Applies old skills and concepts to new material

C.     Describes problems and solutions using multiple representations

D.    Finds and exploits similarities between problems (invariants, isomorphisms)

9.    Listen and Collaborate

A.     Respectful to others when they are talking

B.     Asks for clarification when necessary

C.     Challenges others in a respectful way when there is disagreement

D.    Participates

E.     Ensures that everyone else has the chance to participate

F.     Willing to ask questions when needed

G.     Willing to help others when needed

H.    Shares work in an equitable way

I.      Gives others the opportunity to have “aha” moments

10. Contextualize, Reflect and Persevere

A.     Determines givens

B.     Eliminates unimportant information

C.     Makes and articulates reasonable assumptions

D.    Determines if answer is reasonable by looking at units, magnitudes, shape, limiting cases, etc.

E.     Determines if there are additional or easier explanations

F.     Continuously reflects on process

G.     Works on one problem for greater and greater lengths of time

H.    Spends more and more time stuck without giving up

My first day

It's taken me a little while to find the time to post this, but I wanted to post my first day experience.  Why?  Because it's great to be back!

The context
* 6-12 all girls independent school
* 1:1 laptop school (starting this year)
* 16 students in each class
* 4 desks of 4 students each
* 4 classes of 6th grade
* 50 minute period 3 days a week, one 75 minute period per week (staggered by class)
* most all students have (so far as I can tell at this point) relatively strong, but varying math background
* most students have very strong reading/writing skills
* every student has a (so far) positive feelings towards school

Cushy, I know.  I very much understand and appreciate the challenges that I don't have to face (large transient classes, different home languages, severe learning differences, unstable families, etc).  That being said, I am confident that this lesson would work in any classroom because 2 years ago I did a similar first day lesson in a very very different school; it was probably the best lesson of that year.

The lesson
We were on a special first day schedule so I saw each class for 60 minutes.

Introductions: names and something you enjoy outside of school (5 minutes)

The Desk Problem: Look at the people you are now sitting with.  In a short while, I am going to give you 5 minutes to get up, talk, strategize, and rearrange yourselves so that everyone is sitting with as many new people as possible.  I'll then ask each person individually how many new people they are sitting with.  If you're sitting with 2 new people, you get 2 points.  3 new people, 3 points.  I'll then add up all the points and that's your class score.  Let's see if you can beat my other classes.

This was inspired by a boat problem that I don't remember the details of involving fisherman sharing boats with different people every day of the week.

I tend to assume kids are going to figure things out faster than they actually do, but every one of my groups surprised me on this one and within 30 seconds figured out that one person from each table needed to go to 1 of the 4 tables.  After congratulating everyone on their success and talking a little about different methods of getting the 48 points, we brainstormed ways to "tweak" the problem in order to make it more challenging.  Some of the ideas:

1. Winning class is the class to get 48 points as quickly as possible.
2. Changing the number of people, tables and/or chairs at each table.
3. Conducting multiple rounds where you only score points for people you've never sat with.
4. Forcing every person to change tables

I haven't worked on it at all, but I think the problem of finding a maximum score as a function of # of people, # of tables, and # of chairs at each table would be an interesting problem to work on.

Breaking the code: I then gave a few short directions for quality group work (everyone's contributing, everyone's voice being heard, and not making any decisions without everyone understanding and being on board, no questions for me unless it was a question everyone had) and handed this out with no further directions.  They worked on this for the remainder of the period.

Some notable observations:

• no table asked "What are we supposed to do?"
• no table complained "We haven't learned how to do this."
• every table eventually asked "Do we have to decode the whole thing?" which led to some variation of the following conversation:
  Me: When do you think it would be fair to say you're done?
  Them: When we figure out the code for every letter?
  Me: Sounds good to me.
• Every student was engaged until the end of class
• Not every table did a great job of following the rules of group work

At the end of class, I handed out the homework, which kids quickly figured out was the decrypted version of what they'd been working on.

Great first day!  See you tomorrow.