Sunday, February 5, 2012


Today, one of my students posted to Facebook, "We're mixing food & statistics together, I'm excited."  I'm going to ignore the fact that she posted during class, without permission, and just be happy.  Pi Crust wrote about how happy she was after her class asked to take pictures of some really awesome student work and had several students change it to be their profile photo.   I totally understand her sentiment.  The student that posted is one that has had an up and down relationship with math and school and grades for the two years I've known her.  Being excited about school and showing it and posting it to Facebook is awesome.  Even though her excitement may have been more about food than the statistics, at least stats gets mentioned. 

As for the lesson that prompted the excitement, I wasn't terribly into it.  I had been feeling sick all week and  wasn't mentally all there.  So I found a lesson from a book of hands on math lessons for middle schoolers.   I didn't adapt it at all, I just made copies of the student worksheet.  I felt like a terrible teacher doing that.  I find lessons online all the time for use in my class, but always need to modify and adjust for my students.  Not adapting it felt like I wasn't thinking about MY kids, like I was just one of the many terrible teachers along the way who churn out copy after copy of worksheet after worksheet.  Maybe it was the cold talking. 

Anyways, even though the lesson was for younger grades, most of my students had forgotten the skills.  They needed to estimate how many M&Ms of each color were in a package, then calculate percent of total, make a bar graph, and a pie chart.  After opening the bag, they repeated the process, but this time for the actual number of each color.    

The number of students who wanted me to open the bag before the estimations was shocking.  Some were pretty upset that I wouldn't do it.  Others wanted me to tell them the total and then they would do the guessing.  It wasn't that they wanted to be closest to correct, they just couldn't work under uncertainty.  On the pie chart, they forgot the process, but once I showed them a sample section, they were completely capable.  I remember trying to teach the lesson to my 7th graders a few years ago and wanting to bang my head on the wall.  Figuring out how many degrees each section should be, then measuring it out was a several day set of lessons for 7th graders.  But for high schoolers, it was over in a matter of minutes.  This was one of the first times I really saw them "getting" skills that they didn't remember from earlier grades.  It was still a very procedural understanding, but I'll take what I can get. 

Next the lesson heads into whether a small vs large samples for estimating a population and accuracy.  Here's to hoping M&Ms can capture their attention for more than one day.  

Sunday, January 8, 2012

Pattern Sniffing

During my afternoon time as an EA, I spent most of the class working with an 8th grader yesterday.  The class was practicing solving single and multi step equations.  A particular student finished his sheet of 20 problems in about 10 minutes, whereas his classmates needed closer to 30 minutes.  He had a few errors and pretty consistently missed the negative sign in his final answer.

None of that is particularly interesting, but in watching him do the problems I realized he had no idea of why or what he was doing.  I know this student from last year and he has an amazing ability to see, memorize, and repeat patterns (pattern-sniffing as my PCMI peeps would call it).  He just looked at the numbers and knew "well you add this one away from the x to the one thats alone, then you divide by that other one."  He didn't know anything about "doing the opposite" or "trying to get the x by itself"  he just knew what sequence to enter into his calculator.  He didn't and in fact couldn't write any of his work down.  When he had made a mistake he couldn't see it or correct it and pretty much had a melt-down when I tried to explain and show him the reasons behind why he was doing each step.

Is this a problem?  Is it good as long as he gets the right answer and it doesn't matter if he has a concept of what is fundamentally happening when he is punching the keys?  Will it just self-correct when he gets more complex problems?  I think not.  From what I saw of him last year and from other struggling students, when the complexity rises, without an understanding of the base level, his ability to pattern-sniff is not longer as helpful and he is suddenly in the deep end without any floaties.  There isn't a base to build off of or go back to.  Similarly, when different types of problems are all jumbled together, then the pattern isn't as obvious and if the student doesn't happen to remember "oh this type of problem is done this exact way" then he is out of luck. Maybe that's part of why this brilliant boy, an others like him, does miserably on standardized tests but seems to understand everything at the time.  Had I been the classroom teacher and not had the chance to spend most of the hour sitting and watching and talking, I wouldn't have known how he was solving the problems or that he really knew nothing of the basic concepts.  

Bonus Story:  The reason I know this kid is an excellent pattern sniffer is because he is the fastest to ever beat me at the "Block Game."  I put out 15 pieces.  My opponent can choose to go first or second.  On each turn, we can choose to take 1, 2, or 3 pieces.  The person to take the last piece loses.  Over time students realize that I have a "strategy" that lets me win almost every time.  If a student wins, it is typically by dumb luck.  Using backward deduction will reveal that there is a set of moves that allows the first player to win 100% of the time.  The second player is able to win as long as the first player doesn't know the string of moves.  

Most kids take a few days and many many rounds to figure out how to win. This student figured it out in about 30 minutes.  When I asked him to explain how he knew what to do he explained that he just watched me.  All the other students watch the pieces, or their own plays.  He knew enough to watch the person who controlled the game.  From there, he memorized and analyzed every move I made to figure out what to do in each situation.  He didn't know the nice "summary" of the moves or realize that there was a particular string that I was playing each time.  He just knew that if my opponent pulled 3, then I pulled 3 and so on.  I was very impressed by him and his ingenuity.  He has become very adapted to finding and exploiting patterns, but isn't yet capable of taking the next step and coming up with a clear set of rules.

Back from Break

Last week was the first week back from break and we only saw each class three times because of having Monday off.  Thank goodness it was a short week.  It was fairly easy week, but draining.  Even though I kept a similar sleep schedule over break, by 7PM I was exhausted.

In my statistics class with the juniors and seniors I did two longer application problems. One (found here) on misleading graphs and statements.  I was a bit nervous with the political tone of the assignment, but no one really noticed it.  Only one student picked up on the fact that the ACS might be trying to mislead with the graph in order to help out Giuliani.  On one hand, it was good because there were no political battles in class, on the other hand, it meant that the students still aren't totally thinking.

My biggest issue with my statistics class is that the students are so used to being led through problems and worksheets that anything requiring actual thought is really difficult.  One girl even commented, "You really come up with good questions.  I have to think.  I don't like it.  Why do you do that to us?"  My whole goal is for them to think, especially when it comes to presentation of graphs and data and arguments.  I've been pretty disappointed in this unit and haven't figured out how to remedy it.    They still can't tell the different between a causal and a correlational argument unless the title says "X causes Y."  They only see the articles at face value.  I gave them a series of 5 articles on soda and teen violence where some articles said soda caused it, others explained a correlation, and the last put down the articles that claimed causation.  On the whole, the class didn't see the differences between the articles and thought it was all totally dumb because adults were trying to say that soda made them violent.  Even the attempt at a discussion after went no where.  The one or two kids that "got it" didn't really say anything and the others just stayed with their own initial perception rather than evaluating the article.

I think the biggest issue with my lessons on them identifying inappropriate causation in articles is that many of the students do not read well.  They only had to read two of the 5 articles and each article was a page or two, but it was still tough for them.  Others probably never did the reading.  I want to do an interdisciplinary math class, but how can I get around, differentiate, scaffold, or support the low levels of reading?    The students want to do well. They want to learn.  They like the class and they try.  But without the ability to read, interpret, and analyze newspaper articles they are at a big deficit.  I want to say "I can't correct it all" and "It's stats class, I don't have time to teach reading too," but I can't have it both ways.  I can't demand reading skills and ask them to apply knowledge they don't have.  I need to make time for it.  

Wednesday, December 14, 2011

TIES Presentation

Yesterday, after spending my morning teaching like usual, I headed over to join my colleagues in a presentation about the use of Google Docs in the classroom.  This year I've changed the way my class works and been doing more with Moodle, so I haven't used Google Docs as much.  Searching through all of my old notes reminded me how great they are.

Use 1: Warm-Ups
I used to have students write their answer to a problem I had on the board.  Then I'd walk around, making sure they all had the answer, then go over the answer.  I spent most of my time managing the scoring or trying to get students started  There was no time to work with students.

Enter Google Forms.  Students walk into class then open the form.  It looks the same way every day.
When they are finished, they hit send.  I can choose to have a response pop up with the answer I was looking for or directions about what to do next.  I can spend my time walking around working with students on the problems.  I don't collect any papers, don't have to pass anything back.

Better yet- I can have instant answer analysis.  On my screen it looks like this: 

I can see who turned it in on time, who has a complete and thorough answer, who copied and pasted from a website, and who really needs some more help on isosceles triangles (like the student who said a triangle with two equal sides).  I can scan through all the answers in about a minute.  When it comes time to teach the class or do the classwork, I can decide who to go work with first because I know they are missing the previous lesson. 

Use 2:  Pre-tests
I don't always love them.  I feel like kids know more than they show, but sometimes, when there is a topic I am convinced they know, or, one they think they know but I'm pretty positive they don't, I use them.  For me, pre-tests are best when they kids know where they are and I get a sense where the class is and individuals are.   But, that doesn't mean I love grading them (and I don't really need to, since it is just to inform my own teaching) and I always want to reduce my paper pile.  

Enter: Google Forms 
I just set up a Google Form with fill in the blanks for each question.  I still give the paper test, or a pdf of the paper test, and the kids can do work on paper, but the final answers are entered into the form.  This means I get a quick idea of whether or not they ended up getting the answer, and can look at their work if I want to. 

I can look at the spreadsheet or the summary.  I also liked letting the students see the summary on pre-tests too.  Then the one kid who put 18 instead of 17 knows right away that he was probably just making a simple error.  The student who put 34 realizes that they forgot something.  Or if the whole class gets the question wrong, then they can see that.  When I look at the summary, if I scroll over a response, it tells me the name associated with it, so forms fits my goals of knowing where the whole class is, and checking on individual students. 

None of this is revolutionary, but it made/makes my class a lot more streamlined and easier for me.  And I rarely get stuck under piles of grading. 

Wednesday, November 30, 2011

An Awesome Day

Today was one of those days.  Everything went well, and even the things that didn't, well I don't even remember them.

My Geometry classes are diving into proofs, only they don't really seem to know it.  Last year proofs went horribly.  The kids hated it and fought me every step of the process. "TWO Columns?  WHY!"  "FLOW CHARTS! Those are stupid. "  "PARAGRAPHS! You want me to write, like actual sentences, in math class?  No way."   This year, we slowly crept into proofs with the points lines, planes project.  Then on Monday, we reviewed the best slides for each one.  Students who had slides selected were beaming with pride.  Students who weren't selected were still trying to argue why their interpretation was correct or why their photo was better.  (I got to them! They cared! In math class of all places. And they cared about whether or not their justification and representation of a postulate about undefined terms was best.  Whodathunkit).

Yesterday, we did some more Always, Sometimes, Never statements and had the "Best of PLP Project" and the list of postulates as a guide.  Explaining that what we are doing today is just like the project, except instead of taking pictures and making diagrams as proof, we have a set of statements that mathematicians have all agreed are true.   It went fantastically. The students who were still shaky after the project were rock solid with the postulates to ground them in what was always true.

Today, we kept with the postulates and added in two column proofs and talked about algebraic statements.  Not one student even made a peep to complain about the two columns.  No one asked why we needed to write a reason for every step.  They just bought into the idea of really being precise and explaining everything.  I really like coming from a place where they are confident (solving for x) and adding in the new things.  I made sure to have a problem set that grew in difficulty.  It started with x+2=9, then added in coefficients (6x+3=15), then multiple x terms on one side, then x's on both side, then some challenge problems of solving systems of equations.  They haven't formally seen solving systems of equations, but the students who reached those problems were able to figure out what to do in order to solve by using their list of properties of equality.  I know they are motivated and engaged learners, but they figured out systems on their own!  That was awesome to see.  And students who did not have a strong algebra foundation were able to pick up on the pattern quickly.  That brought a big smile to my face.

In Stats, a student who I used to have a great connection with has been battling with me all November.  I'm part of the problem, it's probably easy for her to tell that I'm frustrated and angry because I am.  With all the time we've put in together (including several Saturdays), she seemed to be throwing it in my face, with a lot of foul language and attitude to top it off.  Today, magically, it was all better.  She worked all class long, asked questions, and was generally pleasant.

The other students in the class finally saw the point of this cardiovascular project we've been working on all week (will post more later).  "Ohhh this isn't so bad.  The questions are short and help us to figure out what is important in the article.  And the math parts are mostly review.  The only new part is clicking on a different type of graph in Excel.  I guess we have to do a lot of thinking too, but that's not so bad.  We have all the pieces, we just need to decide how they go together."  I couldn't have said it better myself.  Way to go guys!

Last week I was sad about the number of students with missing assignments or who needed to retake quizzes, but this week, they are getting their act together.  Yesterday a student stayed after who I keep trying to work with in class, but he brushes me off with, "I'm good.  No thanks."  We went over a quiz and he said, "Man, if it could just be this fast in class, I'd get everything."  When I said, it can be like this, thats what I'm asking to do when I say, "How's it going?" or "Need any help?"  He got this confused look, and just said, "Oh.  Well then definitely.  Let's do it!"

Similarly, in geometry, I have two students who I can see are struggling, one of whom just get angry and talks about how class sucks, the other comes to class late and avoids work by trying to get kicked out.  All year I've been trying to engage with them but they weren't ready.  Today, both of them called me over and asked for help, and totally got what was going on after some one on one time.  I ignored the other kids for a good chunk of class, but the others were ok.   Those two were on task the entire work time.  

Another geometry student who almost failed last quarter and squeaked by with a D has an A.  He turned in one of the best Points, Lines, Planes projects and came today to study for a quiz that isn't until Friday.  He didn't realize he had an A and was beaming when I told him.  Then it was my turn to blush when he showed me how he was using Notebook and the way I taught him to make 3-D drawings for a history project.  Another student who failed last quarter with about 7% has a 97%!   He finished his homework on time, caught up on the work from when he was absent, and left with a big, smile.

All in all a great great day.

Sunday, November 20, 2011

Process vs. Product

In the last post I wrote about one of my students getting pretty upset about me not telling him if he was right or not.  When he told me, "I've never had a math teacher be like this,"  I wondered if there was an emphasis on math.  Are his other teachers like this?  Is it just math that he expects to be delivered in neat, compartmentalized chunks of notes, problem sets, and correct answers?  This was a student who aces every test and project, so his reluctance wasn't an issue of not comprehending the statements or assignment.  The discomfort he felt with not having me tell the correct answer is exactly the reason I wanted to do the project this way.  I could have done the same project but hinted, and suggested, and told them if they were right.  If done that way, I'd guess that they students would earn better marks on a multiple choice test about points, lines, and planes, but then they would still think that there was one correct answer.  They wouldn't have that same feeling as when Ms. A and I debated if there was such a thing as two coinciding planes.  If there was, would that count as an intersection?  

In forcing them to actually invest, to think, to struggle, the math was getting to them.  I had more kids be frustrated, throw fits, and tell me they were quitting.  But they didn't.  I told them to pick an answer, and move forward with it.  If they got more stuck, they needed to back up and try a different answer.  Or talk it out with a classmate.

It was also really eye-opening to me to see how different the process focused approach is from being product oriented.  In the past I have claimed to be process oriented.  My school is art focused so I hear a lot of discussion about being process oriented.  Typically it goes hand in hand with a discussion about how students work at different paces and having rigid deadlines or only accepting one format of answer is stymying creative output.  As a result I decided to allow students to retake tests, to emphasize that the process of learning is important, not learning by a deadline.  In actuality, that is still product oriented.  There is a right answer (a product) that I want them to reach, and I don't care how they get there or how long it takes, just that they get there.  I also decided to allow for multiple formates on most of my projects.  During reviews students can turn in videos or word problems or a review game or a set of textbook problems.  Again, I felt like I was being process oriented, but there was still an outcome fixed in my mind.  I wanted them to review.  I didn't care how they did that, just that they did.

The execution of the project was the first time I feel like I have actually been process focused.  I fought every urge to tell them "No! That's not right!"  I fought (almost) every urge to hint, "Hmmm.  Maybe look at number 2 again,"  or "Well, do you really think a plane never ends?

Ultimately, with the recap next week, there is a product.  I want the students to know that if two planes intersect, they always intersect in a line.  In a math class, I can't deny that there is content, some nugget of information I am hoping that they will learn.  In showing a "best" answer, I hope that I am not diminishing my initial goal, for the students to learn that it's ok to struggle in math, but that they need to push forward and argue for why they think their answer is correct.

The one time I broke down was when a student had a fantastic photo of why a plane never had an edge (I'll upload it once he turns his project in).  He thought the statement "Planes have an edge" was always true.  I asked why and he said, "well it's impossible for something to not end."  I asked if lines ended or went on forever and ever.  He said, "well lines go on forever and ever," so I countered that it was possible for things to not end.  He thought for a minute and said, "So if two lines cross in an x and they both go on forever and ever and you fill in the space in between the lines with like paper or more lines or something, then is that a plane?  Wait!  That's like the graphs we make.  OH! that's called the coordinate plane isn't it?  Is that why it's called the coordinate plane? Is that a plane?"    I initially felt bad that I was nudging him, but after that grand epiphany I was grinning ear to ear.   He was visualizing and making connections and it was beautiful.  Not bad for last period on a Friday.

Teaching Points, Lines, and Planes

The time has finally come! I get to teach the lesson I spent weeks developing!  For a bit of background:  see my previous post about the PCMI project.  Also: I teach 3 sections of geometry.  My co-worker, Ms. A, teaches 2 additional sections.  All of our 9th graders are enrolled in geometry except for a few students who took it online last year.  There are a few sophomores, juniors, and seniors in the classes as well, typically students from other districts who still have Algebra in the 9th grade.   We both taught the project at the same time.

When Ms. A and I decided to teach this lesson we wanted it to be as an introduction to proofs.   As a result, we wanted to emphasize them arguing for whatever they felt was correct.  The photos, diagrams, and sentences would be their supporting statements.

I started the activity by going through the definitions worksheet (their givens).  I changed the third column to be "real-world approximations."  I have a number of students who would need the scaffolding of having a list of what points, lines, and planes could be represented with.  I was really glad we did that because when students were struggling I just sent them back to the definitions sheet.  I was originally going to have the students use geogebra to make the diagrams, but I didn't ever really teach them how.  I left it open as an option, but encourages them to do it by hand or use the shapes/lines tools in SMART notebook (which they all have installed on their computers).   I really wish I had spent more pre-activity time teaching them how to make the drawings and explaining why they were important to know how to do.  Ms. A spent more time on explaining the drawings, so we shall see if she gets better results.

When I was explaining the project I spelled it out really clearly that part of my goal was to introduce them to a new way of thinking about mathematics.  During this unit, and going forward, there might not be one right answer.  I don't really care whether they pick always, sometimes, or never, as long as their photos, diagrams, and sentences back up their argument.  I would not be telling students if they were right or wrong at any point.  In fact, I wouldn't really be offering much help at all.  They have teammates, classmates, a definitions sheet, a textbook, and everything online.  

I explained that there were a few statements that were purposefully unclear.  They needed to choose an interpretation and run with it.  I talked about how my math teacher friend and I argued for hours at math camp and that Ms. A and I got into a fight about the answer to one of the statements the night before.  One smart alek asked if we threw punches or just swore at each other.   When I said, "neither.  It was pretty civilized,"  he turned to the class and said, "So it was both. They totally cussed each other out.  Ms A took a swing, but Ms. W knocked her out just after." At least he was loyal and picked me to be the winner.  

Throughout the project they kept trying to get me to tell them, but I stayed strong. One student was having a hard time with "Two lines that don't intersect are always parallel."  He recognized that with the lines being on separate planes they could never intersect and not be parallel and wanted to know if that was right.  I kept evading his demands to tell him if he was right and he got quite upset.  Eventually he told me, "I've never had a math teacher be like this.  Math has an answer.  You know the answer.  JUST TELL ME! Isn't it your job to teach?  You're not doing anything"  I just reiterated that in mathematics, people don't know if what they are researching is correct or not, they just have to make assertions and proceed.  After he calmed down and said, "Well, we haven't really talked about two planes before, so I'm going to assume it means one plane and go with Always True."  At the time, I was a bit sad that he went that route, but if I'm saying it's about their thought process and argument, I have to be really pleased with that moment.   He got over his frustration, chose a path, came up with an argument, and worked it out, which is what I wanted.

In being more hands-off, I know that a lot of the groups have some major misconceptions.  But I'm ok with that.  Points, Lines, Planes, and those crazy drawings of them intersecting etc are always a confusing topic for my students.  I'd rather them learn that math can have that fuzzy I don't know if I'm right or not, but I'm going to go with it feeling too.  After we come back from break, I will use Kate Nowak's wrap-up activity of presenting the "best" 10 statements and have them correct 4 slides with misconceptions.

Class has been a bit messy, but I'm pleased with the progress and the project.  Of course, the day that the kids are laying on the floor (a line on a plane), rearranging photos on the wall, and wandering the high school with cameras, is the day that the principal wanders by, the vice-principal stops in for an observation, and the tour group of potential families parks itself in front of the math rooms rooms.  It was a controlled chaos.  Ms. A was pretty calm about it and reminded me that with projects, things look a bit messier, and that's part of the point.  Messy is ok.  Neither my vice principal or principal said a word to me about it.  Didn't ask why math class needed 10 digital cameras or why the students were all over the school.   So my worries were unfounded and I love that they were unfounded.