Category: personal

Teaching + Openness = Pay It Forward

I’ve always considered myself a rational person.  I mean, I majored in math and then went on to get a masters in pure math.  I really always loved the logic and thought that went into proof and how I could think rationally about so many things in life.  Even when life threw me curve balls, I could find a way to rationalize most things and think them through in some way.

In 1996, my mother died of colon cancer at age 56 – probability of a female getting colon cancer at that age around that time was about .018%.

In 2015, my father succumbed to a number of cancers (but mainly breast cancer that metasticized in his brain) at the age of 83.  He was quite a fighter – probability of a male getting breast cancer now is about .01%.

In November 2018, my son, at age 21 has been diagnosed with Acute Myeloid Leukemia – probability of this happening is 0.2%. (all from cancerstatisticscenter.cancer.org )

All of the doctors say that none of these are related to each other at all – independent events, right? So the probability of this happening in my life is 3.6 x 10^-5.  Pretty crazy, huh? Next to impossible, probably isn’t supposed to happen to anyone.

After trying to deal with this rationally, I decided to deal with it irrationally.  I wouldn’t talk to anyone about it.  Clearly no one could help – what was the point of discussing it with anyone?  Loss, grief and sadness were always a part of my life and this was nothing new.  I just had to suck it up and get over it.  Do my best and move on.  Get through it, I always did.

My husband refused though.  We deal with things differently.  He wanted to tell the world.  He writes emails giving people updates about how my son is doing – people I haven’t even met and I felt it was a huge invasion of our privacy.  But something happened that totally surprised me.   These people, some I knew and some I didn’t know, started sending us things – cards, notes, emails, gifts cards to Panera, cookies, offers to make us dinner, and many other things I never expected.  I couldn’t fathom why they would do this.  I would say to them, “I don’t know how to repay you.” And their response would always be, “You don’t have to, just pay it forward.”

180 degree turn to the world today – students in MAGA hats harrassing indigenous peoples at a protest rally, supreme court upholding the ban on transgender peoples in the armed forces, our bullying, self-absorbed president keeping federal workers from having their paychecks, Republican elected officials losing their integrity and concept of their job to look out for their constiuents – what is happening?  Where is the kindness and care in the world today?  I was losing it quite regularly.

In all of this confusion in my brain, I felt the need to let people know that I had to cancel some public speaking and courses that I had committed to this spring.  I was nervous about tweeting about something so personal, but I know some folks who follow might have been trying to sign up for my course or looking forward to my NCTM talk.

That’s when it hit me.  I received so much love, support and kindness from the virtual PLC of math teachers that I did not expect.  I could not believe it.  People I knew, people I knew virtually, people I’ve never met – all reaching out and telling me that they cared and wished us the best in this struggle.  It was an interesting comfort at a time when my personal struggle was being exacerbated by the complex chaos of the world today.

I do believe that there is no balance between chaos and calmness.  We just learn to take both as it comes.  The peace that was given to me from the #MTBoS and #iteachmath community definitely allowed me the moment of calm that I needed and I will always remember that.  You all are surely paying it forward from someone who did something for you and I hope that I can do the same someday.  I do think that teaching has something to do with paying it forward as we do every day, honestly.  People say they teach because they love it.  But in some way, the relationality that it takes to be a good teacher, be open with people and understand our students is the way in which we pay it forward every day.

Thank you so much for giving me this moment of calm at this time in my life.

 

Ch-Ch-Ch-Changes….Turn and Face the Strange.

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Strange fascination, fascinating me
Changes are taking the pace I’m going through

Well, you know it’s been a long time since you wrote a blogpost when a perfect stranger sends you an email and says,  “I hope you’re just busy or I’m looking at an old blog you don’t update anymore or something and everything is ok.” It was a very nice email and I often forget when I get busy and wrapped up in what’s going on in my own life that there are actually people that read my blog and my tweets.  I need to keep up with my communications and I feel horrible.  So I have many apolgies to make.

But first let me tell a story that just happened a few weeks ago.  I was doing a 2-day workshop at a school that had made a decision to implement PBL in the first two years of their high school curriculum.  Because it was such a cross-curricular choice, almost everyone in the department was going to be required to teach a PBL course at some level or another.  We had spent a great deal of time talking about ways to facilitate discussion, make best use of class time, allow students the agency to work out understanding together, etc.  At the end of the second day, we were reflecting on what we were still concerned about for the coming fall.

Many teachers had very specific questions about how to put the theory into practice, but then one wonderful veteran teacher spoke up.  He talked about how he was very excited to try this type of teaching and what it really meant for the students’ learning in his class.  However, he then got very serious and sincere – he basically said, “What if I’m not good at it? I am mean I know I am very good at explaining mathematics to students, I’ve been doing it for 20 years this way.  I enjoy helping kids understand by telling them my understanding.  What happens if I am not good at holding back and letting them struggle?”

Of all the questions that teachers have asked me PD workshops, I thought this was the most honest and probably most unasked question that existed.  He was being extremely vulnerable and I think asking what everyone wanted to know.  We had a wonderful conversation about how he coached football for many years and how this type of teaching is very similar to coaching.  We talked about the satisfaction that you see on students’ faces when they are allowed to take ownership of the knowledge in the classroom and each others’ understanding.  It will be a difficult change in his pedagogy but it is worth a try. He’s probably one of the most brave teachers I have met in my work.

If you are one of these people that is interested in learning more about PBL and how it could enhance your classroom pedagogy, I am looking to pilot an online course this summer.  Click here to subscribe to a mailing to find out more and possibly enroll in the class.

So honestly, change is not that bad – it gives me anxiety and makes me worry.  But honestly, communicating and learning all about it is really the best.  I am moving out of the classroom for a year next year to see how that goes.  I’ll be doing Learning Experience Design for Online Learning and also Teacher Support for Math Teachers who are teaching with PBL.  Hopefully, I’ll also be planning the 2018 PBL Summit as well!

Turning to face the strange – and taking the pace I’m going through – please be forgiving of the changes in my life while I get back into blogging!

To Hillary, With Gratitude

This morning as I woke up and found out about the results of last night’s election I was at first filled with despair and finally got myself somewhat out of that funk.  Then I thought about what Hillary Clinton must be feeling – she must be exhausted of course.  What did it take to put all of that energy into this campaign?  And those years of service to this country? And to put up with her husband? And the criticism?  This is not to say she didn’t make mistakes in the public eye of course.  I’m not saying I didn’t disagree with some of her stances, but I just want to look at it from the female perspective.  What I want to say to Hillary right now is thank you.  Thank you for being the first woman to have to go through the ordeal of running for president and dealing with all of the mess that goes with that.  I can’t imagine what that was like.

I have to say that in my career I know what it’s like to be one of two women in a meeting room and have to work extra hard to get a group to listen to your point.  Or perhaps to couch what you want to say in terms that the men will want to hear until they come over to your side in order to get them to vote your way on a certain agenda item or thinking twice about what I wore so as not to get judged.  The diplomatic skills that are acquired in just being a woman in an administrative position are invaluable because of the ways in which you know you need to listen and be heard. Being a woman in mathematics, the message is usually clear at national meetings when the majority of conference-goers are female classroom educators and the presenters are more often male speakers who are not currently classroom teachers.  In my graduate school education in mathematics I had one female professor and I was the only female in the Masters program.  You learn to “blend in” by speaking like them, working like them and going about your business like them.

I wonder if there isn’t a little part of Hillary that this morning just said “Phew, no more of that faking it.”  She was tired of being the male-culture-created part of herself that she had to be in order to run for president.  A few female heads of school that I have spoken with have said that in order to lead, many women are expected to downplay their feminine qualities – to not cry or be emotional, to be sure they are surrounded by male advisors so no one can say you made mistakes because you “are a woman.”  Spending so much time worrying about balancing speaking your mind with being nice to everyone so you are not labeled “bitchy” gets really tiresome.

What this election taught me overall is that misogyny is alive and well in the U.S. (not that Hillary needed to learn that) even more than racism.  My guess at this point is that we will elect a gay man in the future before we elect a woman but either way, I am grateful for Hillary and all she has done to pave the way for each other woman who comes next.  I read that Kamala Harris (CA) was the second black woman to be elected to the Senate, Ilhan Omar (MN) was the first Somali-American woman elected to Congress and Catherine Cortez Masto (NV) was the first Latina Senator to be elected.

I’d like to think that Hillary is waking up today really looking forward to spending some time as a grandmother, writing a book and working on the next great way to help kids, health care reform and education.  Sure that’s just me being idealistic, but as a woman, I would like to think that’s what I would do – well, after crying after losing for a little while.

Need Some Help Looking Forward

So I’m trying to figure out how to reach more people and thinking about the future of my professional development plans with PBL for all levels of teachers.  I’ve gotten some great feedback from people about the PBL Math Summit so far (from the two years we’ve had it) and I have some ideas about how to create some better online resources too.  If you have the time, and are interested in helping me out, would you please fill out this short survey about PD Needs for PBL Math Professional Development.  Also, tell others who could give me insights too.  Thanks so much for reading my blog and for also being inspired to be interested in PBL math teaching!

 

A Math Girl’s Story or the Introduction to My Dissertation

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I am not really a negative blogger but I do have to say how tired I am of research reports that over and over again talk about the way we are not doing enough to support girls in math education (or other underrepresented populations of students).  There is enough evidence now from many research reports (NCTM, 2016, Why so few? AAUW, 2010, Riegle-Crumb, et al, 2012 I could go on…) that show that there is little difference in math ability by gender and that the reasons that girls and women choose to leave STEM fields are culturally related.  And yet, we still need a white male to make statements like:

“I believe that this issue of women’s confidence is cultural, not biological. It fits in with all we know about stereotype threat. When the message is that women are not expected to do as well as men in mathematics, negative signals loom very large. Calculus—as taught in most of our colleges and universities—is filled with negative signals.”

  • David Bressoud, MAA Blogpost, Launchings, October 1, 2016

Now, I don’t know Mr. Bressoud and perhaps this most recent research study really pushed him over the edge to being a believer, so no offense meant.  But I’ve just had enough of it.  My life experience had been based on all of this and it’s enough for me.

We need to do more to change the way math is taught in the U.S. so that more girls (and other underrepresented students) feel connected and desire learning, feel like they belong and their ideas and voices are valued within the context of mathematics and the community of mathematics learning – at the secondary level and the college level. Period.

Here is the introduction to my dissertation, “Dismantling the Birdcage:  Adolescent Girls’ Attitudes towards Learning Mathematics with a Relational Pedagogy in a Problem-Based Environment” (2013) (don’t feel the need to read the whole thing).

“If you look very closely at just one wire in the cage, you cannot see the other wires…You could look one wire up and down the length of it, and be unable to see why a bird would not just fly around the wire any time it wanted go somewhere…There is no physical property of any one wire…that will reveal how a bird could be inhibited or harmed by it except in the most accidental way.  It is only when you step back, stop looking at the wires one by one and take a macroscopic view of the whole cage, that you can see why the bird does not go anywhere; and then you will see it in a moment.  It is perfectly obvious that the bird is surrounded by a network of systematically related barriers, no one of which would be the least hindrance to its flight but which by their relations to each other, are as confining as the solid walls of a dungeon”. (p.5)

-Marilyn Frye, Oppression, in The Politics of Reality (1983)

I will begin with a story.  It is the story of a young girl excited and interested in learning and doing in all aspects of her elementary education.  Luckily, her parents were always encouraging and supportive of her learning goals and her initial schooling included “enrichment” class for which she was chosen to receive out-of-class group instruction in advanced topics – including mathematics and science.  The girl was confident, motivated and eager to move forward in her exploration of new topics and share these ideas with her friends and family.  As middle school approached, it became clearer to the girl that categorizing students by ability became more important and she realized that her work and grades in her classes, as opposed to her interest in mathematics, would begin to determine her path through her education.  The pressure of this realization, and possibly other determinants, affected her performance and she was placed in a pre-algebra course in the eighth grade, which she knew, even then, would set her on a trajectory that somehow indicated less success.

However, the following year, the girl’s work in algebra was so successful that her teacher that year recommended that this adolescent girl now double-up in her mathematics courses and take geometry and a second year algebra course concurrently. Reinvigorated and more confident in her abilities, she regained her momentum and faith in herself as a mathematics student, although the fun with her peers and connections with the teacher from her “enrichment” classes were now a thing of the past.  Mathematics seemed made up of a set of disjointed courses that needed to be passed sequentially in order to fulfill the requirements for graduation.

Finally, the ultimate course in mathematics came during her senior year of high school where she would be able to truly show that she had made it to the top – Advanced Placement Calculus.  However, difficulties arose when little interaction occurred between the teacher and the students surrounding mathematics in the classroom.  Utilizing a textbook that was published almost 25 years earlier, the now young woman felt isolated and alone in a class where asking questions seemed to signify weakness and demanding an explanation also showed that a student was incompetent. Students who could easily and quickly replicate the mathematical exercises performed by the teacher were praised and favored whereas those with difficulties were dismissed and even asked not to take the Advanced Placement exam at the end of the year.  Sadly, our young lady was among those disinvited to be part of the elite exam takers.  This was a turning point in her desire to continue with mathematics as an intellectual endeavor.  She vowed to never take a math class again and moved on to college to pursue music as a major field of study.

On arriving at her chosen college in the fall, the young woman was required to take a mathematics placement exam in order to fulfill her natural science portfolio requirement.  Begrudgingly, she took the short test and a few days later she was told she could register for Calculus III.  How was this possible?  She did not even take the AP exam in May and barely passed the course in high school.  Would this roller coaster ride of messages of encouragement and discouragement ever end?  Who did they think were, telling her to move into Calculus III?  She would show them and just retake Calculus I and be done with it – get that natural science requirement out of the way and move onto much more interesting and meaningful courses so that she could leave mathematics in the dust.

However, something surprising happened in that basic Calculus I course that fall.  The young woman had an interested professor that saw her potential and talents.  The professor engaged her in conversation about mathematical justification and questioning. Citing the young woman’s exceptional ability in Calculus, the professor questioned why she was even in the class.  At the end of the term, the professor had convinced the young woman to continue on and even elect to take a computer programming course to see if she liked it. “Why not?” the professor said, “and it’s required just in case you decide to be a math major someday.”  The young woman laughed out loud.

One by one mathematics courses came and went.  The smaller seminar style of the upper level mathematics electives worked extremely well for her learning style.  Although she was often one of two, or the only female in the course, the girl believed that she was supported and encouraged by the professors she met.  There was a community of mathematicians who allowed her to grow and develop her skills, as opposed to suppress and discourage them.  Abstract courses like Linear Algebra, Number Theory and Topology connected much of the mathematics that for too long seemed discrete and disconnected. After serving as a teaching assistant for much of the department and receiving honors on her senior thesis, the girl was encouraged to apply to graduate programs.

In graduate school, once again the girl found isolation among a male-dominated community of academics and senior mathematics professors seemed to look differently at her, wondering why she was not in the Master of Arts in Teaching program with the women.  After two years of struggling with the environment, but quite enjoying and thriving in the teaching classroom, the young woman realized her gift and decided to find a way to make her journey complete.  Combining her love of mathematics and her talents for teaching was the way to make a life worth living while also bringing the consistent support and encouragement to students that she so greatly needed all those years.  Although it took her 20 years to realize this direction, ultimately it became a passion and lifelong commitment.

At this point in stories like these it is generally tradition to state “The End.”  However, at this point, I would change the phrase and say “The Beginning.”  Yes, it was just this story that has led me to this place and passion in my research for gender equity in mathematics education.  Now, 22 years into my teaching career I can look back and see how it began with this personal experience, but when I started my teaching and my doctoral program, I am not sure I was as aware of the implications my own story had for my research and teaching interests.  As my career brought me in and out of single-sex schools, my research interests led me towards a relational pedagogy.  As individual students that I crossed paths with shared their own hopes and fears about their relationship with mathematics, it began to be clear to me that it was more than a coincidence that my dissertation research, and perhaps my life’s work, would be centered around finding ways to improve the education of marginalized students in mathematics education, if possible.

And so it was the beginning – the beginning of a long journey with this question about how it might happen – how to improve the learning of students who feel marginalized in the world of mathematics as I once did.  But I would begin with one group of marginalized students in the mathematics classroom to whom I could relate most readily; adolescent girls.

One of the Original “Makers”

Apologies to any faithful readers out there – I have had a heck of a summer – way too much going on.  Usually during the summer, I keep up with my blog much more because I am doing such interesting readings and teaching conferences, etc. (although I’m running a conference for the first time in my life!) However, this summer I was dealing with one of my biggest losses – the passing of my father after his 8 year battle with breast cancer.  I thought I would honor him by writing a post talking about a problem that I wrote a few years ago, well actually a series of problems that utilized his work when teachers of algebra I asked me how I taught the concept of slope.  So dad, this one’s for you.

In 1986, my dad, Francesco (Frank) Schettino, was asked to work on the renovations for the centennial project for the Statue of Liberty.  He was a structural steel detailer (also known as a draftsman) but he was really good at his job.  Everywhere we went with my dad when I was younger, he would stop and comment about the way buildings were built or if the structure of some stairs, windows or door frames was out of wack.  He could tell you if something was going to fall down in 10 years, just by looking at it.  At his wake last week, one of the project managers from a steel construction company that he worked on jobs for told me that they would save the interesting, most challenging jobs for him because they knew he would love it and do it right.
photo (1)I remember sitting with my dad at his huge drafting desk and seeing the drawings of the spiral stairs in the Statue of Liberty.  He talked to me about the trigonometry and the geometry of the circles that were necessary for the widths that were regulated for the number of people that they needed to walk up and down the stairs.  This all blew my mind at the time – that he needed to consider all of this.  So to be able to write problems that introduce slope to students about this was just a bit simpler to me.

If you take a look at my motivational problems on slope and equations of lines I believe it’s numbers 2 and 3 that refer to his work (excuse the small typo).  Over the years I’ve meant to go back and edit these a number of times.  If you are someone who has taken my course at the Anja S. Greer Math, Science and Technology Conference at Exeter, you are probably familiar with this series of questions because we have discussed these at length and talked about how students have reacted to them (and how different adult teacher-students have as well).  We have assumed no prior knowledge of slope (especially the formula) or the terminology at all.

Some questions that have come up: (with both students and the teacher-students I’ve worked with)

1. What does a graphical representation of “stairs” mean to students?
2. What does “steeper” mean and what causes stairs to be steep?
3.  Why are we given the “average” horizontal run for the spiral stairs? Would another measurement be better?
4. Why does the problem ask for the rise/run ratios?  Is there a better way to measure steepness?
5. (from a teacher perspective) why introduce the term “slope” in #3? can we just keep calling it steepness?

These are such rich and interesting questions. The questions of scaffolding terminology and when and how to introduce concepts are always the most difficult.  Those we grapple with specifically for our own students.  I always err on the side of allowing them to keep calling it steepness as long as they want, but as soon as we need to start generalizing to the abstract idea of the equation of the line or coming up with how to calculate that “steepness” a common language of mathematics will be necessary.  This is also where I take a lesson from my dad in terms of my teaching.  His great parenting style was to listen to me and my sisters and see where we were at – how much did we know about a certain situation and how we were going to handle it.  If he felt like we knew what we were doing, he might wait and see how it turned out instead of jumping in and giving advice.  However, if he was really worried about what was going to happen, he wouldn’t hesitate to say something like “Well, I don’t know…”  His subtle concern but growing wisdom always let us know that there was something wrong in our logic but that he also trusted us to think things through – but we knew that he was always there to support and guide.  There’s definitely been a bit of his influence in my career and maybe now in yours too.

NCTM 2015 – Reflections

I know I’m a little late but I did want to post my own handouts and talk a little bit about my experiences at NCTM Boston this year.  I want to thank all of the great speakers  that I saw including Robert Kaplinsky, Ron Lancaster, Maria Hernandez, Dan Teague, The Young People’s Project (Bob Moses’ Group), Deborah Ball, Elham Kazemi and of course the inspiring Jo Boaler.  One of the things I thought was great about Jo Boaler’s talk on Thursday night was that even though I had heard a great deal of what she had said before, there was a different tone in the room.  I’ve been a fan of Jo’s since I first read her research in 2001 when I started my doctoral work on girls’ attitudes towards mathematics learning.  What I felt that was different that night was that she was no longer trying to convince people of anything.  There was a different message and that was “join the revolution” and the audience seemed to be on board and excited.  It made me feel very energized and empowered that a huge ballroom full of mathematics educators had bought into her ideas and were enthusiastic to make change happen.

Some of the best times I had were spent just connecting and reconnecting with people – some who I met for the first time (MTBoS folks and other Twitter folks I met F2F which was really nice) and others who were old friends who mean a great deal to me.  I forget how much the mathematical community of professionals enriches my life and makes me proud to do what I do.  Thanks to everyone who reached out to find me and say hi – or tell me a story, talk to me about what they are doing or ask a question about what I am doing.  You are all inspiring to me.

I left the conference with exciting ideas about teacher observation for PD, how teachers can share problems with each other better on the internet (awesome resources at Robert Kaplinsky’s problem-based lesson site), great ideas about agent-based models to add to courses, and ways in which teachers can talk to people about the Common Core and gain respect about the difficult work we do in teaching.  Overall, I felt like it was an amazing time.

I want to thank everyone that came to my session.  Although I had an unfortunate technological snafu and was unable to do an exercise I had planned where we were going to analyze a segment of discourse from my classroom using the framework of the MP standards (which would’ve been great), I felt that at least the resources that I shared were worthwhile for the people that came.  Here is a link to the powerpoint presentation and the handouts I gave.

 

Handout 1 NCTM 2015 Schettino

Weekly Learning Reflection Sheet

Handout 2 Schettino

I’ll just put in one more plug for our PBL Summit from July 16-19 this summer – we still have a few more spots and would love to have anyone interested in attending!

Inspirational colleagues? Wow…

OK, so I’m not really doing the full blog challenge – This weekend was nutso and blogging everyday is really tough – enough with the excuses.  But this question, “Who was or is your most inspirational colleague and why?” just really struck me at my core.  There have been so many, probably for all of us in education, it would be extremely difficult to pinpoint just one who was MOST inspirational.  I continue to be inspired rather regularly by my past professor (now friend) Carol Rodgers (SUNY) who is just one of the most amazing writers, Dewey Scholars and researchers and reflective practice I have ever met.  She is an amazing teacher mentor and has taught me a great deal. Ron Lancaster (OISE – Toronto) continues to show me how to be a true teacher of teachers every time I see him.  Nils Ahbel (Deerfield) and Maria Hernandez (NCSSM) and two of the most passionate mathematics educators I have ever met and every time I speak with them about my practice, I learn something new – period.  If you all ever get a chance to hear any of them speak, I highly recommend it.

I’ve already written about my inspiration and admiration for Rick Parris and the amazing life he led as a an educator, so I won’t go into that again, but I do feel that if I had to name someone who was not only inspiring, a major role model, caring, patient and kind, and truly changed my life, it would have to be Anja Greer.  If there is anyone to whom I have to attribute my work and lifelong love of teaching mathematics with problem-based learning, it would be Anja, mostly because I would not have had the opportunities and the courage to have taken the risks and to work with people who intimidated the heck out of me when I was only 26 years old.  She was a woman at school that had a very male-dominated history and she always spoke up for the students that were underserved and underrepresented.  She gave of herself in every way and gave me a job opportunity in 1996 that changed my life.

In the classroom, she was a teacher, mentor, innovator and amazing administrator.  To watch her handle a room full of very opinionated and argumentative mathematics faculty was amazing – never losing her grace and determination.  She took her time finding the words that she wanted to say and to this day, when I feel that I am pressured to quickly say something I think of her, take a breath, and rethink my words in my head.

The day I met Anja she frankly explained that she had to put a wig on in order to take me to campus because the students hadn’t seen her with her hair so short.  You see, she was battling cancer at the time that she was serving as department chair, implementing a new curriculum and hiring 4 new teachers that year.  The courage she had to “put on that wig” and move through her days for the next few years inspired me so much.  My son was born the year she lost her battle to cancer and she still had the compassion to let me know how happy she was for me that January.

I am so grateful for Anja’s influence on my life and I continue, in her memory, to teach annually at the conference that was named for her.  If I can even remotely come close to influencing another teacher in the way she has for me, I will have just started to repay her.

Encouraging Student Voice without Knowing It

I’d like to think of myself as a master teacher.  I’ve always thought of myself as very aware of student perspectives in my classroom, but today after a weekend of being in bed with a bad cold, all I wanted to do was get through the problems and get back home – I admit it.  I was not being very reflective and deliberate in my teaching.  However, even with all that, something amazing happened today.  Of course, I’d like to take all the credit, but I have to say I think the credit goes to the method of PBL, relational pedagogy and the students in my class.  So here’s what happened.

We had gone through about half the problems and I think they could tell I was in a pretty bad mood.  We got to one of my favorite geometry problems that starts an interesting strand of thinking that has to do with which polygons tessellate with others.  One girl goes to the board (I’ll call her Robin) and she presents her solution to this problem:

The diagram at the right shows three regular pentagons that share a common vertex at P.  The three pentagons do not quite surround P.  Find the size of the uncovered acute angle at P.

So Robin does what I expect (and in my fuzzy state of mind I am just happy that someone knows what’s going on).  She writes on the board:

Her argument being that the three angles in the pentagons were congruent since they were all regular and the leftover part would be the difference between those three and 360.  So, I was ready to move on.  She was right, after all?  Let’s go and do the next problem.  But no, Tye speaks up and says, “Hey that’s what I got but that’s not what I did.  That’s so cool I got the right answer.”  So, as tired and sick as I was, I said, “What did you do Tye?”  He says all I did was do 108 divided by 3 and it worked!”  He was so proud of himself.  I sat there and was like, OK, this isn’t going to fly, but I was so exhausted that ….but wait, another student says, “Hey that’s cool.  You just take the angle measure and divide it by how many polygons you have.” I’m thinking, oh no, this is gonna get out of hand fast….

Then another student says, “wait a minute, lets see if it works with hexagons: ”  So before they  know it they realize that it can’t work with hexagons and Helen says, “but that’s because there are too many, you need something with room left over, like a square.  What if you only use 3 squares? Is the angle leftover 90 divided by 3?”

So they soon have disproved the theory that if they just take the number of interior angle of the regular polygon and divide by how many polygons there are and divide them, they’ll get the leftover angle.  But Tye is still adamant that he’s all proud he got the right answer.  I am, however, still struggling with the fact that he can’t justify to himself why it works.  I say,”Listen, why don’t you think about it some more and we’ll come back to it?” but guess what, they don’t let it go.

Luke says, “Well, what I did was just draw a triangle down there.”

He says that he knew that empty space was really an Isosceles triangle and because the base angles were supplementary to the interior angles of the pentagon, he could find the angle at the top.  At this point, I’m like “will this ever end? Will I be able to get some Tylenol?” (I know can you believe me?  what a role model…)  A few other kids really liked what Luke did and said they did that too and thought they had been wrong, but now see that it was a valid method.  I mean, could I ask for more?  This was awesome stuff going on!

So at this point, we move on and do a few more problems, but then towards the end of class, I notice there are about three kids who aren’t really paying attention to the problem at hand.  I couldn’t figure out what was going on because they are usually right in the thick of our discussions.  So finally, one girls practically yells (and I mean, with arms flailing and everything) “I got it!”  Alanna had been working on a justification for Tye’s idea of dividing the interior angle by three the whole class period, as had two other students.  It was so interesting a problem to them that they just couldn’t stop.  Alanna said, “I knew they were vertical angles, but I just couldn’t see how they could be the same.”  She and the two other students had been playing with some isosceles triangles and vertical angles and come up with this solution:

By finding that the base angles of the isosceles triangles on the sides were both 36, and that the one in the middle was also 36, they had seen the reason why Tye could just divide 108 by 3.  They knew it wasn’t just pure luck that it worked and it made them all so happy.  It was so satisfying and I could just feel the excitement in the room.

It was so funny to me because everything that I did to try to discourage them from going to that place of curiosity or demanding the reason didn’t help.  They went there anyway. Their voices were heard – again and again.  The culture that we had set was there and no matter what I did now, at this point in the year, they knew what was expected of them – asking questions, not giving up, being inquisitive & creative.  This class helped me realize that even on my bad days I need to see each student for who they are and be just as excited for each of them to realize their own potential.  And if I don’t have the energy or strength to push them through, maybe, just maybe, they will.

30-Year-Old Wisdom, Not Recent Rhetoric

Recently, the Exeter Bulletin published an amazing Memorial Minute in honor of Rick Parris just this past week which I believe was wonderfully written.  In it they use a quote that Rick stated back in 1984 which shows his wisdom and insight into student learning of mathematics and the basis of my interest in PBL.

“My interest in such problems is due in part to the pleasure I get from working them myself, but it also stems from my belief that the only students who really learn mathematics well are the ones who develop the staying power and imagination that it takes to be problem-solvers. Such students will have thus learned that being accomplished in mathematics is not simply a matter of learning enough formulas to pass tests; that creative, original thought requires living with some questions for extended periods of time, and that academic adventure can be found in the pursuit and discovery of patterns, more so than in the mere mastery of known formulas.”

In this one paragraph is the whole of what you can find in so many blogposts, writings of so-called “experts” and “thought-leaders” in education nowadays.  I’m not so sure that the recent trend of promoting curiosity, innovation, creativity, perseverance and ‘grit’ are such original ideas.  It just has taken a long time to catch on in any type of mainstream educational jargon.

Rick Parris knew the truth over 30 years ago and led the charge in curriculum writing, pedagogical study, and leadership in student learning.  Always humble, but deeply interested in discussion, he would never shy away from the chance to discuss teaching mathematics and so I was lucky enough to have him as a colleague in the early years of my career.  He helped form my teaching philosophy and I owe a huge debt to the wisdom he imparted to me.  I seek to help students “live with some questions” every day in my classroom and I join them daily on the “academic adventure” of problem-based learning.  I can only hope that the mathematics community and society as a whole in the U.S. can catch up with his wisdom and we can eventually change the way we view learning mathematics.