Category: Conferences

How do you use empathy to teach math?

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This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.

 

One of the things that is interesting about teaching with PBL is how students often describe enjoying this type of math class more than others they have had in the past. It’s hard for students to paint a picture of what it is that produced their enjoyment.  The interesting thing is that it is often not the mathematics they enjoy, but the class itself – the interactions and relationships between the people in the class, and should they be solving some interesting problems that pertain to mathematics, that’s pretty great, too.

What one girl, Isabelle, described enjoying about my class once, was the way in which she saw mathematics as no longer black and white – with only the teacher’s information as what counts.  In a research interview, I asked her to describe for me what that was like:

Isabelle:  Like it’s, if you have a question you can just ask it and then that can lead into, like, some conversation or [the teacher] can ask a question and then kind of leaves it out there for us, the kids, to answer it, so…

Ms. S:  OK, and why do, why do you like that better?

Isabelle:  Um, because it’s not so uptight and [laughs], like it’s not like focused, “memorize all of this stuff…”

Ms. S:  Hmm

Isabelle:  It’s more relaxed, and that helps me learn better I think.

Isabelle’s more traditional view of the mathematics classroom with its “uptight” and rigid nature reminds her of memorizing facts and formulas and she stated that she responds better to a classroom that, in her eyes, is more “relaxed” and interactive allowing her views and responses to matter.  This is extremely consistent with Frances Maher and Mary Kay Thompson’s (2001) view of the feminist classroom’s responsibility to “deliberately position students as academic authorities” in order to allow them the input for the feeling that their responses matter, but so that that they do not “dismiss their own emerging sense of themselves.”  Also, Isabelle’s feelings are consistent with what Fox Keller (1985) once called “dynamic objectivity” which she defined in terms of how we might be inclined to think about the idea of integrating student input with factual mathematical knowledge.

Dynamic objectivity is a form of knowledge that grants to the world around us its independent integrity but does so in a way that remains cognizant of, indeed relies on, our connectivity with that world.  In this, dynamic objectivity is not unlike empathy, a form of knowledge of other persons that draws explicitly on the commonality of feelings and experience in order to enrich ones’ understanding of another in his or her own right (Fox Keller, p.117).

We can view this more flexible way of viewing knowledge as necessary for including students like Isabelle who find the more rigid mathematics classroom not conducive to learning.  She would rather remain connected to the material and the persons in the classroom with her in order to facilitate learning for herself.  Many students truly enjoy the fact that students are the contributors to the knowledge and part of the authority presence in the classroom.  Because of the openness to the dynamic objectivity of the knowledge, the students are able to accept that their input is valuable.  When I asked Isabelle why she thought the students felt so compelled to participate in a PBL classroom, she had this to say:

Ms. S:  Yeah, there’s almost a guarantee that people will… I wonder why? I wonder what guarantees that everyone will have something to say.

Isabelle:  Well [both laugh] it’s probably just because geometry has like twenty… like a lot of different ways to do certain problems so there’s a lot of variations in the way that people do them, so…

Ms. S:  Hmm.

Isabelle:  That might be it, or it might just be that people feel comfortable in the situation they’re in to participate and it’s not like, “OK nobody ask questions so we can leave now.”

Ms. S: [laughs]  Yeah. Ok. So there’s a certain amount of like motivation to want to talk about it?

Isabelle:  Yeah.

Ms. S:  because it’s like interesting to hear what other people did? [pause] Um, yeah, I can’t figure that out.

Isabelle:  I think everybody like shares the same curiosity level and like when somebody… like I know in our physics class he never tells us the answer to questions and it drives everybody crazy…

Ms. S:  Huh…

Isabelle:  And then we all start talking about it to try and figure out if like we can find out the answer ourselves so and the same thing happens in my math class so…

Ms. S:  Yeah?

Isabelle:  I think it’s just the motivation to find the right answer and like, because I know everybody in my class wants to understand.

Isabelle had described a mathematics classsroom culture with a tacit understanding of the dynamic objectivity of the part that students play in the formation of knowledge.  When presented with a problem where the solution is unknown and the teacher presumes a certain lower level of authority than the students, the students take on a higher level of responsibility and curiosity in finding solutions and methods for those solutions.  Being open to a view of dynamic objectivity and the empathy that it needs, allows many students to have their comfort in this type of learning environment and fosters more equity in learning for students who have previously been disenfranchised in mathematics and science classes.

 

Fox Keller, Evelyn, (1985). Gender and Science. New Have: Yale University Press.

Maher, F. A., & Thompson Tetreault, M. K. (2001). The Feminist Classroom. New York: Rowman & Littlefield Publishers, Inc.

Earning Your Status…and Eating it too.

When I was in elementary school, I was lucky enough to have a teacher named Mrs. Bayles who believed that what it meant to be “cool” was enjoy solving really interesting problems. I remember one time she gave everyone in class a piece of pie and asked us all “What’s the best way to start eating this piece of pie?” and everyone else in the classroom immediately took their fork and stabbed it right in that pointy corner, where, they argued, they would get the most of the juicy center of whatever type of pie they had. I was sitting with my group of friends (mostly girls) who were self-defined math “geeks” (although I think back in 1976, that’s not what they were called). We all kept thinking about that and eventually came to the conclusion that we wanted to start with the crust because thought saving the middle for last was a great idea.

Mrs. Bayles thought that was so awesome and asked the four us to come up to the front of the room, draw a diagram on the board and give evidence as to why eating pie from the back of the piece of pie was somehow better than eating it from the tip.  We thought we were the Albert Einsteins of pie-eating.  We just loved it. Even though the other kids in the class thought it was kind of weird, since we could justify our choice with a good argument, we stood together and most importantly, Mrs. Bayles respected our evidence and let us have our authority in our say.

One of the things in recent years that has become a passion of mine in the mathematics classroom, and more precisely, the mathematics probem-based learning classroom, is the idea of status and positioning of students in the discourse and learning that occurs.  This has become such an important issue that I invited one of our keynote speakers this year at the PBL Math Teaching Summit to speak on this very subject.  After many, many years of hearing teachers’ concerns about how to handle a student who tends to dominate a conversation, or who doesn’t speak enough, or what happens when kids get off on the wrong the track when discussing a problem – it is about time that this socio-emotional topic (which includes race, gender, privilege, equity, and all things relational that made me start studying this pedagogy in the first place years ago) be moved to forefront of the mathematics classroom once and for all. (Aside – huge thank you to Teresa Dunleavy who gave an awesome talk on this BTW!)

I have shared this specific story with so many people at this point in time, but I find it so important that I want to repeat it here for Sam Shah’s “How does your class move the needle on what your kids think about …. who can do math?” prompt for this “Virtual Conference on Mathematical Flavors.” I feel it is something I’ve worked on for almost twenty or so years and I still don’t have it down to a science, I just know that I can’t let it go anymore.

In my classroom, I allow students to use dynamic geometry apps or technology as much as they want to justify their answers or as evidence for their thinking, as long as it doesn’t go awry (and of course, as long as it is correct and they can describe their thinking).  Two years ago, I had a student (for whom I will use a pseudonym here because I’ve used his real name in the story, but not on the Internet), I’ll call Ernie.  Ernie was one of those kids who could do no wrong – very popular in his current class, very successful academically, which made him very outspoken in his ideas, very good-looking in our white, hetero-normative, social class acceptable way and to top it off – (what Dr. Dunleavy says is usually one of the highest privileges in white schooling) – an exceptional athlete.  Mix all of these privileges together and what automatically came with him into this class? Mathematical status.

Mathematical status doesn’t mean that he was not a good math student, that’s for sure.  Ernie worked very hard and had excellent intuition, as well as good retention from his past math courses (–hmm, I feel like I’m writing comments from the fall term here..) These were neither here nor there however to the rest of the class.  When students bring their own thoughts and impressions of a student into the class with them, its the class itself that priveleges that other student (in this case Ernie) the high mathematical status that he had.  There might have been other students in the class who should have had higher status but because they were not as outspoken, had different relationships with others, were messy or not as articulate about their ideas, asked “stupid” questions (you know that’s not what I mean) or whatever the behavior that was exhibited – the other students in the class would assign a low mathematical status to other students by the things they say, brushing aside questions or simply by just not listening.

So one day we were discussing a question about the congruence of two triangles that were in the different orientation, plotted with coordinates.  Students were supposed to come up with some triangle congruence criteria (I believe it was supposed to be SSS) for why the two triangles were congruent – this was at the very beginning of the concept of Triangle Congruence.  Ernie had plotted the triangles on GeoGebra and simply said, “These two triangles are not congruent because all of the correponding sides are not equal” stepped back, matter-of-factly with pride in a job well done.

There was thoughtful silence in the room as the class looked at this diagram up on the board projected from his laptop.  There was no arguing with the fact that the sides of his triangles were not all the same.  However, there was still confusion I could see in some of the students’ faces.  Some kids asked him to find the lengths of the sides.  “That’s what I got,” “Oh I see what I did wrong,” and “Thanks for clarifying” were some of the comments that Ernie received.  Under her breath, I heard one girl just whipsering to herself, “That doesn’t make sense” and I tried to follow up on the comment, but she would have none of it.  We spent maybe 5 more minutes of me trying to get anyone else to make a comment. It got to the point where I even got out my solutions because even I was doubting myself (the power of Ernie’s status) because I had sworn that those two triangles were supposed to come out congruent.  I knew some of those kids knew it too.  Why weren’t they all saying something?  It was as simple as a misplaced point.  Not a huge problem, why couldn’t anyone call him on it?  I decided to do a little experiment.  “OK, well let’s move on then, but I really think there’s a way to show that these two triangles are congruent.”  Ernie was intrigued.  He couldn’t be wrong so tried to start finding his error, but couldn’t.  I said, “no, no, I want everyone to go home tonight and try to see if we can find a way to show that these triangles are congruent.”

Jump to the next day in class and kids are sharing their solutions from the previous nights struggle problems.  Before we start discussing them, I say, “Did anyone think about the problem that Ernie presented yesterday?” Radio silence….I wasn’t sure that anyone would actually do it, so I had come prepared with a geogebra diagram of my own.  I projected it on the whiteboard and asked if they noticed anything.  Still no one said anything (outloud so everyone could hear, but I could tell that some students were at least talking to each other).  Suddenly, Ernie says, “Oh my gosh, I plotted the wrong point! It was supposed to be (6,1).” There was this huge metaphorical sigh of relief from the whole class at this moment that could be felt by everyone.  I just coudn’t understand it.  Although no one was willing to speak up that they knew Ernie had been wrong, they were all relieved that that he realized his own error.

I expressed my concern with this dynamic in my classroom.  Simply asking them why didn’t anyone help Ernie with the problem yesterday in class? or what kept anyone from speaking up when they thought the triangles were congruent? wasn’t getting us anywhere.  So what I did was tried to let them know how much I wanted to hear their ideas – similar to what Mrs. Bayles did with the pie.  If students can see and hear evidence that the teacher values all voices equally, not just those that the students have given high status, can truly make a difference in how they start placing their status beliefs.

What I saw change in the class slowly, wasn’t the status that the kids all gave Ernie.  In fact, if anything he got even more from finding his own error – but what happened was that girl who had spoken under her breath, spoke a little more loudly.  Students who presumed that Ernie was correct, asked an interesting question that Ernie had to justify.  These other students were growing in the status that the others were giving them. I believe that it is very hard for us as teachers to control what the students come into the classroom believing about each other, but we can have an impact on what they believe is valuable and meaningful about what they do in the classroom.

 

 

 

Resources for my NCTM Conference Talk Washington 2018

I hope there’s lots of interest in the lessons that I’ve learned from my years of having students journal. Here are some resources that you could use if you are interested in trying journals in your math classes.

Handout for NCTM Session Handout Schettino NCTM 2018

Blogposts about Journaling:

journals-paper-vs-digital-the-pros-and-cons/

what-i-get-out-of-student-writing/

revisiting-journals-getting-kids-to-look-back/

does-journaling-in-pbl-promote-resilience/

using-journal-writing-in-pbl/

Page: metacognitive-journaling/

Slides for Talk: Slides from Journal Presentation

Looking at PBL Practice from a Thematic Perspective

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So I’m here down in Florida – loving it (all sing-songy like Oprah would say).  I’ve been to so many talks that have been great learning experiences so far.  The weather is beautiful – I went for a very long walk and tried to think about what my talk was missing.  I did a bunch of edits and now I think I’m ready to post it.

Here’s the powerpoint of the talk:

Here’s the document that I handed out with some “threads” of themed topics:

Three Threads Document

Please contact me with any questions, comments or concerns – I love talking to people about PBL and my work.

I’ve looked at life from both sides now…

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This past July, I spent a few days at the MAA Mathfest in Chicago for the first time. The main reason I went was because the Academy of Inquiry-Based Learning was having a Conference within the Mathfest with the theme of “Diversity in IBL.”  IBL is generally what college faculty call the type of teaching and learning that many of us at the secondary level has been calling PBL for years.  I was so interested to hear many mathematics professors talking about the struggles of writing curriculum, dealing with facilitating discussion, using writing – all of the same parts of this type of teaching that we may have been talking about for so any years.  I highly suggest that we could benefit from talking to each other.  If you would like to get involved with this movement, Stan Yoshinobu, the Director of the Academy of Inquiry-Based Learning, has put forth some challenges for his community.  Check them out.

One of the most interesting talks that I attended was by a professor from Denison University, Lew Ludwig, titled, “Applying Cognitive Psychology to the Mathematics Classroom.”  As a devout social constructivist, I generally like going to talks where I can learn more about other views of education.  Seeing both sides definitely helps me understand many of the views of my colleagues and see if evidence supports my own perspective. Ludwig had published a review of another article that was titled, “Inexpensive techniques to improve education:  Applying cognitive psychology to enhance educational practice”(Roediger and Pye, 2012).

Basically, the presentation summarized three simple techniques that cognitive psychology had evidence helped student learning. The three techniques were called

  1. The distribution and interleaving of material and practice during learning.
  2. Frequent assessment of learning (test-enhanced learning, continual assessment)
  3. Explanatory questioning (elaborative interrogation and self explanation; having students ask themselves questions and provide answers or to explain to themselves why certain points are true).

In the original article, the authors write:

“Repetition of information improves learning and memory. No
surprise there. However, how information is repeated determines
the amount of improvement. If information is repeated back to back
(massed or blocked presentation), it is often learned quickly but
not very securely (i.e., the knowledge fades fast). If information is
repeated in a distributed fashion or spaced over time, it is learned
more slowly but is retained for much longer”

When this was reported, I was first in shock.  I couldn’t believe I was hearing something in a presentation about Cognitive Psychology that was actually supported by the definition of PBL that I use.  The curriculum I use takes the idea of looking at topics and teaching them over a longer time span, but distributed among other topics.  I have called this decompartmentalization of topics, which helps students see the connectedness of mathematics.

The second idea, consistent assessment, is based on the concept that testing is not really a great measure of how much a student has learned, but it actually solidifies the learning that has occurred.  So three groups of students were given different ways of learning by reading a passage of information. The first group read a passage four times. The second group read the passage three times and had test.  The third group read the passage once and was tested three times.  Their performances on tests on the information in the passage 5 minutes later and then one week later.

Diagram of retention testing research
from Roediger & Pye (2012) p.245

So if we connect the idea that testing is not the best method of seeing how much students have learned and the fact that consistent assessment actually helps students retain their knowledge, what I do in my PBL classrooms, is not only “test” but do all sort of forms of assessment (writing, oral assessment, hand-in homework with feedback, labs, quizzes, problem sets, self-assessment, etc.) alternately throughout the term.  There is probably not a week where students are not assessed in at least 2 ways. I feel that this has led students to have good retention of material and the assessments are strong measurements of their learning.

The last one was the one I was most excited to hear about – explanatory questioning.  This seemed to give students so much more responsibility for their own learning than traditional cognitive psychology as I had understood it.  The authors of this study claim that explanatory questioning can be broken in to two areas:

Elaborative Interrogation – students generating plausible explanations to statements while they are studying and learning.  This speaks directly to the idea of mathematical discussion and how students generate explanations when they ask themselves “why?”

Self-Explanation – students monitoring their learning and describing, either aloud or silently some features of their learning.  This idea can be found all over the PBL classroom but in mine, it’s generally found most in metacognitive journaling where students use self-explanation the most.

“Obviously, the elaborative interrogation and self explanation are related because both strategies encourage or even require students to be active learners, explaining the information to themselves (perhaps rephrasing in language they understand better) or asking themselves why the information is true.”

I honestly couldn’t believe what I was reading – this is an article on educational methods based on cognitive psychology that is suggesting that we require students to be active learners and “explain the infomation to themselves”?  This is lunacy.  I have been teaching for 25 years where students have been complaining to their parents that they have had to explain things to themselves – who would’ve known that I was applying cognitive psychology?

My guess is that these ideas are only enhanced by the social aspect of the classroom and other constructivist ideas – clearly the constructivitst classroom in enhanced by or agrees with some of these cognitive psychology methods as well.  Listening to both sides of the theories is actually helpful and I’m seriously going to continue doing this! Although I never thought that there might be strong connections between cognitive psychology theories and PBL, I do know that it’s life’s illusions I recall and I “really don’t know life, at all.”

Documents for CwiC Sessions at Anja Greer MST Conference 2016

Instead of passing out photocopies, I tried to think of a way that participants could access the “hand-outs” virtually while attending a session.  What I’ve done in the past a conferences is have them just access them on their tablet devices.  You can also go and access copies on the Conference Server if you do not have a device with you (you should be able to use your phone too).

These link to This is an Adobe Acrobat Documentpdf documents that I will refer to in the presentation about “Assessment in PBL”

Information on Spring Term Project and Spring Term Project Varignon 2015 (this document includes rubric)
Keeping a Journal for Math Class
Revised Problem Set Grading Rubric new
Rubric for Sliceform project and Sliceforms Information Packet
Weekly-Learning-Reflection-Sheet

Page at my website with Rubrics and other guides for Assessment

Connections Between IBL and PBL

At the PBL Summit a few weeks ago, we had two wonderful speakers, Julian Fleron and Phil Hotchkiss from Westfield State University who are founding members of the Discovering the Art of Mathematics Project.  They gave a great key note address on Friday night about Inquiry-Based Learning and motivating students in an IBL classroom.  You can find their talk at our Summit Resources website if you are interested.  I wrote a blogpost a few years ago about my interest in IBL and the commonalities between PBL and IBL and I thought I’d reshare in honor of them.  Enjoy!

A number of years ago, I needed some kind of suport text for a Number Theory tutorial that I was doing with two rather advanced students who had gone through the curriculum at the school where I was teaching.  These two girls were advanced enough that I knew that if I used my notes and problems from my wonderful Number Theory course from college (some many years ago) we would have a great time.  I looked online and found a great book called “Number Theory Through Inquiry” published by the MAA which came with an instructor’s supplement including pedagogical discussion and some solutions.  It sounded so much like what I was doing with my other classes that I couldn’t turn down the opportunity to see what it was like.  So I ordered the book and while I was reading the instructor’s supplement I came across something that I had not heard about before (and now I am so embarrassed to admit this). The authors described what they called the “Modified Moore Method” of instruction or Inquiry-Based Learning and went on to describe what sounded interestingly so much like what I was doing in my classroom.  I had to learn about this Moore Method.

I ended up researching R.L. Moore online and it seems that he was one of the first math teachers – ever – to think about and act on this idea of not teaching mathematics with direct instruction.  He did it all the way back in 1948, but at the college level – and it was radical there!  The idea of Inquiry-Based Learning has expanded from there, but it has really only stayed at the college level in mathematics for a very long time.  There are many initiatives at the college level, including the folks at Westfield State University who are writing a wonderful curriculum project funded by the NSF called Discovering the Art of Mathematics with is a math for liberal arts curriculum at the college level.  I think it could be used at the secondary level as well for an alternative elective in the senior year for those students who still want to take a college-level math course but aren’t ready for or interested in an AP course in Calculus or Stats.  If there are any secondary teachers interested in beta-testing this unique curriculum please contact me and let me know.  I am on the advisory board for this project.

What made me think about the connection between IBL and PBL was this wonderful blogpost I just read by Dana Ernst, of Northern Arizona University in which he describes, in such wonderful ways, the pedagogy and nature of IBL.  The similarities between the definition of IBL he cites (by E. Lee May) and my definition of PBL are eerie – and it is one of the only ones that I’ve seen that stresses a reference to teacher authority being diminished.  Many wonderful resources are given by Ernst at the end of his post as well.

I do remember back in 2003, when I published my first article on my experiences at Emma Willard, after I left Exeter (where they called in Harkness teaching because of the table), in attempting to teach the way I wished to.  I had no idea what to call what I was doing.  I believe in my first article I called it teaching with a Problem-Solving Curriculum (PSC).  After I started my doctoral work, I found PBL and I realized that’s what it was.  Then I read more and more and realized that others thought PBL was project-based learning and called what I did discovery learning.  After reading about R.L. Moore, it sounds like he was doing it all along since 1948 and called it IBL.  In whatever branch of the pedagogical family tree you find yourself, if you are asking students to look at mathematics with wonder and question what they know – you should know that you are supported, know that you are doing good work and know that there is someone out there who has done it before and wants to discuss it with you.

PS – I’m hoping to attend the Legacy of R.L. Moore Conference next year in Austin, if anyone is interested!

Think about where the learning happens in PBL

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After a few weeks of recovery, I wanted to write about having a BLAST of a time at our first attempt of putting together the PBL Summit my friend Nils Ahbel and I organized from July 16-19.  I wanted to thank all of those who came and participated in the discussions and talks and who shared their ideas so freely.  It’s such a great reminder of the huge resources we all are to each other as math teachers.  I know that I at least tripled my Professional Learning Network and hope that all of the participants did too.

I’d also like to thank everyone who gave feedback and the amazing ideas for next year – including a pre-conference session for those of you who might have been PBL “newbies” and might have needed more of an intro, topic-level groups, more in-depth SIGs for people who want to dive deeply into writing or assessment writing too.  The ideas just kept flowing and I think we will have a wonderful plan for next year too.

One of the take-aways that I left the PBL Summit with was how differently people view what “learning” means in PBL.  From my long career both teaching and studying PBL, I have had a lot of time to form my own frameworks for student knowledge construction and pedagogical theory and often take for granted that all of us are on the same page. As I have traveled and talked to many other math teachers and heard others who are experts in PBL (both PjBL and PrBL) speak, I realize more and more that we are often NOT on the same page.  This does not mean that any one of us is more right or wrong.  We just need to understand each other more.

My big question to everyone I talk to is “where/when does the learning happen?” or “where/when do the students construct their knowledge and understanding of the mathematics?”  If students are presented with a problem, for example they watch a wonderful interesting video of a basketball player shooting at a basket or watching someone fill a water tank and they come up with their own question based on a real-life phenomenon from the video, how do those students know the mathematics to answer those interesting questions?  If students are sitting through direct instruction lessons to be exposed to the mathematics but using them to answer their own questions, this is definitely an improvement than passive mathematics classes of the past.  Having students take ownership of the material in this way is is a powerful method of creating agency for mathematics learning.  The problems that they are solving and from where they are posed are extremely relevant to the motivation and agency in learning.

I would posit that PBL can be more and mean more and in more ways to student learning. Even when posed with a good problem (one they did not come up with themselves).  In PBL, students can:

  • see the need for a new method without the teacher introducing it
  • see the need for discussing other students’ ideas
  • find their own organizational strategies for problem solving
  • access prior knowledge that they did not realize they needed before
  • use their resources to discuss the problem with each other
  • use resources to find new solutions and follow their own thinking
  • make connections between topics in mathematics that they might not have realized before
  • create community in the mathematics classroom (like in other disciplines – humanities, fine arts and science)
  • realize that reflection is one of the most important parts of the learning process
  • learn to relate to others in math class
  • see mathematics as a creative endeavor

and so much more. I’d love to hear from people some that I have left out.  In my mind, even the mathematical learning happens in these contexts and students are the shapers of where and when this happens.  Robert Kaplinsky is one of those amazing PBL teacher/speakers who has a somewhat different approach than I do, but is very similar in many ways and I heard him say this April, “Don’t teach what students need to know before they do a problem-based lesson.” In that way, we are all on the same page, for sure.

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.