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.

Keeping the Dice Rolling: Questioning in PBL

Returning from a week-long conference is always invigorating for me – not for the reasons that many people think.  I do appreciate the great feedback I get from my “teacher-students” that I interact with during the week who are so extremely eager to learn about PBL – this truly invigorates me and allows me to do so much work over the summer myself.  However, what I always look forward to is how much I personally learn from the interactions with my students that week.  At this point, PBL is so popular in its use in mathematics classrooms across the country, although people see me as an expert in the field, I gain so much from the questions and process of those who are learning that it is so useful for me to move through that process with them all the time.  I believe this is why they call it “professional development”!  So I just wanted to give a HUGE shout-out of thanks to everyone who took my workshops, came to my Cwic sessions, had conversations with me or interacted in some way – it might have been one of the best professional weeks I’ve ever had!

Since that week in New Hampshire, I’ve done a lot of reading, editing of my own materials, and catching up with my own work.  I recently read a blogpost on edutopia entitled “The Importance of Asking Questions to Promote Higher-Order Competencies” which stood out to me as something that we talked a great deal about in my own PBL classes, although this blogpost was not specifically about PBL or math at all.  It was written by a professor at Rutgers University in the Psychology Department, Maurice Elias, who is part of the Rutgers Social-Emotional Learning Lab, and made me wonder if he had done any work with Cindy Hmelo-Silver, who is also at Rutgers and does work with PBL in Psychology.  The concept of asking questions is something that we discuss and practice in my workshops because Hmelo-Silver says that it is a characteristic of an experienced PBL teacher to ask probing questions that are metacognitive and at a higher-order level.  Interestingly, the four areas that Elias discuss are often not linked to higher-order thinking (for example, yes/no questions) so I thought I might take his “Goldilocks” example and try it through the lens of math PBL.  Elias’ four questioning techniques are 1)Suggest 2) Ask a Closed Question 3)Ask an open question and 4)The Two Question Rule.

The idea of “suggestion” is one that I always tried to stay away from since student voice and experience is first and foremost in my mind as a pillar of the PBL pedagogy.  Allowing students to make first attempts at making those connections on their own I believe takes precedence over critical thinking skills of choosing from alternatives.  However, that concept of making a choice between alternatives is important as well and might be a very good skill to have them practice every now and then deliberately.  I think I will begin to try this in class.  The next time when it seems like no one has an idea or when the student at the board is going in the wrong direction, I may decide to say something like “Should Joe go with the method of completing the square or factoring here?”

The second idea of asking the closed question (yes or no) is also one that I have always tried to stay away from.  In my experience it’s kind of a conversation staller, but the way it’s explained by Elias in his blogpost is actually a very interesting twist on the closed question.  It takes a yes or no question but embeds an opinion in it, so almost forces a justification of the closed question with the yes or no.  It makes the teacher find a way for the student to continue (well, the teacher must make sure the student follows up).  So for example, if the teacher asks asks, “Do you think the quadrilateral is a rhombus?” it might seem very obvious that a student could just say yes or no and the conversation could just end there.  Everything I’ve read about closed questions say that you should not phrase the question that way but be sure that the question has within it some interest in the student’s opinion. “Why do you think it’s a good idea to argue that this quadrilateral is a rhombus?” (Which is a closed question in disguise but opens up the conversation).

Then there’s the Open-Ended Question (or what Bingham calls a True Question) which I have written about before.  I talk about this in my workshops as well and real open-ended questions are questions that the teacher doesn’t really know the answer to.  I love Bingham’s analogy of trying to predict with your students what the sum of two dice will be (the answer)  but trying to keep the dice rolling for as long as possible without knowing the answer.

Dice Metaphor

What’s an example of this type of question in mathematics?  This is a tough one because as we know so well, there are definitely right and wrong answers in mathematics.  However, we can ask questions like “Why did you chose that method?” and “What do you think of Sara’s argument? Do you agree with her?” These types of questions can make mathematics teachers very uncomfortable but we can keep the box wiggling for great deal longer than we could before with these questions and they allow us to work towards the CCSS Mathematics Practice Standards of persevering and critiquing other students’ work.

Elias’ Two Question Rule isn’t just as simple as asking a follow-up question, but makes the assumption that students want to see if when you ask a question the first time, you really wanted to know what they wanted to say.  For example, in most mathematics classrooms, students are accustomed to the I-R-E form of dialogue which is short for Initiation-Response-Evaluation (Teacher-Student-Teacher) where the teacher generally knows that answer that they want for the question they have asked (kids know this, they’re not dumb).  So when the same old kids do the response part of this, instead of just doing the evaluation part, why not blindside them and actually rephrase the question and ask it again in a different way, or ask one kid themselves individually in order for them to know that you really want to hear from them?  I think that’s what Elias is talking about.  (or even better don’t use IRE, break that darn habit, I know I’m still trying to!)

We had some great fun during my workshops role modeling and just trying out different ways of questioning the mock student who was at the board – it’s hard to break old habits.  But the more we are aware of what we are trying to do and do it deliberately, the more important it becomes and bigger agents of change we can be as well. If you have any thoughts on these questioning techniques in math PBL classroom – please let me know

Hmelo-Silver & Barrows (2006). Goals and strategies of a PBL Facilitator. Interdisciplinary Journal of Problem-Based Learning , 1(1), 21-39

What I learned over my NCTM break!: Part 1

Wow!  What an amazing three days I spent at the NCTM annual conference in New Orleans!  I can’t believe how much I learned (which actually never amazes me and always humbles me – one of the many reasons I love going to these conferences.)  I also hate leaving and knowing that I missed at least 20 sessions that conflicted with ones that I did go to, so now I’m catching up and trying to email the speakers that I didn’t quite get to see or get in contact with while there.

One of my major a-ha moments was in Gail Burrill’s session on logarithms.  You’d think that after 25 years of teaching that you’d understand how much you understand about logs right? Oh, no!  So she had us all have a very large number and we were doing an exercise where we had to put a post it note with that number (mine was 72, 753) on a scale of powers of 10 {10, 10^2, 10^3, 10^4, 10^5…}, her argument being that one of the main reasons to teach logs is to have a different scale for very large numbers.  So after all of these teachers did this, we analyzed where all of our numbers were on the scale – particularly between these numbers.  Since my number should’ve been between 10^4 and 10^5, I knew I put it in the right place – but oh no, I had it in the wrong place relative to the middle.  She asked us to calculate the middle of those two – 10^4.5 which was 31,622 and yes, I admit that’s very close to where I put my post-it.  I could blame the person who put their’s up first which said 75,289 and I just put mine by there’s but I won’t.  I just didn’t really think.  But I know this was a light bulb moment for many of the teachers in the room.  Students don’t really understand how a logarithm is an exponent in the first place and we were doing this exercise without even using the word “logarithm.”

Then we went down to the section below that was between 10^3 and 10^4 and checked some of those numbers.  They were very off too and Gail asked us what number we expected to be in the middle.  At this point, some of us pulled out our calculator (yes, I admit, I did) but some of the smartees in the room just said “3,162” and I finally got it.  By just dividing by 10 and looking at the scale in this nonlinear way, students would be able to make the connection between the algebraic properties of exponents and what a logarithm was.  I thought this was an amazing way to introduce logs.  Has anyone done this before?  Thanks so much Gail!!  I think I’m going to write a problem for my curriculum about this, it’s such an insightful experience.

More reflections to come – just can’t do it all at once – catching up on school work!

A Total Win…with lots of understanding

Before I left for the Anjs S. Greer Math Conference last week, I read an amazing blog entry at the Math Ed Matters website by Dana Ernst and Angie Hodge that was talking about Inquiry-Based Learning and the mantra “Try, Fail, Understand, Win.”  The idea came from one of Prof. Ernst’s student course evaluations this past spring as his student summed up his learning experience in such an IBL course.  This blog post was so meaningful to me because for each of these four words, the authors wrote how we as teachers (and teacher educators) can take this student’s perspective towards our own work.  I decided to attempt to take this attitude going off to my own conference with two courses to give and three smaller talks.  It was sure to be a busy week.

And in fact, it really was.  I had very little time to sit and listen to others’ work, which I really was quite sad about.  However, in my own classes I was so impressed with the amount of enthusiasm and excitement my participants had for PBL and their own learning.  As I sat in front of my computer this morning reading the course evaluations and their tremendously helpful input, it finally occurred to me how truly powerful the experience had been for my participants.  Many of them became independent thinkers and knowers about PBL and feel so much more knowledgeable and prepared for the fall.    Part of the class time is spent in “mock PBL class” where I am the teacher/facilitator and they are the students doing problem presentations.  We then sit and talk about specific pedagogical questions and distinctions in classroom practice.  Some of the class time is spent in challenging problem solving which is where I also learn so much from the participant’s different perspectives. “We win when we realize there’s always something we can do better in the classroom” – as Ernst and Hodge write.

The now Infamous ‘French Garden’ Problem

I want to give a huge shout out to all of my participants from last week and encourage them to keep in touch with me.  Many of you wrote in your evaluations that you still have many questions about your practice and how to integrate your vision of PBL in your classroom.  I will always be only an email away and hope that you continue to question your practice throughout the year.

My plan is to try to write some blog posts at the end of the summer/beginning of the year in order to respond to some of the remaining questioning while you plan for the beginning of the school year such as:

  1. How to plan for week one – writing up a syllabus, creating acceptable rules
  2. Helping students who are new to PBL transition to it
  3. Assessment options – when to do what?
  4. Working hard to engage students who might not have the natural curiosity we assume

If you can think of anything else that you might find helpful, please post a comment or send me a message and I’d be happy to write about it too!  Thanks again for all of your feedback from the week and I look forward to further intellectual conversation about teaching and PBL.