Category: Persistence

PBL: High Expectations or Learning to be Self-Directed?

One issue that seems to arise after teachers have been teaching with PBL for some time is the question of how students can remain active learners while listening, taking notes, comparing solutions, being engaged in discussion, etc.  All student-centered mathematics classrooms now have this issue don’t they?  Can a student learn well when they are being active in their learning? How do you allow them to both have agency by being part of the construction of knowledge but also have ownership by taking responsibility for the active part of learning.

Here’s a scenario: Grade 8 class has a student at the board presenting a method of factoring that is obviously confusing everyone – you know, they learned “the box method” somewhere else and are presenting it like it’s just a given that you are supposed to know this.  I’m observing this and I’m seeing at least 5-7 looks of confusion, maybe 1-2 students who are following the student and at least 3-4 who have checked out totally – maybe drawing a tree in their notebook.

How do you maximize this moment?  It is imperative that the teacher move in and ask questions that get at the student presenter’s understanding, especially if the other students are not asking questions.  There may be an air of “oh god, I’m supposed to understand what this kid is talking about” and others will not be asking good questions.

The teacher can ask questions like:

“OK, good work that you have a method that works for you.  Can you back up and explain how this method is showing what the factors of the quadratic are?”

“Let’s slow down a minute and see if there are any questions.”

“Why don’t you explain why you chose the number and variable you did for each box and what those boxes represent?”

“Can everyone else write down a question for …. and then we’ll share out.” (this can include making up a problem for those who do understand and seeing the presenter do another example)

These will bring the moment back to the group, wake them up to the fact that it’s OK to have questions and maybe an alternate method as well.  The kids who have checked out might feel validated in checking out.  But at the same time, checking out shouldn’t be an option. . How can we teach students to remain connected even when they really feel like all is lost?  In PBL, it is most important for student to have the tools in order to do this.

Today I saw this infographic tweeted by Brian Aspinall (@mraspinall) that does an excellent job of summing up ways to have students remain engaged when they want to check out.

Screen Shot 2018-02-23 at 2.59.57 PM

So many of these relate to the expectations for students in the PBL classroom.  Some of my favorites are

1. Reflect in writing – hugely important for the introverts in the PBL classroom and to share the floor and authority.

2. Relate it – not only to “something” you’ve experienced but another problem that you might have done that is connected to it!

3. Control your environment – How engaged you are is really your decision – How can students minimize their own distractions?  Of course day-to-day this will vary, and is developmentally different from grade 8-12 of course, but students, when aware of being distracted, can often find ways to get back into the work.

4. Self-Assess – this is one of my favorites – it keeps them engaged, makes them think critically and thinking ahead.

Allowing kids to know that doing all of these different behaviors in the math classroom is not only OK, but expected and encouraged, is part of teaching with PBL and encouraging the “active” in active learning.

Can you undo an adolescent’s fixed mindset?

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Yes, it is this time of year where I have to stop and wonder – what the heck am I doing wrong? Is it me?  Is it the kids? Is it the combination of us? In the spring, many of the kids are breezing through and finding ways to problem solve and have gotten really comfortable with being uncomfortable in doing their nightly struggle – they’ve learned to trust that when we get together the next day, their questions will get answered and all will come together, if not that day, then the next.

This year is somewhat more frustrating for me and I can’t figure out why.  I feel as if the students are still attempting to get everything right every night.  It’s as if they created habits that I did not see somewhere along the way.  Reading the beginning of Andrew Gael’s blogpost on Productive Struggle  made me realize this was true and I’m more frustrated than ever now.  I’ve noticed that the conversations that I am “facilitating” are actually either one student talking about their ideas (basically the kid who thought they got it right) and everyone listening intently checking if they agree with him/her or everyone remaining silent until the one or two kids who are willing to take the risk speak up and take the risk to see if they are right.  I’m not quite sure what this is about.

In prior years, there have been kids that really felt much more comfortable with attempting something and being wrong.  I am really wondering what I did differently this year.  There is much more of a feeling of holding back – many more caveats of “I don’t think this is right…” before someone puts their ideas on the board (even though I repeatedly stress that that is not important.)

I have in the past few years become very disillusioned with the idea that high school students are capable of undoing 12-14 years of fixed mindset.  I think I tweeted about this last year sometime when, after a conversation and exercise about Fixed vs. Growth Mindset a student said to me “Is this supposed to make us feel bad?”  I was in shock.  I couldn’t figure out what I had done to make him feel bad at all as I had done just what Carol Dweck suggests and presented the two mindsets as a continuum – a journey of learning about yourself and how you learn best.  Some of the kids saw it as a good tool to know about yourself, but many of them saw it as just one more thing they had to “overcome” in order to get in to a good college or to be the “best they can be” – because you know, if you have a fixed mindset, that’s not the “best you can be” – you have to change that too now.  Oh god, what have I done?

So, maybe there’s a little part of me that feels bad for them and truly understands the fear of being wrong. My goals are to prepare them for the thinking, for problem solving in life and their immediate goals are getting good grades, doing the best they can right now to get into a good college, etc.  Sometimes these goals are definitely at odds and it’s really tough to compete with the immediacy of what they perceive as success for them and those people they want so much to make proud. And as always when there are two parties who have goals that are at odds – there is ultimately conflict.  And the battle continues.

Everything Old is New Again…(or why teaching with PBL is so great)

So I heard that what everyone is saying about the new Star Wars Movie, The Force Awakens, is that “Everything Old is New Again” – go ahead google it, there are at least 5 or 6 blog posts or articles about how “BB-8 is the new R2D2” or “Jakku is the new Tattoine” or whatever.  I actually don’t have a problem with J.J. Abrams reusing old themes, character tropes or storylines because I think that really great stories are timeless and have meaning and lessons that surpass the movie that you are watching.  I still thought it was awesome.

This concept of everything old is new again really hit home to me today in my first period class when I was having the students do a classic problem that I probably first did in 1996 while I was under the tutelage of my own Yoda, Rick Parris (who I think wrote the problem, but if someone reading this knows differently, please let me know).  The problem goes like this:

Pat and Chris were out in their rowboat one day and Chris spied a water lily.  Knowing that Pat liked a mathematical challenge, Chris announced that, with the help of the plant, it was possible to calculate the depth of the water under the boat.  When pulled taut, directly over its root, the top of the plant was originally 10 inches above the water surface.  While Pat held the top of the plat, which remained rooted to the lake bottom, Chris gently rowed the boat five feet.  This forced Pat’s hand to the water surface.  Use this information to calculate the depth of the water.

What I usually do is have students get into groups and put them at the board and just let them go at it.  Today was no exception – the first day back from winter break and they were tired and not really into it.  At first they didn’t really know what to draw, how to go about making a diagram but slowly and surely they came up with some good pictures. Some of the common initial errors is not adjusting the units or mislabeling the lengths.  However, one of the toughest things for students to see eventually is that the length of the root is the depth of the water (let’s call it x) plus the ten inches outside of the water’s surface.  Most students end up solving this problem with the Pythagorean Theorem – I’ve been seeing it for almost 20 years done this way.  Although I never tire of the excitement they get in their eyes when they realize that the hypotenuse is x+10 and the leg is x.

However, since everything old is new again, today I had a student who actually is usually a rather quiet kid in class, not confused, just quiet, but in a group of three students he had put his diagram on a coordinate plane instead of just drawing a diagram like everyone else did.  This intrigued me.  He initially wrote an equation on the board like so:

y= 1/6 (x – 0)+10

and I came over and asked him about it.  He was telling me that he was trying to write the equation of one of the sides of the triangle and then I asked him how that was going to help to find the depth of the water.  He thought about that for a while and looked at his partners. They didn’t seem to have any ideas for him or were actually following why he was writing equations at all.  He immediately said something like, “Wait, I have another idea.” and proceeded to talk to his group about this:

Jacksons solution to Pat and Chris
Jackson’s Solution to the Pat & Chris Problem

He had realized from his diagram that the two sides of the triangle would be equal and that if we wrote the equation of the perpendicular bisector of the base of the isosceles triangle and found its y-intercept he would find the depth of the water.  He proceeded to find the midpoint of the base, then the slope of the base, took the opposite reciprocal and then evaluated the line at x=0 to find the y-intercept.  I was pretty impressed – I had never seen a student take this perspective on this problem before.

This made my whole day – I was really dreading going back to work after vacation and honestly, first period was the best class of the day when this wonderful, new method was shown to me and this great experience of this student’s persistence refreshed my hope and interest in this problem.  Perpendicular bisectors are the new Pythagorean Theorem!

The First Followers…how do I get them in the PBL classroom?

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So I have one class this year that is rather frustrating and pretty tough to handle when it comes to buy-in with what I’m doing in the mathematics classroom.  Perhaps it’s because it’s first period, or perhaps it’s the mix of kids (quiet, shy, cynical?) – but I’m having a hard time inspiring them to speak out about their ideas or even be somewhat active in class.  This has made me wonder if I’m doing anything differently?  What’s the difference between first period and second period?  Why would this class be that much different in student make-up and personality than other challenging classes that I’ve had.

These thoughts made me remember a video I saw at a conference talk this summer and how important the “first followers” are.  This video is basically about a guy the narrator calls the “lone nut” who is dancing at a music festival (maybe you’ve seen it, it’s been around for a while) and how his leading becomes a “movement.”

It’s one of my favorites and so true.  But what I am afraid of is that the “first followers” I had in my first period class are not necessarily “followers” but students who realized they better do what I want or they won’t do well in my class.  This is not the same as “buy-in” to PBL.  This led me to think about what I needed to do in order to create first followers who would truly be inspiring and lead to more followers.  I’m not sure about this, but a couple things I tried:

  • talking about the pedagogy and how it’s different with the students
  • discussing the class contribution rubric with them and having them do a self-evaluation of their contributions to class
  • discussing listening skills when learning
  • Being deliberate about asking questions that are more open-ended (not just procedural)
  • Being less “forgiving” that it’s first period and they are tired – keeping my standards up of what I expect from them.
  • Giving praise when students take risks and learn from mistakes at the board
  • Offering a reward (like a Pez Candy) when a student is wrong but has taken a risk

So far my attempts have not been in vain, but I still don’t feel the “movement” as I do in other classes.  This has been an interesting first month with this group and I think many of them are actually learning, but don’t seem to be enjoying themselves.  I think I just need a couple more “first followers” to allow the others to see that what I am asking of them – although harder and requiring more energy and effort on their part – is actually an important part of their journey of learning.  I would love to hear from anyone who has experienced this and what steps can be taken to increase the followers in a “mob” of the whole class!

Teaching Persistence Takes Time….But How Much Time?

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I just read a great story posted on a blog about Malcolm Gladwell’s comments about Alan Schoenfeld’s research on persistence in problem solving in Gladwell’s book Outliers. In this story, a young woman persists for 22 minutes on a problem that had an average persistence time for most students of about two minutes.  Of course we would love to have students be persistent in the face of a problem they couldn’t solve and have some determination and creativity to bat to allow themselves to grapple with the problem (in other words, not just sit there and persist in the feeling of gosh-I-wonder-how-to-do-this).

But at the same time, imagine that you actually have kids who are well-intentioned, pretty smart and actually interested in learning.  Let’s just give them the benefit of the down for a second here – and we’re in a classroom where we have interesting problems that might keep them engaged in the evening with an cool idea with which they must grapple for a while.  I would ask the question, “How persistent do we want them to be?” (and so would they).

Many kids in the PBL classroom wonder this in the beginning of the year and I am asking myself now too.  This student of Schoenfeld’s that persisted for 22 minutes –  is that a good thing?  How long is too long?  When would we want a kid to know to look for resources?  To question their prior knowledge in a different way?  To know to stop and wait to discuss with others the next day?  To try using technology?

So my question would be how do we know when we are teaching persistence as a good  and productive thing or when we are teaching students that their problem solving is just the definition of insanity (repeating the same thing over and over expecting different results?).  My thought is that persistence without a growth mindset (or the belief that you can change your way of thinking and knowing) can be just as dangerous as no persistence at all.