If you’re here it means you like reading blogposts that I write. Check out my guest blogposts and the Mathematics Teacher site at NCTM for the next month or so. I’ll be trying to incorporate some of the advice and challenges I’ve faced in teaching with PBL and some of the ideas that I’ve shared with others.
One of my big “beefs” at my school is the fact that we have three levels of tracking – count ’em, three. There’s the honors track, that of course at a college prep school, most kids think they belong in. There’s the regular track, that which is still pretty quick and difficult, and there’s the track that the kids who are sometimes, I would say, just not very motivated to learn math, or have less interest in math, or maybe come from a school with a less rigorous math program, are placed in. These are the kids who probably all their lives have been told they are not “math people” and have been pigeon-holed as an “artist” or “writer” so they won’t actually need math when they get older. This really, really irks me. But I do it – I go along with a system that has been in place way before I got here. I’m only one person – even though I cite Jo Boaler’s list of research showing why “tracking” in general is just a bad idea and hurtful – I know I can’t win.
Anyway, I really shouldn’t complain because the department has let me do my thing with the geometry curriculum and I have written a PBL curriculum for the three levels. In my 201 book, I have created scaffolded problems that I think work really well with these “low-abiity” kids and often challenge them enough to make them realize how much ability they actually have. We just started talking about dimension and we watch these clips from Flatland: The Movie, where Arthur T. Square meets the King of Pointland and then meets the King of Lineland.
We had a great conversation about why the King of Pointland keeps saying “Hello Me, Hello Me” and can’t really understand why there’s anyone else there. We talked about why the King of Lineland doesn’t understand where the Square is because he only understands the directions of left and right. One of the kids goes, “Is this kind of like what happened in the movie Interstellar? I think he went through a black hole and just appeared in the future or something.” Now, I hadn’t seen that movie but then another kid said, “Well, I’m not sure it was like that.” But then one of the other students says, “No, the King of Pointland is kind of like the kid in the movie Room. Did you see that movie?” I nodded in understanding and so did many of the others in class. The student went on, “In Room, the little boy grew up thinking that “Room” was his whole universe so that was all he understood, and that’s why the King of Pointland seems so nuts. That point with no dimensions is all he can understand – there’s no one else in the world.” I was so blown away by that analogy. She really had an understanding of the idea of the limitations of being alone in the universe of a point. I had never had a kid in a “regular” or “honors” class make a connection like that – but then again, Room just came out!
Back in 2010, I wrote an blogpost comparing teaching with PBL to doing yoga. Since I have been doing Bikram Yoga for almost a year now and still can’t do “standing head to knee pose” *at all* – I thought I would repost this one just to give myself some perspective, and possibly many of you out there who might need a little encouragement at this beginning of the year time. I know that every year when students begin a year in a PBL math class the obstacles return. Parents are questioning “why isn’t the teacher teaching?” Students are questioning “why is my homework taking so long?” Teachers new to the practice are questioning “When is this going to get easier?” and “why aren’t they seeing why this is good for them like I do?” The best thing to remember is that it is a process and to understand how truly different and hard it is for students who are used to a very traditional way of learning mathematics. Give them time, have patience for them and yourself and most of all reiterate all of what you value in their work – making mistakes, taking risks, their ideas (good and bad) and be true to the pedagogy.
Here’s the original blogpost I wrote:
I don’t think my professor, Carol Rodgers, would mind me borrowing her yoga metaphor and adapting it to PBL. I use it often when talking to teachers who are nervous about falling short of their ideal classroom situation or teaching behaviors. I think this can happen often, especially when learning best practices for a new technique like facilitating PBL. There are so many things to remember to try to practice at your best. Be cognizant of how much time you are talking, try to scaffold instead of tell, encourage student to student interaction, turn the questions back onto the students, etc. It really can be a bit overwhelming to expect yourself to live up to the ideal PBL facilitator.
However, it is at these times that I turn to Carol’s yoga metaphor. She says that in the practice of yoga there are all of these ideal poses that you are supposed to be able to attain. You strive to get your arms, legs and back in just the right position, just the right breathing rhythm, just the right posture. But in reality, that’s what you’re really doing – just trying. The ideal is this goal that you’re aiming for. Just like our ideal classroom. I go in everyday with the picture in my head of what I would want to happen – have the students construct the knowledge as a social community without hierarchy in the authority where everyone’s voice is heard. Does that happen for me every day? Heck no. I move the conversation in that direction, I do everything in my power for that to happen, but sometimes those poses just don’t come. Maybe I just wasn’t flexible enough that day, or maybe the students weren’t flexible enough, maybe we didn’t warm up enough, or the breathing wasn’t right. It just wasn’t meant to be. I have exercises to help me attain the goal and I get closer with experience. That’s all I can hope for.
So I tell my colleagues who are just starting out – give yourself a break, be happy for the days you do a nearly perfect downward facing dog, but be kind to yourself on the days when you just fall on your butt from tree pose. We are all just trying to reach that ideal, and we keep it in mind all the time.
Sometimes my ignorance with respect to Twitter just floors me. Today alone I made two huge faux pas (is that plural?) with two people that I really respect and just made a fool out of myself – typos, misinterpretations, and misunderstandings abound in my tweets. But I have to say I press on – because I have found so much that informs my teaching and learning that I can handle the fool-making and embarrassment.
So here’s one thing that I did a few weeks ago – someone tweeted about this great article that I, of course, then went and read, took a picture of the great diagram in the article – but forgot to “like” or “favorite” or whatever it’s called now. So now I can’t give credit to whomever brought me to this wonderful enlightenment about which I will now write. So if you are reading, sir/madam, who tweeted this article, please forgive me.
I read this short blogpost entitled “Brennan’s Hierarchy of Imagination” and immediately made the connection to PBL.
The author, John Maeda, wrote about a conversation he had with Patti Brennan about Maslow’s famous Hierarchy of Needs of students in their learning. These two were talking about the fact that teaching creativity is really hard and Patti Brennan was thinking that it was a bit easier to teach someone to use their imagination. She was talking about this in the context of the health care field – trying to help people empower themselves to help themselves.
Of course, the first thing I thought of when I saw this pyramid was problem solving. I thought this was brilliant! The foundational, lower level of reflex or instinct is analogous to doing problems that you have seen before. Students love this instinct – the idea that if you can do a problem that someone has shown you how to do, that you are problem solving, – it gives them a reaction of completing something, some kind of satisfaction.
The next level which is appropriately called problem-solving is when you are actually solving a problem that has occurred but is constrained and you are executing skills. I love this. It still takes some talent and analysis, but you are still just reacting to a given situation. Perhaps putting together two different methods that someone showed you and seeing what happens?
The third level that she calls creativity is the first step to unique ideas and methods. The first attempt at doing something in a different way – saying what if we tried this? Has anyone every done this before? Why not? It’s still bound by the reality that we experience, but seeks to move past the knowledge that we have.
And finally imagination is what I think happens when my 4-year-old niece explains how the clouds got up in the sky because the stars moved so fast pieces came off and clumped together, or when a student can’t figure out if there are more real numbers between 0 and 1 as there are integers and they try to describe the size of those sets to me with things they imagine.
If I’m lucky, I’d say some of the kids in my classes get to creativity – in fact I think that’s been my goal as of late. To get them at least to experience it with some projects, assignments and good problems. To get them to realize that mathematics is more than just a reflex or even just reiterating a process.
Hopefully, this coming year the number of kids who will get to that third level will increase, but who knows? I do love this framework because at least I know I don’t want them stuck on that bottom level for level for long.
I was totally honored the other day when I saw some tweets from TMC16 from @0mod3 and @Borschtwithanna
— Danielle Reycer (@0mod3) July 17, 2016
— Anna Blinstein (@Borschtwithanna) July 17, 2016
And yes it’s true, I’ve been writing and practicing the use of metacognitive journaling for very long time – probably since 1996 ever since I read Joan Countryman’s book about mathematical journaling and heard about it in many workshops that summer. I wrote a rubric (make sure you scroll to the 3rd page) while I was at the Klingenstein Summer Institute for New Teachers (that’s how long ago it was) and since then I’ve been refining that rubric based on feedback from students and teachers. A few years ago, I finally refined a document called How to Keep a Journal for Math Class to a degree that I really like it now. However, please know that lots of math teachers do journaling differently and without the metacognitive twist. I do believe that metacognitive writing is essential to the PBL classroom (read more here)
So this morning, I was asked this question on twitter
— Kathryn Freed (@kathrynfreed) July 23, 2016
Which is something that many people often ask so I thought I’d respond with a more in-depth answer.
Here are the pros, I’ve found over the years of having students journal digitally:
Speed/complexity: Students are used to typing, using spell-check, inserting pictures, graphics and naturally including documents, links and thinking in the complex way that digital media allows them to. It allows their journal to be more rich in content and sometimes connect problems to each other if their journal is say on a google doc that can connect to other html docs. If they create, for example, iBooks or Explain Everything videos, there is even a lot more richness that can be embedded in the file as well – their creativity is endless.
Grading/Feedback: I found grading in Notability or on Google docs or some other digital platform really nice that allowed you to add comments with a click or audio extremely easy and quick. I did not receive feedback from the students very often about how the feedback helped them though. If you use an LMS like Canvas that integrates a rubric or integrates connection to Google it’s even nicer because you can have those grades go right from your assignment book to your gradebook.
I love having kids use digital platforms for writing/creating in mathematics when it is for a project or big problem that I want them to include many pieces of evidence, graphs, geogebra files and put it together nicely in a presentation or portfolio. Not necessarily for their biweekly journals. Some guys who make use of digital journals in interesting ways are @GibsonEdu and @FrasiermathPBL at the Khabele School in Austin TX.
Here are the cons, in my mind of using digital journals: (which might be the “pros” of paper journals) – which is the side I have come down on.
the “real” writing factor: there is some research about the actual physical process of writing and the time it takes for kids to process their thoughts. I do believe that when i want kids to be metacognitive about their learning and also want them to be thoughtful and take the time think about their initial error, think about what happened in class discussion to clear up their misunderstanding and also then what new understanding they came to. That’s a lot of thinking. So I want them to take the time to write all that down. Sometimes typing (like what I’m doing right now!) is a fast process and I’m not sure I do my best writing this way.
practice in hand-writing problem solving: this is re-enacting doing homework and sitting for assessments (in my class at least) and I want them to do this more regularly. If in your class kids take assessments digitally or do homework nightly digitally then maybe they should do their journal digitally as well. This also give me practice in reading their handwriting, getting to hear their voice through their handwriting and seeing what it looks like on a regular basis. In a time crunch on an assessment it honestly helps me know what they are thinking.
Conversational Feedback: I feel that when I hand write my feedback to them I can draw a smilely face or arrows or circle something that I want to emphasize more easily than when it is on something digitally (this is also true in a digital ink program – so that is something to consider, like Notability for example). I give feedback (see some journal examples on my blog) that is very specific about their writing and want the to improve not only in the math aspect of their writing but in how they are looking at their learning. I want them to respond and I want to respond in the hope that we are starting a mathematical conversation about the problem. I have received more questions about the feedback in the paper journals (like “what did you mean by this?”) than on the electronic feedback – not sure why.
Portability: I find that small composition graph paper notebook is extremely portable and easy for me to carry home to grade. The students bring them to their assessments and there is nothing else in the notebook (no homework at all and no access to the internet) so I am not worried about academic honestly.
There are probably more but this is it in a nutshell – please add your comments below or tweet me to let me know your thoughts!
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).
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
Page at my website with Rubrics and other guides for Assessment
On an assessment students did for me today I gave this question:
An aging father left a triangular plot of land to his two children. When the children saw how the land was to be divided in two parts (Triangle ADC and Triangle BDC), one child felt that the division of the land was not fair, while the other was fine with it. What do you think and why? Support your justification with mathematical evidence.
This idea of where the height of obtuse triangles are is a really tough one for some geometry students. But more than that the idea of sharing a height and what effect that has on the area is also difficult.
We will see tomorrow if this student is able to take my feedback and see what whether the division of the land is fair.
By the way, here’s a response that another student had:
“Because the height is the same, it’s the ratio of the bases that would determine which child would get the most land. I think the division of land was not fair, because the heights are the same so therefore the bases are determining the area of the plot. If x=5 then child one would get A=20, child 2 would get 12.5 and that makes the original plot of land 37.5. This means child 2 has a third of the land (12.5:25) (part:part) and half of child 1’s) Even without x=5, the child 2 would only get a third of the land.”
We’ll see what happens!
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.
It doesn’t always work this way, but it would be awesome if it did. When PBL is perfect or ideal, the students are the ones who make the natural connections or at least see the need or motivation for the problems that we are doing. Yeah, some of them are just really interesting problems and the get pulled in by their own curiosity, but as all math teachers know, we have a responsibility to make sure that students learn a certain amount of topics, it is quite that simple. If students from my geometry class are going into an algebra II class with trigonometry the next year where their teacher will expect them to know certain topics, I better do my job and make sure they have learned it.
So how do I, as a PBL teacher, foster the values for the problem-based learning that I have while at the same being true to the curriculum that I know I have a responsibility to? This is probably one of the biggest dilemmas I face on a daily basis. Where’s the balance between the time that I can spend allowing the students to struggle, explore, enjoy, move through difficulty, etc. – all that stuff that I know is good for them – while at the same being sure that that darn “coverage” is also happening?
So here’s a little story – I have a colleague sitting in on my classes just to see how I teach – because he is interested in creating an atmosphere like I have in my classes in his. We have just introduced and worked on problems relating to the tangent function in right triangle trigonometry in the past week and now it was time to introduce inverse tangent. I do this with a problem from our curriculum that hopefully allows students to realize that the tangent function only is useful when you know the angle.
So as students realize they can’t get the angle from their calculator nor can they get it exactly from the measurement on their protractor (students had values ranging from 35 to 38 degrees when we compared), one of the students in my class says, “Ms. Schettino, wouldn’t it be great if there was a way to undo the tangent?” and the other kids are kind of interested in what she said. She continues, “Yeah, like if the calculator could just give us the angle if we put in the slope. That’s what we want.” I stood there in amazement because that was exactly what I wanted someone to crave or see the need for. It was one of those “holy crap, this is working” moments where you can see that the kids are taking over the learning. I turned to the kids and just said, “yeah, that would be awesome, wouldn’t it? Why don’t you keep working on the next problem?” and that had them try to figure out what the inverse tangent button did on their calculator. They ended up pressing this magical button and taking inverse tangent of 0.75 (without telling them why they were using 0.75 from the previous problem) to see if they could recognize the connection between what they had just done and what they were doing.
At the end of the class, the colleague who was observing came up to me and said, “How did you do that?” and I said, “What do you mean?” and he said, “How did you get the kids to want to learn about inverse tangent? I mean they just fell right into the thing you wanted them to learn about. That was crazy.” I really had to think about that. I didn’t feel like I did anything honestly, the kids did it all. I mean what made them all of a sudden care about getting the angle? Why were they invested? It doesn’t always happen in my classroom that’s for sure. This is not a perfect science – there’s no recipe for it to work – take a great curriculum, interested kids, an open, respectful learning environment and mix well?
I do think however that a huge part of it is the culture that has been created throughout the year and the investment that they have made in their ownership and authorship in their own learning. We have valued their ideas so much that they have come to realize that it is their ideas and not mine that can end up driving the learning – and yes, I do end up feeling a little guilty because I do have a plan. I do have something that I want them to learn, but somehow have created enough interest, excitement and curiosity that they feel like they did it. It is pretty crazy.
It’s always refreshing when someone can put into words so eloquently what you have been thinking inside your head and believing for so long. That’s what Darryl Yong did in his recent blogpost entitled Explanatory Power of the Hierarchy of Student Needs. I feel like while I was reading that blogpost I was reading everything that I had been thinking for so long but had been unable to articulate (probably because of being a full time secondary teacher, living in a dorm with 16 teenage boys, being a mother of two teenagers of my own and all the other things I’m doing, I guess I just didn’t have the time, but no excuses). Darryl had already been my “inclusive math idol” from a previous post he wrote about radical inclusivity in the math classroom, but this one really spoke to a specific framework for inclusion in the classroom and how in math it is necessary.
— Carmel Schettino (@SchettinoPBL) February 3, 2016
In my dissertation research, I took this idea from the perspective of adolescent girls (which, as I think towards further research could perhaps be generalized to many marginalized groups in mathematics education) and how they may feel excluded in the math classroom. These girls were in a PBL classroom that was being taught with a relational pedagogy which focuses on the many types of relationships in the classroom (relationship between ideas, people, concepts, etc.) – I did not look at it from the perspective of Maslow’s Hierarchy of Student Needs and this is really a great tool.
Interestingly, I came up with many of the same results. My RPBL framework includes the following (full article in press):
- Connected Curriculum– a curriculum with scaffolded problems that are decompartmentalized such that students can appreciate the connected nature of mathematics
- Ownership of Knowledge – encouragement of individual and group ownership by use of journals, student presentation, teacher wait time, revoicing and other discourse moves
- Justification not Prescription– focus on the “why” in solutions, foster inquiry with interesting questions, value curiosity, assess creativity
- Shared Authority – dissolution of authoritarian hierarchy with deliberate discourse moves to improve equity, send message of valuing risk-taking and all students’ ideas
These four main tenets were what came out of the girls’ stories. Sure many classrooms have one or two of these ideas. Many teachers try to do these in student-centered or inquiry-based classrooms. But it was the combination of all four that made them feel safe enough and valued enough to actually enjoy learning mathematics and that their voice was heard. These four are just a mere outline and there is so much more to go into detail about like the types of assessment (like Darryl was talking about in his post and have lots of blogposts about) the ways in which you have students work and speak to each other – how do you get them to share that authority when they want to work on a problem together or when one kid thinks they are always right?
The most important thing to remember in PBL is that if we do not consider inclusion in PBL then honestly, there is little benefit in it over a traditional classroom, in my view. The roles of inequity in our society can easily be perpetuated in the PBL classroom and without deliberate thought given to discussion and encouragement given to student voice and agency, students without the practice will not know what to do. If we do consider inclusion in the PBL classroom, it opens up a wondrous world of mathematical learning with the freedom of creativity that many students have not experienced before and could truly change the way they view themselves and math in general.