Need Some Help Looking Forward

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What is “Low Ability” Anyway? Comparing a Point to “Room”

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!

Repost: Always Striving for the Perfect Pose

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.

Being Imaginative in Problem Solving

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.

Journals: Paper vs Digital: The Pros and Cons

I was totally honored the other day when I saw some tweets from TMC16 from @0mod3 and @Borschtwithanna


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


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!


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

Page at my website with Rubrics and other guides for Assessment

Adventures in Feedback 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.

 So this student had a hard time with this question. Since there was no height given and the bases were different, she was unable to think about how to compare the areas. She was however able to say that it would be a fair split if the areas were the same. So since I am doing this work this year with giving feedback first and then grades (see past blogpost “Why teachers don’t give feedback before grades and why they should”) I wrote this feedback on the problem set: 
 I am trying to get her to remember a problem we did in class where there was a similar problem we did with an acute triangle and obtuse triangle that shared the same height:

The area of the shaded triangle is 15. Find the area of the unshaded triangle.

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:

Just in case you can’t read it:

“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!

Can you undo an adolescent’s fixed mindset?

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.