How do you justify the time that PBL takes?

I just wanted to respond to a really great question that someone asked on Twitter the other day.

This is a common concern of teachers starting out with the idea of PBL. What does “Class Discussion” mean, first of all? I would agree that discussion does “eat up valuable” time in class on a daily basis, for sure. But what is actually happening in that discussion where something else would be normally happening in the math classroom? What does the discussion replace?

In my mind the discussion itself replaces the lecture, teachers ‘doing of problems” for the kids to then repeat, then kids often sitting on their own or in pairs doing problems that were just like the ones the teacher showed them how to do. The importance of the class discussion (which honestly is the main idea of PBL) is for students to share their ideas of prior knowledge, connections between problems, where they are confused and see where others were not confused and what prior knowledge and experience they brought to the problem.

Here’s a diagram that I use when doing PD work with PBL teachers to help explain all of what is supposed to be happening during class (it’s a lot!)


The student presentations are really just a jumping-off point. It is not just for students to explain “how they did a problem” – as they say – or they think what they’re supposed to do. The steps of Hmelo-Silver’s “process of learning in PBL” diagram that I’ve circled in pink is what students would/should do for homework. However, the part that is circled in blue is actually the learning process that happens in the class discussion – so is this time that has been “eaten up” in class or is it actually a very necessary part of the important learning, reflection and self-regulation of the process that needs to happen?

Is this harder for students? Heck, Yeah. There is so much more focus, listening, questioning and reflection that is needed in order for this process to be successful and productive. But there are ways to make it easier for students and that’s what the “class discussion” time is for. It takes a lot of practice and mastery on the teachers’ part to realize what is needed. Making mathematics discussion productive is a very important part of teaching in PBL and definitely not a part that should be seen as subtle, intuitive or straightforward.  There is so much more to this that I can not put in a single blog entry, but it’s definitely worth beginning the discussion.  Would love to hear others’ thoughts.

I’ve looked at life from both sides now…

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

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

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

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

In the original article, the authors write:

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

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

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

Diagram of retention testing research

from Roediger & Pye (2012) p.245

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

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

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

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

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

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

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

PBL and second language learners

As I am not going to be in the classroom next year, I have been going through some old boxes from my study and as many people who have been teaching for a long time have, I have boxes and bags full of cards from past students.  I spent the afternoon one day going through these, reminiscing about so many great kids that I remember.  One of them I had a card from the beginning of her freshman year and also one from the end of her senior year.  Crazy!!

I don’t claim to be an expert in emergent English language learners and mathematics at all.  I did have 10 years of teaching experience at a school (Emma Willard) where they had an ESL program and many students came into my mathematics classes who were not proficient in the English language.  I do think those girls knew what they were getting themselves into and were up to the challenge, but some of them were very frightened.

Since she has now been out of college for a while, I would assume it’s ok for me to share this on my blog.  Here is the card she gave me as a new student in 2001:

Jinsup's card from freshman year

Jinsup’s card from freshman year

This card was written with the voice of a student who was used to a very structured, repetitive mathematics class and I believe she knew that coming into the U.S. things would be different, but possibly not as different as they were in my class.  When she said, “I’m so nervous that you will let me to talk a lot in the class” I’m sure she was saying that she was nervous that I would expect her to contribute to the class discussion.  What I did with many of those students, including Jinsup, was I focused in the beginning on letting them listen and write.  I gave them lots of feedback on their journals and made sure they had the correct vocabulary and that their grammar in their writing made sense.  I allowed them to ask more questions initially than to present their ideas until their confidence became stronger.  Jinsup, as most Korean and Japanese students did, had excellent skills, as that was what their math education had focused on since elementary school.  However, she was not very good at reasoning, sense-making or critical thinking on her own.  It was almost as if she had not been asked to communicate about mathematics, as she was trying to say in her note to me.

However, she ended up doing very well in that first class and then I taught her again in precalculus (which we called Advanced Math) and then in BC Calculus her senior year.  Her excellent background allowed her to focus on the reasoning aspects of all of these courses and in the end, I was very impressed with her growth.  She really got the best of both worlds – the skills from her Asian mathematics education and the collaboration, communication and reasoning skills from the PBL here.

This is the note she wrote me at the end of her senior year:

Jinsup's card at end of her senior year

Jinsup’s card at end of her senior year

Although I know this is only an anecdote and I don’t really have research evidence that PBL totally works with ELLs I do have confidence that with the right environment and patience, it is actually a great way of teaching for many of these second language learners.  It allows them to find their voice in a language that is already new to them but at the same time have some practice in terminology that they may have heard before.  I think this might be my next interesting research project – if anyone has some thoughts on this I’d love to hear them.

Ch-Ch-Ch-Changes….Turn and Face the Strange.

Strange fascination, fascinating me
Changes are taking the pace I’m going through

Well, you know it’s been a long time since you wrote a blogpost when a perfect stranger sends you an email and says,  “I hope you’re just busy or I’m looking at an old blog you don’t update anymore or something and everything is ok.” It was a very nice email and I often forget when I get busy and wrapped up in what’s going on in my own life that there are actually people that read my blog and my tweets.  I need to keep up with my communications and I feel horrible.  So I have many apolgies to make.

But first let me tell a story that just happened a few weeks ago.  I was doing a 2-day workshop at a school that had made a decision to implement PBL in the first two years of their high school curriculum.  Because it was such a cross-curricular choice, almost everyone in the department was going to be required to teach a PBL course at some level or another.  We had spent a great deal of time talking about ways to facilitate discussion, make best use of class time, allow students the agency to work out understanding together, etc.  At the end of the second day, we were reflecting on what we were still concerned about for the coming fall.

Many teachers had very specific questions about how to put the theory into practice, but then one wonderful veteran teacher spoke up.  He talked about how he was very excited to try this type of teaching and what it really meant for the students’ learning in his class.  However, he then got very serious and sincere – he basically said, “What if I’m not good at it? I am mean I know I am very good at explaining mathematics to students, I’ve been doing it for 20 years this way.  I enjoy helping kids understand by telling them my understanding.  What happens if I am not good at holding back and letting them struggle?”

Of all the questions that teachers have asked me PD workshops, I thought this was the most honest and probably most unasked question that existed.  He was being extremely vulnerable and I think asking what everyone wanted to know.  We had a wonderful conversation about how he coached football for many years and how this type of teaching is very similar to coaching.  We talked about the satisfaction that you see on students’ faces when they are allowed to take ownership of the knowledge in the classroom and each others’ understanding.  It will be a difficult change in his pedagogy but it is worth a try. He’s probably one of the most brave teachers I have met in my work.

If you are one of these people that is interested in learning more about PBL and how it could enhance your classroom pedagogy, I am looking to pilot an online course this summer.  Click here to subscribe to a mailing to find out more and possibly enroll in the class.

So honestly, change is not that bad – it gives me anxiety and makes me worry.  But honestly, communicating and learning all about it is really the best.  I am moving out of the classroom for a year next year to see how that goes.  I’ll be doing Learning Experience Design for Online Learning and also Teacher Support for Math Teachers who are teaching with PBL.  Hopefully, I’ll also be planning the 2018 PBL Summit as well!

Turning to face the strange – and taking the pace I’m going through – please be forgiving of the changes in my life while I get back into blogging!

What I get out of Student Writing

I have been using journaling in math class since 1996 – which was a really important year in my teaching career for lots of reasons, but it was definitely because I was introduced to the idea of math journals.  Since then I’ve done many different iterations for what my expectations are.  Even this year I did something new where I allowed students to write about errors they made on assessments in order to attempt to compare their assessment problems to what they did on homework in the hope of reflecting on the work pre-assessment for future problem sets.

However, a lot of students still use their journal almost like a problem-solving conversation with me, especially after we have already gone over a problem and they still don’t understand a method.  Here is one I ran across just the other day in my lower-level geometry class and thought it just perfectly expressed some of the goals I am hoping to accomplish with journaling.

I’ll call this student Cindy and we had just introduced the theorems about parallel lines through a geogebra lab and this had been the first problem they looked at that took the concepts out of the context of the lines and threw it into a triangle.  For many students this might be an easy transfer of skills (including the algebra, other theorems, etc.) but for the kids I have – not necessarily.  Here is what Cindy wrote:file_001-1

The first thing that Cindy does in her journaling is make her own thinking explicit (which I love).  She is stepping me through her thinking and the questions that arose for her.  This is actually a major step for many students who are confused – are they able to even know what they are confused about?

She writes: “I know the problem probably deals with the parallel line theories that we dealt with.” and then lists the types of angles we studied and then with a big “OR” says “maybe it has something to do with the sum of the angles of parallelograms and triangles?”  Little does she know that what she is doing is practicing synthesizing different pieces of prior knowledge – is it overwhelming to her? – possibly, but she went there and that’s so great!  I wanted her to know that I was excited that that she even thought about the sum of the angles so I gave her some feedback about those ideas.

She wrote down what she knew about the sums of the angles which we had also studied.

She writes her first equation to think about: “5x-5/=180” using one of the angles in the top triangle.  I would’ve loved to know where that was coming from.  What made her write that?  She then notes that “but it wouldn’t work because if x is the measure of the angle than the equation should be set to 180”

There is so much that this tells me about her confusion.  She is not understanding what the expression 5x-5 is supposed to be representing in the diagram I think, or she isn’t connecting what x is “not representing” (the angle) and the whole expression is representing too.  She also is confused about between the sum of the whole triangle’s angles and just that one angle.

She then looks at the two expressions she is given, 5x-5 and 4x+10 and I think makes a guess that they are corresponding angles – she doesn’t give any reason why they are corresponding.  She just asks the question.  But the cool thing is she says “Let’s try it.”  I love that.  Why not – I am always encouraging them to go with their ideas and the fact that she tries it is wonderful.  The funny thing is she does end up getting the same value for the two angles so she asks: “Does this mean that this is correct?” and then “What do I do for “6y-4?” and still has not connected many of the ideas line the fact that these angles are a linear pair and that’s where the 180 comes into play, or even why the angles were corresponding in the first place.  So many questions that she still has, although I am encouraged by her thinking and risk-taking.

This journal entry allowed me to have a great follow-up conversation with Cindy and she was able to talk to me about these misconceptions.  I’m not sure I would’ve had this opportunity to clarify these with her if she had not written this journal entry and then she would not have done so well on the problem set the following week.  I just love it!  Let me know if you use journals and if you feel the same clarifying or communicative way about them too.

See my website for lots of sample entries and also other blogposts and resources about journaling if you are interested.

 

Yours, Mine and Ours

Yesterday we had a speaker in our faculty meeting who came to talk to us about decision-making process in our school.  He spoke about the way some colleges, universities, independent schools are very different from businesses, the military, and other governing bodies that have to make decisions because we are made up of “loosely-coupled systems.” These are relationships that are not well-defined and don’t necessarily have a “chain of command” or know where the top or bottom may be.  They also don’t necessarily have a “go-to” person where, when a problem arises, the solution resides in that location.  The speaker said that this actually allows for more creativity and generally more interesting solution methods.

About mid-way through his presentation he said something that just resonated with me fully as he was talking about the way these systems come to a decision cooperatively.

“The difference between mine and ours is the difference between the absence and presence of process.”

Wow, I thought, he’s talking about PBL.  Right here in faculty meeting.  I wonder if anyone else can see this.  He’s talking about the difference between ownership of knowledge in PBL and the passive acceptance of the material in a direct instruction classroom.

Part of my own research had to do with how girls felt empowered by the ownership that occurred through the process of sharing ideas, becoming a community of learners and allowing themselves to see others’ vulnerability in the risk-taking that occurred in the problem solving.  The presence of the process in the learning for these students was a huge part of their enjoyment, empowerment and increase in their own agency in learning.

I think it was Tim Rowland who wrote about pronoun use in mathematics class (I think Pimm originally called it the Mathematics Register). The idea of using the inclusive “our” instead of “your” might seem like a good idea, but instead students sometimes think that “our” implies the people who wrote the textbook, or the “our” who are the people who are allowed to use mathematics – not “your” the actual kids in the room.  If the kids use “our” then they are including themselves.  If the teacher is talking, the teacher should talk about the mathematics like the are including the students with “your” or including the students and the teacher with “our”, but making sure to use “our” by making a hand gesture around the classroom.  These might seem like silly actions, but could really make a difference in the process.

Anyway,  I really liked that quote and made me feel like somehow making the process present was validated in a huge way!

End of Term Reflections

Phew…exams given…check…exams graded…check…comments written…check…kids on bus…check.  Now I can relax.  Oh wait, don’t I leave tomorrow to drive to my sister’s for Thanksgiving?

Such is the life of a teacher, no?  Just when you think you are on “vacation” there’s always something else to do.  I had an exam on Saturday then worked the rest of Saturday and Sunday finishing up that grading and writing my comments that were due this morning at 9 am.  But wait, I told some people I would write a blogpost about what my classroom is like, so I really wanted to do that too.  That’s OK though, I think it’s important for me to reflect back on this fall term – what worked and what didn’t for my classes.

I have three sections of geometry this year that I teach with PBL and a calculus class that I would say is something of a hybrid because we do have a textbook (as an AP class I needed to do what the other teachers were doing), but I do many problems throughout the lessons.

In my geometry classes, the student have iPads on which they have GeoGebra, Desmos and Notability where they have a pdf of their text (the problems we use) and where they do all of their homework digitally.  My class period for that course alternate between small group discussions in the Innovation Classroom in the library on Mondays and Thursdays and whole class discussions with student presentations of partial solutions (a la Jo Boaler or Harkness) on Tuesdays and Fridays. (We meet four times a week 3 45-minute periods and 1 70-minute period.)  Because my curriculum is a whole-curriculum PBL model, we spend most of the time discussing the attempts that the students made at the problems from the night before.  However, in class the discussion centers around seeing what the prior knowledge was that the presenter brought to the problem and making sure they understood what the question was asking.

classroom-shot1

Whole Class Discussion in regular classroom

 

geom-class-2

Small Group Discussions in Innovation Lab

If this didn’t happen we end up hearing from others that can add to the discussion by asking clarifying questions or connecting the question to another problem we have done (see Student Analysis of Contribution sheet).

One of the things that I had noticed this fall in the whole class discussion was that the students were focusing more on if the student doing the presentation was right immediately as opposed to the quality or attributes of the solution method.  There was little curiosity about how they arrived at their solution, the process of problem solving or the process of using their prior knowledge.  Unfortunately, it took me a while to figure this pattern out and I felt that it had also weeded itself into the small group discussion as well.

One day in the small group discussions, it became clear to me that the students were just looking for the one student who had the “right” answer and they thought they were “done” with the question.  This spurred a huge conversation about what they were supposed to be doing in the conversation as a whole.  I felt totally irresponsible in my teaching and that I had not done a good enough job in describing to them the types of conversations they were supposed to be having.

This raised so many questions for me:

  1. How did I fail to communicate what the objectives of discussing the problems was to the students?
  2. Why is this class so different from classes in the past (even my current period 7 class)?
  3. How can I change this now at this point in the year?
  4. How can I stress the importance of valuing the multiple perspectives again when they didn’t hear it the first time?

In my experience, sometimes when students are moving forward with the fixed mindset of getting to the right answer and moving on, it is very difficult to change that to a more inquiry-valued mindset that allows them to see how understanding a problem or method from a different view (graphical vs algebraic for example) will actually be helpful for them.

My plan right now is to start the winter term with an interesting problem next Tuesday.

“A circular table is pushed into the corner of the room so that it touches both walls. A mark is made on the table that is exactly 18 inches from one wall and 25 inches from the other.  What is the radius of the table?”

table-picture-problem

I have done this problem for many years with students and I have found the it works best when they are in groups.  I usually give them the whole period to discuss it and I also give them this Problem Solving Framework that I adapted from Robert Kaplinsky’s wonderful one from his website.  I am hoping to have a discussion before they do this problem about listening to each other’s ideas in order to maximize their productivity time in class together.  We’ll see how it goes.

To Hillary, With Gratitude

This morning as I woke up and found out about the results of last night’s election I was at first filled with despair and finally got myself somewhat out of that funk.  Then I thought about what Hillary Clinton must be feeling – she must be exhausted of course.  What did it take to put all of that energy into this campaign?  And those years of service to this country? And to put up with her husband? And the criticism?  This is not to say she didn’t make mistakes in the public eye of course.  I’m not saying I didn’t disagree with some of her stances, but I just want to look at it from the female perspective.  What I want to say to Hillary right now is thank you.  Thank you for being the first woman to have to go through the ordeal of running for president and dealing with all of the mess that goes with that.  I can’t imagine what that was like.

I have to say that in my career I know what it’s like to be one of two women in a meeting room and have to work extra hard to get a group to listen to your point.  Or perhaps to couch what you want to say in terms that the men will want to hear until they come over to your side in order to get them to vote your way on a certain agenda item or thinking twice about what I wore so as not to get judged.  The diplomatic skills that are acquired in just being a woman in an administrative position are invaluable because of the ways in which you know you need to listen and be heard. Being a woman in mathematics, the message is usually clear at national meetings when the majority of conference-goers are female classroom educators and the presenters are more often male speakers who are not currently classroom teachers.  In my graduate school education in mathematics I had one female professor and I was the only female in the Masters program.  You learn to “blend in” by speaking like them, working like them and going about your business like them.

I wonder if there isn’t a little part of Hillary that this morning just said “Phew, no more of that faking it.”  She was tired of being the male-culture-created part of herself that she had to be in order to run for president.  A few female heads of school that I have spoken with have said that in order to lead, many women are expected to downplay their feminine qualities – to not cry or be emotional, to be sure they are surrounded by male advisors so no one can say you made mistakes because you “are a woman.”  Spending so much time worrying about balancing speaking your mind with being nice to everyone so you are not labeled “bitchy” gets really tiresome.

What this election taught me overall is that misogyny is alive and well in the U.S. (not that Hillary needed to learn that) even more than racism.  My guess at this point is that we will elect a gay man in the future before we elect a woman but either way, I am grateful for Hillary and all she has done to pave the way for each other woman who comes next.  I read that Kamala Harris (CA) was the second black woman to be elected to the Senate, Ilhan Omar (MN) was the first Somali-American woman elected to Congress and Catherine Cortez Masto (NV) was the first Latina Senator to be elected.

I’d like to think that Hillary is waking up today really looking forward to spending some time as a grandmother, writing a book and working on the next great way to help kids, health care reform and education.  Sure that’s just me being idealistic, but as a woman, I would like to think that’s what I would do – well, after crying after losing for a little while.

Modeling with Soap Bubbles

I am so very lucky to have a guest teacher with me this year at my school.  Maria Hernandez (from the North Carolina School of Science and Math) is probably one of the most energetic and knowledgeable teachers, speakers and mathematicians you could ever find – and we got her for the whole year!  We are so excited.  I am working with her and she is so much fun to work with.  I have been teaching calculus with PBL for almost 20 years now and thought I had all the fun I could but no!  Maria is bringing modeling into my curriculum and I’m enjoying every minute of it.

As we started teaching optimization this week, Maria had this wonderful idea that she had done before where we want to find the shortest path that connects four houses.

picture-of-houses

I let the kids play with this for about 10 minutes and then did this wonderful demonstration with some liquid soap bubbles and glycerin.  We had two pieces of plastic and four screws that represented the houses.  As the kids watched, I dipped the plastic frame into the liquid and voila-file_000

Right away the students saw what they were looking for in the shortest path.  Now they had to come up with the function and do some calculus. As they talked and worked in groups, It was clear that using a variable or one that would help them create the right function was not as easy as they thought.  However,  I was requiring them to write up what they were doing and find a solution so they were working hard.

file_000-1

We have been doing a lot of writing in Calculus this fall so far and they are getting used to being deliberate about their words and articulating their ideas in mathematical ways.

Here is the outline of the work they did in class: Shortest Path Lab

and here is the rubric that I will be using to grade it.

rubric-for-lab-3-2

The engagement of students and the buzz of the classroom was enough to let me know that this type of problem was interesting enough to them – more than the traditional “fold up the sides of the box.”  The experience they had in conjecturing, viewing, writing the algebra and solving with calculus was a true modeling experience.

If you decide to do this problem or have done something like it before, please share – I’d love to do more like this.  I am very lucky to have a live-in PD person with me this year and am grateful every day for Maria!

 

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!