PBL for Engagement and Empowerment in the Mathematics Classroom

OK, I admit it. Someone just asked me about my blog and I thought – what blog? It has been since last August since I sat down and felt the urge to write. But in reality I had been writing, just not online. I have been immensely busy in my life with many other things and honestly had not been inspired to put words down on electronic paper. So here I am again thinking and reorganizing my thoughts. I have put together all of the past entries on the year that was the transition to the PBL in our geometry class. They are archived into the “Transition to PBL Entries” which can be searched by tag, if you like. I will still be using those entries in my workshops and hopefully they will be useful for people who will be making that transition as well. I still hope someday to use those entries to put together some sort of journal or book that might be useful about the process itself.

However, right now as my colleagues and I move forward in the editing process, we are working on integrating the technology (the labs, the SMARTboard use, etc.) into the curriculum more and more, and I feel my focus is leading me in that direction. The students this year have come such a long way with the problems and we are more than pleased with the classroom dynamic, the learning process and the curriuclum itself. Some other goals I have are to use some type of visualization software to create a curriculum map (which some of my particpants last summer asked me for). I will be offering courses again this summer if people are interested, let me know by posting a comment.

Another immediate goal of mine is to begin garnering support to begin work on another level of curricular reform. I’m not sure which course, but I’m hoping perhaps Algebra II or Trigonometry. This is an extremely skill-based course at our school, which many students find very dull and repetitive. If anyone has any insights on which level might be better I’d like to hear that as well.

I have a renewed sense of interest in this discussion as my graduate work is getting more serious in the upcoming years, so I foresee my blogging getting revitalized as well as my excitement about technology. So hopefully, I will keep this up a little bit more!

Hi everyone! It’s almost September and I’ve got a new theme. I always love September because so much is new and there’s always hope for change. I have had the most wonderful summer talking to people about Problem-Based Learning and sharing my thoughts with so many interesting educators. I decided to change the theme on this blog – not because I didn’t like the pretty green theme anymore, but because I thought, why not start fresh? My colleagues and I have revised all of our materials – they are brighter and better and we are very excited. I thought I’d talk a little about my summer and the changes that we made to our curriculum. I hope that many of you will add your comments to my blog this year (now that some of you know that I’m here!)

Back in June I offered a workshop at the Technology conference at the PEA at Exeter, NH. I had such a wonderful group of teachers in my workshop who were all so curious about Problem-Based Learning. Our conversations really made me think about what I do and had me rethinking so much of what I say and do about how I discuss Problem-Based learning. I want to thank every one of those participants! I think the most exciting part for me was to see how much interest there really was out there which made me realize that I’m not doing all this in a vacuum. It’s really exciting to think that we could start a real professional discussion about all of this. It also helped me pinpoint what some of the questions, concerns and cares are out there from all of you.

In July, my colleagues and I took a long, hard look at our own materials and did some great work of our own. We spent some time rearranging the topics in our materials. For one thing, we were not happy with the way we discussed congruence of triangles. We had originally used some problems where students were asked to construct triangles with specific angle and/or side lengths. For example, construct a triangle with a side length of 8 cm, 6 cm and an angle of 24 degrees. We were hoping that they would use a protractor and measure the angle and use a ruler and measure the sides. We also hoped that some kids would come to class with the sides surrounding the angle and other kids with the sides next to each other, right? Well, those hopes were often dashed. We found that in maybe 4 of our 5 sections this exercise actually did what we wanted it to. Some students couldn’t find their protractor, so didn’t do it. Some kids used inches by accident, so they weren’t congruent anyway. Some kids used the excuse that they didn’t know how to use the protractor so couldn’t do it. This just didn’t work at all, so the idea of talking about triangle congruence in the problem went out the window altogher. The teacher ended up artificially discussing ASA or SSA anyway. UGH!

So, one of us said they had liked what they had seen somewhere else in a textbook a while back and we discussed the idea of asking the question “What’s the minimum amount of information needed to say that two triangles are congruent?” and we took it from that standpoint. On our campus, there is a big triangle in the middle of the main campus that is called the “senior triangle” which means that only the seniors are allowed to cut across is and no one else is allowed on it at all (we really do). We created a scenario where the juniors were jealous of the seniors and measured the “senior triangle” – part by part. So we took this “story line” and wrote a strand of questions where the juniors took one part of their measurements, say one angle, and copied the senior triangle – could they have had the same triangle? That is supposed to motivate the discussion (hopefully, a short one) of why having only one angle the same is not enough. If you have our materials, you can follow that strand all the way through and tell me what you think of it. We’ll see how it goes with our classes too.

There were many other changes we made, but I will save those for another entry. I have many more preparations to make. The year is beginning, whether I am ready or not, right?

Yes, I am a huge LOTR fan but can’t help feeling a little like Sam Gamgee and stating, here at the end of all things, “Well, I’m back.”  It has been a very stressful year for many reasons, one of which was dealing with the new curriculum in our Geometry class.  May zoomed by so fast that I didn’t even have time to think about what I could post.  Here are a few comments.

We definitely wrote and planned for too many problems.  Unfortunately we did not get to the end of the curriculum which had volume and surface area of three dimensional solids.  We feel OK omitting this mostly because in general, students have seen this material before.  It was much more important to include a solid foundation of Trigonometry and exposure to arcs and angles in circles.

We have resigned ourselves to the fact that some students really didn’t like this method of teaching.  We did course evaluations and some students were pretty frank about their feelings on how this class was run.  However, the good thing is that almost for every student who didn’t like it, there were 2 or more that did.  Also, even the students who didn’t “like” it, wrote that they came to appreciate the teaching method.

I also had had my class video recorded one day in order for me to reflect on my teaching. I was amazed at how much work was getting done without my knowledge.  But I was also amazed at how much of other behaviors goes on in my class without my knowledge.  I highly recommend having a class videotaped at least once in your career.

I also would like to reflect on how important attitude is to the success of the student in this course.  I know I’ve said this before in my blog, but I think that once a student has buy-in and investment into the process, it is truly an amazing thing to experience and watch.  However, that can be three months for most students, 6 months for others, ( for a smaller minority and for a few, it just doesn’t happen. 

I’ll use the example of a student, Bob, who has not been able to take down his barrier of pessimism and bad attitude.  Just last week, at one of our last classes of the year, Bob could not control his anger towards the class and stormed out of the room when the bell rang.  I am sure that this anger was not necessarily directed at me, but most obviously directed at himself for not being able to accept his own responsibility for the lack of work put into this course.  However, on the final problem set, I happened to observe Bob describing to some members of his group how to find the area of a sector of a circle, without prior knowledge of a formula.  I was very proud and smiled at him quietly.  He smiled back.

Another student, I’ll call Sam, had real reservations about a team final exam.  She said in class one day,”I just feel badly for my group because I don’t think there’s much I can contribute”.  She seriously did not see what she might bring to a group discussion of a problem.  However, afterwards, while I was looking at the course evaluations, I could tell Sam’s by her handwriting and noted that she said “I was surprised at how much I had to add to the discussion.”  I believe this was a turning point for her, and may have been a place where her confidence in her abilities has changed for the better and for good.

There’s so much more that I could say and my hope is to somehow organize my thoughts from this blog so that I can form chapters in a book.  Hopefully, I can affect more than just the classrooms at my own school, but possibly model what is possible at any other school.  I would like to thank my two colleagues who embarked on this challenge this year with me.  They were continually an inspiration and without them, this would not have been possible.  Also, I owe a debt of gratitude to the head of my school for allowing us the freedom to “think outside the box” with our curriculum .  The autonomy and respect she afforded us and the faith she had in our abilities really was a support throughout this year.

And so I come to the end of this blog (at least for this year), I hope if anyone is reading this that they leave me comments so that I can give further information that might be helpful.  Thanks for reading!!

One of the biggest challenges we’ve faced this year is what to do with our “final exam”.  Our mostly traditional math department has always had cumulative and shared final exams where students cram material to show their teachers how much of the year’s work they have retainou caned.  This was always interesting to me since I never really understood the point this type of final exam.  Does learning mathematics mean that you are able to repeat material that you learned nine months ago? Does it mean that you can study very hard and remember how to do problems becasue you’ve memorized the process?  Or does it mean that you have a certain bank of skills and concepts that you can access in order to solve problems given to you?

We geometry teachers agreed with my last definition of the purpose of a “final exam”.  We felt that if we were really teaching this course towards better problem solving skills for students, we better find a way to emulate that experience for the students in order to most authentically assess their problem solving skills.  This was a daunting task and a very difficult one to craft.

What we decided to do was create a situation most like the problem sets they had done for us all year, but also to emulate the classroom environment of discussion and sharing of ideas in order to best assess their independent and group problem solving skills.  The first piece of this was to come up with five problems that we thought were challenging enough for a group to take on, but at the same time included skills and concepts that were threaded through the curriculum throughout the year.  Students were allowed to use their journals, which served as their resource throughout the year as well.

But how to replicate the classroom environment?  We decided to construct groups that would be well-balanced with respect to ability, in order to see these students work together and use their skills.  Assessing that type of work proved to be very difficult indeed.  The teachers walked around and attempted to observe students in their classes and rated them on how well they:

• Listen well to others
• Focus on Task
• Participate in Discussion
• Productively discuss ideas
• Help others understand
• Contribute to the solution. 

Not only did the teachers assess each student in their section, but we had each group members assess the others in their group.  The average rating in each category (1-5) was then added up for a total number of points out of 30.  This percentage then counted as 50% of their final exam grade, with the other 50% being the points for the 5 written solutions solved as a group.

In retrospect, it may have been a slightly-less-than-perfect model to use, but we were glad we tried it.  Some problems that arose were that a few groups (2 or 3 out of 16) did not take this rating system seriously and the teachers were forced to balance their scores with the students’.  We also found that there was so much feedback about how the groups worked together that it was difficult to accurately score how a certain individual did.  Interestingly, some students did feel comfortable writing their comments on the sheet we gave them and one emailed us an explanation of her scoring later that day.  Some ideas for next year would be to have 2-3 questions done as a group and 2-3 questions done as individuals and see if that worked out more fairly.

Overall, I am proud of my colleagues who trired something different, but extremely deliberate.  We tried to model for the students the classroom process that we held so important all year long.  For that, I feel that it was authentic, even if we learned a great deal from it as well.

 Well, the end of my transition year with this new curriculum for my colleagues is almost done.  We have realized that we wrote way more problems than we needed in order to get through our old geometry curriculum content-wise.  It is an interesting thing, though, to have to go through and cut problems that you really loved when they were written.

 Anyway, something else that comes to mind at this point in the year is how far the students may or may not have come in their work with problem solving.  Clearly, the stronger mathematics students basically just eat up these problems.  They are confident in their skills and ideas, so being part of the conversation is not difficult.  However, the students who might have come into this course with the preconceived or realistic notions that they are pretty weak mathematics students are much more of a concern.  The concerns range from how attentive they can be during a problem discussion to how they can possibly be metacognitive enough to write a journal entry.  All of these issues surfaced throughout the year with different students and I thought I’d share a few with you.

One student, I’ll call John, came into this course with a barely passing grade in Algebra I.  In fact, there was serious rethinking of him moving on at all into this course.  The year began for him as a studet who was barely engaged in the problems, but because of his positive attitude, persevered.  John had severe retention and learning issues, which a few months into the year were diagnosed by a professional.  Clearly, there was no way this student could remember to solve a linear equation from Algebra I, if he couldn’t recall the properties of parallelograms from the previous problem, two days ago.  This was clearly an extreme case, but worth noting.

John never stopped working and half-way through the year finally was linked with a tutor who specialized in his special needs.  John started meeting with his english teacher to help in writing his journal entries, and finally something clicked.  The entries became much more connected to class problems and ideas.  The work in class kept him engaged and sometimes even volunteering to share a solution that he knew was wrong.  The growth in this student since September has been tremendous, even if grade-wise he’s still in the C- range.

Another success story is a girl I’ll call Sheila.  This is a student who came into the course with very little confidence in her ability to speak her mind(with respect to math) or have good ideas.  She was a B-/C+ student in Algebra I and definitely retained some of her skills from that course.  She started the year mainly complaining blatantly in class about how much she did not enjoy the pedagogical framework of the course (my summarizing of course), and frequently stopped class to comment about how a problem could’ve been more easily stated or how unfair it was that we would expect her to draw her own diagram.  I allowed this to happen a number of times since I wanted to create a sense of freedom of expression in the classroom.  However, at mid-year, I made it clear to her that this type of sharing with the class will no longer be tolerated.  Interestingly, once that happened there was a change in her work and attitude.

Now, I’m no psychologist, so I can’t claim to understand what made Sheila change the way she worked, but all of a sudden so much of this course mattered to her.  It came across in her contributions to class, which were often surface in her understanding, but sometimes showed great insight.  She once described the reasoning behind the Triangle Inequality brilliantly with a piece of her hair.  At the end of the fall semester, she had a C-, and most recently earned an A- on a problem set.  Again, I can’t exactly put my finger on what has changed for this girl, but I can hypothesize.

It might be that Sheila finally stopped complaining enough to see that she did have important things to say in class besides her complaining.  It might also be that once she did that, she actually enjoyed being a productive part of a solution process with her peers.  I sometimes catch her explaining something to another student and think that her confidence in her abilities has grown tremendously.

The more important question that follows is twofold 1) are these changes because of the coursework or are they just the normal intellectual maturity that happens to teen-agers and 2)what will happen next year when they move on to a more traditional course in Algebra II/Trig.  It is the second question that has my colleagues most troubled, and for now we will have to just wait and see.  Although our head of school, would be happy to have us writing another year’s worth of curriculum this summer, we really feel that we need to edit and really solidify the year we have right now.  Hopefully, in the future, we’ll be able to bring this experience to other courses as well.

I’ve recently read an article that discussed a study relating writing-to-learn programs and their effectiveness on academic achievement.  This article (Bangert-Drowns, et al, 2004) states that using a writing-to-learn program has a small positive impact on students learning (measured by testing) in a specific class.  I should mention here, that we have been using an informal writing-to-learn program in our Problem-Based Curriculum this year in the form of journal writing.  We have required every geometry student to keep a journal which has become a compilation of their questions, ideas and problem-solving processes that the students use for reflection.  Since this was the first year that all of us geometry teachers were trying this, we decided to do it very casually and not require certain entries on specific topics.  We left it up to the students what problems they would write about and how detailed they would be about their solution methods.

However, we graded the entries on the depth of understanding shown in the entry.  Not only understanding of the problem’s solution, but of the context of the problem, how it relates to other problems, how it would be categorized, what relevant new material was explored, etc.  Students’ journal entries have ranged a great deal in quality and quantity.  As the year progressed, practically every single student has improved in their ability to write clear, concise and thoughtful explanations and sometimes even made connections to other problems.  We felt that this part of the curriculum was very successful.

- Until I read more into this article I received.  It seems that the authors who were writing a meta-analysis (a study of studies concerning writing-to-learn programs’ effectiveness) across disciplines.  It seems that in order to be most effective, these programs needed to be used over a long treatment length.  Well, this is good, I thought to myself.  We’ve been doing it all year, generally one or two entries a week, this must mean that it’s being effective right?  Hmmm, well some of the factors that created negative effects were introducing the writing in middle school (phew!) and having longer individual writing assignments.  OK, well also didn’t do either of those.  However, the other factor that enhanced the positive effects was the introduction of metacognitive prompts.  This we were not doing (well, 3 out of 4 ain’t bad, right?)

 Well, let’s think about it.  The role of the metacognitive prompts would be to get the students to begin to think about how they were learning the material they learned in each problem.  These prompts would be questions such as “Describe your thought process when you first attempted this problem and comment on your misunderstandings or misconceptions” or even “How does the theorem discovered in this problem relate to a previous problem on a procedural level?”  I think I could come up with some of these prompts and actually make them more a part of the journal writing process next year.  If it has shown to help students’ academic achievement it is hard to argue with.

It does take a lot of effort to put all the pieces in place, so why would it be worth it to me?  It seems to me that one of the premises in which PBL is rooted, is the value of procedural and conceptual thinking.  Writing on the metacognitive skills needed for the problem solving processes seems to be one way in which to focus the students’ attention in a reflective mode of thinking.  This reinforces their responsibility in their own learning, creates authorship of their own ideas and invests them in the processes of which they write.  For example, one student wrote this about learning an algorithm to use to find the coordinates of a point reflected over a line.

“What did we just do? – We know that the perpendicular bisector of a segment is the mirror of a reflection.  In order to reflect point K over the mirror line we made a perpendicular line going through point K.  Then by finding the intersection point we were finding the midpoint of the segment KN and then adding the vector of K to the midpoint to the midpoint, was finding the actual point B which was the image of K by reflection”

OK, so it’s not Pulitzer Prize writing, but this high school freshman was able to articulate a very complex process.  Here’s an added comment she wrote below her more “algorithmic” message:

“On the homework, when adding the vector to the original point, I was so used to finding vectors from the origin that I just doubled [the vector] instead of thinking I had to put the actual coordinates of the point into perspective.  It wasn’t until class that I realized the actual vector from point K to the intersection was [half of it].”

This was quite an a-ha moment for this student who made a very conceptual jump in her head from the process she wrote about before.  She realized that the vector she found from the point K to the mirror line would be half the vector from K to its image – something she came to because of the practice of thinking metacognitively about her work.

My plan for next year is to have a list of metacognitive prompts to help enhance this program.  I know the work of providing these prompts will take some time over the summer, but at this point, I’m so invested I can only hope to improve on what we’ve created this year.

Bangert-Drowns, R, et al (2004). The Effects of School-Based Writing-to-Learn Interventions on Academic Achievement: A Meta-Analysis. Review of Educational Research, 74(1), 29-58.

Ahh, and herein lies the rub…practice or not to practice – that is the question. Do you feed into the tradition of repetition that perpetuates students’ perceived notions of success in learning mathematics? Do you continue to lead them to believe that if they can do 25 of the same type of problem that they are now successful problem solvers? Where does our responsibility lie in being mathematics educators – to the standardized tests to which our students must submit or to the larger skills that will move them forward in all aspects of their lives? Or can there be a balance?

We read in newspapers almost monthly about school districts that have initiated a “progressive” mathematics program with all good intentions to foster creativity and problem solving, higher order thinking skills. But the article inevitably is not about the good that is being done, but the outcry of angry parents or unsatisfied board members is usually the point of the article and how we all need to just “go back to basics” (see NY Times Education section article by Tamar Lewin, As Math Scores Lag, a New Push for the Basics, November 14, 2006 or Samuel Freedman’s ‘Innovative’ Math, but Can You Count? November 9, 2005). Do we just ignore these issues and pound ahead as if only our agenda matters?

We’ve had to find a balance this year in our geometry curriculum that allows the students who need it, the practice to gain certain important skills. We would make the decision to do practice only when we all felt that not only the majority of students needed the practice, but we also saw the skill as a necessary tool to move onto their next course. Let me see if I can give an example. We spent a great deal of time between October and December having students write equations of medians, perpendicular bisectors and altitudes. Although we don’t feel that the ideas of the points of concurrency is important for further study, we definitely felt it was worth it for them to have the practice in order to move onto Algebra II/Trigonometry next year. We also practiced factoring and solving quadratics too.

However, we are going on the assumption that it is not the specific content that we cover that is important, but the practicing of problem solving skills through the content itself. From my experience and from readings of research, I believe that the process of solving problems can be practiced and with that important repetition of the risk-taking and potentially being wrong, it becomes much more easy to solve problems simply because you are open to trying. I’ve seen this in over 16 years of teaching, that students don’t want to try because they don’t want to be wrong. Their definition of success in problem solving is one that is based on always using a method that a teacher has shown them and getting the right answer the teacher tells them is right. This is what routine practice gets you, students who can match a teacher’s method, not necessarily students who can be independent problem solvers.

So, I think that the right type of practice can be an extremely helpful friend to PBL. As long as what the students are practicing is a balance between necessary skills and independence in problem solving. Both types will move students forward in their confidence and ability to take risks.

Throughout this year, my colleagues and I have been using a dynamic geometry software package that many teachers are familiar with called Geometer’s Sketchpad. It has been interesting to find the best ways in which to use the software in the context of PBL since we are, of course, attempting to foster problem-solving skills first and foremost. I have written a number of lab worksheets in order for this to be accessible during class time. Our school is in the middle of transitioning to a laptop computer program where next year all new students will be required to lease or buy a tablet PC computer that students will then use in their classes. This year in my section of Geometry, I have around 9 out of 13 students who have a tablet, so that when I wish to do a lab, I simply alert the students to bring them to class. I have 3 or 4 students who use their tablet in lieu of their notebook and always have it with them in class.

This is all well in good if the point of a tablet program is for students to use their computers. High school students will always find a use for their computer, the majority of which is to store their music, pictures and movies. However, what is the best way to use technology in order to promote the improvement of problem solving skills? One thing I have found that does not work, is the idea of a general conversation centered around one computer projected on the screen. When I’ve tried this method, discussion quickly breaks off into smaller subgroups with one or two students focused on the sketch on the projector, but the others talking about other possibilities.

Just last week, I attempted to have a class discussion about constructing a square as opposed to drawing it on GSP. Throughout the approximately 20 minute discussion, there were spurts of engagement from various students. It was clear that trying to have a problem-solving conversation in a large group like this was not going to work. However, in the previous week, when I told the students to bring in their laptops and there was no more than two students working together, it was the picture of efficiency and good discussion. These experiences were a valuable lesson for me.

I am still grappling with the idea of using technology to enhance problem solving and am trying to find ways in which to do so. One of my goals this spring is to do a bit of research to see what others’ experiences have been in this arena.

Just when it seems like you’ve given it your all and no ones seems to be listening or watching…

Last week I was getting fed up with a tough cookie in my class, I’ll call Susie. This girl has convinced herself that she can’t and won’t put herself out there and try to learn. She has decided it is really not worth her time. After a number of attempts at getting Susie to see that she has potential and that it would be worth her time to invest herself in the learning process, I decided to pull out the big guns and call her mother. I was very nervous about this conversation – thinking that her mom would try to blame her daughter’s poor grades and even poorer attitude on the curriculum itself, or on me as a teacher.

This is not how the conversation went. The mother was entirely grateful for my time and concern for her daughter’s learning. She asked for my advice on what we could do together to get Susie to be more invested in the process. Susie just happens to be a basketball player and her mom had been at a game the previous night, she told me. She had found herself talking to other parents at the game that had students in this geometry course. She relayed to me that the conversations generally were saying “well, it seems like this curriculum is working.” I can’t tell you how happy I was to realize that this was the case. These were some of our toughest critics at the beginning of the year, and now half-way through, they were admitting that they saw their kids learning.

I am so impressed with my colleagues and my school for allowing us the faith and trust it took for us to embark on this project. Having the approval of the parents was never my hope or goal, or even care- however, the smile on my face may have proved otherwise.

As we have completed our first week back to school, I feel the need to write something else, but I’m not quite sure what to discuss. I thought this entry could be just a summary of what we are doing right now.

The geometry teachers met to discuss a bunch of problems that we were planning to do very soon. Many of them included more difficult proof problems. Interestingly, we found that for some of them, we had all done them differently. I love when this happens, because it shows how differently we, as adults, all think. It helps to keep us in perspective about the students’ work. It’s so easy from our perspective, to know where a problem is going and what its point it. Can you imagine how the kids feel? It must be totally disconcerting to think you are going in one direction and then come to class and realizing your thought process was off track. It can be very discouraging.

That’s why attitude plays such a big part in this type of curriculum. A few of my students are definitely getting the hang of coming to class and being open to being wrong. They do not see it as a bad thing anymore, but as their learning experience enhanced. Yesterday in my class, we had a little time at the end of the period and I had them all start on one problem that was very abstract. The first part of this problem gave them four general points A(0,0), Q(a,b) and S (c,d) (where a>c and b<d). These three points were supposed to be three vertices of a parallelogram. They were asked to find the coordinates of R, the vertex not shown. I had them think alone for a little while and then I walked around the table and asked what they got. It was an interesting discussion and I thought it worked well, because now everyone had that part of the problem done and correct and could go back to their homework later and feel that they had accomplished at least a part of it. As the bell rang, one student said, “I really liked that” and she meant that it helped her feel better about her homework when we started discussing the problems in class. I think I might consider being much more deliberate about this in the future.

Another thing that worked really well this week was doing a Think/Pair/Share exercise. This is where I give them a problem and first have them think alone for 3-5 minutes. Pair up with another student (randomly by some silly rule that I come up with) and then the pairs share with the whole class. I believe I had everyone talking in the class at least once during this exercise, which is generally my rule of thumb for a good discussion.

I am encouraged by the positive feedback that I have received from not only my department colleagues, but colleagues in other deisciplines as well. I was approached by a teacher in the science department who said that they were considering teaching Biology with this method in the future and could we meet to discuss this. The Algebra I teacher is considering moving in a direction where there is much more problem-based learning going on. I feel proud that others are finding value in what we are doing and that we had the courage and deliberateness to create such a good curriculum that it has impressed others with its value.

Of course, no curriculum is perfect and my fellow geometry teachers and I are already looking to the summer to plan our work on our revisions of this curriculum…but that’s another entry!