Author: Carmel

Thanks for a great week!

Thanks to everyone who was in my PBL class this week. I had a wonderful time at the Anja S. Greer PEA Math Conference and met lots of wonderful people. For those of you in my week-long class, please feel free to fill out the course evaluation at:

Schettino Course Evaluation

So many teachers that I’ve met were extremely inspiring – As usual I learned so much from everyone about new ways to view technology, certain types of curriculum, ways to incorporate different topics in the classroom and even how to do a Rubik’s cube. I appreciate this converence so much and keep coming back every year. Thanks again to everyone. Special thank you goes out to Ron Lancaster for his special gift of the DVD movie version of The Housekeeper and the Professor which is a wonderful story of relationship through mathematics and creativity. I highly recommend it. Thanks so much Ron!

Wrong is not always bad

I recently read an article in Education Week that was proclaiming the benefits of discussing student mistakes in class. The author, Alina Tugend who has recently published a book entitled, Better by Mistake: The Unexpected Benefits of Being Wrong, cited that in some asian cultures students can be asked to work out “math problems in front of the whole class for a healthy period of time…even if [they] are doing it wrong.” She goes on to discuss that the teacher might ask the student to discuss her thought process and why they chose to do the problem that way and the decisions they made at certain points in order for the class to see the choices that were made at certain crossroads in the problem solving process. Some researchers believe that this type of discussion allows students to help create a sort of “index” of what still needs to be learned or what has already been learned.

In other words, the class is actually viewing the errors and misunderstandings as a helpful thing. They’re using the opportunity of the mistake, of being wrong, as kind of a check point to see what else they need to know. Perhaps someone else in the room might have did something differently that might have led them in a direction that was more fruitful and everyone can learn from that as well. So there is much more to be learned from this type of environment. On the one hand, the students learn the material, but on the other hand, they are learning that they can learn from each other and they learn that being wrong in the first place was actually helpful.

This also goes to the idea of how problem-based learning is ideal for this type of learning. Posing the problem in the context of a prior knowledge base, allows students to think that they have a background that is sufficient for them to do the problem, they just need to recall what that was with a little push. It also fosters was researcher Carol Dweck calls the “Growth Mindset” allowing students to believe that their intelligence and ability to succeed to flexible and not pre-determined.

I am getting excited for my course next week at the Math and Technology Conference in Exeter, NH which I believe is full. I love this conference because I always meet lots of people who are so eager to engage in mathematics and learning. It should be a great time of dialogue and I look forward to a great time!

Many thanks

I have returned from my trip to Indianapolis and I would like to thank everyone that I met there for turning out to both of the presentations that I gave. The talk I gave on Saturday, which was with my colleague was more about our Problem-Based Learning curriculum at our school. The turn-out there was amazing and we were so impressed with the questions and comments from the group. Some of the feedback was great food for thought, especially some specific questions about our definition of PBL. It was also extremely useful to hear what teachers would feel are the challenges of implementing PBL in the classroom. I would direct some teachers to the blog of a teacher in Massachusetts named Mark Vasicek who has attempted to you PBL pretty consistently for a number of years West Side Geometry. It’s good reading.

I think that one of the reasons that so many people might be interested in even thinking about changing the way that they teach right not is because of recent work through the CCSS. There were so many opportunities at this conference to read about, talk about and learn about the details of the Common Core State Standards that I think by Saturday many people were almost tired of hearing about them. However, PBL definitely directly relates to at least a few of the CCSS standards of Mathematical Practice:

Making sense of problems and persevering in solving them.
Reasoning abstractly and quantitatively.
Constructing arguments and critiquing the reasoning of others.
Looking for and expressing the patterns used in reasoning.

We tried to give examples of how we see these standards coming to life as outcomes in student work on a regular basis in the PBL classroom. Having so many people come up to us afterwards for more information, or with interest in getting in touch was really exciting. Sunshine and I truly hope that you do. I really look forward to it.

First Day at Indy

What a great day at the NCTM national conference in Indianapolis. My colleague and I arrived late last night so I missed registration. However, since my talk was today at 9:30 am, I was supposed to be registered at least two hours before I was supposed to speak, so I needed to be up pretty early to get there to register in the exhibit hall. That was not a problem since I was so excited that I was up at 5:30 anyway. After registering, I went to a presentation of orchestrating successful class discussions in the math classroom which seas geared towards elementary and middle school teachers. This was very interesting because most of what I have read has been about secondary level reaching. It was interesting to sees the framework that they used and how important it was to try to anticipate what you. Thought students would say. Many of the ideas they shared were very similar to what I would have said.

I showed up to my room about 15 minutes early and people started coming in. I was very excited that there was interest in my topic. Because this one was a research session I had not planned many interactive activities because I had so much information to share. However, I encouraged people to be a part of the conversation. I think it went very well because many people shared thoughts during the talk and also stayed afterwards. I ran out of handouts and am hoping that those people with questions will contact me and we can keep the conversation going.

I got some great input from some members of the audience that I think will really help me improve my article. One fellow graduate student told me that he thought I would have a stronger argument for the self-referencing pronoun use being a positive sign of empowerment in the discourse if I used a chi-squared test inn the data I had instead of just looking at it qualitatively. Another preservice teacher told me that I should tr to make a distinction between social norms in the use of the pronouns and sociomathematical norms. I think this is a good point and something that I need to look into more in the current research.

Overall, this first talk was a great experience and I’m so glad it was so well received by those who participated. Hopefully, tomorrow will be just as fun and we’ll get some good responses from the crowd. Thanks to everyone that attended.

How do you measure success?

Last week I was being observed by a colleague and my class was doing an exercise in GeoGebra about circles, arcs and inscribed angles. I don’t think I can do the experience I had justice as I try to describe to you what happened in this class, but strangely, I just can’t believe that someone else was there to witness it. Have you ever attempted to scaffold learning in a way such that the questions you asked would move the students forward so that they came to the conclusions themselves? Well, this is what I do everyday in the PBL classroom and sometimes it’s a success, sometimes it’s in between and I do more “telling” than I’d like, but on this day I couldn’t believe what happened.

I had the students construct a circle with a central angle and measure the arc and the angle and see that they had the same angular size. This was no surprise to them. My plan was then for them to extend one side of a radius of the central angle and make an inscribed angle that intercepted the same arc so that they would measure that one and see that it was half the central angle and the arc it intercepted. Often when I do this students don’t understand that the angles intercept the same arc, or something else goes wrong. However, one this day, I wished I had been recording the flow of the conversation that went flawlessly after my simple question, “What do you observe about the two angles?” From one student to the next around the table it went:

“Well, it’s definitely smaller…”
“Mine’s almost ninety and the other one looks like it’s almost 45.”
“Maybe it’s supposed to be half?”
“Yeah because the side is a diameter and the other one’s sides a radius – like it’s in a proportion?”
“No, that can’t happen, you know the angles of a triangle don’t work like that..”
“move it around and see if you can get it to be exactly a half…”

After a while, they all agree that it seems like the inscribed angle is half the intercepted arc. So I prompt them again,”So why do you think it might be exactly half? Is there any relationship between these two angles that might make it that way?”

“Is it like because of the midsegment theorem? The radius is half the diameter so it makes them parallel?”
“Well, it doesn’t seem like the other one is parallel though…”
“It seems like the other central angle next to that one adds up to 180 with the one that intercepts the arc.”
“Hey wasn’t there a theorem about that?”
“Oh my gosh, yes”
“It was like about outside angles or something like that being …like if you add the two inside you get the one outside”
“Oh I see, the triangle on the other side is an Isosceles triangle because it’s a circle..”

At this point, I almost freaked out because I hadn’t said anything in almost 15 minutes or so, they weren’t doing it all themselves and almost every student (except maybe 2 or 3, who were still engaged at least) had contributed something to the conversation. I mean, I was in teacher ecstasy, and to top it all off, I had somewhere there to see it all. I couldn’t believe it. Besides that I had been having a really bad day, and this just turned it all around. I don’t know if this was all a function of the practice of PBL, or a function of the kids in the class, but it was truly amazing. I heard at least three of students leaving class that day say “that was a great class!”, what more could I ask for?

Documents for NCTM National Conference

Here are some links to documents that I will make public for my talks at the conference in Indianapolis this week:

Powerpoint Presentation for
Improving Classroom Discourse to Support Communication, Equity, and Students’ Agency

Handout for
Improving Classroom Discourse to Support Communication, Equity, and Students’ Agency

Powerpoint Presentation for
Problem-Based Learning (PBL): A Transformed Perspective for Standards-Based Geometry

Handouts for
Problem-Based Learning (PBL): A Transformed Perspective for Standards-Based Geometry
Emma Willard M225 Course Syllabus with Problem-Based Learning
Emma Willard M225 Course Curriculum Map

Emma Willard School Algebraic Geometry Problem-Based Learning Curriculum
M225 Curriculum 2010-2011

Modeling Proper Mistake-Making

Whether it be a small arithmetic error, or correcting a student when they were actually doing something right, we always make mistakes in class. The other day, I wrote the parametrization of the unit circle as x=cos(t) and y=sin(t) and took the derivatives as dx/dt=sin(t) and dy/dt=cos(t). It wasn’t until a student humbly interrupted me saying, “Um, Ms. Schettino, don’t you mean -sin(t)?” and I looked at it for a little while, trying to figure out what she was saying, and then I realized what I had just done. I knew I was calculating the arclength, using the arclength formula, and that I was going to square it anyway, so I just left off the sign, but they didn’t know that – skipping steps in my head is a really bad habit. So I said, “Yes, oh yes, sorry – thanks so much for fixing that for me.”

I do things like this all the time, and hopefully, I am a big enough person to admit my mistakes and give students credit for finding mine, especially if it might affect some students’ understanding. I was talking to a colleague about this with respect to PBL the other day. I asked her why it’s so important to her to admit to students when she makes mistakes and to fix them in front of the students. She told me that she likes to “model proper mistake-making” for her classes so that when they do it, they can see what she does and use the same humor, self-confidence, risk-taking and humility to fix their own mistakes, learn something and move on. I actually see this in her classes and have heard her students say that they do this too. I believe that without this attribute students do not fully take advantage of a problem-based curriculum because they cannot find the way to learn from their mistakes. I even heard a student once say that “There was one time during class that I put a problem up at the board and got the entire thing correct. I was actually, in a way, disappointed because I feel like I learn better from my mistakes.” I was amazed that a student could see that in her own learning, that the growth happened for her when she was wrong, as opposed to when she was right.

Clearly, being wrong in front of students can be somewhat embarrassing, but for me, it allows me to have bit of solidarity with my students, if even for a moment. It allows me to feel, what I ask them to do every day, to move out of their comfort zone and attempt a problem that they cannot do, and perhaps not live up to their expectations of themselves. It reveals my human side, which I do end up feeling the relational part of my teaching is all about.

Receiving Feedback

I received an email from a colleague a few weeks ago, that was amazingly touching. She had been meeting with an advisee and asked the thoughtful question, “Can you think of a course or a moment that changed your academic experience in a significant way.” One would think that most high school sophomores would either take that question lightly, or would at least need to pause and reflect on the depth of this question. My colleague said that her advisee responded without hesitation, “Geometry last year. I hated it at first because I couldn’t do what I had always done and do well. But by the beginning of the second semester, I had started to figure out what it was all about. And this year in my other classes, like English and history, I’m THINKING better, I’m analyzing differently because of that Geometry class.” This student seemed to be able to connect improvement in her critical thinking skills in other disciplines to the work she had done in her problem-based learning class. Did she have empirical evidence that this was the cause? Of course not, but something in her intuition and learning process was telling her that the struggle she had undergone to move through a course that challenged her in so many different ways, allowed her to grow intellectually like no other course had. For this student to even recognize this was very mature, and for her to attribute her success and skill in other courses to the learning that had occurred in this course was remarkable.

Throughout the year when teaching with PBL, I struggle with the comments I receive from students. I often wonder when they ask me to go up to the board and give more direct notes, “Why don’t I just go up and make them happy” instead of asking another scaffolding question? Why do I continue to push them out of their comfort zone and let them sit with the unknown for just that extra amount of time grappling with their peers, instead of relieving the tension and anxiety by giving them want they desire? And then a moment comes when they realize something on their own, and clarity, true understanding takes over. It is a moment of true joy on both our parts, because I know not only did they come to that understanding through their own agency and empowerment, but the way they came to it has allowed them to be the relievers of their own anxiety. They have transformed their vision of what is possible in coming to make meaning in mathematics. It is in that moment that we are together changing their understanding of what a mathematics classroom is, just a little more.

And then the next week, they are back to asking me for more direct instruction, while at the same time there is just a glimmer of increased curiosity and I know that by the end of the year, there will be a changed student sitting in front of me who may have a different belief system about learning mathematics.

A Moment from Class

The other day in my Algebraic Geometry class, we were doing this problem:

An airplane is flying 36,000 feet directly above Lincoln, Nebraska. A little later a plane is flying at 28,000 feet directly above Des Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of descent, predict how far from Des Moines the airplane will be when it lands.

This is one of the original problems from the PEA materials that we use in our PBL curriculum and I love using it for many reasons. This problem is on a page in the book where we are discussing slope and points that are collinear. So many students’ first idea is to think of the rate of change of the plane as it descends – at least that how I expect them to think about it. However, the student in my class who presented this problem, I’ll call her Robin, had a similar algebraic perspective. Robin realized that since the plane dropped 8,000 miles of altitude for every 160 miles across, she could just see how many times she needed to subtract 8,000 from 36,000 in order to get to the ground, then multiply that by 160. This was crystal clear to Robin, but other students were a bit confused.

So Sandy chimed in. Sandy drew a picture where the airplane was at a height of 36,000 feet and proceeded to subtract 8,000 a number of times drawing triangles as she did this. She did this until she got down to 4,000 (which was 4 times of course), and then realized she only needed another half of 8,000, so realized it was a total of 4.5 triangles that would go 8,000 down and 160 across to get down to the ground. So she multiplied 4.5 x 160 which of course was the total distance across the ground or 720 miles. However, this was not the answer that other students got.

So then Noa, who really likes algebra, says, “Isn’t 8,000/160 just the slope of the line?” Many of the other students agree with her and nod their heads. “So I just wrote the equation of the line as y=36,000-50x and graphed it on GeoGebra. Then I just found the x-intercept. But I knew that we were only looking for the distance from Des Moines to the landing point, so I subtracted 160 from 720, so the answer is 560.” This then inspired Sandy and Robin to check if their answers agreed with Noa and it did.

Just then, Anna said, “Can’t you just plug in zero for y in Noa’s equation? Why do you have to find the x-intercept on the graph? I just plugged in zero and solved for x.” Noa replies,” That’s the same thing…” which created a debate about finding x-intercepts of lines. Which then inspired another student to say that she saw it a completely different way and compared to triangles that had the same slope and set up a proportion giving her an equation that said 28,000/x=8,000/160, which of course set off a bunch of students writing other proportions that were also true.

After this discussion died down, and it seemed we had exhausted that problem, Sandy looked thoughtfully at the board and all of the different methods. She said, “That’s really cool. I can’t believe we all looked at it in so many different ways and we were all right.” And just having a student say that in a spontaneous way made the whole discussion worthwhile for me. It was such an amazing moment, that I sat and paused and let them all accept the pride in their own creativity and ability to use their own knowledge to solve the problem the way they saw fit. I was so proud of them.