Tag Archives: math

Math Portfolio - Sharing my own story.

In Calculus, I use the third edition of Finney, Demana, Waits, and Kennedy. I love the selection of activities and explorations that are used to get students where they need to be for the Calculus AB exam. A colleague recommended that I check out Dan Kennedy's website as a treasure trove of resources both mathematical and philosophical about teaching. One of the things I found there that I decided to bite the bullet and do this year is having students put together a math portfolio detailing their work over the year.

The reasons for doing this are many, some of them more selfish than others, but they include the following:

  • By having a record of student work, I can easily look back and remind myself of some of the major mistakes and misconceptions that students have at a particular moment in time.
  • I like reading and seeing how students respond to their own work. I often have students reflect on their work on short time scales ("I should have studied X or Y to do better on the unit test") but don't do as much over long periods of time ("I've become much better at graphing lines in comparison to when we first met linear functions in class.") Part of this is because my students don't tend to hold on to their papers for very long. I take partial responsibility for this, never holding them accountable for it, though I do occasionally remind them that the easiest way to study for a final exam is to look at old exams.
  • I think students selecting what work represents their progress often means things that are very different than what teachers see as their best work. Sometimes students are afraid of sharing their failures, though we as teachers see those as being the most meaningful learning experiences. Whichever is right, having students actively evaluating their own work and thinking about their own learning process is valuable for being able to identify how they learn best.

My introduction to the concept of the portfolio took a lot from Dan Kennedy's document describing them, and I am incredibly thankful for his decision to publish his document online. My own document describing the content of the portfolio and how it is integrated into the grade is here.

At the beginning of the year I introduced the idea, and the response wasn't applause. It was, incidentally, very similar to the introduction this year of true student-led conferences. The students wanted to know why we were demanding they do much more work just for parents and teachers that see their work anyway on the report card. My responses, fully sincere, included the ones I gave above: portfolios are opportunities to highlight not the grade that was received, but the learning process that it describes. Conferences, however, went extremely well as reported by teachers, parents, and most impressively, the students. Since requiring students to also produce the portfolio, I have been equally impressed by some of the thoughts shared by students about what they do and do not understand, the mistakes they tend to make, and also some of the things that go through their minds when thinking about learning.

One of my requirements is that students write a reflection and scan in their skills quizzes any time they want to retake a quiz. This is my current implementation of standards-based-grading, though I am considering expanding it significantly soon. This raises the bar somewhat for what students have to do to retake, but I don't object to this requirement at all. Sometimes I have to tell them to do the reflections a second time - in this situation, they usually look something like "I didn't get it but now I studied and I get it" without any detail as to what it is, what "not getting it" means, or what "studied" actually looks like. Once I get them past this point to do some serious thinking about what they have difficulty understanding, I am very pleased with the responses.

I tried handling the start of the portfolio myself since I wanted to make sure they all looked similar in case these did become official school documents at some point. This was a lot of work keeping track of quiz retakes, reflections, scanning them in, etc - I finally turned over the files as they were last week and have given them to students to keep up to date. Some strong students, however, have nothing in their portfolios because they weren't retaking quizzes, and the only thing I had time to really check up on before the end of the first quarter was that the bios were in place.

What I decided to do to show ALL students what I was looking for in the math reflection portion (with the mathematics exploration to be added soon) is to share my own portfolio with some artifacts from high school that I still happen to have. I've always guarded my test, quiz, and project papers from high school as really authentic sources of material not only to use with my own classes, but also to show students that might not believe I ever had any difficulty in math.

Here is my own math portfolio, complete with biography and student (namely my own) work:  Weinberg portfolio example

I shared this today with students and had some really interesting responses:

  • "This is really your work from high school? Why in the world did you save it?"
  • "You had a 63 on a math test?"
  • "That looks like really hard math"

I got to tell them (1) to read it all the way through to see my comments and (2) that I was proud to show them some of my work along the way to becoming the math student that I was when I left high school. If nothing else, I am hoping that they will read it first because of the inherent fascination students have with their teachers as actual people (I love when they say things like 'It's cool to know you are a real person) and second to get some inspiration for the sort of thinking and reflection I want them to put together.

I know it is difficult to expect reflection to be a perfect process when it is new - it takes time and effort and it doesn't immediately pay dividends. I want students to understand that reflection is not only a really beneficial process, but that over time becomes enjoyable. It shows that learning is a continual process, that you don't just suddenly "get it". This is the same process that I am enjoying about writing on this blog. It takes time, I have to make time to do it - in the end, I really enjoy looking back at my thoughts and holding myself to the commitments I make to my own practice and my students.

So I am leading by example. This group of students continues to really impress me when I expect great things out of them - here's just one more way I am hoping to help them grow.

Your students might not be cursing at you...

One of the students I had the pleasure of teaching in AP physics in the Bronx started with quite a reputation. As a student that spoke Chinese and little English in the 9th grade, he was placed in the entry level math class. It took only a short time for his teacher to notice that, given his background and obvious mathematical skills, this probably wasn't the right place for him. He was quickly moved up the sequence of courses until he ended up in a Math B course that included trigonometry as I recall.

This was not just a case of this student having memorized mathematical concepts from his time in China, though he had seen a lot of math by the time he arrived at Lehman. In his junior and senior years, the quality of his insights and ability to predict, comprehend, and connect ideas in both math and physics were truly impressive and indicative of a strong talent. As his teacher in physics, the greatest challenge I had was not in teaching him how to solve a physics problem, but to write down his line of reasoning that scattered together with frightening speed in his head. My favorite teaching moments with him came on the rare occasion when he had an actual misunderstanding and I witnessed the exact moment of his realization of what he did not get; the physical change in his face was unforgettable.

I was brought back to a story I heard a while back from colleagues about his early times in the classroom. He had a tendency to mutter to himself during class. On an occasion when a student made a comment that was an oversimplification of a concept, this student started saying at a noticeable volume something that sounded like 'bull-shit'.

The teacher, clearly shocked by this, reacted softly with a word after class. Given the student's limited English ability, the message had little chance of making it across. The outburst happened again under more unlucky circumstances when the assistant principal and principal were both in the room observing the teacher - this time, the consequences were a bit more serious. The fact was that, given his personality and the directness associated with translation into a second language, it didn't seem completely out of character for him to call out a teacher on glossing over a math concept. He saw past the simplification for the sake of his classmates. Calling a teacher out publicly like that, though clearly inappropriate to all of us, might have just been a side effect of being in a new place with new people.

If math was the only language he understood well, and he witnessed math being communicated in an way that was not fully clear to him, of course those moments would attract such a reaction. Over time, we learned to react constructively to these reactions and counsel him into more appropriate ways to ask questions or address his usually correct abstractions of the ideas presented in class.

Fast forward eight yearsto when I was with my ninth graders on our class trip to Shandong province a week ago. As a reward for a hike up thousands of stairs the day before, we spent the final night of the trip visiting a hot springs pool. While the students were splashing around, our tour guide was having a conversation with one of the other tourists in the pool. I was relaxing my eyes staring out at the rocks around the pool when I heard something strangely familiar in their conversation.

"Bu shi...Bu shi..."

I knew both of these words now with my limited experience, but had never thought of them together before. The character bu (不) negates whatever comes after it, and shi (是)is essentially the verb 'to be'. Putting it together in my head while getting prune fingers at the time, I realized that the phrase bu shi must then mean 'isn't'. I confirmed my reasoning with the guide: she was saying that something the tourist was saying wasn't true.

There I was, seven thousand miles away, realizing long after the fact that this student we all came to admire was probably not cursing at us. He was just saying he thought something he was being taught wasn't entirely true. It's the sort of thing we hope our students are thinking about during lessons, questioning their understanding of the content of a lesson. I've had students do this in English and never felt threatened by it.

There are many different lessons to take from this. I have been cursed at as a teacher, and I knew it was happening when it was happening because, well, it's pretty hard to ignore it when it's happening to you. The fact that this student was having a fairly normal reaction when something wasn't making sense to him was overshadowed by our misunderstanding of what HE was saying. We assumed he was being out of line. He was innocently saying what was on his mind.

How often do we assume we know what our students are saying without really listening? I'm guilty of wanting to hear an answer that moves a lesson along, but it's not right, especially when the understanding isn't there. My students in the Chinese student's physics class would say an answer they thought was right, and I would on occasion fill in the gaps and go on as if I had heard the correct answer I wanted to hear, even though what the students actually said wasn't even close to what I wanted. Over the years since they called me out on that, I've worked to make that not happen.

In an international school like the one at which I am now teaching, there are languages on top of ideas on top of personalities in my classroom that mix together every day. It is incredibly important to make sure that with such a complex mix of factors, you really know what your students are saying to you and each other.

Climbing Mount Tai - #wcydwt edition

I am spending an amazing few days with students on this year's class trip to Shandong province in China. We spent a couple days wandering around Qufu, the home of Confucius, and the location of the temple and mansion constructed for his relatives. There were some cool opportunities to think about mathematical thinking in Chinese architecture (more on that later) but nothing ready for prime time.

Today's trek led us to the foot of Mount Tai, China's #1 mountain for it's cultural significance (not due to it's height.) we decided as a group to trek up the mountain from the Heaven's Gate which reduced the climb somewhat, but will descend the full height of the mountain in the morning after watching the sunrise.

From Wikipedia (to be replaced by my own pics when I get home, I promise.)


The realization that I might be able to do something really cool with this came after regretting that I had decided to leave two of my favorite data collection devices (heart rate monitor and hiker's GPS) at home being unsure during packing if they would really be worth bringing. I had done this hike in March and had several conflicting reports of the exact height we climbed up and down. The students were asking me how many steps there were, and I vaguely recalled something around 7,000, but I wasn't sure. This question actually popped out from a few different students as we passed the first set of steps. It got me thinking. Is it possible to take either one of these numbers (height or number of steps) and try to calculate or estimate the other? If the students were asking it standing at the bottom looking up, there might be a possibility they would be interested in answering it on their own if posed the right way.

I grabbed my camera and grabbed the best standard length measure I had on me: my iPhone.


(It probably isn't necessary to say this, but this is just an example I took in the hotel.)

I took a number of pictures like the one above on way up the steps, trying to come up with a fairly random sampling of the size of stairs compared to the phone along the entire height. Through some combination of Geogebra, pencil & paper calculations, and some group discussion, I can see some height calculations for the climb coming out of this.

On the way up, there was also a perfect "answer" to this challenge posted in the form of a placard fixed to the wall that says both the vertical height and the number of steps - again, I will include a picture of this when I can transfer photos from the camera I used to take the good photos. I could see cropping this photo in a way that hides the answer, though I'm sure there is a more dramatic Act 3 to this challenge out there.

I think there is some potential here for some fun, as well as for good student discussion and writing about how close the number actually gets to the right answer. This is the second time in a week that I've been able to find something good that could work for a class activity, and I wanted to get the details out while still buzzed about its prospects.