Author: David Wees (page 13 of 97)

First, give an exit slip to your students based on a critical math concept for which you want to check for understanding.

After class, sort the exit slips into piles based on the method students chose to use (whether they used it perfectly or not). Choose two examples from the student work that highlight one or two probable misconceptions students still have on the chosen critical math concept.

Remove identifying information from the student work, photograph it (or use a document camera) and show it the next day in your class. Ask students a question about the work that requires them to think about the work. “Which one of these two examples is correct?” is not a very good question because it can be answered by guessing. “Why do I really want these two students to talk to each other about their solution?” is a better question because if students answer it, they will have to think about the concept a bit differently.

Ask students to think about their own answer, write it down (if necessary), then turn and talk and share their work with a partner while you circulate and listen in on student discussions. Select 1 – 3 students to share their thinking with the whole class.

Repeat this every day.

The objective of my presentation at NCTM in New Orleans was to introduce participants to social media, which was made difficult because participants did not have Internet access. As it turns out, this ended up forcing me into a couple of activities which were pivotal experiences for participants.

Here are my slides from my presentation.

Instead of trying to bring my participants to the Internet, I brought the Internet (or at least a portion of it) to my participants, and in doing so, provided them with concrete examples of how people use social media to interact.

I started my presentation by sharing some of the stories I have from my use of it, and who I have been able to interact with and how this has enriched my professional learning. If you use social media as a professional tool, then you have some of these stories too.

Next, I gave them an experience of what it might be like to participate in a live Twitter session. Participants were given a question, 30 seconds to find new group members, and 140 seconds (suggested by Dvora) to discuss the question in their small groups. This highlights that Twitter conversations are often with people you don’t know very well, and can be brief and short interactions.

I then asked participants to describe the attributes of our face to face conversations, and then speculate as to how these might transfer to an online conversation. I then highlighted for participants some of the features of these kinds of conversations. In particular, the conversations parallel conversations you would have with people face to face, but that conversations online can take place between participants who are separated by vast geographic (and cultural) differences.

Participants went around the room and read one or two of the four blog posts I had printed and put up on the wall, and put sticky notes up to comment on the blog posts. We then debriefed the experience with the main observation being that blogging is a lot like reading and responding to a letter from the editor.

Finally I wrapped up by talking about some of the specific projects that have been created through collaboration with other people in the online mathematics education community, and how our participation online has resulted in resources of real value in our teaching.

In the final questions at the end, one participant astutely observed that it would be easy to find a “how-to” guide online, but that he felt my “why-to” session was more helpful. There’s no reason to tell people how to do a bunch of technical details if they don’t see a reason to do them.

Everything in education these days is black and white, with no possible shades of grey.

Standardized tests are evil. Except when we agree with their findings. Or when they are used in research about teaching methods. Or when they highlight inequity in our society.

Lecturing to students is evil. Except when it’s coupled with peer instruction. Or when it is used to talk about peer instruction. Or when the lectures are short and have animations added. Or when the lecture is about creativity.

Common Core is evil. Except when it raises standards for students. Or when it communicates a common definition of what it means for students to do mathematics. Or when it encourages multiple ways of solving mathematical problems.

I recently read an essay published by the Carnegie Foundation for the Advancement of Teaching called “Getting Ideas into Action: Building Networked Improvement Communities in Education.” The basic argument of the essay is that while traditional research can be effective for finding out what works in specific circumstances, it frequently is too far away from practice to be useful in an educational setting, but that many of the ideas of research can be applied in practical ways through the creation of networked improvement communities (NIC) which use the design cycle to study and solve shared problems of practice. This to me does not say that research is useless, as it can inform this process, only that research as a tool for improvement is insufficient.

After reading the article, I thought to myself, in what ways is our online community of math teachers like or unlike a network improvement community? A clear and extremely important difference is that as a community we have no central objective. We also do not have many obvious structures which lead to sustained improvement and accountability back to the community. However, we do have a community of people who are all obviously interested in improving our own individual practice; quite simply it would otherwise not be worth the effort we put into our online participation.

Here’s a proposal. Let’s form some small groups of math educators (or alternatively, educators of math educators) who carefully study our practice together and think about improvements, using a process I learned about here at New Visions called Inquiry. This process is detailed in a book called Strategic Inquiry by Panero and Talbert (reading this book could be a useful starting place). We would meet once every couple of weeks (using a Google+ hangout, I guess unless there is alternate technology out there that is better) and take turns presenting on our work, and asking for feedback from the group. This will allow these small groups to collaborate around our teaching, our students, and mathematics, which Ilana Horn describes as among the most effective learning experiences for math teachers.

Anyone interested in participating?

Watch this video. Every time you feel you are confused, just pause and rewind the video and rewatch it. Do this until the video makes sense.

Can’t do it can you? It doesn’t matter how much you pause and rewind a video, if it doesn’t make any sense to you, watching it again isn’t going to suddenly change the video so it makes sense.

One critical job of the teacher is to find out where students are, and help them at that stage. Explanations that don’t make any sense aren’t particularly helpful. In fact, since our goal is to help students learn the mathematics, and not just the words that represent the mathematics, explanations may not be helpful at all.

PARCC recently released some sample computer-based test items for ELA and high school mathematics, so I thought I would check them out since NY state is still officially planning (eventually?) to use the PARCC assessments.

First, some kudoes to the team that created these assessment questions themselves. In general I found that the questions were looking for evidence of mathematical reasoning, and would be difficult to game with classroom test-preparation. What I think is missing is an opportunity for students to demonstrate the complete range of what it means to do mathematics, including asking questions themselves that they answer, but for an assessment with this function, this seems much better than the current generation of standardized tests.

If you want to stop reading this and preview the questions yourself, feel free to do so (if you are only interested in looking at the sample math questions, you’ll have to skip through the sample ELA questions).

Here’s my preliminary thoughts from attempting the first few problems myself.

1. Use of space is critical. The first assessment question does not do this very well. Look at the video below that explains my reasoning on this.

   
    

2. The second question has two issues, one of which is really very minor, the other of which is something PARCC should make an effort to fix.

   I’m okay with questions that use approximate models for mathematics, but it might be at least worth noting to students that these models are approximations.

   Taking a test on a computer is already extremely distracting as compared to taking a test on pencil and paper. Given that there is research that shows that people generally read slower and with less understanding on a computer, PARCC should make an effort to mitigate the platform issues as much as possible. Imagine if your work flickered in and out of existence while you were writing it down on paper?
    

3. I put the wording for the third question through a reading level estimation calculator, and it estimated that the reading level was grade 10. While it is reasonable to expect a certain amount of competence from students in this respect, we have to be careful that our assessments of the mathematical thinking of students aren’t actually measuring whether or not they can read the prompts in our assessment.
    
4. Question 5 assumes a certain amount of cultural knowledge, specifically knowledge of playing golf. Having worked with students who do not have the this sport in their cultural background, I found assessment items like this frustrating. Usually, the questions are doable without the cultural knowledge, but imagine you are a student who comes across a question that contains an idea with which you know nothing. Regardless of whether or not the knowledge is required to do the mathematics of the problem, it impacts student confidence and therefore their performance.
    
5. The sixth question assumes that students have some minor technical knowledge, which I would classify in the same genre as my fourth point; students with a minimal technical background may struggle with the mechanics of this task. This may not affect a huge number of students, but the assessment instrument should be as neutral as possible to allow the greatest number of students to interact with the mathematics of the task, not the mechanics of the task.
    
6. The seventh question has a video. It’s probably between 4 and 10 megabytes in size. Can you imagine what this will do to your school’s bandwidth if every student in a particular grade is accessing the resource at the same time?

There are some things which I think are obvious to me about the computer based assessment that PARCC is working on.

The first is that many of these questions are still going to require actual math teachers, with some experience looking at student work, to look at. Most of these questions are not just reformated multiple choice questions (although some of them are). While this increases the per-student cost of the assessment, I do not think that there are computer programs (yet) that exist that can accurately capture and understand the full range of possible mathematical reasonings of students.

Next, some of the more adaptive and social aspects of the work Dan Meyer and company have put together, are not present in this work. This assessment is intended to capture what students think now, rather than what students are able to do once given more information. This is still an assessment of the content students have learned, and does not appear to do an ideal job of making sense of how students make sense of problems and persevere in solving them, attend to precision, or any of the other standards for math practice (SMP). While it is clear to me that students will have to use these standards when doing this assessment, I do not see how anyone looking at the resulting student work is going to be able to say with any accuracy what is evidence of each of the SMP.

Unfortunately, unless the standards for math practice get captured somehow by an assessment (a goal of ours during next year with our project is to make an attempt to do this systematically), it is unlikely that teachers will use them.

Words are not ideas, anymore than the picture above is a pipe (it’s a picture of a pipe).

When we communicate about ideas we are forced to use words (or gestures or images, which are also not the ideas themselves), and so consequently we are never communicating ideas directly. We communicate about ideas through the medium of language.

It is possible for someone to learn the words that represent an idea without learning the idea themselves, even so much as to be able to mimic the output expected of someone who understands the idea.

This happened to me. I learned about linear functions in school. I learned about “constant rates of change” and “y = mx + b” and “find the rise over the run” and many other phrases which were connected to finding and subsequently graphing the equation of a line. I even later learned how to think of a specific line as a transformation through a translation and a rotation from any other line.

However, in a fourth year topology class, my professor used a completely different set of words. He said, “Let’s do an easy example. We have line segment AB and line segment CD, and our objective is to find the mapping function from AB to CD. Think about that problem and bring your work to next class.” Well, I spent a week thinking about that problem and could not do it. I spoke to my professor about it, and he gave me a look and told me that it was “not my fault, this is unfortunately due to how you were taught.” Note how I was earlier able to articulate “transform one line into another”, but not able to repurpose this information to create a function. It turns out, the solution is a linear function.

Here I am, many years later, with a much better understanding of the wide range of things in our world which are linear functions. In fact, in only a few minutes, I am now able to “prove” to myself (at least experimentally) a more general version of what my professor asked me to prove all those years ago. I have moved past a “just the words” understanding of linear functions into a more flexible and useful understanding.

This is the objective of good mathematics teaching, and the reason why I think that we need far fewer lists of things to achieve with our students, and far more time spent looking at very similar things but in a very wide range of contexts. We need to move past students learning about the words that represent mathematics and giving them enough time to actually learn about the corresponding ideas.

The house my wife and I live in was recently sold, and so we have started looking for another apartment. Our current lease expires in a year and a half, and so we decided that, given how challenging the rental market is in NYC, that we should start looking right away. We also decided, somewhat arbitrarily, that we would attempt to find an apartment in the next six months, if only because we knew we would get sick of looking pretty quickly.

During the first 5 weeks of our apartment hunt, we found five apartments that we thought were worth looking at. Over six months, I extrapolated that we would get to see about 26 apartments that would satisfy our requirements. What we discovered, with each apartment, is that we basically got to see the apartment and then pretty much decide immediately if we wanted to apply to rent the apartment or not.

It turns out that there is a nifty mathematical algorithm that one can use to optimize one’s chances at picking the best apartment possible. We expect to have 26 apartments to look at, each of which we inspect and then either accept or reject immediately, and we want to optimize our chances of picking the best apartment possible, from the 26 that meet our minimum criteria. This exactly matches a solved problem in mathematics; the Secretary Problem.

In the Secretary Problem, where one has to decide on the best applicant between n randomly ordered applications, an optimal solution is to reject the first n/e applicants, and then accept the next applicant that is better than any applicant you have seen before. The proof of this particular solution is here. I couldn’t reproduce this proof if asked, and there are details in it which are fuzzy for me, but I am pretty sure I understand why it works. Informally, the first n/e applicants act as a sampling space, and this gives you information on how good applicants will be, and that n/e happens to be where you achieve an optimal amount of information on applicant strength, allowing you to make the best determination you can of which next applicant to choose, without raising the probability too high that you’ve already rejected the best applicant.

For our specific apartment hunting problem, with 26 total apartments to view, 26/e (e ≈ 2.71828) ≈ 9.6 ≈ 10 apartments. So, my wife and I looked at 10 different apartments, and while we did this, I informally ranked the apartments based on the criteria my wife and I agreed were important to think about (space, apartment lay-out, cost, location, quality of neighbourhood school, commute time). The 11th apartment either she or I looked was superior in many ways on all of these criteria than any of the others we had looked at, so I whole-heartedly threw in my support for it, knowing that this specific apartment is most likely to be the best apartment we will see.

We’ve submitted an (overly lengthy) application for the apartment. Wish us luck.