Year: 2013 (page 6 of 15)

I am reading Disrupting class, by Clay Christensen, Michael Horn, and Curtis Johnson. In the introduction of Disrupting Class, Clay Christensen, et al., give six possible reasons for why the United States education system does not appear to be doing well when compared with other OECD countries:

1. Schools are underfunded,
2. There aren’t enough computers in the classroom,
3. Students and parents are to blame,
4. The U.S. model of teaching is broken when compared to other models in other countries,
5. Teachers’ unions,
6. The way we measure schools’ performance is fundamentally flawed.

Unfortunately, this is a good example of how the question asked frames the answer (Question asked: Why isn’t the United States doing better in education compared as a whole to other OECD countries?).

When you compare the United States education system as a whole to other OECD countries (using PISA data, for example), you find the United States appears to be in about the middle of the pack. However, if you disaggregate PISA data and look at a state by state comparsion, and then compare each state to other similar OECD populations (in terms of average income), you find quite a different story. The high income and low income areas of the United States hold their own against comparison OECD countries. but the United States has a much higher poverty rate than most OECD countries, which results in an over-representation of low-income students in the aggregate U.S. data.

We know that all students can learn, when they are given the supports they need, so a better question for Christensen, et al., to ask is; why isn’t the United States doing more to combat the effects of poverty on its most vulnerable population?

Other notes:

In chapter 1, Christensen, et al., use the idea of multiple intelligences and learning styles to suggest that it is incredibly challenging for a teacher in a classroom to succeed at teaching all of his/her students. Unfortunately, recent research on learning styles and multiple intelligences suggests that they are not a useful framework for understanding student learning (except that all students probably benefit from experiencing ideas in different mediums).

In chapter 5, Christensen, et al., make the claim that their calculations show that by the year 2014 student centric learning (aided by technology) will become mainstream. The authors of Disrupting Class should probably have talked to teacher colleges before writing their book, where training teachers to use student centric approaches has been mainstream for decades.

(Source: Complex Variables with Applications, Second Edition, A. David Wunsch)

I started transporting my professional books to my new office this week, five books at a time. One of the books I brought today was my old textbook from when I took Complex Analysis in university, about 20 years ago. I decided to skim through the book to see how much of those two courses I remember, and as it turns out, the answer is not much.

I stumbled upon this notation that I was unfamiliar with very early on in the textbook, and decided to work my way back through the textbook to see if I could find out where this notation was introduced, and found the only reference to a definition of the notation above.

It occurred to me that this is not so much a failure of the author, but a failure of the medium, and one that could be addressed in a digital medium much more easily. One cannot easily link to notation, especially notation used often in a traditional textbook, back to it’s origin. However in a digital textbook every single instance of this notation could be made linkable (perhaps in an unobtrussive way so as not to be distracting) back to not only the first instance of the notation, but to carefully constructed examples of the notation in use.

On a related note, in my classroom, I try my best to introduce vocabulary and notation as it is needed to describe mathematical (or otherwise) objects that the students have been gaining some familiarity with. This way the vocabulary or notation is meeting a need; labelling something that we want to discuss, rather than being artifically introduced "because we will need to know this later."

"If some one say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times." Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts."

~ Muḥammad ibn Mūsā al-Khwārizmī (Source: Wikipedia)

The tools for doing algebra have evolved over the years. When Muḥammad ibn Mūsā al-Khwārizmī was working on algebra, he did all of his work in words (see above). The symbols we have invented are a different tool we use for solving algebra problems. The fundamental structure of algebra is therefore something different than either of these tools.

Can we do algebra with a computer (which is today’s new tool for doing algebra) and preserve the underlying qualities that are algebra? How does access to a computer, and knowledge of programming, change what we can do with algebra? 

No one is born hating math. Our attitudes about it, positive or negative, are a result of our culture, our interactions with math, our experiences with other people while doing math, and the messages we see daily about mathematics.

What can we do as teachers, and as parents, to address negative stereotypes about mathematics?

I’m participating in Dr. Jo Boaler’s course "How to learn mathematics" which started two days ago. Here are my observations so far:

1. I like the structure of the ideas Dr. Boaler has presented so far. The "quizzes" we have done so far seem not to have right answers, and are more designed to make us think. The videos are short and engaging and easy to follow.
    
2. The premise of the course is excellent, and I think that this kind of course is best held in a discussion style, with some ideas being seeded by the instructor, which looks like the purpose so far of the course.
    
3. I read every single introduction people posted, and I was very impressed with people’s willingness to share that they have had poor experiences in math. I know that happens quite a bit (almost everyone I meet tells me they were terrible in math after I tell them what I do), but not so often in print, and I suspect not so often on the first day of a course. This almost certainly has something to do with the way Dr. Boaler framed the course which has clearly made people willing to start the course by candidly sharing their experiences.
    
4. Unfortunately, the user interface for discussion is awful, which I know has nothing to do with Dr. Boaler, since she is very likely constrained in what platform she uses (given that she works at Stanford). This is also an issue that I brought up with respect to Dr. Keith Devlin’s course on Mathematical Reasoning.

Once I participate in a conversation, I can see no way of finding out if anyone has responded to my conversation without looking up the conversation in the long, long, long list of other conversations that have happened. I have no "home base" with which to find conversations for later. I actually used CTRL + F to search for my name after loading the very long welcome thread to find my name! My first recommendation for improving this is that the designers of Stanford’s course software should look at other forum software, much of which has evolved over at least a decade of use, and not try to recreate a new user interface for a standard forum discussion. My next recommendation is to offer a way for users to see, in one place, who has responded to something they have posted, and to be able to choose to receive notifications when someone responds. The discussion space should be more like Facebook, and less like Moodle.

Another issue I have noticed is that I seem to have to scroll through the conversations and load them as I scroll. This means that if I am interested in finding an older conversation, I may potentially have to spend many minutes scrolling through unwanted conversations looking for the one I want.

This issue around the usability of the discussion space is an important one, but this course is very interesting to me, and I intend to work with the discussion space as offered.

Designing an interesting and open-ended task is relatively easy. The challenging part comes when you attempt to use the task and learn something as a teacher about what your students understand.
 

This graph represents the main issue that comes up when designing open-ended tasks for students to use; the more open-ended a task is, the less information you are likely to be able to gain from using the task. You can gain insight into what your students are thinking when they are working on an open-ended task, but chances are much greater that they have a much wider variety of insights and misconceptions that will come up during their time working on the task.

If we define "formative assessment" as "any tool used by teachers to gain insight into their student’s thinking and use that information for future planning of teaching and learning activities," then the open nature of these tasks helps with gaining insight into student thinking, but it makes planning future activities more challenging since we could potentially end up with much more information about what each student knows how to do, but potentially less information about what each student does not know how to do.

The very task that gives students the freedom to try many different potentially successful mathematics techniques to solve the problem unfortunately also limits how much of what we know of those paths that students chose not to follow. Did a student not use an algebraic approach because they don’t know algebra? Or did they use a non-algebraic approach because they didn’t think of an algebraic technique? Did they use a table to solve the problem because they love using tables? We have no time-efficient way of answering these questions.

One possible solution is to make sure that each time students work on a task, they have the opportunity to share their different solutions with each other. This way students who tend to use graphs to solve problems will see algebraic solutions to those same problems. Students who generally solve a problem using a table of values will see how other students used a diagram instead. By sharing different solutions created from the students amongst your class, students may be able to add to their own toolbox of methods they can use to approach problems.

The problem with this last solution is that this only works when the students are working on essentially the same problem; it will fail to work when the task is so open-ended that students have sufficient influence over the questions they ask. Maybe there are some other solutions to this problem?

In a future post, I will take a task and create different versions of it, sliding along the scale of open-endedness, and hopefully this will lead to some insights as to the challenge involved in open-ended tasks.