The Reflective Educator

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Tag: mathematics (page 1 of 4)

Keynote on Formative Assessment

I recorded some video and the audio from a keynote presentation I gave a couple of weeks ago. It turns out the video wasn’t all that useful, but I did a screencast of my presentation notes, and added the audio from my keynote to it.

 

 

Were I to do this again, I would definitely do a better job of summarizing my main points at the end, and I would probably explore more closely some of the different concrete methods through which one can do formative assessment.

 

 

Ways to use technology in math class

Here are some ways you can use technology in your math class which are more interesting and innovative than using an interactive white board or having students watch instructional videos. Note that these ideas are all examples of potential student uses of technology.

Students could:

 

Any other suggestions of ways students can use technology in order to improve their mathematical reasoning?

Six things about math education which do not work

There are six things (at least!) about mathematics education which do not work:

  1. pacing for coverage of curriculum rather than focusing on effective student learning,
  2. fear that if students take more than five seconds to solve a problem, they will give up,
  3. teachers spending more time talking than students get to spend thinking,
  4. teachers working in isolation to plan lessons, units, and understand their students,
  5. students being forced to work in isolation from their peers as potential resources,
  6. and an obsession with procedural fluency over conceptual understanding.

The objective of my current work is (collaboratively with the rest of the members of my team at New Visions) to develop tools for teachers that will help address as many of these issues as we can. These tools will be used collaboratively with teachers to look at student work and try to address the question, "What were these students probably thinking?" and "How can I help this student further their understanding of mathematics?"

Ambiguity in mathematical notation

I’m reading Dylan Wiliam’s "Embedded Formative Assessment" book (which I highly recommend) and this paragraph jumped out at me:

"To illustrate this, I often ask teachers to write 4x and 4½. I then ask them what the mathematical operation is between the 4 and the x, which most realize is multiplication. I then ask what the mathematical operation is between the 4 and the ½, which is, of course, addition. I then ask whether any of them had previously noticed this inconsistency in mathematical notation — that when numbers are next to each other, sometimes it means multiply, sometimes it means add, some times it means something completely different, as when we write a two-digit number like 43. Most teachers have never noticed this inconsistency, which presumably is how they were able to be successful at school. The student who worries about this and asks the teacher why mathematical notation is inconsistent in this regard may be told not to ask stupid questions, even though this is a rather intelligent question and displays exactly the kind of curiousity that might be useful for a mathematician — but he has to get through school first!" ~ Dylan Wiliam, Embedded Formative Assessment, 2011, p53

Mathematical notation has been developing since the introduction of writing and has largely grown organically with new notation added as it is needed. In fact, if a mathematical concept is developed in different cultures, it is entirely likely that each culture will develop its own mathematical notation to describe the concept, and these mathematical notations inevitably end up competing with each other, sometimes for centuries.

This observation by Dylan Wiliam suggests to me that difficulties in mathematics for some students are almost certainly related to the notation that we use to represent it (especially in classrooms where mathematics is largely presented to students in completed form, rather than being constructed with students), and that people who end up good at math in school may be good at being able to switch meaning based on context.

Can you think of any other examples of mathematical notation which are potentially inconsistent with other mathematical notation? I’ll add one to get the list going:  which is clearly inconsistent with algebraic notation, and potentially with fractions too.

130 years of climate change data

 Average temperature graph

Daniel Crawford is working on an interesting problem; how can we represent data about climate change in other ways. Each note he plays represents the average temperature for a year, with higher pitched notes representing higher temperatures. While I wouldn’t call this piece very musical, it is a very interesting and useful way to represent data about average temperatures.

Once people understand that our world is getting warmer, and significantly so, the next step is to wonder why. I’m interested to see what it would sound like if they overlaid these graphs with graphs of the average parts per million of various gases known to be associated with the greenhouse effect, and played by different instruments.

Two views of mathematics

Beautiful fractal image
(Image credit: DanCentury)

As usual, there is an argument going on Reddit on mathematics education. There is a statement from that argument that I would like to highlight here, and a related discussion on Reddit with a related comment.

"I solemnly declare that no kid ever learned math by watching a video OR by reading a paragraph, since math is an action [emphasis mine], not an exposure." deadletter

In a related discussion on resources for a 4th grade student wishing to explore mathematics further, a commenter made this bold claim, in response to someone suggesting that the Khan Academy would not be a good resource:

"Why not? If it’s to pass state Core Standards, it’s more than enough. If it’s to give his daughter who loves math more to learn, it’s more than enough. If it’s to showcase the power and beauty of math, it does that, too." misplaced_my_pants 

(Aside: I struggle to see how the Khan Academy "showcase[s] the power and beauty of math." At best, this is a side show on their site, although the introduction of Vi Hart and her videos to their team at least indicates that they are aware of this deficit.)

These are very different views about what it means to learn mathematics.  One person holds that mathematics is an activity that people undertake, the other believes that mathematics is a specific set of knowledge that ones gains through exposure. These are very different definitions of mathematics, and have very different consequences of what would be required to learn mathematics.

I tend to lean toward the mathematics as activity definition, but understand that my responsibility as a teacher is to ensure that my students also know some specific set of mathematics. It’s not a line I’m particularly comfortable straddling and I feel a lot of tension as a teacher as a result. Whenever I have some freedom from the curriculum, I lean heavily toward explorations of math, either as independent activities, or as a group activity.

At the very least, I want my students to know that there are two views of mathematics (which some may consider to be opposing views), and that they should have the ability to make an informed choice between them (or to choose both, if that is even possible).

 

Math in the real world: Train tracks

This is another in a series of posts about how one could find mathematics in the world around us.

Train tracks

My son loves to play with train tracks. A few days ago, while playing with his train tracks, he observed, "Daddy, I can’t turn a train around." I asked him what he meant. "No matter which way I go on this track, I can’t get my train to start facing in the other direction. I’d have to pick it up, but that’s cheating." (Note: I’m paraphrasing here)

Observations like this are mathematical observations about the world. He has abstracted from his train tracks to a property of his train tracks, specifically the direction his train is able to travel. He has then attempted, and I watched him do this, to verify this statement is true by running his trains around the track in every possible comination.

My wife and I spoke about this later, and she came to the observation that in order to be able to turn around his train on the track (without "cheating" by lifting it up), he needs a closed loop with a single entrance and exit point included in his track somewhere, and this entrance and exit point has to connect to the rest of the track in a certain way. So I asked the question, does he have the right track to be able to create a closed loop? If you look at the picture above, you may be able to answer this question yourself.

The area of mathematics that deals with these kinds of issues is called graph theory, and it was invented by Euler for a very different purpose many years ago. It is unfortunately not in most school curriculums, but it is certainly an interesting area of exploration, and one which is accessible to students.

Automaticity in programming and math

I’ve been learning how to program for a long time, a task that has much in common with mathematics. Both programming and mathematics involve being able to solve problems. Some of the problems in programming and mathematics have well established solutions and other problems do not. On a micro-level, programming involves manipulating code, a task much like the symbolic manipulation often used in mathematics. On a macro-level, programmers and mathematicians both need to be able to trouble-shoot, organize, and communicate their solutions.

Sample code:

Sample code

When I learned how to program, I taught myself, and I know that as a result, the code I create does not always follow the most appropriate industry standards. I have some unconventional solutions to some of the standard problems in programming, and I have less than optimal solutions for some basic problems in programming. I’ve yet to develop my own library of solutions, a standard practice in the industry.

On the other hand, I’m not a professional programmer. I program to solve problems I run into in life, and I program for fun. I have many programming projects that I’ve started and not completed. I’m an amateur programmer. I don’t need my work to look exactly what professional programmers’ work looks like because I rarely, if ever, share my programming with other people. I often share the results of my programming though, and this has helped build some useful tools for my students.

There are many low-level tasks that I no longer need to reference. I don’t need to look up how to define variables or functions, and I don’t need to look up loops, conditionals, and other basic parts of the structure of the programming languages that I know. I still need to look up the methods and properties of some higher level objects in the programming languages I know though, and when I program in PHP, I have a reference manual for the hundreds of functions available in PHP always open. I could be said to have developed a certain amount of automaticity in learning how to program, especially for the more basic tasks.

This automaticity was not learned by memorizing programming structures. I didn’t develop automaticity by doing practice exercises. I didn’t develop automaticity by reading books on programming. I developed automaticity in the low-level programming tasks by programming, by giving myself projects to work on that required me to build my skill, and by repeatedly looking up reference material when I got stuck. I developed automaticity because it is frustrating to write code that doesn’t work. It’s frustrating to get error messages that are nearly incomprehensible back from the computer when you make a mistake in the structure of your code.

If we look at mathematics education, we see that many, many of the problems given to students which have standard solutions. We expect students to develop fluency in these problems, often before they ever get to see any of the non-standard problems. In fact, in k to 12 education, students can potentially never be given an open-ended non-standard problem. Unfortunately, I believe this approach has failed our students in the past, and I’m not alone.

I’d like to see a system without a focus on fluency and automaticity in mathematics. These are the wrong drivers of mathematics education. Instead of focusing on the lowest level tasks mathematicians do, and assuming that fluency in these tasks leads to mathematical reasoning, we should focus on the most interesting and challenging tasks, and expect that a certain degree of fluency and automaticity will be developed as a result of these tasks. Instead of expecting students to memorize recipes and algorithms, we should allow them to develop toolkits and libraries to use of their own that they can reference as needed. Instead of feeling that every problem students need to do has to be solved quickly or efficiently, we should allow for alternate solutions and methods to be used.

Presentation: Programming in Mathematics Class

This presentation is based in part on the TED talk Conrad Wolfram gave a couple of years ago, and on some insights gained at the Computer Based Math summit I attended in November. The below presentation is slightly abreviated to make it easier to share on the web.
 

Exploring algebraic complexity

Here is an idea I am exploring.

I’d like some feedback on this idea. If anyone can point me at research already done in this area, that would be appreciated. My objective is to use this to justify the use of technology in mathematics as a way of reducing algorithmic complexity so that deeper concepts can be more readily understood.