Education ∪ Math ∪ Technology

Year: 2011 (page 13 of 28)

Assessment during professional development

Presentation

Workshop

Image credit: uconnlibrariesmagic

 

If you are running a professional development session for teachers, and you recognize that teachers are learners, how are you assessing their learning? Are you embedding formative assessment within your workshop? Are you providing an option for summative assessment of the learning, either at the end of the workshop, or in a follow-up session?

While I don’t think you should be giving grades to teachers for workshops, you do need to provide some way for your participants to receive feedback on what they’ve learned. Feedback in some form while learning is critical. Otherwise, how do you know your participants are learning anything? 

Toolkit model of math instruction

(exec talking to IT person) Apparently our open API is empowering our customers with unprecedented control over their destinies. So please shut it down.

Image credit: Rob Cottingham

I’ve been doing some programming recently for a friend of mine, and while programming I made a realization. Every time I needed to remember how to do a particular algorithm, or use a tool with which I’m less familiar, I look it up online (or I ask someone for help). In fact, I spend a lot of time as a programmer looking up things called APIs and core functions in the programming language I’m using. The basic structure is very solid at this stage, since I’ve been using it over and over again, but there are still lots of things I look up frequently.

I wondered to myself, what would this look like in the mathematics classroom?

It’s not ridiculous to compare programming and mathematics. Programming and mathematics have a tonne in common, in fact much of programming a computer itself is deeply rooted in mathematics. They are both domains of knowledge which allow for high levels of creativity (provided you are given the freedom to be creative) and rely on an ability to construct algorithms and perform computations. Being good at constructing algorithms is useful, but not sufficient, in both domains for creating complexity and solving difficult problems. It takes more than just knowing your stuff to be good at these two areas of knowledge.

What I imagine would happen, were one to follow this model, is that students would have a resource available to them, whenever they needed it, which had very simple and short explanations of each mathematical computation in their toolkit. Whenever they ran into a computation, and forgot how to do it, they could look it up. Perhaps this toolkit (or mathematical API reference) would be in paper form, perhaps it could be in digital form. Whatever it is, it should be easy to search through. In digital form, it should include short (2 minutes ideally) video snippets showing how to do the computations. Perhaps it could even be searchable by entering an example of the computation itself. Students could add to their personal toolkit as they discovered or encountered mathematical techniques that they found useful, which is very similar to the process programmers go through as they build a code library.

Over time as the students used the reference material, the computations that students used often would be things they would naturally memorize. The less frequent computations might be things they looked up a lot. Students could spend more time working on highly engaging and personalized problems or activities, and a bit less time memorizing all of the computations. Most importantly, they would spend a lot more time practicing an important skill, recognizing what type of computation is useful in a given situation, and being able to relearn that computation as needed.

In Keith Devlin’s book, "The Math Instinct" he makes a lot of interesting points about mathematical ability. One thing he points out is that many people use mathematical strategies successfully in life, for example to do their shopping, but almost no one uses the highly efficient "school math" algorithms they learned. The problem is one of transfer of knowledge. People just generally don’t know how to transfer stuff they’ve learned in school to their lives.

The hope is that these types of API references for mathematics would be something so useful, kids would keep them year after year, and potentially use them in their lives. We already give students formula sheets for many exams, this is just one step further.

Note here that I have a premise which I should make explicit. It is not the learning of algorithms or specific computations which is mathematics, it is the learning of how to use these algorithms and apply them to problems in creative ways, and then extend them as necessary which in my mind is what defines mathematics.

New meme: If we taught _____ like we teach math, _____

There is a new meme out there, suggested by @r_w_wright. "If we taught _____ like we teach math, kids would _____."

Here are some examples people have posted so far.

If we taught construction like we teach math, kids would bang nails into boards but never actually build anything. ~ @davidwees

I’ve frequently said similar about statistics: "We show kids the screwdriver, but never show them a screw." ~ @heyprofbow

If we taught driver’s ed like we teach math, students would feel no shame in announcing "I can’t drive" ~ @datadiva

…and as we integrate technology, let’s be sure kids aren’t just banging virtual nails into virtual boards ~ @ChrisHunter36

If we taught English the way we teach Math, students would be able to punctuate a sentence but have no appreciation of literature. ~ @ChrisHunter36

What if we taught math like Umbridge teaches defense against the dark arts? Oh wait, we do. ~ @rjallain

If we taught mathematics the way we taught music, everyone over 11 would have 1:1 tuition outside school… oh wait… ~ @ColinTGraham

If we taught teaching the way we teach math, then most teachers would only teach the way they were taught…oh wait…  ~ @mathhombre

If we taught science the way we teach math, then people would think it’s only for the smart kids…oh wait… ~ @mathhombre

If we taught music the way we teach math, then most people would not be able to play an instrument…oh wait…  ~ @mathhombre..

If we taught videogames… wait, we don’t teach videogames? Why do so many people play? ~ @mathhombre

 

 

Want to add your own examples? Add them either here as comments, blog about them, or post them to Twitter with the #ifwetaught hashtag.

If we taught carpentry like we teach math

Your instructor brings you a board. Before you can use the board and play with it yourself, she tells you how to properly line it up on your table. Next, you practice this over and over again with everyone in your class practicing the same number of times regardless of when they master the skill. When you line it up in creative or fun ways, you get scolded, and sometimes even have your board taken away from you. You look around the room and notice that everyone’s boards all look the same.

Finally, after you are considered to have mastered the skill of lining a board up, your instructor takes your board away and gives you a nail. She shows you for 10 minutes all of the various ways you can line up a nail, but never shows you how this relates to the board, or any other possible tools. You want a chance to practice lining up the nail properly, but your instructor says that time is up, and assigns it for homework, and takes away the nail. "You’ll have to find your own nail to do this for homework," she says. You wonder if that is fair for the students who don’t have nails at home.

The next day, your instructor checks to see that all of you have practiced lining up the nail. She then gives you a chance to practice with the nail for a few minutes, before she again takes away the nail and gives you a hammer. You spend some time learning about the history of the hammer, and finally you learn some of the possible uses of the hammer. You ask if you can play with the hammer, but your teacher says, "That’s much too dangerous for you now, you’ll learn more about hammers when you are older, and then you can use them."

You never get a chance to see how the board, the nail, and the hammer relate to each other before the unit finishes. You don’t really understand how to use a board, and you’ve forgotten how a nail works by the time the test is given and so you fail the final assessment. You want a chance to practice some more with these skills your teacher says are "vitally important" but she moves onto another unit.

"Okay class, in our next unit we are going to learn about sanding wood. Everyone take out their boards and practice lining them up again…"

At no point in your learning of carpentry do you ever find out why people might want to use carpentry, how beautiful some works of carpentry look, or how to put it all together and make your own buildings.

 

You might think that this would be a ridiculous way to teach, but this is exactly how we teach mathematics today. Each unit is separated from one another and the connections between the units, and often the lessons, are virtually never taught. Students almost never have the opportunity to play with mathematics, and never get a chance to use some mathematics once they have mastered it. If we even connect mathematics to the real world, we do it in arbitrary and often nonsensical ways. We teach mathematics as a bunch of discrete tools and not as a holistic study of patterns and our world. In fact, we don’t even really have a consensus as to what mathematics actually is!

It’s no wonder kids usually hate mathematics. They say, "Math makes no sense," and they are right.

I am a feminist

"Feminism is a collection of movements aimed at defining, establishing, and defending equal political, economic, and social rights and equal opportunities for women." Source: http://en.wikipedia.org/wiki/Feminism

Under this definition of feminism, I am a feminist. I hope that many educators would count themselves as feminists under this definition. However, we still have many issues around gender equality in our education system which we need to fix.

When we spend unequal amounts of money on our sports teams, and promote the boys’ sports more than the girls’ sports, we send the message to our girls that what they do is unimportant.

When we call on the boys in our class more than the girls and encourage reflection and deeper thinking from the boys, we are implicitly telling the girls that what they think is unimportant.

When we hire disproportionally more male administrators in schools than female administrators, we tell girls that they aren’t supposed to be in charge. In BC, for example, female teachers outnumber male teachers by a 2 to 1 ratio, but are nearly evenly split at the administrative level. While this is changing, we have a long way to go. Can anyone tell me why female teachers make, on average, $4000 a year less than their male counterparts in British Columbia?

When our prescribed learning outcomes do not specifically talk about the role of media in defining gender for our society, there is little hope that we can counteract the effect of media. Fortunately, gender is discussed in our curriculum in British Columbia, all of 14 times.

It is critical that each of us who are educators, who are helping shape the role of gender in our society, publicly identify ourselves as feminists. We must actively work to break down the rigid gender roles our society defines, because one of the places change happens in our society is in education, and if the inequality between the genders persists, it will be at least partially because of our inaction.

I worked at a school that cheated

When I worked in NYC, in one of those small academies created in the old Chancellor’s district, I worked at a school which cheated in many different ways to improve our test scores.

  • Our principal would trade away his worst performing students to his friends’ schools in other parts of Brooklyn, thus improving his odds of raising his, I mean our school’s, test scores. I don’t know exactly what his friends got in return, but our principal got his $15,000 bonus three years in a row for raising test scores, and then he retired (in NYC, at the time, a Principal’s retirement income was based on his final 3 years of service).
     
  • We didn’t choose students based on test scores, that would be too obvious. Instead, our principal relied on average attendance and word-of-mouth about good programs from which to choose students. Our ninth grade class in my second year of teaching was the strongest class academically ever to attend my old school in NYC as a result. As soon as our principal retired, we got an influx of students from the poorer performing neighbourhood schools, along with a string of awful principals — one after the other.
     
  • When we needed three out of our four weakest students to pass the state Regent’s in math in order to avoid being classified as a failing school, each of whom was classified as a "Special Education" student, they all got readers for the exam (one of them also got a scribe). "Are you sure you want to pick B?"
     
  • We regularly "scrubbed" our test scores and any of them that were close to passing we reread until we had found creative ways to award them points so that they passed. No one got an almost passing score: not one single child. I thought it was common practice; I had no idea this was even frowned upon. When one of our students had forgotten to draw a line in her diagram, we all left the room and when we came back — mysteriously — we realized she hadn’t actually forgotten to draw the line — how lucky!
     
  • Every single question I was supposed to share with the students had to look like a Regent’s exam question. I was instructed to quiz the students using past exam paper questions, give them homework assignments involving past paper exam questions, and all of my exams were supposed to look like Regent’s exams in format. In the final two months before the exams, our students would see nothing but Regent’s style exam questions.
     
  • Our students never seemed to get suspended in September or October, but after whatever that magical date was in November when we got our allotment of Title I funds based on our average attendance, all of a sudden our worst performing students would get suspended in droves. Out of 34 kids in remedial math, six of them remained to take the Regents exam at the end of the year. The rest had dropped out of school. Not surprisingly, I had three out of six of my students pass, which is an amazing 50% pass rate!
     
  • Two years after I left the school, our newest principal (who started during my last year at the school) was fired (or asked to resign) after it was discovered she had modified test scores for students.

I feel bad about what happened at my school, but I was an rookie teacher in a foreign education system that made no sense to me. I did not have enough control in that school other than to do my best to provide my students with an enriching and relevant math curriculum.

The point is, when the stakes are high enough, people cheat. The recent problems that have been discovered in Atlanta are just the tip of the iceberg. There are a thousand other ways schools are cheating: they just haven’t been caught yet.

The Shapes of Stories

This is a fabulous video of Kurt Vonnegutt talking about the shapes of stories. Notice the different types of graphs he produces? This is a fairly mathematical procedure, and I love both the idea of embedding everyday variables on the axis, and of representing the rise and fall of the fortunes of the protagonist in the graph. I remember my 11th grade English teacher sharing with us the common graphs of Shakespeare’s trajedy and comedies in an effort to help us understand the two distinct genres.

 

 

Wouldn’t this be an interesting activity to do with your students?

6 years and 630 posts later

About 6 and a half years ago, I started a blog. The purpose of the blog was to provide updates for my family back home while my (new) wife and I were living abroad in London, England. At the time, Facebook was still a closed site for college students, and Myspace was too messy to share with my parents. I needed an electronic space to share pictures, thoughts, and stories from our lives.

As I was jumping into learning programming, and educational technology, I decided to switch from the free hosting offered by Blogspot to my own server. I first installed Movable Type, then WordPress, and finally settled on Drupal for my blog, learning each of the technologies as I transfered between them. My first choice of domains, "UnitOrganizer.com" was intended to be a place for teachers to share their ideas, in the form of unit plans. That never actually came together, and a couple of years later I switched to my own name for my domain.

During the past few years I’ve switched from blogging about my personal life, to blogging about my programming projects, to blogging as a form of eportfolio, to blogging about my interests in education almost exclusively. I’ve had more opportunities come to me than I can count through my blog including having the opportunity to write a textbook, be published in 3 different magazines, have a short series of articles in a newspaper, and to present at conferences all over the world.

I read through some of my earliest writing, and I can see a big difference between both the strength of my writing and the quality of the ideas behind the writing. My earliest writing on education is fairly naive sounding to me today. I wonder what I will think of my writing in 10 years.

To those of you considering a blog, I say go ahead and do it. The opportunity for reflection and personal growth have been tremendous.

I need to rethink my practices

I got paid the highest compliment one can get from a fellow teacher after my presentation at ISTE Unplugged.

"I know I’ll be changing some of my teaching decisions." Tom Grant

After reading many posts on standards based grading, it seems clear to me that I too need to change my assessment practices. I need to ensure that I have mastery from my students, rather than a superficial level of understanding. I also need to do more to engage my students with more problems which are relevant to their lives.

I need to rethink my practices. Hopefully, that will always be true. Keeping my teaching practices completely static in a changing world seems like a pretty foolish thing to do.