The Reflective Educator

Education ∪ Math ∪ Technology

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Month: June 2011 (page 3 of 4)

Let’s run a greener ISTE conference

(Image credit: auntiep)

Last year at ISTE, I saw a lot of waste, and I feel like we should be able to run a greener conference this year. Here are some recommendations.

First, take advantage of the watering stations.  If you are planning on attending ISTE, bring a water bottle (or two). Make sure to fill up your water bottle rather than using the disposable paper cups, or even worse, plastic water bottles. You need to be hydrated, but you don’t need to be producing extra waste. In fact, you could bring your own utensils and a plate as well, and rinse these off in the bathrooms so that you don’t have to throw away paper or plastic plates.

Next, participants, please avoid vendor hand-outs. You aren’t going to look at them. Use your electronic device, and take pictures, or take notes instead. Take the time to get the contact information on the vendors whose products you are really interested in exploring further, and say no thanks to their brochures. Vendors, please learn about QR codes. Provide short urls to resources which are deep-linked if you must, but don’t be fooled into thinking that your little pieces of paper are useful; they aren’t.

ISTE organizers don’t give every participant a copy of the program guide to look at.  Post maps at strategic places within the venue itself, and your already out-of-date schedule, is up already in electronic form on the ISTE conference website.  This is a technology conference, use the technology effectively! Demonstrate best practices.  There were hundreds of laptops with internet connections set up all over the place; these could have been better utilized. Again, nearly every participant will have a portable electronic device with them. Instead of your newspaper, use a blog on the website with an RSS feed.

Everyone at the conference should put their devices to sleep when not in use. The power output at ISTE will be enormous, and given that in the United States some of this power comes from coal, it behoves those of us involved to be aware of our power use. The ISTE laptops, while useful, should go to sleep, with instructions beside each one available so that people know they can wake them up from hibernation mode and use them. The LCD projectors that nearly every vendor feels like they need to use are expensive in terms of power use. Vendors, your products aren’t more likely to sell because you have a flashy presentation, they  will sell if they are useful.

What are some other things we can do to run a greener conference?


Do you feel respected?

Here are the results from a simple poll I posted through Twitter.

Do you feel respect poll?

A flaw with this poll is that I’ve not defined community, so respondents were free to decide if this meant their local school community, or the community in which their school is located.

The results speak for themselves though. A profession which dedicates itself to long hours and relatively low pay (when compared to other professions where a Masters degree is nearly mandatory) should receive more respect than this.

The good news? Many teachers do feel respected in their community, myself included.

Must reads for new teachers?

One of my grade 11 students wants to be a mathematics teacher. She has asked me a for a list of books she should read. Here is a list of books I would recommend, along with some recommendations from members of my Twitter network.

What else would you add to this list?

Lemonade stand

Lemonade stand

I saw some little kids and their parents at a lemonade stand on Sunday, and I thought to myself, "how cute!" I found out that the proceeds from the lemonade stand were going to be used to help fund sports equipment and other necessary supplies at a local elementary school, and I thought to myself, "how sad."

Wouldn’t it be nice to live in a world where lemonade stands were set up by kids for "extra money"? Shouldn’t we fund education sufficiently so that first graders can worry about other things?

Derek Stolp on “A Mathematician’s Lament”

I recently contacted Derek Stolp, author of Mathematics Miseducation: The Case Against a Tired Tradition and shared with him Paul Lockhart’s essay entitled "A Mathematician’s Lament." With his permission, I’m sharing his thoughts below.

Lockhart’s argument is very compelling, and I certainly share his concerns about the state of mathematics education in this country, if not the world! The traditional approach to teaching mathematics, however, is only part of a much larger problem: the widespread acceptance of the transmission model of learning (as opposed to the constructivist model). Educational practices continue to be dominated by the principle that, if I may state it baldly, children just need to be told what to know and how to think, and this is true to some extent in nearly every academic discipline. (And I believe that this authoritarian model is, in the long run, inimical to the development of democratic values and practices!) I won’t belabor these points here but I do discuss constructivism and the tool metaphor on about pages 50 – 60 in my chapter entitled “Whose Knowledge Is It?”

Despite the power of Lockhart’s argument, I do think he needs to go beyond a lamentation. When I wrote my own book, Mathematics Miseducation, my wife read the opening chapter and said that she found it to be too negative. (As a kindergarten teacher, her preference is for the positive approach.) I told her that, as a math teacher, my habit is to identify the problem and to be certain I understand its contours before trying to solve it. In subsequent sections of the book, I do try to present possible solutions that solve the problem. But the book itself does not provide enough detail for the average teacher to change her/his practice so I decided to set up a web-site with lessons that teachers could use that would be consistent with my approach. I’m not entirely happy with all of the contents of that site – I’ve had to compromise my ideals with the practical reality of the school in which I teach. My 8th graders go on to secondary schools so I have to teach them algebraic fractions and radical expressions in my algebra course, as distasteful as it is to do so. And when they ask me, “Why do we have to learn this?” I try to be honest and say something to the effect, “If it were up to me, I wouldn’t teach these things. Math education is very slow to change, and it’s stuck on 19th century practices. But you’re going on to a school that will require you to learn these things, so I’m trying to get you started in a way that will diminish the pain for you next year.” 

Here’s another example of a compromise I’ve had to make: About 15 years ago, when I was chairman of the math department at Milton Academy in Massachusetts, some of my colleagues and I introduced mathematical modeling courses as alternatives to traditional symbol-manipulating abstract pre-calculus and calculus courses. We taught the same fields of concepts – in trigonometry, for example, modeling studied the rotation of a Ferris wheel and the shifting of tides whereas the traditional course found it more important to develop all those unnecessary trig identities – but at the end of the year, we devoted three weeks to SAT Math Level II review because these kids and their parents, while they loved the course, were not about to sacrifice their standardized test scores.

So here’s my point: Dr. Lockhart has, in his paper, identified the problem very clearly. His next step is to offer an alternative that will remain true to his ideals but that will realistically fit in a more traditional environment. He does suggest a way:

“So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.”

But how could he be more specific for those teachers who are sympathetic to his arguments but need guidance and resources? One way he could do that is to create a web-site with problems he has posed, indicating the levels at which they are appropriate. He’s been teaching high school, I gather, so he must be developing some materials. (I would love to see the day when textbook publishers are out of business because real teachers put their materials on-line for other teachers to access for free.) This, of course, will require considerable compromise on his part, compromise that he might feel destroys his objective. But I see no alternative.

The reality is that our schools are captive to an authoritarian trend to top-down standardization, and neither political party believes in a democratic form of education – Arnie Duncan has merely put old wine in new bottles – because we don’t really have a progressive political party – just a Republican Party and a Republican Lite Party. Progressives in education will have to chip away at the edges a little at a time and that will require distasteful compromises.

I’ve been asked to look at the Upper Elementary (Grades 4 – 6) math program at our school, and I’ve recommended using a program called Everyday Math. While I am not a textbook fan and haven’t used any for more than fifteen years, these teachers are teaching every subject and they need a resource. Here are excerpts from my memo to them:

"I know that it’s normal for every teacher to think about what needs to be accomplished by the end of the year so that his/her students will be ready for the next level and, since the kids are coming up to middle school, I thought I should let you know what skills, habits, and understandings I’d love to see among the incoming children. At is a paper entitled A Mathematician’s Lament which has been expanded into a book. This article was sent to me just this week from a math teacher in Vancouver and, while you’re certainly welcome to read the whole thing, I’d encourage you to read at least the first two or three pages – it will give you a flavor for the author’s point of view and, since I agree with him in principle, mine as well. His goals are lofty if unrealistic in today’s educational context, but they are certainly worth keeping in the back of one’s mind.

So, here are some of the important qualities that I would hope each child would possess coming into the middle school:

– Enjoys doing math
– Feels confident about approaching new problems
– Likes to make conjectures about patterns and is comfortable about being wrong
– Can talk about methods of solution with classmates
– Is resourceful about finding a solution method if he/she had learned it before and has forgotten it.

You’ll notice that there’s nothing here about knowing his/her times tables or knowing how to add fractions. Naturally, I’d love for each child to know these things but, to me, they are secondary. If a child has to take timed quizzes on multiplication facts (mad minutes?) or do pages of drills on adding fractions to master these, then the price is too high because it hampers the development of the qualities that I believe to be more important. So, that’s my bias. In the process of teaching math, we want to expose them to the many concepts and skills in your curriculum but, if they aren’t ready to master some of them, that’s fine – they’ll master them later.

One thing to keep in mind about the program: it’s not important for each and every child to achieve mastery before moving on. Some will, some won’t, and those who won’t will have chances later on to achieve it. And if they don’t by the seventh grade, don’t worry about it – I’ll address it then, and maybe they still won’t achieve mastery. (Several years ago, I had a conversation with a parent at Milton who was also a child psychologist working at MIT’s Department of Brain and Cognitive Sciences and she told me that several of her colleagues used to joke about the fact that they had never been able to learn their times tables. Somehow, these people managed to get by!!) The program has a good balance between skills practice and review on the one hand, and investigations and games, on the other. And I would lean heavily in the direction of investigations and games."

This memo to my colleagues reflects the kinds of compromises that I believe are necessary; the road to progressive education can be walked only one step at a time. Anyway, these are my thoughts regarding Dr. Lockhart’s excellent article, and thanks for allowing me to share them with you.

Derek Stolp

A Mathematician’s Lament – The online book study

Is this mathematics?

Math worksheet 

Join us this coming Thursday, at noon Pacific time, Richard DeMerchant, and I will be hosting an online book study of "A Mathematician’s Lament" which is an absolute must read for all mathematics educators. One of the questions we will try to answer is the one above, and you can ask more questions either on this blog post, or on Richard’s original announcement

‘I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them.’ (emphasis mine) ~ Paul Lockhart, A Mathematician’s Lament, p33

Paul Lockhart, first in his essay, and then in the extended version of his argument in his book, makes the case that current practices in mathematics education are fundamentally flawed because students spend much time learning the language of mathematics, without ever getting to actually do mathematics. I recommend reading his work in full, even if you are unable to participate in the upcoming book study.

Our plan is to work through his book and discuss the ideas in it, using quotes from the book as ways to kick-start conversation. Here’s my favourite quote from his book. What do you think of it?

‘Everyone knows that something is wrong. The politicians say, “We need higher standards.” The schools say, “We need more money and equipment.” Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “Math class is stupid and boring,” and they are right.’ ~ Paul Lockhart, A Mathematician’s Lament, p21

Assess what students can do

An exam
(Image credit: miss_lolita)

I’ve worked on a few programming projects in the past few days, and as I’ve been doing it, I’ve realized that if someone gave me a test on JavaScript, or PHP, or creating Drupal modules, I’d likely not do very well. In fact, I might fail that test.

The problem is, when I program, I spend at least a quarter of my time looking up functions, finding solutions to problems I’m having, and just generally muddling about. My code virtually never works the first time I run through it, and I spend another half of my time trouble shooting and finding errors in my code, some of which are dead stupid. The final quarter of my time is spent on inspiration and actually writing down the code I’m producing, as well as checking that it works. Some of the projects I work on take hundreds of hours, many of which are spent when I should be sleeping. 

I’d fail that test because I don’t have enough of the programming knowledge in my head to be able to regurgitate it as needed. I might be able to rough out solutions to algorithmic problems, but I’d never produce a masterpiece as part of my exam, and I certainly would have to look up many functions and ideas to be able to finish any program.

As it happens, I do well enough on my programming to make money on the side. I actually have made many thousands of dollars over the past few years doing small programming jobs (although less recently). Why? I solve the problems people need fixing. I’m tenacious. I don’t stop when my code doesn’t work, I forge on and find a solution. I know how to find the pieces of information I need to fit into the code puzzle I’m solving.

When I think about my success in programming, and how I couldn’t easily represent my knowledge in an exam, I wonder how useful exams (particularly timed closed book exams) are for finding out about what people know. How many of us have things we "know" but for which we often have to look up additional information? A chef has a recipe book, a mechanic has a car manual, and a chemist has a periodic table of the elements (as well as countless reference books). Everyone who works professionally has some resource they access to remind them of things they know, and to fill in the gaps in what they don’t know. 

Yes, there is specific information related to programming which I know without having to look it up, but if that was all that I knew, I wouldn’t be able to successfully create anything but the simplest of programs. I’d be completely unsuccessful as a programmer based on the sum of the knowledge currently locked in my head.

We must find ways to assess what students "know" how to do which represent the actual conditions of the life they will enter. While there are a small number of times in our lives where we will have to do something without access to information outside of our head, the vast majority of the time we will be able to fill in the gaps in our knowledge whenever we need.

Should we assess students on  what they can memorize? Or should we assess them on what information they can access?

A presentation on why educators should blog

Here’s a presentation I created tonight on why I think educators should blog. I still have some work to do on it, but I thought I’d share what I have so far.

Research on effective math education programs

Richard DeMerchant shared this study with me today. It is a review of studies done in mathematics education research, specifically 189 studies which used good research practices. Here are the key findings from this review.

"Programs designed to change daily teaching practices – particularly through the use of cooperative learning, classroom management, and motivations programs – have larger impacts on student achievement than programs that emphasize textbooks or technology alone.

The most successful math programs encourage student interaction."

So in other words, programs like what I recommend, and a constructivist approach to math education, are better than just changing how students access information

An interesting finding of the study was "there was very little evidence that it mattered which curriculum was used, as none of the curricula showed any strong evidence of effectiveness." In other words, a focus on changing curriculum isn’t going to have much of an impact on improving student understanding of mathematics.

This finding to me suggests that Bill Gates attempts to use the Khan Academy as a replacement for teachers are particularly misguided. At least Sal has come out and indicated that he doesn’t see himself as a replacement for a teacher.

In the 40 qualifying studies [of computer assisted learning programs] that looked at these various programs, there was little evidence of effectiveness. No program stood out as having large and replicated effects, and computer managed learning systems were particularly ineffective.

The implications for mathematics education, as the authors state, are as follows:

  1. Teachers can significantly enhance mathematics learning by adopting cooperative learning.
  2. Teachers can change their classroom management and motivation strategies to improve student outcomes.
  3. Professional development works.
  4. The evidence did not support the idea that different curricula give different outcomes in terms of mathematics achievement.
  5. There is limited evidence in elementary schools – and even less evidence in middle and high schools – that CAI math programs are effective.

One of the things I love about this study is the list of programs found to be effective, along with contact information (or a link to a website) so that you can follow up and try out these programs. There is also a list of programs for which the research that exists does not meet the reviewer’s qualifications for inclusion, and a list of programs which the reviews found to the research suggests are ineffective.

What implications does this have for your math program?

Let’s Abolish Elementary School Mathematics

First watch this video, right to the end.

I love the analogies Gordon uses to describe how we should reform elementary school mathematics. He also has an excellent argument against "real world" mathematics. What he prefers to use instead is "engaging" mathematics and I am happy to reform my own use of "real world" to "engaging" as it is a more inclusive term, and includes more mathematical ideas than the more simplistic real world focus. For more information on Gordon’s approach to mathematics, see

How can we make mathematics at all levels more engaging? Gordon suggests games, problems which are open ended and apply to a wide variety of learners, and abolishing our need for students to be "fast calculators" when computers can do this so much better.

What would you suggest?