First watch this short video created by Dan Meyer so you understand the problem.

[WCYDWT] Coke v. Sprite from Dan Meyer on Vimeo.

I was having trouble wrapping my head around this problem. I saw people’s algebraic proofs, and I just felt there was something wrong with them. So I decided to construct a geomtric proof instead to make it more clear in my head.

At step 1, both glasses have the same amount of liquid. At step 2, one glass has some liquid poured into the other glass. At step 3, we pour out the same amount of liquid from the right most glass, hence the area of the red rectangle is equal to the area of the vertical rectangle with the mixture of the two types of soda. Note that I’ve made sure to go the other direction in the diagram, so as to represent the fact that assuming the two liquids are mixed equally, essentially the soda I pour back is a mixture. In step 4, I note that area A + B is the same as B + C, because the two liquids are the same, and that the area of B is the same in both pictures, and so hence the area of A is the same as the area of C, which means that the amount of the first soda moved back and forth is the same.

Here is the 8th Mathematics and Multimedia Blog Carnival for February (sorry it’s a few days late) with 8 interesting posts about mathematics.

Evolution of the symbol for 8

8 is an especially interesting number. It is the number of bits in a byte for example, and the number of sides on a chess board. It is the smallest composite Fibonacci number, and the only number (besides 1) in this sequence which is perfect cube. 8 = 2³ and 8 in binary form is 1000.

This month we have a number of interesting articles, which I’m diving into sub-categories. You can check them out below.

Pure Math

John Cook presents an interesting relationship between the Twin prime conjecture and the Pentium division bug — The Endeavour posted at The Endeavour.

Guillermo P. Bautista Jr. presents a proof of The Infinitude of Pythagorean Triples « Mathematics and Multimedia which is posted at Mathematics and Multimedia.

Gary Ernest Davis talks about near misses in mathematics, or how computational errors from calculators can result in "disproving" famous theorems on his Republic of Mathematics blog.

History of Mathematics

Romeo Vitelli writes about The Mathematician In The Asylum posted at Providentia. "As it stand, Andre Bloch’s case represents a fascinating example of how even institutionalized psychiatric patients can continue having an influence on the world."

Mathematics Education

Earl Samuelson describes how he teachers an Introduction to Differential Calculus posted at Samuelson Mathxp.

I talk about how it doesn’t matter if students struggle understanding how algebra works, or by using a calculator, either way it’s still Mumbo Jumbo to them.

Maria Droujkova talks about how "[her] multiplicative, non-linear kid" is coping with learning about curvy and non-curvy functions and struggling with the vocabulary of linear vs non-linear. She makes a good point about how much of what we teach is a social construct, rather than a function of the mathematics itself.

Dan Meyer shows that you can get some deep, interesting and challenging mathematics just by asking questions about how well cheese melts in a microwave.

Beth Still asks in a recent blog post

After being on Twitter for nearly 3 years I have learned to spot a bandwagon from a mile away. The latest bandwagon to come along has the words, “Let’s Abolish Grades!” written on the side of it. Maybe I am not as forward thinking or as innovative as I thought I was, but I don’t get this movement. Grades, whether they are letter grades, percentages, scales, or something else help students, parents, and teachers measure growth and progress and also indicate the level at which a student is performing (average, below average, above average). Students are admitted or denied access to certain programs, classes, and other things based on grades. Many times grades dictate scholarships and scholarships dictate where a student will attend college. This decision will have a lifelong impact on a person.

I responded

"I have no choice but to put zeros in their gradebook." Surely this isn’t true? A zero on an assignment is a slap in the face, and it certainly does nothing to encourage learning. You are not grading the student’s achievement, you are grading their behaviour. Those two areas need to be kept distinct from each other.

Here are my arguments against grading:

1. They are a shallow measure, they don’t tell the whole story. As a parent, I’d much rather have a complete picture of my son’s abilities rather than a brief summary.

2. They have a very high margin of error but are used for ranking students. Different teachers grade differently, use different assignments, but then students are all measured as if this wasn’t true. This is the equivalent of one scientist measuring in feet, and another in inches, and having them compare results without converting units.

3. They are so final! They end the learning rather than being an indicator of where further growth is. "Okay, well we finished that end of unit exam, here’s your grade. We’ll never think about that topic again."

4. They communicate the wrong message to students. Here is your F, clearly you know nothing about American History, or here is your A, now you know everything. Really? You can tell that a student knows nothing because of a letter they received?

5. Generally you know within a few DAYS of working with students what grades they will end up with at the end of the semester. To me, that says that the system must be rigged, because how could we possibly know what our students are likely to learn in advance of them learning? It’s like knowing the entire outcome of an experiment you’ve never done before.

6. They are used to rank students, schools, parents, communities. If you don’t think ranking is all that bad, remember that some educators careers, and student’s aspirations for the future are often dependent on a tiny variation between a "passing grade" and a failing one.

7. Grades are a cheap and easy substitute for parental involvement. "We absolve you of actually getting to know your child’s strengths and weaknesses within a subject area because we are going to give you the final summary of what they know in a single grade." Parents don’t have to feel guilty that they don’t know their children because they know their grades instead. This might be a little far-fetched, but think of the parents who act upon the final grades that their kids receive with punitive measures, rather than taking the time during the semester to be part of the learning process with their children.

I grade my students at my school, while recognizing that it is not a perfect system. It is a substitute for having sufficient time with parents and students to communicate the detail I think they deserve about their learning. We split the grade into "approaches to learning" and a summative score, which does help mitigate at least one of the issues Beth brings up in her post but I still find students searching for the final grade rather than being more curious about their learning progress.

I had a student recently who came to me after an exam, and told me she felt like she hadn’t done well. We sat down and went through her exam, and I gave her feedback on each question. We clarified what she understood, and where she had difficulty. It took about 30 minutes to go through the entire exam. I didn’t give her a mark, and say, "yep you did poorly now go away," I took the time to listen to her, find out what she felt like she understood, and give her the feedback she needs to improve her learning.

I recognize that I can do this because I have very small classes, and spending 30 minutes here or there with students to communicate feedback isn’t as big an issue as it would be with gigantic classes. To me, the biggest argument for a reduced class size is not that the students are "easier to manage" or that there is less grading, it is that you actually have the time to give students real feedback about their learning rather than a quick and dirty summary grade.

A colleague asked me today if I could find a documentary we watched last year in Science. She has students working in her Global Challenges elective course, and they told her about the video we watched in Science last year. I told her sure, and started Googling for the documentary. About 20 minutes later, I’m about to give up because I can’t find it. I remember then that I probably linked to the documentary on my lesson plan website from last year, and sure enough, I find it in a substitute lesson plan. Total search time on my lesson plan website? About 2 minutes.

Keeping track of your lesson plans and ideas in a searchable format is a must. At the very least you should have folders organized by topic and grade level. However, if you are looking for a resource and you remember something about the resource but not how it was organized, this type of organization can be cumbersome. If you’ve ever tried to use Windows search to find something on your computer, you’ll know how awful it really is. I’ve tested the Mac OS’s search function, and it is far superior, but neither search function is completely satisfactory.

Storing your lesson plans and resources you’ve created online instead of on your personal computer has a number of benefits.

• Your lesson plans can be easily shared with students, colleagues, parents, substitute teachers, administrators, and teachers from elsewhere in the world.
   
• Virtually all web hosts include backups of your data, which means that your work will never be lost. Still, it is worth keeping multiple backups of everything, but information stored in 3rd party systems is far safer than on your likely-to-fail-eventually hard drive or USB stick.
• Your lesson plans and resources are become easily searchable, especially if you are storing them as text & media in webpages, rather than uploading them in some other file format.
   
• You can get feedback on what you’ve done from sources other than your students. Feedback is what helps us improve, and professional feedback from our peers is more valuable than informal feedback from our students. We normally get much more feedback from our students and so it feels more valuable, but try talking about one of your lesson plan ideas with your colleagues and you’ll notice the difference.
   
• More altruistically, your resources and lesson plans help other teachers improve. While I don’t think that actually using another teacher’s lesson plans verbatim is all that useful, one can get ideas on improving their own lesson plans and learn from how another teacher constructs lesson plans.
   
• You have a chance to have your work noticed by others and be offered other opportunities. Self-publishing your work leads to personal benefits in your career as well.
   
• Your work becomes part of a professional portfolio. In a world of digital media, what you have already published online, and the online reputation you have built, says a lot more about you than what you can cram into your résumé. 50% of companies do Google searches and check Facebook profiles. What do you want them to find?

George Haines has constructed an interesting argument refuting the idea that because we live in exponential times that we need to change how our schools operate.

George says:

Last year Google CEO Eric Schmidt made a big splash by telling us that more information is created every two days than was created from the dawn of time until 2003. This is an alarming find, even if the numbers are fudged quite a bit. The quote wasn’t aimed at educators, but many in the EdTech community took this quote and ran with it. The message Schmidt delivered fit very neatly into the narrative many radical educators subscribe to– that teaching specific factual knowledge is "20th century" and we should be teaching "how to find knowledge" in real time or whatever.

It doesn’t take an expert critical thinker to see the huge hole in this line of reasoning. The reason this is a somewhat meaningless factoid is that there has always been more knowledge in the world than we could possibly teach to students. I can remember sitting in the library on SUNY Stony Brook’s campus and looking around at the over-stuffed shelves of books on just one bookshelf on one floor and thinking "I will never be able to read even a respectable fraction of the books in here."

My response was:

George, I do have one observation that I would like to make and we’ll see if it pokes a hole in your argument or not.

Things we agree upon:

There has always been more knowledge available to know than what can possibly be taught to kids in schools.

Someone needs to select a subset of the available knowledge to show to kids. Kids cannot possibly become completely self-directed. I would like to see much more self-direction than currently occurs, but I don’t see kids as being able to learn how to read, or even adopt most critical thinking skills without a lot of interaction and support from adults.

Things that we do not agree upon:

It will always be possible for a small team of educators to choose the best possible subset of skills or content for our students to learn, using our current systems of determining curriculum goals. 

The process of curriculum construction is linear. A bunch of people get together, they look at what is available to be known, they might examine market trends, read some research about future predictions, and then they carefully select a subset of that total knowledge to share with kids. The rate of change of the subset of knowledge is directly dependent on how many people are examining the curriculum. Mathematically, it is a linear function. While much of this base knowledge remains constant, some of it must change.

If you buy the argument that the total amount of knowledge is increasing exponentially, then you must see that there is a serious problem here. An exponential amount of knowledge cannot be effectively processed using a linear method! 

This has already resulted, in my opinion, of some of what we are teaching kids, particularly in math and science, to be largely irrelevant. Why do we spend so much time teaching algebra when even professional mathematicians hardly do any algebra at all? If our objective is to teach logical thinking skills, that could be just as effectively done using computer programming skills which are vastly more important in even today’s economy and society than algebra is.

You can’t produce long-lasting solutions to problems by alienating the people who would have the biggest impact in effecting change. Want to see what happens when you try?

(Thanks to @ProgresivTeacher for sharing this)

This is the lesson of the protests in Egypt, the protests in Madison, and the unravelling of the changes Michelle Rhee made in DC. If your change involves repressing a large group of people, you might be able to make some faux headway, but as soon as you are gone, all of your changes will revert.

Want to really change education? Listen to educators.

This video explains the process of crowd-sourcing, but there are a couple of important pre-steps:

• Build a strong PLN
• Make sure your question is interesting enough that other people will want to help you

You can view the document in the video here.

So I’m finally able to share an interesting story which has developed over the past 3 months. It started with TEDxVancouver, which I saw Nazanin Afshin-Jam speak about her work as an activist.

(photo credit: Rick Chung)

By a strange coincidence, we had a student at our school who was working on a film project about the stonings that were happening in Iran, and trying to understand the situation. Since I had just heard Nazanin talk about the same issue, I decided to look her up on Facebook. Turns out she has a public Facebook page, and she had enabled messages from anyone in her privacy settings. So I sent her a message.

She responded! Well, actually one of the volunteers who helps her organize and manage her Facebook responded, but he did so with Nazanin’s blessing. After several communications back and forth, and one miscommunication, we finally managed to arrange a telephone interview for today between Nazanin and my student. It took a few months, largely because Nazanin is an incredibly busy person, and travels a lot in her activist work so finding time to meet was challenging. Eventually Nazanin herself handled the last minute timing of the interview.

She called the school and I said hello, and thanked her for her time, and for the opportunity she was giving our student. I then passed the phone over to our student, because as I said, "this was her show."  We didn’t turn on the speaker phone even though we were dying to hear what she had to say, we just sat and listened to the questions our student asked, and I have to say that they were hard questions, and I really was curious to know what the responses were.

"What is the background of stoning," our student asked, after exchanging pleasantries, jumping right into the heart of what she wanted to know. Twenty minutes and 10 questions later, we had a very satisfied student in front of us, one who seemed really impressed by the conversation. My colleague, who sat in on the interview out of curiousity, said to me after our student left, "we’ll never know what Nazanin said." I agreed, "It doesn’t matter though, she obviously learned a huge amount from it. You saw her face."

You can’t help your students have experiences like this unless you ask.

Here are some things that I either don’t use anymore, or almost never use anymore. I can remember using all of these things often, but they just don’t seem useful anymore.

The last time I burned a DVD was for a colleague at work. I think the time before that was at least a year ago.

I have a calculator on my smartphone or on my computer. I do use a graphing calculator, but only at school. Why would I want a single use device?

I don’t mail things anymore, or at least not often enough that I can remember the last time I did it.

I don’t use pencils (and I’ve never used mechanical pencils), except for attendance, or to write a quick note when my phone is out of reach.

This particular Yellow pages came to our apartment unrequested a couple of months ago, and we still haven’t removed the wrapper… 

My paper address book is totally useless to me, now that all of my contacts are online.

The only reason I listen to CDs anymore is because we take car rides, and our son likes to listen to his music. Otherwise, music I listen to is on Youtube, or on my iPhone.

What’s your list of things that you’ve stopped using?