Education ∪ Math ∪ Technology
This is another post in my series on math in the real world.
Vi Hart explains much of the mathematics behind noise in great detail, so watch her awesome video below. Thanks to @delta_dc for sharing it with me.
Notice her use of Audacity? I think we could quite easily turn this into a lesson plan… perhaps related to fractions, or to sine and cosine waves.
A binary number is a number written in base 2 format, like 101010101111. The binary number system is handy because it can be easily related to logical operators used in circuitry, and so almost all modern computers use this format for communication.
We use the decimal system for communication in our day to day lives because it is related to our original numbering systems, and this entire system was developed because of we have ten fingers between our two hands.
To convert a decimal number to a binary number, we want to rewrite the decimal number as a sum of powers of 2. For example, the number 5 is equal to 4 + 1 or 22 + 20 which is the same as 1×22 + 0×21 + 1×20. In binary, we write 5 as 101, since those are the coefficients of the powers of 2 (Try out this application which lets you switch between binary and decimal numbers).
Here is the basic multiplication table for binary, which only includes 0 and 1, since those are the only digits you have to multiply in binary (in a decimal system, you need a much larger multiplication table, since you need to be able to multiply each of 10 different digits, 0 – 9 by each of 10 different digits).
| × | 0 | 1 |
| 0 | 0 | 0 |
| 1 | 0 | 1 |
Compare this to the traditional 10 by 10 multiplication table for decimal numbers.
(Image credit: valilouve)
If you want to multiply numbers in binary, you could use some similar strategies to regular decimal multiplication. For example, 10101 times 101 looks like this:
If you want to double check, 10101 is the same as 1×24 + 0×23 + 1×22 + 0×21 + 1×20 = 16 + 0 + 4 + 0 + 1 = 21 and 101 is 5 (as we noted before) so this multiplication in decimal is 21×5, which is 105. 1101001 = 1×26 + 1×25 + 0×24 + 1×23 + 0×22 + 0×21 + 1×20 = 64 + 32 + 8 + 1 = 105. See this website for a more detailed example of binary multiplication.
The point of this activity is that you have taken something which is hard to do (memorizing a 10 by 10 times table) and switched it to something which is conceptually more difficult, but easier to memorize. For smaller numbers, it is faster to multiply directly in decimal, but for larger numbers, it will actually take less time to convert them to binary, do the multiplication, and convert back. You may notice that the multiplication step itself is much easier than decimal multiplication, since it’s just a matter of remembering 2 facts (0×0 = 0 and 0×1 = 1) and lining up the numbers correctly so that the place value matches. Check this page out for more information on binary number operations.
If all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication.
(Image credit: Christopher Policarpio)
I’ve been using LinkedIn a bit more recently, and thought I should join a couple of groups. I looked for a group for a Canadian Educator group, and found one with a few people who had joined it, but it looked like it was being sponsored by a recruiting agency, and I’d prefer to steer clear of those kinds of groups.
I’ve decided to create a new group for Canadian educators on LinkedIn. Let your Canadian educator friends know about it. I’m not concerned about your job title, just that you are connected to Canadian education. It’s currently in moderated mode, but depending on how it grows, I may open it up completely (I’ve had problems with spam with open groups on other networks).
Find it here: http://www.linkedin.com/groups?about=&gid=4052478&trk=anet_ug_grppro
"If we all learned mathematics in math land, we would all learn mathematics perfectly well." ~ Seymour Papert.
What does math land look like in your classroom? Can we create a space where kids think mathematically, and where the language of the classroom is mathematics?
This is an amazing discussion between Seymour Papert and Paulo Freire. Watch the videos below.
They discussed a fundamental issue in education; should the institution of school, which they call the second phase in learning, continue as it is? Both men agree that this second phase has an enormous problem, which is that kids learn during it to seek knowledge exclusively from adults, rather than exploring it on their own. Seymour Papert believed that access to computers would inevitably lead to over-throwing this second stage, and Paulo Freire disagrees. Paulo Freire suggested that the historical context of schools, and the political willpower to keep them the same, cannot be ignored when looking at their future.
It is an amazing conversation, and rich with information and ideas and worth watching to the very end. I recommend watching all of the video below as some of the most clarity in the conversation happens in the later part of their conversation as the two men dive into the distinction between their philosophies.
This conversation happened in the late 1980s (transcribed here). In my opinion, nothing has changed in most schools. We still have kids in schools learning that adults are the gatekeepers of knowledge. We still have kids who learn during the second stage not to question, but to accept.
The Internet has great potential to do away with the necessity of the second stage of learning, or at least radically alter it, but the political will-power to keep it the same has increased. The current standardization movement sweeping across the United States will do nothing to help kids develop a self-sustaining love of learning. The personalization of education movement in British Columbia is exciting because it has the potential to allow kids to chart their own course through the more formal second stage of learning, but if by personalization of learning we end up with all kids learning the same stuff, but at their own pace, we will have failed miserably to change schools.
Paulo Freire and Seymour Papert
Thanks to Joe Bower for pointing out the existence of this exchange between Paulo Freire and Seymour Papert.
I tweeted out the following yesterday during the #edchat discussion. So far 72 people have retweeted it (4 more since I took that screen-shot).
Every teacher is very likely to have been part of ineffective professional development at some point in their career, either as the organizer, the presenter, or the recipient of the professional development. Bad professional development, while fortunately not the norm, is very common.
I’ve been in professional development sessions were totally inappropriate for me as a math teacher, and sessions where I already knew everything that was being presented. I’ve presented sessions where I had participants literally falling asleep (although not recently!) and I’ve fallen asleep (nearly) in a presentation. I attended virtually the same algebra tiles session at least 3 times while working in NYC.
There are a few reasons I can think of why this happens.
(Do these reasons remind you of the reasons why students sometimes struggle with school?)
What can we do to ensure that we develop and participate in meaningful professional development? What can we do to convert professional development into professional learning?
This is another post in a series I’m doing on math in the real world.
When my son and I were on the roller coaster, I was again in awe about how quickly even a small roller coaster like this travels, and how it doesn’t drive right off the tracks.
Roller coasters have to be constructed fairly carefully, and follow some mathematical rules in their construction. They need to first be concerned about how to make the roller coaster safe. They need to calculate exactly how fast it will travel through the loops and turns, and how much of an angle they will need to prevent the roller coaster from taking a dive during those turns. They need to watch out that they don’t cause the participants of the roller coaster to pass out during a turn as they experience additional forces on their bodies!
The various costs associated with a roller coaster need to be calculated as well. There’s the cost to build, maintain, and operate the roller coaster. There’s an additional cost to pay for insurance for the roller coaster, which means an actuary needs to examine the probability of a problem occurring for any given roller coaster. The operator of the roller coaster needs to determine, given the cost to operate the roller coaster, etc… what they should charge to make a return on their investment, and attempt to maximize their profits.
While you could use a roller coaster simulator to explore some of this math, it’s a lot more fun to experience it in person…
Pierre de Fermat was a mathematician in the 17th century who often doodled and wrote down mathematical ideas in the margins of his notebooks. Once he wrote down the theorem that bears his name, Fermat’s Last Theorem, shown in the Google Doodle above.
The doodle itself has a flaw I wish to point out. You see, Fermat never wrote down his theorem on a chalkboard. He couldn’t have since a chalkboard wasn’t invented until about 200 years after he had his insight and wrote down his theorem. He wrote his theorem in a notebook.
The reason a chalkboard is depicted in the doodle is that mathematics today is seen as an activity done on a chalkboard, probably by a professor or a teacher, and not something that students do.
I’d like to change that.
If you know of another Edcamp happening in Canada let me know. If you want to plan an Edcamp yourself, I recommend reading Mary Beth Hertz’s excellent description of what an Edcamp is, and how to plan it here. All you really need to plan an Edcamp is a small team of dedicated professionals, and someone willing to provide some space.
Heidi Siwak and I were chatting, and she pointed out that, quite often, beginning teachers don’t know what the first day of school looks like. When I did my teaching degree, we talked about the rituals teachers use on the first day of school, but never got to see them actually practices. By the time we observed any classrooms, the rituals and procedures of the classroom had been firmly established.
I suggested that we should all share what do on the first day of school and then provide these resources to new teachers. There are a number of ways we can share these ideas.
To share links to these first day of school resources on Twitter, Heidi has suggested we use the hashtag #newteacher1styear. I’d also recommend cross-posting your resources to #ntchat and #edchat.
Update: In discussion on Twitter, Jana Scott Linday thought of expanding this project to the first year. Lisa Dabbs is setting up has set up a YouTube account to host the videos, and we will be continuing discussions on how we can share ideas for the first year of teaching.
If someone has already started a similar project, please let me know…