According to A Maths Dictionary for Kids 2008 a proportion can be defined as:

-a part to whole comparison

-the equality of two ratios, written as an equation

How do we know when to use a proportion and when not to. Consider these two problems:

Memphis and Texas were shooting an equal number of free throws. Memphis shot first. When they had shot 9 free throws, Texas had shot 3 free throws. When Texas shot 15 free throws, how many had Memphis shot?

My in-laws just returned from Spain. The exchange rate for 3 U.S. dollars with the euro in Spain is 3 dollars for 2 euros. How many euros for 21 U.S. dollars?

Both would seem to be proportion-type problems. They each have 3 numbers and one piece of information that is missing. So you could set up a proportion in the a/b = c/d format for each problem. There is a difference though. In the first problem there is no multiplicative relationship between the numbers whereas in problem two there is a relationship. For every 3 dollars you can receive 2 euros (3:2, 3 to 2, or 3/2). If you are exchanging 21 U.S. dollars you would get 14 euros. You could set up a proportion such as:


In order to solve this problem you can treat it as an equivalent fraction/ratio type problem and see that 21 is 7 times bigger than 3, so x has to be 7 times bigger than 2 (which is 14). As well, you could cross multiply


and end up with 3x = 21 x 2.

3x = 42
3 3
x = 42/3
x = 14

So why wouldn’t the Memphis (who at half did actual shoot 9 free throws!) /Texas problem be solved proportionally? The number of free throws that Memphis shot compared to Texas can be explained with addition or subtraction. Memphis free throws equal Texas free throws + 6 or Texas free throws equal Memphis free throws -6. If Texas shot 15 free throws, then Memphis must have shot 21. The difference between the Memphis and Texas free throws will be + or – 6 (depending on which way you look at it).

In the in-law trip problem, they were exchanging 3 dollars for 2 euros. Therefore we could find any number of dollars for euros by multiplying the dollars by 2/3 (21 x 2/3= 42/3 which equals 14). And the opposite can be calculated as well, we can multiply the euros by 3/2 (3/2 x 14 = 42/2 which equals 21).

In order to understand proportions, one needs to understand ratios (comparing two things) and rates (a ratio that compares quantities of different units). Ratios can be look at as part-to-part or even part-to-whole. For example to make a banana smoothie you might use 1 cup of milk and 2 bananas (1:2). What if I wanted to make smoothies for myself and three friends? Since that is 4 people, keeping the rate the same I would need 4 cups of milk and 8 bananas.

Hi-Ho-Hi-Ho it’s off to grade tests I go…

I will be spending the next three school days grading N.Y.S. Mathematics Exams. Tomorrow is fourth grade, then fifth and on Tuesday sixth. Time well spent?!?!?! For the students sake, I hope every student hits a home-run. But in the end, it was, is, and will continue to be about the students learning. Wherever they are at, it is my responsibility to take them to the next level.

Just out of curiosity is it more important to know how to pick the right answer to:


or be able to work through a problem such as:

The Field Trip Problem (from Catherine Twomey FosnotContexts in Learning):

A fifth-grade class went on field trip. They took four different cars. The school provided each group with heroes (subs, hoagies, wedges in other parts of the country). The subs were cut and shared as follows:

  • Group 1 had 4 people and shared 3 heroes
  • Group 2 had 5 people and shared 3 heroes
  • Group 3 had 5 people and shared 4 heroes
  • Group 4 had 8 people and shared 7 heroes

When the students returned from the trip, they were talking about the sandwiches and they felt that the way the school handed them out was not fair. Did everyone get the same amount of sandwich?

What do you think: Will students get more out of the first or the second problem? Will they take away understanding that will last with the first or the second problem? Which is easier to measure? Which is messy? Which gives the students and the educators more to work with?

Diigo-Bookmarking and a whole lot more

Let me start off by saying that I have been using for some time now as my bookmarking tool. It is functional and it serves me well. Recently I discovered Diigo which has a lot more features available to the user. Wow is what I have to say right off the bat. Not only can you bookmark a site for future reference, you can hightlight passages and add sticky notes right to the page. You can add notes as well. All of your stickies, notes, and I believe highlights can be made public or private. In the Diigo sidebar, you can choose from annotations which I just mentioned or readers. In the readers section you can see who is reading (or read) that particular post as well as who is a reader of the site.

Like, you can access your bookmarks from any computer and even an iphone. You can add tags or lists to keep track of similar findings. If you have a blog, you can send what you find right to it. They have a “send to blog” feature that will allow others to see what you are blogging about. I haven’t tried that feature yet, but will have to give that a try in the near future.

For classrooms, you can create groups that are private, semi-private or public. So if the class is doing a research project, anything that you highlight can be shared with the group. Stickies can be left to discuss the topic, questions people have, or notes for future reference. The group leader can also set up a tag dictionary, so the group can be consistent in it’s use of tags.

Diigo also has a social piece to it. As you are reading, in the sidebar you will see other people who have bookmarked the site and other related content as well. You can subscribe to tags to keep current with specific topics. Also, you can subscribe to sites and other Diigo users. The creators of Diigo will also find content based on your bookmarks.

There is also a “people like me” piece that allows you to find people with similar interests. People can be followed, invited, and sent messages. You have the capability to create an online presence, allowing people to know as much or little about you as you like. This piece of Diigo has a Facebook feel to it.

This obviously can be used for students who are doing research. I would be interested to see how educators and students are using it to improve their learning.

The 2.0 Riptide

Web 2.0. I get it (I think). Our students are using it outside of school. We need it inside our schools. The question is how. I was reading the blog post, The Embedded Practitioner, from the blog “blog of proximal development” by Konrad Glogowski. He posed three questions that have me thinking about POB and what we can do to create a 21st century learning environment. The questions from the post are:

“1. How do we prepare teachers (my change) all educators to teach 21st century learners whose lives are based on rich interactions in multiple on-line environments?

2. How do we help new teachers move away from what Marshall McLuhan once called the “imposing of stencils” and adopt a practice of probing and exploration?

3. How do we help new teachers acquire the courage to transform their classrooms into communities of learners and transform themselves into participants who can embed themselves in those communities?”

How do we turn the tide in POB? Is it like a riptide where you need to find the middle and go through it instead of trying to fight it? If so, what is the middle? By middle I don’t mean middle ground (settling) I mean the middle of the riptide.

(Picture from Wikipedia)

To educate 21st century learners we, as educators (teachers, aides, administrators, board members), need to be 21st century learners. We need to stop using the phrase life-long learning and start living life-long learning. Personal Learning Networks should be a way of life for educators. It shouldn’t be imposed like our 18-hour staff development. We need to get rid of this 18-hours of mandatory staff development (which can only be 2-hours contiguous to the end of the school day). All educators should be developing PLN’s to meet their needs. Overtime, PLN’s change as your needs change.

Besides PLN’s, staff development should be replaced with professional learning communities. What are you interested in? Form a group of like-minded individuals. Chase it (the learning) down. Apply it (what you have gleaned from the PLC) to the learning environment you are in. Assess it. Refine it as needed to meet the needs of all learners. We need to move away from it being about us, the educators at the front of the room, to being about students and listening to what they have to say. Create the learning around their needs. Check your ego at the door. Pick it up when the day is through if you must.

Yesterday, I was reading a post at Teaching College Math and I found this quote and follow up that I think summarizes what I am trying to say. “Quote from Hunter Lovins: “What is the purpose of education if not for future generations?” Now that’s a quote I can sink my teeth into. As educators we can’t dwell on “how we learned it” – we’ve already been educated and have moved into the world community, but the students we teach need to be prepared for the world they will enter. If that means that instructors will have to continue to be learners themselves – so be it.” There is no way around it. To have students become 21st century learners we need support each other in our own learning in order to facilitate student learning.

Questions 2 and 3, I feel help move question 1 along the path of where we need to go. Where are the early-adopters, pushing the envelope? I believe there are some that already exist in the schools and we need to bring them to the forefront of every building, school district, state, and country. I also believe that every new hire needs to have the capability and desire to be a 21st century learner. How about an intensive course in the summer of the new hire to teach tools (Thanks Tom Schwartz)? How about a mandatory blog, website and a wiki that is created and maintained by their students. How about as part of receiving tenure, a teacher should have a portfolio of these creations to share with other educators to demonstrate their own growth and learning?

Teachers should not have to be asking for the hardware; it should already by in place. You really can get by with a minimum amount of hardware, as long as it is functioning and filters do not restrict you. I recently heard a quote from a podcast by the “techchicks” and I may be paraphrasing; “Technology can no longer be integral integrated, it needs to be integrated integral.” This is why administrators and board members need to be aware of what is going on in the 2.0 world. They do not need to be aware of every tool, but they need to see the ways in which it helps students and teachers. If the see it, I believe they will support it and provide the necessary funding.

Educators need to be steadfast in their beliefs, willing to learn and always have at the forefront the students. The new educators will need to seek out and find (and vice-versa) the established teachers who are doing things to promote 21st century learning. No longer can teachers go into their “dimple” (classroom) of the “egg carton” (school) and not be aware, share, and be a part of what is going on around them. Educators need to take risks and stand up for what they believe in.

Do we want this: Teach different

or this: Sir Ken Robinson.

I am going to spend some more time thinking about these things but in the meantime I want to leave you with this: