Yes, James. Fractions are hard, but they can be fun too!
I love my annual exploration of fractions with my fourth graders every spring. This is uncharted territory for them, as we intentionally delay introducing the concept of fractions until they have enough cognitive development to really appreciate and understand these strange numbers. I tell them from the start, “This is probably the hardest thing I’m going to ask you to do all year.” while I sip coffee from my 3 out of 2 people have trouble with fractions mug. We spend time thinking about what fractions mean, building fractions with manipulatives, making observations and looking for patterns. We look at equivalents to “benchmarks” like one-whole, one-half, one-third. All the while, building, drawing, observing, sharing ideas, asking questions and most importantly, letting my students set their course on this adventure.
After about a week of working with fractions, one of my girls asks today, “What about a fraction like eleven-twelfths?” “What about it Alexa?” I ask. “Well, it seems like you can get really, really close to one whole, but never get there.” The rest of the class has been sucked into the intrigue by now, so I suggest we build all the fractions we can like what Alexa is talking about. And then the magic begins!
One says, “She’s right, we’re never gonna get there!” Another notices, “Look, the differences between the fractions are getting smaller!” And, “Hey, even though the pieces are smaller, the bigger denominators are bigger fractions this time!”
After several minutes of playing around, I tell them what unit fractions are, and how we’re building numbers that are one unit fraction away from one-whole. Then it gets serious.
“Wait a minute, is one-whole a unit fraction?” and “Does this mean that ∞-1/∞ is really 1?”
These are the questions that get asked when your students are really thinking about the concept and they are really getting it. Pushing their understanding to the limits, joyfully, of their own accord.
To wrap things up, I asked them to think about fractions that were really close to one-half, but less than one-half and fractions that were really close to one-half, but greater than one-half. Now.This.Was.Fun! Shouts from around the room, each trying to get closer to the mark than the other... “Fifty-one hundredths!” “Twenty-five fifty-firsts!” “Twenty thirty-ninths!” “Ninety-nine two-hundredths” “Five hundred one - thousandths!”
I cannot wait until class time tomorrow morning. What uncharted territory will we explore? Only my students know. I look forward to the adventure.
Give your students room to play with concepts and the tools to do so, and they will dig deeper and learn more than any procedural instruction will provide them. Yeah, fractions are hard, but they’re really cool numbers too!
In his prologue to The Man Who Knew Infinity, Robert Kanigel writes,
"In a way, then, this is also a story about social and educational systems, and about how they matter, and how they can sometimes nurture talent and sometimes crush it. How many Ramanujans, his life begs us to ask, dwell in India today, unknown and unrecognized? And how many in America and Britain, locked away in racial or economic ghettos, scarcely aware of worlds outside their own?"
If that is not a call to arms, fellow educators and sociologists, to seek out and nurture young minds and spirits, I'm not sure what is. People often ask, "Why?" When I tell them I teach math and science to 9, 10, and 11 year olds. "When you could do so much more at upper school or university." Why? Because if no one is there to ignite their passion for the subject and its adventure, my colleagues of older students will have nothing to work with. My students' innocence and naïvety remind me of Ramanujan who came to Trinity with his inspiration, creativity and passion for numbers. Littlewood once said of Ramanujan, "Every natural number is a personal friend of his." My students come to me already mathematicians and scientists. What I bring to them is organization and communication skills while celebrating their awe and discovery.
I didn't start out sharing the quote to be quite so philosophical, really just wanted to put it out there, but it spoke to me and of the responsibility we have as educators to seek out and nurture those gifts of each and everyone of the students in our charge. And for us as "civilized, enlightened society" to ensure access to all minds and spirits to worlds they might otherwise not even know of.
Humbly submitted for your consideration and comment. mw
No, this is not new news, but with the Cicada fever taking over the Mid-Atlantic states of the. US, it is a great opportunity for the average person to gain a bit of an appreciation for the beauty of the mathematical world. My students are anxiously anticipating the emergence of Magicicada Brood II in the Bronx. We've been monitoring the soil temperature since returning from Spring Break, sampling from six locations in the school yard and calculating the mean temperature. Which, by the way, yesterday was a full 2°C above the cicada trigger of 17°C! My Science 4 group spent a period yesterrday surveying the tree trucks for nymphs and molts and the ground around them for emergence holes. Despite several false alarms, no positive signs yet.
Marcus du Sautoy uses the 17 yr Cicada as an example of the pervasiveness of prime numbers (and his favorite number) in Music of the Primes and Num8er My5teries. These are both regular resources that I use in my Math 5 Number Theory unit, so my 5th graders are excited to experience something they've already learned about, and it will be a wonderful first hand refence for my 5th graders next fall!
Recently, in response to #Swarmageddon, The New Yorker posted an article on the role of prime numbers in natural selection. Here's how I shared it with my students via our Edmodo STEM Forum: As we have often explored in class, and will continue to explore through this forum, Maths are EVERYWHERE! I always say that the numbers have always been there, humans have only found ways to decode some of their mysteries through Maths. Most exciting, we keep discovering new Maths, new ways of describing and understanding the universe in which we live. http://www.newyorker.com/online/blogs/elements/2013/05/why-cicadas-love-prime-numbers.html
I sometimes find myself surprised by my students' response to lessons or activities. This week in my 4th Grade Math class I have been experiencing one of those times. I began the week of reviewing the multiplication algorithm with the following prompt:
How many times does your heart beat?
...in an hour?...in a day?...in a month?...in a year?...in your life?
While I expected some interesting discussion about how the set up the problem, I did not expect the incredible level of discussion before doing a single calculation. And not by just the one or two über insightful students, but by the entire class. This simple prompt to initiate practice of a mechanical skill with real data has turned into a incredibly rich investigation. My students have engaged the school nurse, as well as outside experts via Twitter. After presenting the prompt above, my students launched into an animated half hour long discussion, considering a wide range of associated questions:
how many times does it beat in a minute
How do we know how fast it beats?
It matters if you're exercising or still.
It depends on what you're doing.
It might matter what you're eating.
If you're running it beats fast.
It depends how long you live.
The months have different numbers of days and there are leap years.
Does gravity matter?
Does your mass/weight matter?
Does our heart slow down while sleeping?
Does fatigue or dehydration effect it?
Does age matter?
The rate is always changing.
We need an estimate.
Does your height matter?
We need to get resting pulse, then exercising, get an average, something in middle.
We need a rate between the highs and lows.
Everyone will have a different answer.
Does respiration rate matter?
As the discussion started to wind down, I asked for suggestions of what to do. Again, the idea of findinsg a median resting pulse was suggested. Not by name, but in concept: "Some people will have a slower pulse and some will be faster. We need to figure out what the middle [pulse rate] will be." We arrived at a median resting pulse of 76 beats per minute.
For homework, I asked them, "How many times will your heart beat in 2013?" And that brings us to the title of this post. Take care of your hearts my friends, it is a busy muscle.
The investigation has continued to build. I will share more in my next post.
Arranged in no particular order, this is a list of some of the math(s) books on my shelf, piles actually, that I am currently reading, have read, or to be read. Ok, so its the pile within reach of my favorite chair. As time allows, I will create a more complete database and share it. These are the books that I read and reread as I am planning my lessons or simply want to escape into the world of math literature. There are fun problems, human stories, histories, theory... These are some of the books I routinely pull excerpts from to share with my math students to help the see the beauty of math and learn that math is more than arithmetic. They are the source of inspiration for challenging problems for my students, and me, to chew on. They are inspiration. I thought my followers might find something interesting here. Please use the comment box below to share some of the titles in your piles, or comment on how you make use of books like this with your students.
The Math Book From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Clifford A Pickover
After exploring the concepts of equivalent fractions and the fraction, decimal and percent forms of rational numbers, I asked my 5th grade math students to create a brief video that explained or demonstrated one of these concepts of their choosing.
The first thing we did, was take a look at a video (Ma & Pa Kettle Do Math: see below) that used humor to explore a math concept. In this case it was the concept of place value, percents, fractions, and division.
We discussed the clip, how humor was used to illustrate the key concept of place value. We discussed the use of a concise story, simple dialog, and the brevity of the piece. Also part of the discussion was video as a visual medium and the considerations necessary in translating a written script to video.
Students broke themselves into groups of two or three and were given the following outline of steps to get past the producer (me, the teacher):
Treatment & Script Writing
They first had to write a brief summary of the math concept being presented in their MathCast. Each team met with me to go over their concept to ensure that they had an understanding of it. Once that was approved, they were given permission to write a one-paragraph treatment of the video. After approval of the treatment, they set to work scripting out their production. Scripts were written in a collaborative fashion using GoogleDocs on iPads.
Filming and Editing with iMovie
After approval of scripts by the producer/teacher, teams created props, sets, etc. and began to rehearse. When each team felt they were ready, they filmed their MathCasts using the video camera on an iPad. Scenes were shot as individual clips. In some cases, multiple takes were filmed. In other cases, the MathCast was shot as one continuous clip. Using iMovie on the iPads, clips were assembled, transitions and still frames were added and final productions completed. Each MathCast was uploaded to a class Vimeo site and embedded in pages on the class Moodle site.
Screening & Evaluation
Students were notified by email when their video was posted and given an access code to view their MathCast. As a nightly assignment, all students were asked to view one MathCast and respond on the Moodle to the following questions:
What is the math concept presented in this MathChat?
What about the MathChat helped you understand the math concept being presented? Be specific.
What suggestions do you have that might make this MathCast better?
A class discussion also took place to evaluate the process as a means of math learning. Notes from that meeting are included below along with the guiding questions.
Project Timeline
Week 1: Production
Day 1: Introduction, Brainstorming, Treatments
Day 2: Treatment approval, Script writing
Day 3: Script writing, Script Approval, Start Filming
Day 4: Script Approval, Filming
Day 5: Filming, Editing
Week 2:
Days 1-4: Nightly homework, view and respond to one MathCast each night
Day 5: Class discussion evaluating process
Here is one of the finished productions:
Class Discussion Evaluating Process
How did the MathCast help you deepen your understanding of rational number concepts?
I got to listen to the ideas of the others in my group. Get their ideas and put them together with my own.
A different way to learn the concept. You get to learn how other people beside yourself say and understand it.
Having to perform it helps me understand it. Easier to get the understanding than just copying notes.
You have to make sure you really know it or you risk presenting false information. Makes your explanations better.
In our notes we don't always record all that we know and understand. This forces us to explain everything completely, as if to a younger student.
Helps me remember the information, rehearsal of script.
We think we know it, so thinking, "how can I explain this and have it make sense?"
Have to put the ideas into our own words, forces you to really think about it.
What did you like about the project?
The process of making a movie: creating characters, writing script, using iMovie...
Not just putting down notes. Not JUST math, have to learn to use the technology. Cool to learn how to link the math to the movie idea.
I can use the technology to say what I've learned. Don't really do that in other [academic] areas.
Like doing something creative in math.
Most fun thing I've ever done in math. Had to do a lot of math, but also got to include the arts.
Liked that there weren't so many boundaries. We just had to include the math.
Liked that it was all our own ideas.
Fun way to express what I know and creative.
Would like to do this in EVERY unit. I think it should be LAW.
It's cool to watch others' projects.
Why should we do MathCasts? How do they benefit your learning?
Learned a lot from other groups movies.
Makes math more fun.
Like freedom of choosing my topic.
Getting to work with others and hear their ideas. Learn how they thought about the topic.
Collaborative group process. Have to work together. Have to learn how to compromise.
Helps me think about more ways to understand a topic by listening to others' ideas.
Hearing the ideas of others and putting them together with my own. I talked.
More freedom, being able to use humor and entertainment, but still making sure we communicated some math.
Closing Thoughts...
I don't know what more I can say that my students didn't say above to illustrate the power of this project. Yes, I made a deliberate decision not to go on to a new topic and spend another week on these concepts. However, this was incredibly powerful and worth every class minute. Whatever items I don't get to this year are insignificant in comparison to how much they learned in creating these MathCasts and how it deepened their understanding of basic rational number concepts. Using the iPads and the various apps that students employed in their productions was a perfect vehicle for the project. It could of course be done with nothing more than pencil, paper, and a video camera, but having everything needed in one convenient tool helped make it more accessible to the students. It encouraged them to experiment. And, while they were learning, they were having a lot of fun! I encourage any of my followers to give this project a try. Feel free to send me an email if you have any questions about the specifics. Special thanks to my curriculum supervisor for sharing the MathCast idea, and thanks to our two wonderful instruction technologies specialists for helping me think through the technical details of the project.
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