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.
November 2011 has had some interesting dates. There are two dates this month that are "all ones" days: The first: 11/1/11 and the eleventh: 11/11/11. The second was an interesting palindrome: 11/02/2011.
I am taking a bit of a departure from my regular math posts to share a conversation that came up in our family recently...
My ten year old son is Lego fanatic. Building Lego models and making his own creations is one of his favorite pastimes. He particularly enjoys the Lego Mini-Figures and has an extensive collection of them. In fact, the presence or absence of Mini-Figures in a model kit is part of how he evaluates if he wants to purchase a particular model or not. Last night, after playing with his Mini-Figures, he came to me to ask, "Why is Mace Windu the only black Mini-Figure? There are all kinds of yellow-heads, and even several white ones, but he's the only black one." This would be an important question, even if my son weren't African-American.
JD: "Why are there no black minifigs?"
Lego: "When the minifigure was first introduced 30 years ago, it was given the iconic yellow skin tone to reflect the non-specific and transcendental quality of a child's imagination. In 2002, as more licensed properties were added to the assortment, the decision was made to introduce ethnic and skin tones more in keeping with the actual characters and personalities who were being replicated. This included the introduction of black minifigures. However, these ethnic minifigures are only used in our licensed sets, all Lego playthemes continue to use the generic yellow face."
I would challenge the assumption that Lego's generic yellow face somehow connects to "the non-specific and transcendental quality of a child's imagination." As I watch my son play, I think his imagination actually is hampered by the lack of ethnic diversity. He has compensated for the lack of gender diversity, by collecting heads and hair pieces so that he can create additional female characters. He can create Nicole Stott, but he cannot create Alvin Drew of the STS-133 Space Shuttle Discovery crew. He can create Jim, Willy, and Casey for his Mission Impossible team, but not Barney (the black engineer and the resident tech expert on the team --not the purple dinosaur). And what about being able to include Barak Obama in his Presidential motorcade?
As an African-American boy, he identifies with these cool, smart, black men. "I can dress MiniFigures a particular way and pretend they are specific black people, but it's not the same." "You can only buy black, or white, faces in sets. And they're only movie characters, not real people. You can't buy non-yellow heads to create your own people. I can't make black firemen or policemen, a black man driving a boat, or piloting a plane or helicopter. I can't make the kids black."
I asked him about the racial-neutral argument for yellow MiniFigure skin color. His response was "Yeah, it's sort of neutral. I don't want people to be offended by someone [of a particular race] looking weird as a MiniFigure, but I'd like more brown-skinned heads to play with." As an adult, I see the yellow, "racial-neutral" argument as another color-blind argument. Humans are not color-blind. Children won't care about skin-color until they are taught to feel one way or another about it. Yet they most definitely notice it. "She looks like me." "Her skin is darker/lighter than mine." In their creative play they want to recreate the world around them and explore it.
In his article, he spoke of the perpetuation of stereotypes and vilification of a people: "When Lego does make minority figures, these minorities are often stereotyped. The Wild West sets of a few years ago contained Native Americans depicted as headdress-wearing, hair-braiding, Union-Soldier-hating primitives... With these play sets, Lego promoted the United States’ persecution of the Native American Tribes throughout history."
I want my children to be able to see themselves reflected in society in positive ways. In their play, on television, in magazines, in advertising... I want them, as Martin Luther King, Jr. so beautifully said, "to be judged by the content of their character and not the color of their skin." We live in a fast-paced, ever evolving world. Our children have connections to the world far beyond their family or their classroom. They are engaged in a diverse, dynamic, global culture. An argument for "race-neutral yellow" may have made sense in the days of failed attempts to create a color-blind society that ignored the unique identity of individuals. Though I would have questioned it then as well, the conversation wasn't being had then. But we are having it today, and we can no longer ignore the beautiful palate of skin color that is humanity. All colors need to be reflected and celebrated. I challenge Lego to step up to the challenge and provide more tools for our children so they can explore, imagine, and create the world they want to be a part of without limits.
I welcome the reader's comments. I only ask that the conversation remain respectful and appropriate for my ten year old son to follow. After all this is his story and his request for us to discuss these issues.
I fear that far too often we have math instruction turned around backwards. We teach concepts and skills as an end unto themselves, when really these are tools to solve problems. They were human inventions created to understand the world around us and to be able to articulate that understanding to others. When math concepts are presented within the context of a problem, students are much more eager to engage in deep thinking. Furthermore, they will invent the math necessary to solve the problem if given the time and resources to do so.
A few weeks ago, as the Space Shuttle Endeavour was being prepared to move from the Vehicle Assembly Building (VAB) to Launch Pad 39A, I couldn't help but present an intriguing problem to my 5th grade math students: How fast does the crawler move as it transports the shuttle "stack" to the launch pad?
This particular math group is in a rational numbers unit. In this unit, rational numbers, in all their forms, are explored as different ways to notate the same ideas, whether using any non-zero denominator (fractions), denominators that are powers of ten (decimals), or denominators of one hundred (percents). They are comfortable with these concepts and understand that, operationally, if they can work with the number in one form, they have to be able to work with it in another form. When this particular problem was presented, they had not yet encountered divisors that were not whole numbers, though they had explored decimals in the quotient and have moved from expressing remainders as R# to R/div to a decimal expansion. They have also worked with non-terminating decimal expansions and know how to use a vinculum to express these ratios.
The problem was not presented as, ok, let's divide with decimals. Rather, it came in the midst of our following the final three flights of the NASA shuttle program. As Endeavour was moved from the VAB to launch pad, the question came from them.
Students: "How far is the launch pad?"
MW: "A few miles."
Students: "How fast does the crawler move?"
MW: "Not very fast."
Students: "Why?"
MW: "It's extremely heavy, and vibrations could tear things apart."
Students: "So HOW fast?"
MW: "Let's see if we can figure it out?"
We then set out to collect the data. From a couple of NASA Tweets, we found that Endeavour started its rollout at 7:56 p.m. and arrived at the launch pad at 3:49 a.m. We also found that it was 3.4 miles between the VAB and pad 39A. We talked about the fact that speed was in miles per hour, meters per second, kilometers per hour, feet per second, etc. In all cases a unit of distance over a unit of time. Aha! A ratio! They immediately recognized they needed to find how much time had elapsed. They also recognized that the time was not in a decimal form. Hmmm. The first subtraction was start time minus end time. Well that doesn't make sense... Ah, there's that pesky midnight in the middle of it all. They decided to use what we term "subtraction by adding up" to determine the elapsed time. 7:56 to 8:00, 8:00 to 12:00, 12:00 to 3:00, 3:00 to 3:49. 0:04 + 4:00 + 3:00 + 0:49. Now the task of turning a hr:min into a decimal. Equivalent fractions! Eureka! n/60 = x/1000.
Now we have the time as a decimal and the distance as a decimal. Set up the ratio 3.4/7.883. Ew! Decimal numerators and denominators. Equivalent fractions to rescue again! The students already know that the decimal moves when multiplied or divided by powers of ten. So by multiplying both numerator and denominator by 1000, they have two whole numbers in their ratio. Brilliant! They are also familiar with n/d = n ÷ d. With a couple of reminders of these basics, they quickly set up the problem and find that the crawler moves at a rate of approximately...
I'll leave it for you to finish up.
This was an exciting math period where my students identified the math that they needed to solve the problem. Some of which they had at their ready disposal, some of which needed to be discovered. At the end of it, they now had a model for how to work with these numbers that could be translated to a wide variety of situations. Because they now OWN the methods, they are able to hold onto the concepts and reconstruct the math as needed.
This comes by way of @NCTM daily problem tweets. It is a great investigation for probability. By constructing outcome grids to determine the all the possible outcomes vs all the desirable outcomes for each of the situations, students can then compare fractions using benchmarks and concrete materials or drawings to determine which is more probable.
When three 6-sided fair dice are rolled, which type of sum is more likely: a multiple of 3; one more than a multiple of 3; or two more than a multiple of 3?
Fractions are among the most abstract concepts presented in elementary school math. We need to ensure that a sound conceptual base is established. This means taking time and providing numerous, concrete experiences for children to construct an understanding of these concepts before every formalizing them.
Here's an example of work adding fractions in a 5th grade classroom.
Students were asked to find the sum of the fractions 1/4 + 1/2 + 2/8. Using materials, they were able to find that these three fractions had a sum of 1 whole.
Students also noticed that they could trade in the fourth in for two eighths and the half in for four eighths, leaving them with 8/8, another form of 1 whole. Likewise, they could trade the eighths and half in for fourths, or trade the fourth and eighths in to make a half. Thus proving that it was equivalent to 2/2, 4/4, or 8/8... all equivalents of 1.
Without ever thinking about "common denominators" the students were dealing with the concept.
In a more challenging problem, they were asked to think about the sum of 4/5, 7/8, and 2/3. Without worrying about the exact sum, they reasoned that 2 < sum < 3. From earlier explorations, they already know that when a fraction was missing a unit fraction, it was as close to 1 as possible. Thus, three fractions close to one added together, must get close to three. It would have to be close to two because each of the fractions was at least 1/2.
Once assembled, children were easily able to see that these three fractions had a sum of about 2 1/3.
Powerful reasoning, with no need for "common denominators."
Yes, they will be exposed to the algorithm for adding fractions, but they will first construct it through many more investigations such as these. In the last example above, the student noted, "I could find the answer by thinking about 3 - 1/3 - 1/5 - 1/8. That's like taking away about 2/3 from three."
This article is more than a year old, nothing new, but here is some research to back up what many of us who teach math have long known. Teaching math is different than teaching other subjects. One has to really love it and have a deep understanding and appreciation for the subject in order to be effective in the classroom.
ScienceDaily (Jan. 26, 2010) — Female elementary school teachers who are anxious about math pass on to female students the stereotype that boys, not girls, are good at math. Girls who endorse this belief then do worse at math, research at the University of Chicago shows. <more...>
So I'm sitting here at Giants Stadium, waiting for my Redskins to take the field. Thinking about the numbers in football. Like all sports it's full of statistics.
Football scores are quite complex. Six points for touch down. One point for the extra point after a touch down. Or, you could run the ball in after a touchdown for two points. Then of course there is the three points for a field goal. Finally, the saftey for two points. 6, 3, 2, 2, 1.
So I'm thinking, what kinds of numbers can the final score be? Could it be prime? What about square or cubic? What would the score look like written in Binary?
Well, I suspected the Redskins would lose. They have a lot of players out with injuries in key places and the quarterback has been struggling. That, with the fact that the Giants have been doing very well lately.
So the final score was 7-31. Both prime numbers. How many different ways could these scores have been achieved using the posibilities given above?
So boil it all down to one and zero and you have the simplest possible system. Two digits. Yes or no.
The Indian scholar Pingala is credited with the first known use of a binary system. He used a system of dashes, a short dash and a long dash to represent 1 and 0. The Chinese used hexagrams to represent numbers. These are squares made of a stack of 6 lines that are either solid or broken in the middle to represent 1 and 0. Then finally in 1703, Leibnitz uses the 1 and 0 digits as we know them and formalized the binary system as a practical device for computation.
Well, it's the Thanksgiving Break from school and I have time to ponder some of those many ideas that have been rattling around in the back of my brain. Lately I've been thinking a lot about zero. We've been talking about how powerful this digit is in my math classes. This idea of a place-holder is so powerful. It allows us to create numbers of unimaginable magnitude.
"How can something be nothing?" asked the ancient Greeks. But those brave Indian mathematicians about 2200-2500 years ago started using a symbol to represent this idea of nothing. Not just nothing in terms of counting, but nothing of a particular place value in a numeral! This was a huge improvement over the Babylonian's empty space between digits. By the 9th Century AD, the Indian mathematicians were making use of zero in calculations, using it like any other number.
Of course the Maya had them all beat, using a symbol for zero as a placeholder in their base 20 system. However, this system was a parallel evolution and didn't seem to influence the systems we use today.
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