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?
