Sunday, November 6, 2011

Calendar Curiosities

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.

Tuesday, August 23, 2011

Lego Racial Diversity

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.

Jesus Diaz wrote in his interview with Lego in June 2008

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.

Dan Burke of the DoG Street Journal wrote of Lego MiniFigure lack of racial diversity in January 2005.

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.

Thursday, March 24, 2011

When are we gonna use this?

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.

Wednesday, March 16, 2011

Fractions & Probability

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?

Monday, February 28, 2011


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."

Wednesday, February 9, 2011

Female Math Stereotypes

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. 

Believing Stereotype Undermines Girls' Math Performance: Elementary School Women Teachers Transfer Their Fear of Doing Math to Girls, Study Finds

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.

Sunday, January 2, 2011

Happy New Year

2011 is a prime number year.
What's more, it's the sum of eleven consecutive primes!