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Monthly Archives: January 2013

Gender Equity… Dance vs. STEM

In my spare time, okay, not that I have spare time, but in the couple of hours a week that I carve out for being responsible for my own health, I take clogging lessons.  I got interested in it while helping my daughters practice, and when their studio offered an adult class, I jumped on it and haven’t looked back.  I like making exercise and rhythm part of my life.  I’m well aware of how beneficial the arts are to a well-rounded education, and I feel it’s made me a more creative, thoughtful, energetic, and resourceful person.

The gender gap in dance is one that is obvious, and glaring, and hasn’t budged for decades.  My adult dance class is one of the few with a male dancer in it.  Most classes have no boys at all.  The hip-hop dance classes are the least unbalanced, but that means there will be two or three boys in the class with seven or so girls.

For me, it leaves a question in the air.  Why don’t we care?

The gender gap in STEM education and tech jobs spurs a call to action.  We create girls’ computer camps.  We create science field trips just for girls.  We analyze our data and fret over the unfulfilled potential of our girls.  Why is there no similar call to action to get more boys involved in dance?

I did just a tiny bit of research today online to find out about the gender gap in dance.  The few tidbits I found out are that the gender gap in dance is wide and hasn’t budged in decades, and that interestingly, and these pieces of info were fascinating, the wage gap in the performing arts favors men, and boys tend to get more attention and are called on more often in dance class.

This contrasts to the STEM gender gap, in which the boys outnumber the girls, especially in computer science and physics, but are still paid more and get more attention and are called on more often.  Or at least, that’s the way it used to be.  Is it still true?

We don’t apply the same level of urgency to getting boys into the arts as we do to getting girls into STEM.  Presumably, it’s because the job market in the arts isn’t perceived as growing as quickly or having as much earning potential.  Does that mean we’re right to apply no energy into getting boys to enjoy dance?

As a point of reflection, I’ve done a project a couple of times when I felt my classes (and I) were in need of a movement break.  Instead of doing math warmups, we planned a flash mob.  We’d find a funny dance on youtube and practice it for ten minutes a day, and then plan a secret day and time to play the music and have the math classes coalesce and just start dancing.  The students LOVED it – boys and girls alike.  It helped create community, a sense of purpose, got us some much-needed physical activity, and made us laugh.  I have no data to inform whether this is a good educational practice or not – so is it?  How important is a dance break for a student’s educational well-being?

I welcome your thoughts!

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Computer Science and Math

Our school has a 40-minute intervention/enrichment period at the end of every day, and students can take a different class each quarter.  This quarter, we made a decision for me to teach a computer science enrichment.  We invited students to join it, with the intent of targeting certain needs.  We wanted to target students who had potential to grow in math if they learned it in a little different way.  The invitation list consisted of some partially-proficient math students who might make a big leap if they learned math in a non-traditional class, and some gifted students who might enjoy an extension to what they knew.  We created the class to be balanced between girls and boys.

I’m enjoying writing a curriculum that gives kids an intro to programming but also weaves in math topics. I am making things up a little as I go, based on what kids want to learn next.  Here is what we have done so far, and I’ll include the links to the demo programs I made.

1)  Introduction to drawing.  Students learned to create shapes on the screen, and how to plot objects, thinking of the screen as a coordinate plane. We also explored RGB color – how colors on a screen are a combination of Red, Green, and Blue lights, and by turning on the lights in different proportions, you can make any color of the rainbow.
http://www.khanacademy.org/cs/flower/1285404246

2)  Variables and Expressions.  Students learned how to “anchor” a small drawing at certain x,y coordinates, and how to create the rest of the shape around those anchor points using variables.  We discussed the relationship between these expressions and algebra expressions, and how the computer uses substitution to replace a variable with a value.
http://www.khanacademy.org/cs/here-kitty/1298014903

3) Binary Numbers.  I really appreciated place value when I learned and understood the world of binary, so we spent a day and a half working on building that understanding.  I mainly used worksheets from Computer Science Unplugged.  The worksheet on this webpage is great and very kid-friendly.
http://csunplugged.org/binary-numbers
I demonstrated the conversions with a Javascript program, although Javascript doesn’t have simple ways of representing binary numbers.  I haven’t used this in class yet, because I wanted to get into data types and functions before getting here.
http://www.khanacademy.org/cs/printbinary/1289019650

4)  Functions.  In elementary school, students get used to a representation of a function as an “in/out” machine.  In middle school, we extend that to equations that include variables, and they are related to these old “in/out” rules of the past.  In computer science, a function is also an in/out machine.  It takes parameters as inputs, does something to them, and outputs something new.  I showed the students how they could make their little character from Lesson 2 appear on the screen multiple times by wrapping it in a function.  The computer again uses substitution to use the input parameters.  For advanced students, they learned how to make their drawing appear randomly by substituting a random number instead of a fixed one.
http://www.khanacademy.org/cs/multiple-kitties/1298760197

Next, we will work on creating algebra expressions with our variables to make our drawings animate.  We’ll also learn about “if” statements and how they can be used to make decisions.

Loads of fun!  I’m loving it.

 

 

Levels of Comprehension, and Math!

I had a lightbulb during class this week that I wanted to share.  In context, we’re in the middle of a unit on rates and ratios, but our math team is simultaneously working with the kids on creating data displays and writeups for a big science project.  I decided to extend our rates unit into graph/table/equation relationships, because it would dovetail nicely into the science discussion.

The students solved a rate problem involving calculator prices, and I tasked them with creating a line graph.  Many students produced something like this.

Pretty straightforward line graph.

Pretty straightforward line graph.

Some students finished early while others were still working, and the work extended over a second day.  Due to procrastination on my part, I ended up at my desk five minutes before students were to come in, asking myself, “okay, what do I have the fast workers do while the other kids are finishing their graphs??”  So I decided to have everyone write their own questions that could be answered from their graphs.  Remembering a little from staff meetings past, I asked kids to write some Level 1, Level 2, and Level 3 questions.  They shared their questions in whole-class discussion.

Level 1:  These questions could be answered right from the graph. The answers were literally right there.  They wrote questions such as:
– How much do 20 graphing calculators cost?
– How much is 1 fraction calculator?
– How many scientific calculators could you buy for $80?
– What is on the x axis?

Level 2:  These questions could be answered with a little problem-solving if you used the graph to start with.  They definitely had a right answer, but they weren’t right in front of you as the Level 1 questions were.  They had to be inferred.  Students wrote:
– How much would 100 graphing calculators cost?
– What’s the difference in price between 20 graphing calculators and 20 fraction calculators?
– How much would 10 graphing calculators cost in pesos?
– How much would you spend on 10 fraction and 5 scientific calculators?

Level 3:  These questions didn’t necessarily have a right answer.  They are opinion or judgment questions that make real-world connections with the graph, or they extend the thinking beyond the graph.  Students had a lot of fun with these and came up with some clever and off-the-wall questions.
– Who is my audience for this graph?
– What kind of features do graphing calculators have that make them so expensive?
– Why isn’t there a calculator that does all three functions?
– Who would buy 20 graphing calculators?
– What happens if you buy half a calculator?
– Do infinity graphing calculators cost more than infinity fraction calculators?
– Why don’t you go to the dollar store?
– Why not just use the app on your phone?
– Can I have a discount if I buy more than 20 calculators?
– If we save money by getting scientific calculators instead of graphing calculators, can we use the saved money for donuts?

I considered how this would blend into their science report task ahead of them, and I like where it’s headed.

In a science fair report, you generally have your data table / graph, a “results” writeup, and a “conclusions” writeup.  I know now how I am going to structure the kids’ writing for “results” and “conclusions”.

“Results”:  This section should be Level 1 and some Level 2 observations about their science data.  They should start with specific, low-level observations, and then move into general patterns they notice by problem-solving with their graph and data table.

“Conclusions”:  In this section, you start with Level 2 inferences about how well your data supports your hypothesis, and you move into Level 3 references about what your data means in a real-life context, what your sources of error could be, and what you should study in the future.

That five-minute “oh shoot what do I next” moment ended up giving me the gift of some pretty good structure for the kids’ science lab reports.  After they create their graphs, they can craft some statements in each level, and then weave them into a narrative.  Hooray!

 
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Posted by on January 19, 2013 in Lesson Reflections

 

Long Division and Decimals

Fraction-decimal-percent conversions are a fundamental building block for seventh-grade math, and after I browsed the Common Core standards and noticed this in 7.NS.2.d, I decided to do things a little differently this year.

d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

Don’t judge, but I didn’t fully understand why long division worked until well into my adult years.  I suspected my students didn’t either.  So, I created this lesson with these targets in mind – students will be able to:
– Explain why long division results in a decimal equivalent of a fraction.
– Predict when a fraction will result in a repeating decimal or terminating decimal.

The lesson is a workshop model, with a mini-lesson, cooperative investigation, catch-and-release, and summary.

I have a small set of base-ten blocks, but for this lesson, I found some terrific virtual base-ten blocks here. I liked them because with a click on the corner, the bigger blocks can be broken into ten smaller blocks.
http://www.glencoe.com/sites/common_assets/mathematics/ebook_assets/vmf/VMF-Interface.html

I asked the students if they liked to understand why things work, or if they just like to know how to do something and don’t care how it works.  The poll results were interestingly split about fifty-fifty (why??) – but I said “for those of you that like to know why things work, this lesson is for you.”

I put a base-ten block on my head and told them “You need to know something about me.  I have a super power.  My secret identity is Decimal Woman.  My super power is that I can break things into ten equal pieces.  LOOK! It’s our special guest today, PEYTON MANNING!”  And I switched the bigger block out for ten flats. “Oh, Sorry you missed it. See how that works?  Ten equal pieces, each a tenth of the original.  Unfortunately, I’m a one-trick pony.  I can break anything, but I can ONLY break things into ten equal pieces.  It’s all I do.  Cakes, pizzas, enemies… ten equal pieces only.”

I put a place-value chart on the board and reviewed decimal place values.  Thousands, hundreds, tens, ones.  Each was explained in words and exponential form.  I moved to the tenths (10 to the negative first power), hundredths (10 to the negative second), thousandths (10 to the negative third), and so on down.  I told them my super power is useful in a base-ten world, where every place value is ten times as small as the one next to it.  Every time I break something into ten pieces, I can think of it one place value down.  I did some quick random questioning:  what’s a tenth of one cake?  A tenth of a tenth of a brownie?  A tenth of a hundredth of my enemy?

I explained that long division is a manifestation of my power, and I used the virtual manipulatives to put a cake on the board.  I picked four students.. let’s call them, Emma, Jackson, Caleb, and Grace… and offered to split the cake evenly between them.  How much cake does each person get?

One cake - would love to cut it into fourths!  However, I can only cut it into tenths.

One cake – would love to cut it into fourths! However, I can only cut it into tenths.

The students were excited. “One fourth!  One fourth!”  I stopped them. “I would love to give them one fourth of the cake, but I can’t!  I can only break things into ten equal pieces.  So I’m going to cut the cake, but cut it into tenths.  How many tenths does each person get?” The students thought about it and told me each person gets two tenths, but there would be two left over.  I divided up the tenths, put the results in the division problem above, and had the two tenths left over.

base_ten_fraction2

Everyone has two tenths, for a total of 8 that have been given away. But there are two tenths left over!

The students were excited again.  “Cut them in half!”
I stopped them again.  “I can’t cut them in half.  Remember my super power?   I can only cut them into…”
“Ten pieces!  Then give everyone five!”  They started yelling out the answer.  So I broke the tenths into twenty hundredths and gave each student five of them, adding the results to the long division problem.

The cake is successfully divided into fourths, simply by breaking it into tenths and then hundredths and dividing them up.

The cake is successfully divided into fourths, simply by breaking it into tenths and then hundredths and dividing them up.

I pointed out that even though I can only break the cake into ten pieces each time, I could successfully divide up the cake evenly with my method.  Everyone gets two tenths, and I broke the leftovers into hundredths, which everyone got five of.  This is the heart of how long division works.  You divide up the pieces and then break the leftovers into smaller pieces.. ten times smaller each time.

We did the same experiment with a cake and only three students – poor Grace would not get any cake this time.  I had to use my super power to break a cake into thirds among Emma, Jackson, and Caleb, using the same method.  I could only break it into ten equal pieces every time.  We started with the tenths of a cake.

One cake.. attempting to divide into thirds using Decimal Woman's power of breaking pieces into tenths.

One cake.. attempting to divide into thirds using Decimal Woman’s power of breaking pieces into tenths.

We went through the same procedure as before. Everyone got three tenths, and there was one tenth left over.  I broke the tenth into tenths, which the students identified as hundredths.  Everyone got three of those, and there was one left over.  Even though everyone could see where the pattern was going and some students were yelling it out, I divided the leftover hundredth into thousandths and divided them up… each student got three of them, with one left over.  At this point, the base-ten blocks can’t be broken anymore, but the students saw where the pattern was going.

Three tenths and three hundredths and three thousandths... one left over every time!

Three tenths and three hundredths and three thousandths… one left over every time!

This left a question hanging in the air, so Decimal Woman asked it.  Can you predict when a decimal will repeat and when it will terminate?  What kinds of numbers produce each kind of decimal?  I introduced the word “Conjecture” as something between an educated guess and proven statement.  It’s a well-reasoned statement that has patterns and evidence to back it up.  Their group’s product would be a conjecture on what makes a repeating decimal vs. a terminating decimal.

I passed out the group work, which I’ll attach.  The students were asked to use long division to find decimal equivalents of a lot of fractions, from halves all the way through twelfths.  I told them to divide up the work and look for interesting patterns, which would help them create their conjecture.

Worksheet:patterns_of_decimals

We had been practicing long division for a day or two, but many students were rusty on converting remainders into decimal equivalents.  Most students picked this up fairly quickly and the teaching was minimal.  The classroom management was not as big a challenge as I expected, because most students found the task really engaging, to my surprise!  They got excited when they found an interesting pattern, and when they discovered that a decimal repeated or ended.  A few students did not pull their fair share of work, but it was a better cooperative activity than usual, all things considered.  After I let them churn through decimals for 15-20 minutes, I did a “catch” and had them report out some of the things they found.  They reported:

– Eighths are interesting because you always end up with forty of something left over, which divides evenly into five – so all the eighths terminate with a 5.
– Tenths and fifths simply have one terminating decimal.
– Sixths all repeat except for three-sixths.
– Most twelfths start with a sequence and then repeat a single digit.
– Elevenths all have a two-digit repeating sequence, and that sequence is a multiple of 9.
– Ninths are cool because the numerator is the repeating decimal.  This led to a great discussion on whether nine-ninths was 0.999999… or simply 1.
– Sevenths have a six-digit repeating sequence, and the sequence starts from a different number each time.
– If you can reduce a fraction to one of the terminating decimals, then it terminates.

Students determined that halves, fourths, fifths, eighths, and tenths all terminated.  Thirds, sixths, sevenths, ninths, elevenths, and twelfths mostly repeated.  The students were very confused about the general trend that would make a decimal repeat vs. terminate.  It didn’t seem to be related to odds vs. evens, or primes.  I asked them for factors of ten, and the told me the factors were 1, 2, 5, and 10.  I pointed out to them that it was interesting that if I broke something into tenths, I could only divide it up evenly if I could then split it up into groups of 10, 2, or 5 – nothing else – and I prodded them to use that idea in coming up with their conjecture.

I gave them a few minutes more to finalize their conjectures and then we finished with a summary and a round-robin of what they learned.

I know not all students achieved the second objective, and could not immediately tell me without calculating if they had a terminating or repeating decimal.  However, I do think they are well on their way to being able to compute decimal equivalents easily and explaining why the long division method works – and they were engaged and excited with what they did.  I was really pleased with the lesson!

 
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Posted by on January 12, 2013 in Lesson Reflections