Talking about Black and White rather than X and Y

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I was out of my classroom yesterday, attending an AP Stats workshop with my coworker and friend, Dianna Hazelton.  Upon my return, per the usual, I learned that students struggled with the assignment I left for them.  Naturally, it would seem that the most important task to be completed today was to address their issues with yesterday’s work.  But something much more important came up:  a discussion on racism and sexism.

The natural opportunities to discuss race in a mathematics classroom in rural Minnesota are not numerous.   I usually need to carefully weave them into the topics and diligently ensure that equity is valued when a student brings up situations of racism.   The minority population in our district is not high but being the only student of color in a classroom is challenging for them.  These students can’t be expected to just blend with their white classmates when their needs aren’t being addressed.

Somehow today, instead of practicing line graphing on whiteboards, we discussed race and gender when a student expressed her discomfort while attending a concert in Minneapolis.  Her comments were respectful, but her concerns legitimate.  As a teenage girl, when at social events in the city, she and her friends feel vulnerable and sometimes threatened by the sexual advances of men.  She started off pointing out black men specifically, but the conversation progressed to a point where she acknowledged that  first, her isolated experience shouldn’t shape a stereotype about all black men and secondly, white men engage in this behavior as well causing the same discomfort for her.  She quickly realized that her assessment of dealing with harassment shouldn’t be examined through the lens of race.

Rafranz Davis, a woman whose fearless, relentless advocacy of kids I highly admire and respect, summed this up perfectly on her blog:

Students carry unique perspectives about their experiences and until these issues, along with the countless others unaddressed, are met head on through discussion and action, these tensions and perspectives will never change.

I was very proud of this student  for acknowledging her initial prejudice, and as a result, we were able to have an equally productive conversation about gender as well.  And something I didn’t expect happened:  the boys just listened.  They just listened to the girls talk about cat calls and being whistled at.  “Just come and say hello, my name is so-and-so,” one girl said, “that’s much more of a turn on than being harassed.”  And at the end of the class period, one of the boys went up to that girl and said, “hi my name is…”  Bingo.

These conversations are difficult, but when a student is willing to admit their prejudice, the teacher doesn’t only have an opportunity, but a duty to help foster positive change.  Graphing 3x + y = 10 can wait until tomorrow.  The real problem of the day, and every day, is that these kids come to our schools for 7, 8, 9 hours a day and we spend such a small percentage of that time listening to their voices and giving value to who they are inside.

 

 

A Conceptual Victory

College Algebra, reviewing the graphs of polynomial functions.  Each student has a whiteboard.  We started with y=(x+2)(x-3) for simplicity.

Me:  What do we know about this graph?

Student(s): It has x-intercepts at -2 and 3 (or something along those lines)

Me:  What else do we know?

Student(s):  It’s a parabola (or some version of that)

Me:  What else do we know?

Student(s): [Crickets] (or owls or frogs or some other creature that makes noises when all else is silent.)

Me:  Does this parabola open up or down?

Student(s): Up. Down. no Up. no Down.

At this point I’m shocked that they do not remember the one polynomial coefficient that they all nail down in algebra 2:  The a value.  But I shouldn’t have been.  Rather than asking “What will the sign on the x^2 term be?” I decided to approach it differently to see if I could garner some conceptual understanding.

Me:  If x is a really big positive number, like a million, what kind of number will we get for y?

Student(s):  A really big number.

Me:  Similarly, if x is a really big negative number, like negative a million, what kind of number will we get for y?

Student(s): [After much thought and group deliberation] A really big positive number…OH, then it opens up.

This wasn’t a huge victory, but it was satisfying.  Because not a single student mentioned an a value even if they were thinking it.  Additionally, when we moved to cubic functions like y=(x-2)(x-3)(x+4), they used the idea of substituting really big negative and positive numbers for x to determine which way the graph was trending in each direction.  We were then able to have a nice discussion about why a graph like y=(8.5-2x)(11-2x)(x) looks similar to y=(x-2)(x-3)(x+4) when the equations have so many differences.

When students learn a procedure, it’s very difficult for them to deviate from the steps in order to solidify their conceptual knowledge.  I’m very glad that on this Friday, their forgetfulness of the “steps” allowed us to have a nice discussion.

The Corn Sandbox

For an entire year, I’ve been anticipating our family’s return to Stade’s Shades of Autumn Festival.  My excitement has been building for one reason:  To estimate the amount of corn contained in their corn play area (or the Corn Sandbox as my 4 year-old has named it).

We went last Friday, armed with a measuring tape and a measuring cup.  The sign seemed to give away the answer of 800 bushels, but I wasn’t satisfied given that their was no mathematics to back up their claim.  We needed to attend to precision. Here are some of the photos I took:

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My favorite is the bottom photo, my parents counting kernels of corn in 1 cup. (There are 579, by the way)

I love when I present a problem to my class, and it takes longer than I anticipate for them to solve. There was supposed to be time for solving inequalities for the group that worked with this, but that will just have to wait until tomorrow.  I’m sure they were crushed.

The essential question we wanted to answer was:  How many kernels of corn are there in this corn sandbox?

Initial estimates were very low.  I let them revise after I revealed that 1 cup contained 579 kernels.

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One approach:  Use the number of bushels to calculate kernels.

After looking up on Wolfram Alpha that 1 bushel = 9.31 gal, we determined that a reasonable calculation of the number of kernels, based on our 1 cup count, was 69,000,000.

Second Approach:  Use the volume of the enclosure to calculate kernels.

This was a little trickier given the irregular shape of the sandbox.  Numerous calculations and conversions later, we arrived at 81,667,931 kernels of corn.

We were uncomfortable with the over 12 million kernel discrepancy between our two methods.  It remains unclear which is more accurate given the fact that one includes actual measurements and assumptions and the other is provided by the farm.  Perhaps Stade’s Farm should expect a call from Mrs. Schmidt’s 2nd hour Algebra 2 class in the near future to clear this up.

Spiders Everywhere!

Steven Leinwand has a huge influence on how I approach a math lesson.  In my experience, one of the easiest ways math can be extrapolated from almost any task is by asking the questions:  How big? How far? and How Much?

This weekend, I came across this picture on social media, posted by David Roberts:

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Of course the question I asked first was “How big is that tarp!?

Luckily, David was willing to make an estimate and allowed me to share his reasoning:

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I thought it might be interesting to present the photo to my first hour and see what questions they would ask about the photo.  Of course, the surface area of the tarp was on their list.

The original article including the photo added more depth to their questions.  As it turns out, the house is being fumigated after a spider infestation.  It seems as though their curiosity surrounded more the spiders than the tarp and legitimately so.  (The article estimates the house was infested with approximately 5,000 spiders)

I was pleased that my students used visual cues in the photo to make their estimates including the average height of a story of a house, the approximate height of the man in the photo, and the size of the window.  Luckily, we found another photo that gave us a better understanding of how much tarp was needed for the other side of the house.

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I’m glad this man’s golf game was not disturbed by a spider problem in the distance.  (sarcasm)

Anyway, after making some calculations, my industrial-minded first hour realized that this type of tarp must have a somewhat standard size.  After doing some Google searching and actually calling one of the companies that manufactures these behemoth plastic coverings (authentic!  Yes!), they decided that there were four 84′ x 25′ tarps covering the house and the space surrounding it. (We were able to have a nice conversation about how multiplying each dimension by 4 was different that multiplying the area of the tarp by 4.)

In the end, their estimate was 8,500 square feet, approximately double of what David had estimated.  We then critiqued David’s argument and decided that based on the picture only, his calculations were reasonable.  Because we were able to dig for more information, my class believes that their estimate may be a tad more accurate.  Thanks, David for sparking our curiosity this morning.

 

 

Procedure in the Driver’s Seat

I’m fired up today.  I’m going to quasi-vent on my blog, and hopefully I will not offend too many people in the process.  You’ve been warned.

When I walk into another teacher’s room and all of the students are silently working individually, I get sad.  Forget the monotonous drudgery of textbook procedural homework.  When students don’t talk to one another about mathematics, they most likely are not experiencing math as richly as they could if they were working together.

I had two separate and seemingly unrelated incidents today that frustrated me extensively so I’m hoping that the online community can give me some perspective in these areas.

First, Algebra 2.  Solving equations.  We were doing some practice on the whiteboards.  We haven’t talked explicitly about solving quadratic equations but I wanted to do a little experiment.

Me:  Solve x^2 + 1 = 37.  [writing on the board]
Class: [Crickets]

A few were able to work to a solution, but it was the same students that I would have expected to do so regardless of how the question was presented.

Me:  Ok, let’s try this a different way.  I think of a number.  I square it, then I add 2.  My result is 27.  What number was I thinking of?

Every single kid in the room was able to arrive and understand that the answer was 5.  Some did this very quickly.  Maybe you’re thinking “well, they didn’t know that -5 was also an answer.”  News Flash!  None of the kids remember that.  None.   They know the procedures they used before:  quadratic formula, factoring, taking the square root, completing the square.  But they have no idea why, they have no connection for what x really means and they have no conceptual understanding of a quadratic equation.

Christopher Danielson said it so beautifully yesterday:

THE STEPS WIN, PEOPLE! The steps trump thinking. The steps trump number sense. The steps triumph over all.

Here’s a second example, equally as frustrating.  I’m helping a student get caught up on his algebra assignments for another teacher’s class.  I don’t teach this class, and I like this kid, so I don’t mind helping him at all.

So systems of linear equations.  8x + 9y = 15 and 5x – 2y = 17 (or some bolonga like that.)

Kid:  I don’t get this, I mean, what is x?  II know I can substitute numbers for it and get y, but what does it connect to?

Me:  Read this word problem:  You work two jobs.  One you make $6/hr and the other $8/her.   Last week you worked 14 hours and made $96.  How long did you work at each job?

Kid: [3 minutes and an ounce of brain-sweat later]   8 at the first job, 6 at the second.

Seriously, to see this kid mentally crunch these numbers was magical.  To him, that was common sense.  To another kid, it might be a table.  To a third, trial and improvement.  Why can’t most kids do that?  Because we (you) insist that they set up a system of linear equations every single time.  And because the title of the section in the book is 3.1 Solving Systems of Linear Equations.  And then we focus on the steps and the methods.  Substitution. Linear combination. Elimination. Graphing.   And then a year later, those are just fancy terms that math teachers use to make easy things difficult.

Procedural fluency is important, but it must be built on a foundation of conceptual understanding. The procedure should never lead the discussion, and in most high school math classrooms, it unfortunately is.

 

What Questions Do They Have?

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I’m always delighted by the extra wave of energy students put forth when they are asked to develop their own question to a scenario.  I love my job, and this year has started amazingly.  But today was probably my favorite day thus far.

College Algebra:  

Since we are working on quadratics, we did the Many or Money scenario from the Math Forum Problems of the Week.  It’s interesting (and almost entertaining) to watch them discover that there is no question.  This is the first time we’ve done an activity where they developed the question so they came up with the questions I would have expected:

  • What price will maximize profit?
  • How many students would go if the price were $8?
  • How many students will attend at the maximum profit?
  • (My favorite) Can you write an equation that models Ticket price and Profit?

They were able to get started on answering some of these questions.  I had them work on one large sheet of paper in order to share their work.  The period ended before they could wrap up their work.  Here is what one group has so far:

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When talking with teachers about using the Notice and Wonder strategy is usually surrounding the unexpected “wonderings” that students will have.  I think it’s important to allow them to have that creativity of asking outlandish questions like, what is the band’s favorite pre-concert meal?  But to make sure that the math goals are met, shifting their focus on what we can mathematically deduce from the scenario.  I usually ask what would I most likely ask about this scenario and what questions do you have about this scenario?  

 

Algebra 2:

Last year, with this same class, we examined Val’s Values.  The authentic, real-world awesomeness of that particular lesson was going to be impossible to re-create, but the scenario was still applicable and intriguing to this new group of students.

Last year, my students insisted that the ages of both Val and Amir were vital to answering the question Who spends more on jackets over their lifetime?  Most fascinating to me was their estimations of Val and Amir’s ages:

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Desmos made up  a nice scatter plot for us that we could also Notice and Wonder about:

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And Val, my students were slightly disappointed that they didn’t get to examine the entire $300 jacket.  They are VERY curious about it.  ;)

 

 

Resurrecting Visual Patterns

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Here is a pattern from Don Steward’s blog that my college algebra students dissected.

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We started a unit on quadratic functions and I wanted to incorporate some quadratic patterns.  Hats off to my co-workers, Teresa and Amy, who introduced most of these kids to Visual Patterns in their Algebra 2 classes last year.  Now in college algebra, these students have a solid foundation in making connections between tabular, algebraic, and pictorial representations of these functions.

I wanted to share some of their work connecting their general rule to the pattern.

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I’ve been able to move to more difficult quadratic patterns more quickly this year, so I’m looking forward to tackling Nrich’s Steel Cables problem very soon with this group.