This is Our Theorem – College Algebra

“We came up with a theorem once at my old school.  The teacher has it in a frame behind his desk.”

This statement from one of my college algebra students made me both elated and sad at the same time.  Thrilled because this is the type of mathematics I believe all students should have the chance to engage in on a regular basis.  Disappointed because this type of discovery happens so infrequently in American mathematics classrooms that the incident warranted a sacred place on the wall of this teacher’s room.

In College Algebra, part of today’s learning objective was to define a polynomial function and determine some key features.  I have the awesome types of students that if I were to write down the surly definition and features of a polynomial function onto the whiteboard, each would follow in lock-step and write it in their notebooks solidifying it’s place among mathematical obscurity.

Today, we were going to break that cycle with something different.

But I needed to know where they were at, so I had them write down what they knew about a polynomial function.

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After some discussion and leading questions, we were sure that linear, quadratic, cubic, quartic, x^5, x^6, and so on were all polynomial functions.  Awesome. We weren’t, however, as sure about functions including negative exponents, roots, sin/cos, or algebraic fractions.

What makes this group we are sure about special?  Last week, we spent a considerable amount of time on features of functions including domains, end behavior, intercepts, intervals, symmetry, and turning points.  In their groups, I had them examine the graphs of these alleged “polynomials” through the lens of the features of functions.

Two similarities emerged as significant:

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Questions:  Was this true of all polynomial functions?  And if both conditions were not met, could we exclude it from our known polynomial functions?  Hiding my initial excitement, I then had them look at our list of “questionable” functions. For example, did “y = 9 + 1/x” meet each of these two criteria?

Christopher Danielson suggested that my class give this new theorem a name, so we could refer back to it with ease:

“Class. We have found that all polynomials blah blah blah…” [while writing the statement of the theorem on the board.]  In mathematics, when we have an important finding like this, and when all mathematicians have agreed the finding is true, it gets a name.  Sometimes it is named for a person, such as ‘Fermat’s Last Theorem’; sometimes it is named for what it says, as in ‘The Triangle Inequality’.  But that name makes it possible to refer to it going forward. It helps us to remember and to use the thing we figured out. So we need to name our theorem. Who has a name they’d like to suggest?”

Alas, the excitement of naming the theorem will have to wait until tomorrow.

Somewhere between Concrete Sequential and Abstract Random

It occurred to be relatively early in during the trimester this past fall that my college algebra students (generally) have no idea what I mean when I say “quadratic function.”  This isn’t because they have never heard it, learned it, or used it.  But that technical of a term simply has not stuck around in their long term memory.

So, similar to linear functions, we start with a pattern:

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from youcubed.org and visualpatterns.org

I then had them make posters including how it is growing, what the 10th, 100th, 0th, and -1st cases would look like, table, graph, expression, and relationship to the pattern.  Instead of one large poster, they use 4 smaller pieces of paper and tape them together.  That way each group member can contribute simultaneously.

I noticed:

  • It is difficult for students to describe how an irregular shape is growing.
  • It is even more difficult for them to describe something abstract like the -1 case.
  • Many of them expressed the overall growth as “exponential.”
  • Most could easily see the two rectangles formed and determine the dimensions with respect to “n.”

I wondered:

  • If they could connect the work with patterns to other quadratics.
  • How to have a meaningful discussion around the “exponential growth” issue.

Their homework was to answer similar questions for this pattern from You Cubed’s Week of Inspirational Math:

growing

Spoiler alert: the rule for the pattern is f(n) = (x + 1)^2 or f(n) = x^2 + 2x + 1

So where do we go from here, two days before Winter Break?  My goals are to review some specifics on quadratic functions and simultaneously help the students make connections between different representations.  I know what I must do.  I must channel my inner Triangleman.

[Backstory:  Christopher Danielson and I go way back. At least to 2014. Maybe even 2013.  Seriously though, I strive to organize my college algebra class the way Professor Danielson describes in his blog.  I have picked his brain on more than a few occasions and he is gracious enough to give me advice in certain curricular areas. In short, his philosophy titled “They’ll Need it for Calculus” is the foundation of my College Algebra course. ]

Ok, back to room C118.

Me: Write down everything you know about the function y = (x+1)^2

(Most write down the expanded form, some start to graph, but not many)

Me: What other ways can we represent this function?

Students: Tables! Graphs! Pictures! Words! Patterns! Licorice!

Me: Sweet!  Let’s do all of that, minus the Licorice.

(I give them a few minutes to create a table and a graph.)

Me: NOW, write down everything you know about this function.

I circulate and hand each group a half sheet of paper.

Me: Write down the most important thing on your groups list.

At first I wasn’t really concerned what exactly they wrote down, but how they defended their choice. Then I came across this in all of my classes:

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We came up with a pleasing list of attributes of a positive parabola that included vertex placement, end behavior, leading coefficients, and rate of change.

Next up for discussion: Parabolas grow exponentially.

Me: Turn to your partner and tell them whether you agree or disagree with this statement and defend your choice.

Students: Yes, words, words, words.

Me: Ok, now the other partner, say how you know something grows exponentially.

Students: Multiplied every time, more words, blah blah blah.

So we agreed that 2, 4, 8, 16, 32, 64… is an example of something that grows exponentially.

Me: Numerically, how can we tell how something is growing?

Students: (eventually) Rate of Change!

We came up with this table and agreed that these two functions were definitely NOT growing in a similar way.

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Now on to helping them understand what it means for something to grow quadratically…

 

An Open Letter and a Note of Gratitude

“I’ve been married for forty-two years which means I’ve been to forty-two state fairs.”  I’m not sure why this little tidbit from the Park-and-Ride attendant didn’t instantly put me in great spirits.

And then I walked up over the hill, took one look at Math-on-a-Stick, and the indifference melted.  The humble smile of satisfaction from Christopher Danielson was enough to warm the cockles of my heart for months.  But the kids and the konversations around shape, number, space, and patterns were nothing short of inspiring.  “This is exactly how I envisioned this,” Christopher said.  And his vision becoming state-fair reality is going to change the way parents and kids talk about math.  I love hearing kids explain their reasoning.  It’s what makes my job as an educator joyous.


 

Dear Adults: (all of us)

The children in your life are creative, driven, passionate, and intelligent.

GET OUT OF THEIR WAY.

Ask them how they see the pattern. Let them experiment with the shapes.  Let them lead the conversation.

Listen to what they notice. Encourage them to say more.

Then ask them how they know.

The “right” answer is so much less important than a child leading you on their own mathematical journey.


 

Thank you, Christopher.  Today you not only added a great new structure to the landscape of the Minnesota State Fair. But tonight, in hundreds of households, the conversation around mathematics is changing. And after nine more days, thousands more will get a chance to embrace this shift.  When my husband and I have been married forty-two years, we’ll go to the fair and stop by and listen to the kids talk at Math-on-a-Stick.

 

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The Stupidity of Number Flexibility (#TMWYK)

I’d compare the struggle between teachers and learners at the end of the year to that of a parent trying to carry a limp child to their bed.  Eventually they will both get there, but the parent is frustrated and the child is attempting to make things as difficult as possible.   In the end, neither party is probably happy.

Rather than focus on that inevitable struggle, I want to detail a fun experience (for me) that I had with my daughter this past weekend. Her four-year-old rebellion has included a resistance to completing her math and language at school and a refusal to engage in those conversations at home.  Grandma and Grandpa were in town this weekend, which gave me an opportunity to exploit her desire to impress them.

At school, Maria is given “problems” similar to the ones on this sheet.  They seem randomly chosen, and the children are given beads to model the problem if needed.  0526150926-1

 

Given the opportunity to exploit the situation, I handed her 12 beads and wrote down 4 + 4, 5+3. 3+5, 2+6, 6+2, 1 + 7, and 7+1.

After protesting that 5 + 5 (her favorite after 4 + 4) wasn’t on there, she started sorting the beads into two piles.

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I noticed:

  • She knew 4 + 4 by memory and did not use the beads. (Same with 5 + 5)
  • She sorted 5 + 3 into a pile of 5 beads and a pile of three beads.
  • She did not grab the beads to do 3 + 5 but rather recognized it was the same two numbers and therefore totaled 8.
  • 6 + 2 required her to count the beads but she did not grab new beads.  She simply rearranged the original piles.
  • At 2 + 6 she was onto me and simply filled in “8” for the remaining answers.

Maria:  Mommy, this math is stupid.

Me:  Why do you think it is stupid, my sweet little bucket of sunshine?

Maria:  All of these are the same answer.

Me:  And what makes that stupid to you?

Maria:  It is just stupid.  Math is stupid. I never want to do math again.

(Awesome.)

This reaction makes me very curious about where her feelings of “this is stupid” comes from.  She’s only 4.  She has much less experience with “answer getting” than, say, a teenager.  Yet, her evaluation of the task being “stupid” seems to stem from the idea that if the answer is always the same, why do the problem in the first place?  Is the mentality of “answer over process” more innate than we think?  Or is it simply so pervasive in our education system that even my 4 year old has picked up on it?

Number flexibility is something I’ve made routine in my classroom as of late.  Detailing different strategies to arriving at the same result gives students a stronger foundation on which to build algebraic thinking.

Sigh.  It might be a long summer, but she’ll learn I don’t give up so easily.

Listening and Learning from Educators of Color

About a month ago, Christopher Danielson offered up a challenge to white educators to listen more and talk less. Specifically, we should be listening often to students/teachers/people of color and the privilege of being a white american that they do not have the opportunity to enjoy.   I took Danielson’s advice and began to really listen intently to these voices.  This blog post is how my listening will impact my teaching practice.

My family upbringing did not include overt racism, and my parents instilled values that included kindness to all.    I was confident growing up (and still am today) that my father worked very hard in order to financially secure his family.  His beginnings weren’t humble, as most would define the term, but coming from a family with 4 children, earning a C average in high school and attending the only college that would accept him weren’t great indicators of the kind of financial well-being that he has achieved.  My mother grew up in a household which included an alcoholic father and a co-dependant mother.   Her resilience allowed her to escape the dysfunction of her upbringing and earn a college degree. So my conclusion was: My family isn’t racist, my parents worked hard to get where they are in life, so anyone (white or black) should be able to do the same.  If they don’t, the problem must be individual.  After all, not all white people discriminate against black people.  

Then I began to listen.  And with that listening came a fuller understanding and acknowledgement of my white privilege and the institutional racism that still affects people of color today.  For example, I listened to Jose Vilson, whose book This is Not a Test explores the effect that race has on school and teacher quality.  His personal narrative allowed me to fully immerse myself into the issues of equality (or lack there of) that plague our inner-city schools.

I listened to Melinda D. Anderson whose unapologetic, relentless support for students and educators of color opened my eyes to how racism is treated as a thing of the past in our country but is a present day dilemma for people of color.   Her voice has helped me to recognize that black students disproportionately attend high poverty schools making segregation a 2014 issue, not a 1954 one.

I listened to Ta-Nehisi Coates whose monumental article The Case for Reparations challenged me to recognize that black americans may have equal opportunities in our country, but their access to those opportunities is anything but equal.  I listened to an hour long interview he did with Vox and one of the most powerful messages I received was this:  Our country had a 250-year policy of slavery plus another 100 years of downright discriminatory, racist laws.  We’ve spent the last 50 years trying to repair it, with many policy makers still not acknowledging that there was anything to repair in the first place.  So Coates asks, if a country spends 350 years seriously mistreating a particular culture and then 50 years sort of trying to fix it, where would you expect that culture to be socio-economically?

I also listened to this:  “Sixty-Three percent of Americans believe ‘blacks who can’t get ahead are mostly responsible for their own condition.'” And for the first time in my life I profoundly disagreed with that statement.  The very idea that blacks ‘who can’t get ahead’ would choose irresponsibility purposefully, over and over again, doesn’t make sense to me.  There are many reasons I find this belief held by a majority of Americans to be lunacy, but one in particular that is close to my heart is education.  As George Washington Carver stated, “Education is the key to unlock the golden door of freedom.”  How do we expect black students to earn that key to freedom when inequality continues to play a key role in schooling opportunities?   Is education a great equalizer when blacks are wildly disproportionately educated in schools that don’t measure up?

And I continue to listen.  The National Association for Multicultural Education published interviews with teachers of color which help white teachers like me “work more effectively and respectfully” with students of color:

  1. Listen to teachers of color
  2. Examine white privilege
  3. Be honest about your knowledge of a culture
  4. Clarify your purpose for teaching
  5. Challenge your students rather than pity them
  6. Be resilient

(Multicultural Perspectives 9(1), 3-9, 2007)

I want to continue to listen because by listening so far, I have been able to learn.  As a white person, I do not experience judgements based on my race, which is why it is so vital that I keep listening to those who do.

 

 

#TMWYK – The Return of the Sand Pool

It’s a tough time of year for teachers, and I don’t say that to garner any sympathy.  But I’m going to take a moment to deviate from the regular musings of my classroom and write about my favorite topic:  my daughter.  The discussion won’t be completely unrelated as I have learned a great deal about my students’ development of mathematical literacy while watching my daughter make sense of numbers, quantities and shape.  And of course, Christopher Danielson’s development and facilitation of Talking Math with Your Kids has encouraged me to continue the conversation with my own child.  Specifically, I appreciate that his daughter is a few years older that Maria so that I know what I’m looking for and what to look forward to.

Maria (3.5 years old) loves to be outside.  As soon as the snow melted, she insisted that it was now summer and hence every activity from that moment forward must be done in the great outdoors.  A personal favorite is the sandbox, with water.  I’m not opposed to the sandbox overall, but mixed with water, it becomes more like a swamp.  Plus, let’s face it.  It’s Minnesota. It’s Spring, not Summer, and taking out the hose just isn’t in the cards just yet.

So we made a deal that when the temperature on my weather app reached 70 or above, we could take out the hose.  In the mind of my three year-old, this meant that the first of the two digits needed to be a seven.  On Saturday, this lucky girl got to take out the hose.

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Results as expected.

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Sunday, I decided to test Maria’s understanding of these numbers.  She again asked “is it seven on the phone?”  I instead showed her Chicago’s temperature which was a balmy 82 degrees.    As expected, her response was “Aww, it’s not seven so we can’t do water.”   I know she knows 8 is bigger than 7, but hasn’t yet connected that a temperature that begins with an 8 represents something warmer than a temperature that begins with a 7.

 

 

Moments from MCTM

My brother wisely told me when he saw who I followed on twitter to stop following dumb celebrities and start following some real people.  The problem was that back then, I didn’t know which real people to follow.  Luckily, I soon discovered that there were math teachers on twitter.  Lots of them.

I’ve been to MCTM a couple of times and NCTM once or twice. I felt energized, and motivated after those conferences definitely, but this year was different than any conference I’d previously attended. The difference was my willingness to make a face-to-face connection with people I knew from twitter.   I’ve loved twitter for a long time for a variety of reasons, but meeting some tweeps in person and getting to talk math and more math was a real thrill.  It mattered less which conference sessions I attended, although they were great,  and mattered more who I took the time to interact with in between.   Although Christopher Danielson says that he doesn’t remember me as a snarky student in one of his math ed courses, I was grateful to get to spend some quality time talking with the man behind the hierarchy of hexagons. I met many others, and truly got to appreciate the wide range of awesomeness that make up Minnesota’s mathematics teachers.  

Next time, though:  book a hotel room right away.  Lesson learned.

 

A Desmosian Gem

I finally had a chance to do the Function Carnival with my classes.  Thank you to Desmos, Christopher Danielson, and Dan Meyer for their work on this project.

As David Cox captured in his blog previously, the real power of this activity is the immediate feedback.

 

When the graph looks like the one below and 8+ rocket men burst out of the cannon, the students see that right away and adjust for it.

Rocketman

 

Dan had mentioned in a blog post a while back that “this stuff is really difficult to do well.”  After seeing students work through this activity today, I can appreciate the difficulty in creating an online math activity that gives both students and teachers detailed feedback in real time.

Some observations:

  • Students don’t realize at first that you can see their work live.  I allowed them to “play” for a minute, but some may need more encouragement.
  • A tool to allow you to communicate digitally with the class would be nice.  Google chat, for example?
  • Some students don’t realize that the bumper car SHOULD crash and make their graph to avoid it.
  • A student or two misunderstood the graph misconception questions and went back and changed their graphs to look like the misconception graphs.
  • It was interesting to see which students wanted their graphs to be perfect versus which ones said there’s was “good enough.”  It would be interesting to have a discussion about which is appropriate in the particular situation.

Bravo, Dan, Christopher and the Desmosians.  Thank you for creating an online math activity that gives me some faith in online math activities for the future.

Olympians, Tweagles, & Friends in my Phone

I started tweeting in 2008, around the Beijing Olympics. It was cool that actual Olympians would respond to my tweets.  When Summer Sanders responded to one of my tweets, I about fainted. Twitter was new, they probably didn’t know any better.  

I followed a few celebrities. I found some of their off-color honesty hilarious and sad at the same time.  In the meantime, my hilarious brother managed to rack up tens of thousands of twitter followers. (@sucittam if you are looking to add some hilariousness to your timeline). Here’s one of his tweets being featured on Ellen:

He opened my eyes to the idea that following actual REAL people is more entertaining and fulfilling. He was absolutely right.

I went through a phase where I followed a bunch of people who tweet as their beagle.  I’m pretty sure I was the first one to use the term Tweagles, although I have no proof of that. 

Then in January 2013, my indifferent view of people on twitter changed forever. My 29-yr old sister in-law, Danielle, suffered a massive brain aneurysm and it wasn’t certain she would recover.  She was in the ICU at the University of Iowa for almost 6 weeks, and while my brother stayed by her side every day, his twitter followers rallied support that went viral. All of these people, most of which he’d never met, wanted to reach out to help. Benefits were organized, gifts were donated, and memorabilia was auctioned all to benefit Danielle whose recover was slow, but steady. 

Rex Huppke (@RexHuppke) wrote a beautiful article illustrating that the people we interact with on twitter are not just cyber-acquaintances.  Danny Zucker makes the best point:

 “We’re willing to accept the concept that cyberbullying is real, and it is. But if you can accept the idea that the negative is real, then you have to accept the idea that the positive is real. If strangers can hurt you, they can be friends as well.”

And just like that I leaped head first into the T of the MBToS. I realized that people like Fawn Nguyen, Andrew Stadel, Kate Nowak, and Christopher Danielson were real teachers just like I was.  They had great blogs, and they were on twitter too. And if I wanted to get a real benefit from all of the resources I had found online, I needed to start posting feedback of how I incorporated them into my classroom.  And then tell the creator of the activity about how it went. Through this I’ve really been able to experience the genuine human behind all of these @ symbols. These are not only great teachers who don’t just shine on their own. They want to freely share what they’ve done so that others can shine just as brightly. 

A Visual Patterns Trifecta

This is my third (and most exciting) post about my new found love for Visual Patterns.  My enthusiasm stems from a growing appreciation of how these patterns can be used in such a wide range of grade-levels, including advanced algebra.  The use in an elementary or lower-level secondary classroom is easy to see.  However, the teacher and student need to dig a bit deeper into the make-up of these patterns in order to generalize them.

For example, here is Pattern #8.  Kudos to Fawn Nguyen on this one.

It’s not immediately apparent what step 4 should be.  But even more so, the quadratic nature of this pattern is not necessarily simple to comprehend.  From yesterday’s pattern #5, the students had a method for finding the number of penguins in the nth step by converting the penguins into a table and creating a system of equations.  I didn’t want to encourage this method, as it is very procedural and tedious.  However, it was a good place for students who liked to work in a more algebraic way to feel successful.

Also, the table allowed them to explore what the difference of differences really told them.  I had a student, let’s call her Kay, ask “I wonder what the constant difference of differences represents in our equation for the nth step.”  She came up with a conjecture by comparing it to our problem from pattern #5.  Kay concluded that the “a” value in the equation ax^2 + bx + c = y is half of the constant difference of differences.  I challenged Kay to continue to examine these values in future problems to see if her conjecture holds true.  I had another student, Em, wonder if that meant that the “a” value in a cubic function is equal to one third of the difference of difference of differences.  This she will investigate as well.  What is very exciting about these questions is that they were non-existent 5 weeks ago.  It wasn’t that the students didn’t WANT to be mathematically curious, they just didn’t know HOW.  It was a huge thrill for me as a teacher to see these kids move from looking at a math problem with a single solution to being able to ask new questions.  A nod to Christopher Danielson for helping me realize that learning is having new questions to ask.

  Back to the problem at hand:  How many penguins are in step n?  A few of the students were able to get the answer without using a table.  These were mostly the students who like to do things in their head.  The ones who want to fully process the problem in their brain, but not write any of it down.  [Side note:  these are usually the ones who are brilliant with numbers but get lower grades in traditional math classes because they don’t want to “show their work.”] Anyway, I wanted to challenge those who used the table method and set up a system of equations to relate their model back to the picture.  Spoiler alert!  The answer is 1/2n^2 + 1/2n + 1, but I wanted my students to be able to relate that back to the picture.  What do the individual pieces of the expression represent in penguins? This way, the students were able to make that connection of a picture or pattern that didn’t seem quadratic to begin with and flesh out its quadratic properties.

When the students figured this out, it was a magical moment.  I had to capture it:

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Another cool experience with this problem:  the same evening that I did this problem, our school hosted parent-teacher conferences.  One of my students came into conferences with her parents and her three little sisters, ranging in age from about 5 to 10.  One of the little sisters sat down and wanted to be part of the conference.  I pulled up the visual pattern and asked her how many penguins would be in the next step.  It was a validation of my initial thoughts of how open and accessible these problems are to all levels of mathematics.  Here was an 8? year-old looking at the same pattern that her 17 year-old sister explored earlier that day.  And it was mathematically applicable to them both.  Beautiful.