Algebraic Anguish

The following prompt presented at Twitter Math Camp by the Mighty  Max Math Forum (aka Max Ray) has been rattling around in my brain for the last few weeks.  Here a grid representing streets in Ursala’s town:

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The problem-solving session, masterfully orchestrated by Max, allowed each group of teachers to develop their own representation of the situation and think about what questions could be asked. For example, if Ursala is at point 1 and needs to get to point 19 along the line segments, without backtracking, how many ways are there for her to travel?  Lots of discussion ensued at our table including the definition of backtracking.

I’ve been at school the last few days and anyone who has sat near me at a meeting in the last few weeks has seen me doodle this scenario, I’m sure wondering what my nerdy math-brain was concocting:

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Simplifying the grid and turning it into a pattern expanded the questions that I wanted to ask.  For instance, how many line segments (or streets) in Ursala’s case) are used in step n?

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What I’m still grappling with is how to expand my wonder about this scenario past the algebraic representations.  In talking with other teachers recently, it seems as though many of us have been programmed to solve these, and many other problems algebraically.  I recognize that many students won’t reach for the algebraic aid.  So my next step is to try to see this situation in other ways, sans algebra to better understand how my students are likely to see it.

 

Confession: I’ve never really been good at math

Here’s a confession of mine:  I’ve never really thought of myself as ‘good at math.’  Yep, I’m a high school math teacher proclaiming my discontent with my mathematical abilities.  Ironic?  Sad?  Make you want to hide your children?  Read on, it’s not as bad as you think.

Being a math teacher was a second career for me, as my undergraduate degree is in accounting.  I dabbled in a minor in mathematics while at the University of Iowa but let a ‘C’ in Linear Algebra from a cold professor change my trajectory for the next 4 years.   When I went to graduate school to earn my masters in Mathematics Education, I was always intimidated by the math undergrads who were much more polished and current on mathematical theory.  Recently I came across this article which shed some light onto what often happens with girls in areas like mathematics. In short, women tend to give up on themselves more quickly because of their strong inner voice.   I know that I was never discouraged from pursuing difficult challenges by my parents, especially academically.  I came from a family that was very supportive of my education.  It was my own inner-voice telling me that I wasn’t as good at pure mathematics, which was the lingering after effect of that C grade.

Recently, Rafranz Davis wrote a blog post about the transformation of twitter admiration into palatable inspiration.   This post was timely for me since as summer conference season reaches its peak, I’ll be attending Twitter Math Camp starting on Thursday with dozens of other math tweeps with whom I’ve admired and been inspired by.  These positive interactions have projected me to a place where I’m comfortable with my mathematical abilities and completely humbled by my ability to participate with such a wonderful group of educators across social media.

 

 

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.

 

 

Crafty Math

I was recently inspired by @mathinyourfeet‘s post: https://twitter.com/mathinyourfeet/status/479332580227964928 Hoping that this was a project that could be adapted for 3-5 year olds, I inquired about the details.  Malke Rosenfeld was one step ahead of me with a blog post.  Some background on my craftiness:  My mom is the crafty one.  Growing up, I’m sure she was frustrated that I never took to sewing or quilting, but her gift in that area is unmatched.  (Luckily, my brother ended up being the artsy one.)  It’s hard to believe I became a math teacher, but I don’t excel in the realms of ‘measuring’ and ‘cutting.’ I knew the 3.5-year-old focus on this was going to be short (9 minutes to be exact), I wanted to maximize our mathematical conversation.  First, she decided that she would be pink and I would be yellow (our respective favorite colors).  She also decided that the strips should be weaved in a pattern of pink/yellow/pink/yellow. IMG_5596

Next, she noticed that the papers were making small squares.  Because of the pink/yellow pattern, she pointed out a face with eyes and a nose.  🙂

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After about 3 pink strips, she groaned, “mommy, I’m getting really tired.  Can you finish it for me?”  Of course I wasn’t going to allow this craft to remain undone, but I think she appreciate the outcome.

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Luckily, the extra strips of paper didn’t go to waste.  They ended up as a wall decoration as well.  IMG_5603

Pair Products – An Nrich Favorite

In a few short weeks, I will be making a presentation at Twitter Math Camp on my favorite Nrich Tasks.  I know a lot of teachers have reservations about integrating rich mathematical tasks into their regular routines so I want to focus on problems that have that “traditional” feel while still allowing students to explore mathematical relationships more deeply.

Pair Products is an amazing offering by Nrich and its low barrier to entry makes it accessible for all students.  After working through the problem myself, Nrich offers additional questions to raise the ceiling.

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Additional Questions to Ask:

  1. What happens when you use 4 consecutive even or odd numbers? 5? 6? n?
  2. What happens when you use 4, 5, 6, n consecutive multiples of 3? Multiples of 4? 5? 6?
  3.  (My Favorite) What happens when you use n consecutive multiples of w?
  4. Does your generalization from #4 hold for numbers that increase by .5?  (For example: 3, 3.5, 4, 4.5)

My favorite Nrich pair, Charlie and Alison, offer two different approaches.  Charlie explains a clear algebraic manipulation to arrive at two expressions with a numerical difference.  Alison, on the other hand, represents the product of numbers with an area model.

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An interesting challenge might be to ask students to show the area model that Alison employs for some of the additional questions.

 

Twiddle dee Twiddla

Yesterday was our first official day of SUMMER.  So after a thunderstorm curtailed my gardening plans, I thought I’d check out some apps that have been on my to-do list for a while.   First up:  Twiddla, an online collaborative whiteboard.  Why a collaborative whiteboard?  Our school district uses Google Apps and there are many beneficial collaborative options through Google docs, sheets, etc. The problem:  Mathematics just doesn’t translate very well when typed or through a computer medium.  If I’d like kids to collaborate in real-time via the web, Twiddla might be a viable option for students to collaborate in real time online, with a blank canvas.

What I like:

  • No login required.  Just post the web address and kids are good to go.
  • PDF’s and images are insertable into the background.
  • There is a grid background as well.
  • Students can “chat” or audio conference while working.
  • A variety of colors, shapes, and line thicknesses can be utilized.
  • The Pro version (usually $14/month) is free for educators and students.
  • The writing is very smooth without a stylus.

What I did not like as much:

  • Annotations are added when writer “pauses” rather than as they are writing.
  • An “undo” button would be helpful.

Some screenshots from my twiddla-created session:

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Now, I’ll have to wait until Fall to test this app out with students, but I’m optimistic about it’s potential.  It could just be one of those things that’s “cool” but in reality, pencil and paper will do.

Brain Sweat

I’ve talked about my Algebra 2 class at length on this blog over the last 2 months, and as the trimester comes to a close, I want to celebrate the positives in this class as much as possible.  They frustrate me sometimes, but the bottom line is I’m willing to fight and fight hard to make their experience with math more positive.  Ultimately, they’ve been dealt an unfair hand:  crammed into giant classes and labeled incapable of high-level mathematics.  They are capable of more than they give, but they also deserve much more than they’ve been given.

The perpetual optimist in me wants to continue to celebrate their achievements and play the hand they’ve been dealt as best we can.  Today we took on Robert Kaplinsky’s Cheeseburger Lesson.  I’m not sure why I’m constantly drawn to this lesson, since the picture of the 100×100 makes me a little ill.  Perhaps it’s the constant student engagement I get from it, time after time.  The intriguing thought that someone actually purchased this godzilla-burger hooks students every time.

What I liked most about my class’s efforts toward this task was the multiple revisions they had before arriving at the correct answer.  I had many students assume that a 3×3 cost the same as three cheeseburgers, only to find that their burger only needed one bun.

Below is a student’s work that I really appreciated.  At the end of the activity, he said,

Mrs. Schmidt, I’m sweating.  I thought so hard on this problem that I’m sweating.  But I believe I have the right answer.”

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If I’m being completely honest overall, this class has tested me, day in and day out.  I’ve worked very hard, but in the end, I’m not sure I taught them much of anything worthwhile.  I hope I have, but I’m not sure I did.  A class size of 36 seemed insurmountable, and perhaps in some ways, I never really overcame it.  Unfortunately, next year’s class size projections promise more of the same.  The silver lining, however, is that I get another crack at teaching this same course, and I’m 100% sure I can do it better the next time around.

Probability Ponderings

It’s been a great week in my probability and statistics classes.  I’m not sure why I’m pleasantly surprised.  This time of year it’s absolutely essential that we engage kids in meaningful mathematics and when we do, they respond well.

Monday, we did expected value and Dan Meyer’s Money Duck.  See Monday’s blog post for details.  Extra Credit if you can find my duck pun in there.

Tuesday, after assessing expected value, we moved to tree diagrams and conditional probability.

Wednesday, I used Nrich’s In a Box problem to create some discussion about dependent and independent events.  

I started with a bag with unifix cubes and had them do some experimenting to see if the game was fair.  What I love about this problem is that the initial answers that the kids come up with are usually completely wrong.  It really allows the teacher to identify the misconceptions.  Additionally, this problem is so easy to extend.  Simply have the students come up with a scenario of ribbons that creates a fair game.  Most will come up with something like 2 red and 2 blue. Have them test their theory, find out it’s wrong and then test another.  Even when they find the magic combination that creates a fair game, there is still the task of generalizing the results that’s challenging.     

Thursday, I totally stole Andrew Stadel’s 4! lesson.  What a great intro to the idea of factorial.  Last trimester I used IMP’s ice cream bowls and cones, which I still might refer to.  I felt like having a few students up in front at the beginning got everyone on the same page at the same time.  It was completely awesome to see the different methods for solving this.  I love the repeated reasoning here:

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Plus, opportunities to use animal counters in HS math are scarce.

What’s the most pleasing about this week is that I think that this group’s conceptual foundation of these concepts is more solid than it has been in any previous year.  We still have practice to do, but I feel like they have made a good connection to what their answers represent.  In the past, my formula driven instruction didn’t bode well for retention of the concepts. I’m more hopeful this time around.

Thanks Ashli for the spectacular idea of sharing what adorns our classroom walls.  I’ve got the regular math posters, sports schedules, school policies, and motivational cliche’s, of course.  A classroom would not be complete without a stock photo along with transformational words like, “the key to success is self discipline.”

What really brings me the most joy in my classroom and truly makes my classroom mine is my dog wall.

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Ok, it’s actually two walls.  Backstory:  I love dogs, beagles in particular.  Duh.  But the reasoning behind my dog wall runs deeper than that.  Yep, the dogs are adorable and the kids love that they can put a picture of their own dog in my room.  I love it when I have younger siblings of former students, and they ask “hey, you have a picture of my dog!”

The real power behind the dog wall is acknowledging what dogs can teach us about love.   In short, no one on earth is capable of loving you as much as your dog.   Oprah gives us a nice example when remembering her cocker spaniel, Sophie.  If you have a dog, you know what I’m talking about.

I recognize that not all students are lucky enough to own a dog.  I also let them bring in a picture of any dog, but I make sure to mention that I like beagles best.

My plans for the expanding dog wall include using them for some estimation and data exploration.  Someday.