Modeling perfect answer is not modeling perfect learning

calculator-scientific
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Last week a student of mine was curious about the quadratic formula.  He had seen it before but didn’t understand why we used it.  “Where does it come from?”, he asked.

I loved this question.

I did the thing any excited math teacher would do and decided to take a couple of minutes to explain to him how we can start with ax^2 +bx + c = 0 and complete the square.  Luckily, this took place after school because five minutes quickly turned into twenty and soon he said “sorry, mister, I need to run!”  Even though it was a concept that I understood and I have taught before, I ran into bump after bump and I found myself thinking collaboratively with my student about the problem but ran out of time and couldn’t quite figure it out in time!

Luckily, I didn’t feel too defeated because I figured it out a few moments after he left, but it got me thinking…  Often as a teacher, I feel the need to be fully prepared – down to the last detail.  I feel like any example I show or any question that I answer must be done to perfection.  And yet, learning is the exact opposite.  Learning is messy and bumpy.

If I would have prepared a lesson and taught the quadratic formula explanation perfectly by detailing every step would more learning have occurred?  What is the value of modeling a perfectly solved problem?  Would it be more meaningful to model good questioning and struggle through a problem?  How do we let students solve questions like that while still allowing them the support to access it?

I don’t know the answer to it all, but definitely some questions worth grappling with.  Curious if you have thoughts or similar situations to this!

Rewiring the Teacher Brain for Growth Mindset

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Teaching growth mindset is not as much “showing students how” as it is a “changing the way we talk about what we already do”.  If we are able to talk about learning differently, it becomes embedded in daily routines.  Here are a few subtle changes I made to my syllabus this year.

Old syllabus

EXTRA HELP:

Get it when you need it!!   After school!  Before school!  During class work time!  From your friends!  From me!  From other teachers!  From parents!  From older siblings!  From a tutor!  From the internet! From Khan Academy! The possibilities are endless – YOU just need to make the effort.  Just like life, math doesn’t come easy for most of us – you have to work at it!

New syllabus

GROWTH MINDSET:

Work hard and ask for help when you need it. After school or before school! During class work time! Browse the internet! The possibilities are endless – YOU just need to make the effort. Just like life, math doesn’t come easy for most of us – math is not about speed, it is about the struggle!

As you can notice, very little changed.  Instead of implying getting help is an extra part of learning (making kids feel dumb if they need to get help), I express that hard work is part of the learning process and that asking for help is the norm.  I like how the original paragraph emphasizes effort, but I tried to take it further by emphasizing that struggling is okay.  If students are content with solving problems quickly we are robbing them of real learning.

I challenge you to choose your words carefully when creating documents and especially as you are talking with students.  It is not easy at first, but soon enough it becomes a normal way to talk about learning.

Building Grit – Being Proactive

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It is the start of another school year and that means in a month the calm of the first few weeks will be overshadowed by the insanity of contacting parents, getting kids caught up/ back on track, and never-ending lesson planning.  Last year I chose to be proactive in hopes that I could at least help kids from falling behind early in the school year.
Here’s what I did.

1.) Ask around to see if it was possible to receive an organized list of my students’ math grade from the previous year.

  • This is important because otherwise I would have had to individually look up each and every student one by one; always find the easiest way

2.) Find all students that ended the previous course with a D or failed the course and are repeating

  • Last year I chose to find students with and C, D, or F.  This year’s students are coming in with more D’s, so I’m not worrying about the C’s to keep the process manageable.  Choose what works best for you.

3.) Email parents asking them to meet the first week of school to get to know their student’s strengths, weakness, and to set goals.

  • Below is the message that I sent to parents:

 

“Hi there!

My name is Casey Ulrich and I will be your daughter/ son’s math teacher this year for Algebra 2. I am excited to start the new year and know that it will be a great one!

I am reaching out to you because looking at last year’s math grades your daughter/ son seemed to struggle with Geometry. Algebra 2 builds on many of the concepts that are presented in geometry and I want to make sure this school year gets kicked off with a positive start.

I would love to set up a meeting with you and your student to get to know some of the strategies that help them learn best and to set a goal for the semester. I am available any day after school this week Tuesday, Wednesday, and Thursday as well as most days the following week. Please let me know if you would like to set up a conference and what times would work for you!

I look forward to the chance to meet you and learn all about your awesome kid.”

 

It turned out to be super effective in helping to understand incoming students.  Knowing a little bit more about them at the start of the year as well as establishing communication with parents led to great relationships and a positive learning atmosphere.  What I saw in these students was an increased level of grit.  They were less afraid to ask questions and often times were the students I spent the most time helping after school.  It allowed them to see that struggles are a part of learning and developing strategies to improve led to more learning. (what a novel idea)

No data to prove that this was a direct success, but I can vouch that it passes the gut check (certainly feels like it helped).  Not all parents responded and not all students that showed up changed their learning habits drastically, but it was conversation that laid out the fact that I believed that they could learn.  Hearing that from the teacher the first week can go a long way.

What else do you do to foster grit at the start of the year?

Best Pi Day EVER

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If you are reading this, there is a good chance you were aware of the level of intensity this year’s pi day brought.  On Saturday, 3/14/15 at 9:26 nerds around the world celebrated and reflected on the beauty that comes from dividing any circle’s circumference by its diameter.  It makes sense, then, that I began 8th hour on Friday asking my students “So are you guys excited for tomorrow?!”  Little did I know, that question would help transform the next 45 minutes of class into one of the best lessons I have ever been a part of.

Let me start by mentioning this 8th hour class is a smaller class filled with students that have struggled with math for one reason or another.  I’ve got freshman, sophomores, juniors, and seniors coming from Algebra 1, Geometry, and Algebra 2.  They are not used to having quality conversations about math. In fact, just the opposite – I actually planned on them continuing their online course work silently.


 Here’s what happened when I winged a class period

 

Part One – The Set Up

Teacher: Are you excited about tomorrow?!

Student 1: You mean Pi Day?

T: Yeah! Pi Day because 3/14/15 (I write it out).  At 9:26 it’s gonna be the bomb!

Student 2: How many digits of pi do you know?

T: Let me show you something! (I walk over to my computer)

Student 3: Can we not do anything today?

T: Just one thing – then we can do whatever. (Bring up my email to show them this banter between some staff members)

email pi day

At this point the students laugh, call me a geek – whatever.  Then one student goes – isn’t it 3.141…and has like 13 digits memorized!  A student I normally have to bargain with to get any work done knows 13 digits!  How cool!

 

Part Two – Intro to Pi

T: Okay, let’s watch this quick video on pi, then I’ll stop with the nerdy stuff.  (I show them this awesome Jo Boaler material)

T: Okay, so C = d* pi right?  Does anyone remember the formula for area of a circle?

S1: Isn’t it  pi*r^2?

T: Yes! It is.  How do you know that?

S1: I don’t know… it’s what we learned.r squared

T: Okay, well how can we show it? (I draw the picture to the right)

S2: That’s a nice circle.

T: So in the circle the radius is always the same length all the way around.  We’ve got r and r so that makes this box the area of…

S1: r squared.

T: Nice! How do you know that?

S1: Because it’s r times r.

T: Okay, so how does r squared relate to A = (pi * r^2)?  Why does that make sense as a formula?

Silence

T: Okay.  How many of these boxes do you think fit in the circle? (pointing to the r squared box)

S1: 3.14. You know it’s going to be less than 4.  You have to cut the edges off.

T: Good.  So you have 3.14 of them right? pi *r^2?  Alright, bare with me ellipse(as I draw the ellipse to the right). Just one more thing. (20 mins of class left at this point). What is this shape?

S2: An oval.

T: Yeah, as you get higher in math we call it an ellipse.  That’s not important.  What I want you to see is that here we have this “a” and this “b”.  What do you think the formula is for the area of an ellipse?

Blank stares

Part Three – Setting Up The Conversation

At this point I take time to lecture students on the process of turning on their brains.  They have never been asked to think and participate in a discussion where the formula hadn’t already been given to them.  “It’s okay to be wrong!  We know our brains grow more when we make mistakes, but you have to try!”

S3: But when I raise my hand and get the answer wrong people will laugh at how wrong I am.

T: Exactly!  That’s terrible, but realize that’s not you; it’s the environment your teacher has allowed.  I doubt they even realize it, so it’s on all of you to be comfortable being wrong.  Let me ask you…how many students tend to answer questions in class?

S4: 2 or 3.

T: That’s a problem.  A teacher often needs to go off of their gut feeling.  If you only let those 2 or 3 people answer the teacher will believe that everyone is good!  You need to be okay with being wrong. Now I want you all to really think!

S1: Would it be pi * (a*b)?

T: Why?

S1: Because its pi * r * r.  Why not make it pi * a * b?

T: What would that look like in a picture?  How could we use area to explain your idea?ellipse2

S1: Draw a box.

T: Okay what’s the area of this box? (students start with their thoughts)

ab^2
a^2*b^2     <= these were the three options that were thrown out by students
ab

I then ask each student to explain their explanation.

S4: (a^2 * b^2) Well I know there are two a’s and two b’s and I know it’s multiplying because of the r squared thing.

S3: (ab^2) That’s what I said for my formula two.

T: Are they the same thing then?

S5: They need to have parentheses so the exponent goes to both.

T: (I draw in parentheses with arrows showing the exponent effecting both the a and the b) Do we understand the difference? (They do!)

S5: (ab). So I just figured before we only had an r and an r so we would only need an a and b, not two of each.

S4: No way! Can’t you see my answer is right.  Mr. Ulrich even just showed the thing with the two!

S5: But the r square would be r and r on the opposite sides and we didn’t use two of them.

The argument went on for about a minute – a few other students chimed in and soon enough we voted.  After all arguments were given the class landed on the formula a^2 * b^2.

So what’s the answer?! – They ask.

T: Well…you’re all wrong. (groans)  But, you were thinking and debating and that’s what is important in learning.  You made a mistake, yes, but you learned and used your brains!

With only about six minutes of class left I tied our findings to pi, recapping that there is 3.14 of those rectangles in an ellipse and… boom – an amazing lesson with two minutes left to spare.


Let’s recap what was covered

  • how many digits of pi are there?
  • what does pi represent?
  • formula for circumference of a circle
  • formula for area of a circle and ellipse
  • why mistakes are important to make

Math practices used

  • Make sense of problems and persevere in solving them
  • Construct viable arguments and critique the reasoning of others

 

A Stressful Moment

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Today was stressful.

It wasn’t really any more stressful than an average day of teaching, but somehow the culmination of a bad unit and reaching the point cluelessness in a new curriculum put me over the edge.  All of the stars of negativity aligned and I left school defeated and thinking about all of the other things I could be doing with my life.

Teaching does not equal learning.

In my Algebra 2 class, we have been learning about “Rules of Exponents”(RoE).  Let me rephrase that – I have been teaching my students about “Rules of Exponents”.  To gain perspective, these students are mostly sophomores and juniors.  They are introduced RoE in Algebra 1 and it is recovered in Geometry.  By the time they get to me one would think that they have a decent foundation of understanding.  To quote another teacher at my school “assume they know nothing”.  Sad quote, but when it comes to understanding  the material it is spot on.  They may recall a few of the tricks, but understanding?  Not even close.

In Algebra 1 students are thrown all of the rules in the first few days and asked to learn and understand them.  The tricks are taught, “anything to the zero is one!” is thrown about, about day 3 or 4 negative exponents are introduced and BOOM show us what you understand…sorry.  I keep writing the word understand like there is even a chance of that happening in six days.  By day 6 we ask students to show us what procedures they were able to memorize.  In geometry, the process is the same, and after looking at assessments might be a scaled back version of Algebra 1 (wtf?).

Anywho…now they come in to Algebra 2.  Day 1 – You know how to do RoE, right?  Day 2- Hope you’re good because here comes simplifying radicals (they have pretty much never seen), multiplying and adding radicals, rational exponents, and exponential functions.  This is my life.  We have been learning these skills for close to two weeks.  Each night I rack my brain for a creative method for students to latch on to these ideas and each day I come to school thinking that they are understanding.  A majority of the students are trying, asking questions of each other and of me, and I am doing my best to help motivate the students that are difficult to motivate.  This chaos and disorder is part of the job and as a teacher you get used to it, but checking in with students today – two days before test day – it seems like the entire class needs to go back to square one.  “So you’re telling me you can’t simplify the square root of 48?  You realize we spent like…3 days just on that?” Talk about defeating for a teacher.

Students laziness is not an excuse for bad teaching – call it what it is.

The next piece of stress came when I tried to share my frustration with other teachers.  Explaining the situation to them, I received the same response: “The kids are being lazy.  None of them do their homework, so you can’t expect them to understand it if they don’t try”.  The reason this angered me was simple: I didn’t believe it.  I have awesome students and I feel like everyone of them wants to learn the material.  There are definitely some that do not realize the benefit of homework as much as they should, but that is not the sole reason they struggle.  Teachers use the myth that students are lazy to shield them from the fact that how they are teaching may not be what is best for learning.

It is uncomfortable and it would be much easier to shift all of the blame to students, but I realize whatever methods I chose to help students learn did not work out as planned.  I see myself as an innovator and a teacher that listens to student input and tries to change my lessons accordingly, but time and again  my results are the same as every other teacher.  I am left wondering if anything I do really makes a difference in student understanding.

A downer for sure, but I am proud of the fact that I am upset.  It means I care about how students learn best and I want all of my students to be successful.  If I chalked all of my shortcomings up to lazy students I would be a much happier teacher, but a teacher that never grew or challenged myself.  I hope that out of this ordeal our department is able to have good discussions about the skills taught in each sequence of courses, and who knows…the kids could still ace the test on Friday. : /