Modeling perfect answer is not modeling perfect learning

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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!

Using Pennies to Estimate Area

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To introduce area, we took some time to estimate the area of our tables.  Our goal was to figure out “how many pennies” it will take to cover the table.  It was so cool to see the different ways students attacked the problems.  Creativity is an underutilized resource in learning, as is student collaboration.

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Understanding the Importance of Fixed Mindset

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Screen Shot 2017-07-13 at 4.57.57 PMNow that grad school has finished up, I have more time to reflect on the multitude of great experiences that took place over the last year.  One of the best fanboy moments came when I was able to meet with Carol Dweck.  I was working on a curriculum centered around growth mindset and my teammate and I were lucky enough to get 30 minutes to bounce ideas off of the celebrity known as Dweck.

My main take away from the meeting was that mindset is much more complex than many educators portray it to be.  For example, the talking points around growth mindset tend to be “growth mindset is good” and “fixed mindset is bad”.  As teachers, we emphasize the importance of developing a growth mindset and communicate they need to have the proper mindset in order to find success.

Walking away from my meeting with Dweck, I realize that fixed mindset gets a bad rap.  She explained that fixed mindset isn’t this awful thing that we need to get rid of at all costs; instead, she talked about how fixed mindset is your mind’s natural reaction to new and challenging situations.  It is your mind’s natural defense mechanism.  By asking students to get rid of a fixed mindset we are asking them to become inhuman and ignore their body’s natural reactions.

Instead, she proposes teaching students to become aware of the moments in which fixed mindset presents itself.  “Give a name to your fixed mindset”, she said.  Recognize that it is a part of you and when it shows up, acknowledge it by name and thank it for trying to protect you.  Tell it that you need to push past that uncomfortable feeling for the moment because there is an opportunity to grow.

As an example, I named my fixed mindset Jeremiah.  I was at IKEA earlier this week.  My first time there. Guys.  It’s super overwhelming.  I’m a small town boy and this building was bigger than my town.  I wandered my way around and finally got to the nightstand/ dresser section, which is what I was looking for.  I was finally there, and I didn’t know how I was supposed to buy the items I really wanted.  I snapped my friends, telling them how dumb IKEA was and seriously considered just leaving and going to Target.  I was overwhelmed and didn’t know what to do.  I was sweating.

I didn’t know it in the moment, but these are the feelings that arise when fixed mindset is afoot:  stress, being overwhelmed, anxiety, frustration.  I didn’t know my way around the store and rather than asking an employee and risking looking stupid I kept to myself for WAY too long.  Finally, I went up to an employee and asked “I’m so confused.  How do I buy a dresser?”.  I hadn’t formally acknowledged Jeremiah, but I did finally decide that to figure this out I needed to risk looking stupid to learn how to buy the furniture I needed.  In the end, they explained it to me and, sure enough, now I know how to buy furniture from IKEA (yay me!)

So next time you talk to students about growth and fixed mindset, don’t hate on fixed mindset.  Instead, have students give their fixed mindset a name and help them become more aware of the moments fixed mindset arises in their life.  You can always start with yourself.  When do you find yourself getting defensive or upset?  Is your body just trying to protect you from failure and/ or looking stupid?  Once you become aware of the moments your mindset is fixed, it’s easier to consciously alter them into moments of growth.

Learning is Hard

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One dark evening a man was on his hands and knees under a street light looking through the grass.
A pedestrian asked what he was looking for.
“The keys to my car.” replied the man.
Having some time and feeling helpful, the pedestrian joined the man in his search.
After a while, with no success, the pedestrian asked: “Where were you when you lost your keys?”
“Over there by my car.” the man gestured.
The pedestrian was puzzled. “Why are you looking for them here?”
The man without keys explained: “The light’s better!”

Why is it in education that we continually look for answers in the wrong place?

We give homework, tests, and assignments then grade students on their work.  When they don’t measure up to our expectations we encourage them to develop better habits and we talk to parents, then we move on to the next unit.  We decide the trouble lies somewhere in the work ethic of the student, the lack of support from home, or the general difficulty the student has “doing school”.

We shine the light on their ability to follow our rules and search for remedies that will allow the student to get “back on track”, neglecting the complexity that is human life.  Rather than meaningfully understanding the needs of students and responding to them, we focus on the limited time we have with students in class and expect them to figure out what they need to improve on their own.  

We seldom look for ways to deeply understand and connect with students and the ways in which they learn.  Once they are beyond the door of our classroom, it’s on them to do the learning, and if they can’t handle it, it’s their own fault.  Why is this the case?

Learning is hard.

True learning is hard and messy and takes a lot of time.  Honestly, thinking about a hypothetical classroom in which my most struggling students receive A’s gives me a panic attack because of the chaos, coordination, and deep focus it would take for me to help them find success.  The same can be true about searching for keys in the dark: it can seem impossible, but if that’s where you need to focus your attention, it appears to be a waste looking anywhere else.

I’m not arguing that it is the job of the teacher to do everything for students.  I’m simply arguing that giving them a C on a paper with comments is not enough for a student to do better on the next paper.  Earning a D on a math test and saying “you need to study harder next time” doesn’t help a student prepare for the next test.

I don’t have an answer to this dilemma, but I think it is important to admit just how difficult learning actually is and take one step toward embracing the messiness that is teaching.  It is time to stop looking at test scores and expecting students will change on their own.  It is time to stop looking under the street light and expecting we will find our keys.

 

Choosing Whichever Comes Last

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We’ve all heard the commercial.  Buy a new vehicle and get some kind of 10 year/ 100,000 mile warranty.  When I was younger, I remember being curious about how it can be both 10 years and 100,000 miles under warranty only to find out that the warranty is covered until whichever happens first.  This makes sense for a business model, but I find this idea of “whichever comes first” creeps into our daily decision making and causes many of us to stop short of our true potential.

These thoughts came to me during a run.  I was out for a quick run and I told myself I was going to run for three songs then turn back around.  As the third song ended, I looked ahead and saw a side road about a quarter mile ahead.  I had the option to reach my goal of running for three songs or to push on for just a bit more.  I chose the latter.

The first issue many people face in reaching their potential is not setting a goal.  It is very unlikely to grow or even feel accomplished if you don’t have a goal in mind.  The second issue is that often people are satisfied with reaching their goal and, once they reach it, stay stagnant.

When given the option of 10 years or 100,000 miles I challenge you to choose whichever comes last.  Run to the next street, then one hill more, then finish that song.  Get your degree, get the job that you’ve dreamed of, but don’t ever stop striving for the next step.  You are bound to fail or come up short sometimes, but its the only way you know you gave it your all.  Learn from it, and get after it again.  Goals are great motivators, but they are just the beginning.  Greatness in school, in work, in relationships, and in life happen when you’re given two options and you choose whichever comes last and whichever takes the most work, reflecting on your journey and constantly preparing for the next challenge.

A Simple Switch Creates Curiosity

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How often have you heard this in a meeting with your department, a parent, or even in your own head:

“Kids are too dependent on calculators!  They can’t do anything in their head any more!

There are countless examples of 17 year old kids not being able to complete simple arithmetic which is distressing, but it is not the argument I am here to make.

Instead, I am concerned the fear of calculator dependence is negatively affecting the way teachers design lessons, structure discussion, and assess their students.  When that fear is in the forefront of our minds, we ask students to put the calculators away.  “We need to understand how to do this by hand before we use the calculators”, you might say.  I ask WHY?

Living in the 21st century, we have technology all around us and if I want to know the answer to something I am going to google it.  If the calculator can do the problem for us, why are we wasting our time?  Maybe you argue you have to understand the process to really get what’s going on.  Okay.  My argument becomes: make a question that demands I understand the process.

I am not arguing that understanding how to complete a problem by hand is a bad thing; instead, I’m arguing it’s a great thing!  But we need to make students feel the need to understand.  We need to show that the calculator can calculate, but only humans can think, dig deep, and discover connections.  We need to design problems where the calculator can’t solve it in one step or at all so it once again becomes a tool in the learning process rather than the process itself.


 

To make my point, I will use the example of a lesson devoted to adding and multiplying 2×2 matrices.

Method 1

  • Give students an example of two matrices adding together, then work on the problem with them and show them how it is done.  Leave time for questions.  Then give them a few problems to try before moving on.  Pause for questions.
  • Now give students an example of two matrices multiplying together.  Don’t forget to warn them that this one is tricky!  Then, go through the process with them and take questions.  Then give them a few problems to try and walk around to help answer questions.
  • Two days later, show them how to complete it on the calculator.

Method 2

  • Show students how to add two matrices together using the calculator.  Have them figure out the pattern. Takes about 1 minute.
  • Show students how to multiply two matrices together using the calculator.  Have them figure out the pattern.  Takes very long.  Eventually after giving some hints and gentle nudging, students (maybe not all of them) figure out the pattern and share it.  Without discussing right or wrong, put up examples for students to try by hand and then check them with the calculators.

 

An amazing thing happens in method 2.  Students begin to view the challenge as a puzzle to figure out rather than an “enter” button to be pressed for an answer.  If you pause long enough after kids first type the multiplication into the calculator someone will ask, “why does that work?”. THAT’S LIKE NEVER ASKED!! YOU WANT TO KNOW WHY?!?  It’s a cool feeling.

Even though we allowed the students an opportunity to struggle, we have to wonder what motivation students have to remember the meaning of a topic and how to complete it by hand.  Why can’t they just go back to using their calculator?  The truth is, they can….if you design problems with simple answers. (Assume these are 2 x 2 matrices below being multiplied)

Easy With Calculator

[  1    2 ]      [ -2   4 ]
[  7   -4 ]      [  0   8 ]         =     ?

Not Easy With Calculator

[ 2     4]       [3      x]                [4      12]
[ 1     1]       [-1    6]        =       [14    9 ]

 

A slight change in thinking renders the calculator powerless or, at most, a guess and check monster that drags out the process.  Instead of fearing the power of a calculator, we need to make kids jealous of the power it has and push them to ask why and how it works!  If we can do that, while creating challenging questions that force students to think deeply, we won’t have to let the fear of calculators cloud our judgement.  A simple switch can lead to more curiosity, discovery, and understanding for students.

The Struggle of Letting Them Struggle

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Today I was teaching an honors level class how to add and subtract rational expressions.  These are the students that have pushed themselves to take Algebra 2 over the summer to advance their potential opportunities as they become upperclassmen.  We’re talking serious students.

To begin the unit, I modeled my teaching after a Dan Meyer inspired idea and, although, not the most exciting thing, it helped students gain a little buy-in.  The previous day we had actually looked at adding fractions, discussed the similarities and differences of adding rational numbers and made significant progress.  So I thought.

We started today by checking over the homework and there were a lot of questions.  THERE SHOULD BE  A LOT OF QUESTIONS.  This is hard stuff and I was fortunate to have curious students, anxious to learn the content at a deep level and ask about their confusion.  If there aren’t any questions after the first day please don’t assume they’re good.  In fact, I could sense the frustration and kept telling them:

This is hard, but that’s why it’s better than many of the other things we look at.  It’s a chance for you to struggle and flex your creativity in solving problems.  Keep trying, keep failing, and keep asking questions.

Instead of moving on, I asked them if they would prefer to practice this a little more; they said yes.  I started by giving them a more simple problem.  Then, I increased the level of difficulty, and finally, gave them this harder problem:

__5__  +         4         –      2  
x(x+2)         -x – 2             5x

The students were able to handle the first two problems fairly well with asking only a few questions.  The final question pushed many of them outside of their comfort zone.  I let the students begin on their own, then as they began to ask questions I went in to help clarify some confusion.  But….it was rough.  I’m talking seriously rough.  Like, why would you do that, how can you even think that after the other things we looked at rough.

What I found myself doing as they asked me questions was becoming overwhelmed.  Student after student asked me questions, and I was having the most difficult time thinking about how to steer them in the right direction without showing them exactly how to do it.  At one point, I told a student:

Just hang on for a minute; I’m going to go through it with the class in a little bit.

It was at this point I realized the struggle of letting kids struggle.  Learning is messy, but when we have a quality problem of difficulty and students ambitious enough to struggle through it, we must ask ourselves whether we are ambitious enough to help them through the struggle rather than re-gain control and show them how it is done step-by-step.

As teachers, we need ways to encourage kids to take risks, while demonstrating that it is okay to do so.  We need to allow kids to make mistakes and fail, but we cannot be there to catch them as soon as it gets a little difficult.  Instead, we need to foster a classroom where creativity is encouraged and wrong answers are explored and shared.  In that moment, I wonder how the learning in my classroom would have changed if I grabbed three students’ notebooks, threw them under the document camera and, as a class, we discussed the math (wrong or right) that the student displayed.

I, like many other teachers, get caught up on the right answer rather than the process of getting there.  Because of this, I get frustrated when students are nowhere near the correct answer.  Instead, we need to embrace the messy process and the learning that is held within.  Maybe the student that struggled and got a problem wrong three times actually ends up learning and understanding the process at a deeper level than the student that got it right on the first attempt.

I ask you to be careful next time you get frustrated because no one in your class is finding the right answer.  Use the opportunity to talk about mistakes and continue to give them chances to flex their creativity and make mistakes.  It’s a struggle.

Using Snapchat in Math – Applying Similar Triangles

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Did you see their face swap?  Did you post it to your story?

Snapchat seems to be the only thing students (and a growing number of adults) worry about most of the day.  Kids are constantly finding creative ways to make funny pictures and send them to friends.  Adults are constantly posting how great their vacation or night on the town is going.  Being the tech-curious person I am, I decided to develop a different way to use it in a math classroom.  Check it out.


 

Lesson: Applying Similar Triangles

Course: Honors Geometry with mostly freshman and a handful of sophomores

Lesson Objective: Determine the height of the school building using only a ruler and Snapchat.

Standard: ACT CCRS (G 603); CCSS (G.SRT.B.5)

Previous Knowledge: By the time I taught this lesson, we had discussed congruent figures and had discussed that corresponding sides of similar triangles have the same scale factor.  We had also discussed the congruence and similarity theorems.  Students had worked out problems and we more or less comfortable with problems like those below.

SnapChat Pre-lesson

Lesson: Ideally, I wanted to take the students outside to measure the height of the building, but it did not work out for whatever reason.  I ended up giving the students the assignment as an extra credit opportunity and asked them to find the height of their house or some other object that was at least two stories tall.  I handed out the following, then did my best to explain to students what I wanted them to do.  I also created this template for students to organize their thoughts and upload a picture of them in action.

SnapChat Lesson

Results: Here are a few examples of student work.

Sample 1:

SnapChat 1I really liked this student’s work.  They seemed to understand the idea well.  I discussed with them to remember that he started a little below the ground (standing in the street).  Also, note this assignment may encourage to stand in the road for a pose.  Probably not the best…

Student’s Reflection:The only thing that can be improved is maybe try it in class too.”

Sample 2:  

SnapChat 2This student may have understood the general premise, but missed the importance of precision (at least in the drawing).

Student’s Reflection:We probably did it wrong. I thought it was kinda a cool project of how you can find the height of an object outside just by using snapchat and math. What can be improved for next time is showing an example of this project.”

Sample 3:

SnapChat 3We had covered many of those “if John has a 6ft shadow and the tree has a 15ft shadow…” problems and this student – along with others – believed they needed to use shadows to solve the problem.  Unlucky for them, they chose to do the project at sun-down.  Whoops.

Student Reflection: “It was nice to apply the concepts learned in class to real life situations. The most difficult part was trying to find where the shadow actually started and ended. Also, it was a challenge to get the whole picture in one entire frame. Next time, it would be better if we could do this kind of activity in school so that we can verify if we are doing it correctly.”

 


My Reflections

I really enjoyed the work that I got back from students.  I am curious if this was only a successful application because it was an extra credit assignment for honors students.  Most likely, it would need to be modeled or carefully explained if it was expected for all students of varying abilities to find success with it.

A question I have is whether the lesson could have been more or less effective with less specific direction.  I am a big fan of 3 Act lessons which leads students to ask the questions and determine what information is necessary to solve a problem.  Rather than giving them problems similar to the situation beforehand, could we somehow introduce the situation and have them draw the conclusions by themselves?  Even if we covered similar triangles deeply before the activity, maybe we could let students struggle with how the task could be completed with just Snapchat and a ruler.

Please let me know if you read this and have any suggestions!  I’d love to build it into an even more meaningful lesson for others to use.

p.s. What if snapchat somehow came up with an educational aspect of its application?  That’d be the bees knees.

Not Tech-Savvy? Be Tech-Curious.

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The year was 2006.  I was voted most tech-savvy among my friends on Myspace.  To that point in my life, I had mastered the portable CD player and creating a Green Bay Packers themed MySpace page.  I borrowed cell phones to call people and used Ask Jeeves past the point a person probably should have.  I don’t think “tech-savvy” was the correct phrase to describe me.

Fast forward six years to December 2012, my first year of teaching, when I purchased my first smart phone.  A co-worker and I both had new smart phones and we were excited to try these “apps” we heard so much about.  “You should check out this app called Snapchat.  All the kids are using it and it’s kind of fun!”.  Sure enough, it’s awesome, and over the past two years I have developed a lesson for students to learn about similar triangles through the use of Snapchat.

Somewhere around the same time, I began using this thing called Twitter.  You may have heard of it.  My college roommate’s pet tortoise had an account and tweeted about eating lettuce, and my friends used it to share tidbits of our 2012 roadtrip.  Little did I know four short years later I would be using it nearly every day to connect, share, and discover innovative ideas centered around education.  Something happened in the years from 2012 to 2016, and it was great.

I share these stories to show that I really started out on the other end of the “tech-savvy” spectrum.  I grew up with a computer, yes, but compared to my peers, I always seemed to lag a few years behind.  Although I would say I was not very tech-savvy when I entered teaching, I would argue that I quickly became “tech-curious“.  This has led to amazing growth for me as an educator and as a leader.

Being “tech-curious” is an old way of thinking wrapped with a 21st look.  Being tech-curious means that you are realistic about the world in which we live and are determined to find the best ways for students to learn and for peers to improve.

Long before the modern era of iPads, Chromebooks, Smartboards, and apps the best teachers still understood their students.  They understood pop-culture, they understood the most popular styles, and they understood the pulse of each generation; they understood what made their students tick.  Today is no different.  A teacher today understands that smart phones and other technology are part of our students’ lives, and it is our job to find a way meaningful way to bring it into our curriculum, pedagogy, and reflective practices.

I believe being tech-curious is important because using technology effectively offers students more opportunities to collaborate, publish their work, and personalize their learning.  If you are a new teacher or experienced, tech-savvy or not, you have the opportunity to say “YES!” to being tech-curious.”  The change is not instantaneous but curiosity begins to drive who you are as an educator, and you will constantly find yourself tweaking, refining, and asking “is there a better way?”.

Once again, I am labeled one of the “tech-savvy” teachers at my school.  This surely has more value than my 2006 Myspace page, yet I still don’t really feel I have earned the title.  “Savvy” really makes it sound like I have a clue what I’m doing.  Perhaps a title more fitting would be “Likely to try new technology tools, probably screw up, share successes and failures and then annoy people with reminders to tweet their work as well”.

Kind of long winded…perhaps I’ll just go with the title – tech-curious.

Being Proactive with Struggling Students

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There has been a growing number of people who have been sharing their “why I teach moment” on twitter.  Maybe you’ve seen them.  I know that I made of one those a while back too, but not remembering what I said before, I would currently say I teach because I enjoy the opportunity it gives me to learn and grow as a human being while allowing me to help others do the same.  It really comes down to helping others improve and be the best they can be.

Personally, I think I nailed the answer to that question.  Yet, in the thick of a day to day struggle “helping others improve” really looks like emails home, silent prayers, and a consistent struggle of getting another human to do ANYTHING productive.  Like seriously, you don’t need to face swap in my class.

Struggle is something that is inevitable in teaching and, when you stop to think about it, is an important part of helping others.  As a teacher, I preach the importance of struggle to my students; using positive strategies to overcome challenges is how I expect them to improve.  Although stressful to the everyday life of a teacher, the struggling students are the ones that push us to be great!

I realized after my first few years of teaching that working with struggling students was something I needed to improve.  With no time to waste, I decided to reach out to my struggling students, but more importantly, their parents.

Before the school year began, I looked up the historical grades of my incoming students.

An easy enough idea but something that was never provided to me at the start of the school year.  Because of this, I knew which students had struggled in previous years before I even met them.  This was good because I knew who may be my struggling learners, but potentially dangerous because now I had preconceived notions of these students as possibly lazy or dangerously unskilled.

I decided to email the parents of any student with a C-, D, or F and set up a conference during the first few weeks of school.

Some of you may be thinking – these are the students with uninvolved parents. Will they even show up?  The answer is YES!  Okay, the majority of emails sent do go unanswered, but when parents of a handful respond, you are given a great opportunity to meet them, get to know more about your students, and set clear expectations from the beginning.

From the meetings, I can tell you that parents are thrilled to see a teacher take the time to really know their son/ daughter from the beginning of the year.  Students, on the other hand, are often aloof because they don’t know if you’re someone that is out to get them or someone who really cares.  Despite a student that appears to be annoyed that a teacher has brought their parent into the mix before the school year even started, you will find as the year progresses you won them over day one.  Showing that you’re more than a teacher that simply goes through the motions to teach students, you win major brownie points.

I try to keep the meeting short – about 20 to 30 minutes.  Here are a few questions/ topics that I like to discuss:

To student (if present):

-tell me about one of your strengths, it doesn’t need to be in school
-why do you say that is a strength?
-how did you get good at it?
-tell me about one of your weaknesses, again it doesn’t need to be in school
-why do you think that is a weakness of yours?
-what do I need to do as a teacher to help you have a successful year?

To parent:

-tell me about your child (parents can run and run with this one, but it all helps paint a picture)
-what have you found your child struggles with in school?
-what strategies do you use at home to help him/ her with learning?

I usually end with a quick mini-lesson on growth mindset and the importance of using the growth mindset language at home and at school.  

Teaming up with the parents won’t mean the year will be perfect, but it does give you a more well-rounded view of the student and sets a positive tone for the start of the year.  I have found success by adding this to my teaching arsenal and hope it could work for you too!