Category Archives: Math

Shooting for the Stars

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On the first day back from Christmas break I decided to jump start the math brains of my Algebra 1 with Math Lab students.  I gave them the following problem:
You have 10 fewer quarters than dimes and 5 fewer nickels than quarters.  The total value of the coins is $4.75.  How many of each coin do you have?
I didn’t tell them until we were almost finished that this is an Algebra 2 Trig problem.  To solve it you need to write and solve a system of three equations with three variables.
I gave them about 10 minutes to work with their group without any help from me.  They could use any method – I even brought in some coins for those that need to “see” it.  You can see some of their efforts on the papers below.  A few solved it by “guess and check.”

After 10 minutes, I started prompting them with the following:
            Which do you have more of: quarters or dimes, nickels or quarters?
            Write an equation that relates the number of quarters and the
                    number of dimes.
            Do the same for nickels and quarters.
            Define variables for the number of each of the coins you have.
            What is the value of one quarter?  What is the value of all of the
                    quarters you have?
            Do the same for dimes and nickels.
            Write an equation that represents the value of all the coins you have.
We ended up finishing the problem together on the board.  And I think most of the students understood the steps and why it worked.
When we finished, I asked them why they thought I chose this problem for today.  They said:
            To get our brains going again.
            To show us that struggle is ok.
            To have us make mistakes and get the synapses firing.
            To encourage us to stick with a really hard problem and not give up.
Yay!  Growth mindset is sinking in!
Notice I didn’t tell you the answer.  Can you figure it out?  I believe in you!  You can do this!

Design-Thinking the Final Exam Review Process

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In an effort to put my students in the driver seat of our Geometry Exam review, and to help them see the connections of what we’ve studied this year, I tried a new review format that mimicked much of the design thinking activities I’ve engaged in, in the past. In advance of our week-long review period, I asked my students to make a list of everything we learned this year.  Then, on the first day of review I divided them into four groups and give each group a large board and lots of sticky notes.  They had 5 minutes to get everything they wrote down on the board (one concept per sticky note). The one rule throughout the process was that they could never refer to a chapter or section. They couldn’t say, “Section 4.3,” for example. They had to know what CONCEPT was covered in that chapter and use real math vocabulary as opposed to artificial chapters and sectioning. See video here.

Then all groups rotated.  Each group ended up at another group’s board full of stickies and they were instructed to group and organize them into larger topics, much like we’ve done with our design thinking work this year.  Again they had five minutes. See video here.

 Then they all rotated again. For this round they could add any stickies that were missing and they were also encouraged to make arrows connecting stickies to multiple topics. See video here.

 Finally they rotated again for five minutes with the same instructions. See video here.

 At the end they went back to their original board and digested what was in front of them. See final boards here, here, here and here.
I chose to do this for the following reasons:


  • So often our teaching, reviews and even assessments are organized by chapter.  In an effort to make sure we cover everything, our reviews and tests follow the chapters of the textbook: Two questions from chapter 1, then a few from chapter 2 and so on and so forth.  A predictable set of unrelated problems where a student might be able to (and a teacher definitely could) draw the lines where one chapter ends and another begins. Instead, I wanted these students to see the interconnectedness between the CONCEPTS (not chapters) we had learned.
  • I wanted them to see topics written by their peers that perhaps they didn’t think about.
  • I wanted them to be at the center of the review process.  Instead of me providing the review content, they had to generate it.  
This was a nice activity for this seventh period class.  They were up, active and very engaged in what they were doing.  Much moreso than if I had led the review or provided a review packet to complete (I did eventually do this).  When I asked how they liked this process, they were overall positive but they did say over and over that they wished I had given them the topics they needed to know.  This is not surprising given the cultural change we are trying to make in the Math department:  away from teacher-centered direct instruction and more toward student-centered discovery.  We still have a lot of work to do but this was a fun way to change up the Final Exam review process and continue to move us in that direction.  

A Mathematician’s Lament

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During our
math department PD (“How to Learn Math for Teachers”) we read an article by
Paul Lockhart titled A Mathematician’s
Lament.  In this article Lockhart
writes about a man who wakes from a terrible nightmare in which music education
and art education were made mandatory.
“We are helping our students become
more competitive in an increasingly sound-filled world.” Educators, school
systems, and the state are put in charge of this vital project. Studies are
commissioned, committees are formed, and decisions are made—all without the
advice or participation of a single working musician or composer. Since
musicians are known to set down their ideas in the form of sheet music, these
curious black dots and lines must constitute the “language of music.” It is
imperative that students become fluent in this language if they are to attain
any degree of musical competence; indeed, it would be ludicrous to expect a
child to sing a song or play an instrument without having a thorough grounding
in music notation and theory. Playing and listening to music, let alone
composing an original piece, are considered very advanced topics and are
generally put off until college, and more often graduate school.
Throughout
the article it becomes increasingly clear that the author is using this music
education analogy to articulate that mathematics instruction is a
“nightmare”.  The author even mentions,
“It is considered quite shameful if one’s third-grader hasn’t completely
memorized his circle of fifths,”—a clear parallel to third-graders’ memorization
of their multiplication tables.  The
author makes a similar analogy to art education in his “nightmare”, writing
that “The really excellent painters—the ones who know their colors and brushes
backwards and forwards—they get to the actual painting a little sooner. …
Nothing looks better that Advanced Paint-by-Numbers on a high school
transcript.”

This article
makes me feel sad and motivated at the same time.  I’m sad because his nightmare is
reality.  Math, which I would argue is a
very creative subject, has been whittled down to the memorization of basic
facts and formulas.  What if that was
done to art and music?  Wouldn’t that be
devastating?  Then why is it not seen as
devastating when its being done to math? 
A book I read over the summer mentioned that many people see math as a
finite subject, in the sense that they feel that everything about “math”   has already been figured out—all you have to
do is memorize the facts and formulas.  Where
is the fun in that?  What if art and
music were seen that way?  That all of
the songs had already been composed and all of the art had already been
created—all that was left to do was memorize the steps. 

Once I get
over my initial sadness, I feel motivated. 
I want to show my students the beauty and creativity that math has to
offer.   I want them to see math used in unconventional
ways and be curious about how they can apply math outside the confines of the
textbook and classroom. I want them to understand WHY certain formulas work—not
just that they do. I want them to see math as a tool for solving questions THEY
have, not as a set of facts for solving problems that have already been figured
out.  Sometimes I feel like I have more
questions than answers for how to make this a reality, but that’s my personal
charge.  I want to figure it out.

The Fibonacci Sequence = Math + Religion

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An Introduction to the Beauty of the Fibonacci Sequence

Recently we’ve been talking a lot about cross-curricular projects and classes: math + science, English + history, religion + history, etc.  What about math and religion?  Where is the intersection?

Last year, a colleague shared a video with me called “The Fingerprint of God”.  In the video the narrator shows many examples of the Fibonacci Sequence (and spiral) in nature, including spirals in nautilus shells matching the spiral curve of a wave matching the spiral curve of our galaxy, and comments that this spiral is like a fingerprint of a common creator.  It blew my mind when I first watched it.  God does math?  Math came from God?  How does this all work?

The Fingerprint of God Video

I desperately wanted my students to have that same enlightening moment so I tried to design a project/activity that would allow them to have that experience.  I asked Adam Chaffey to help me and together we planned our Fingerprint of God activity which spanned the Monday and Tuesday before Thanksgiving break.  The girls were definitely confused when I told them we would be doing a religion + math hybrid activity, but getting to “break the silos” and show them that math and religion can work together was exciting.  On the first day, I showed the girls the Fibonacci Sequence (my students had already “discovered” this earlier in the year) and some examples of the sequence and spiral in nature before sending them out to find examples of their own.  On the second day, Adam showed the Fingerprint of God video and we discussed how seeing this “fingerprint” affected our faith.  Finally, the students were tasked to create a song/video tying together all that we had talked about over the last two days.  (I’ve attached a couple below)


Things I have to figure out how to improve for next time: 

  1. Make it a more discovery-based project instead of a discussion-based project.  I think I was so excited about sharing this with the students that I front-loaded this activity too much.  The discussions were good, but it felt anti-climatic.  There didn’t seem to be any “ah-ha!” moments.  The students did everything we asked of them, but didn’t really figure anything out on their own.  How can I guide them without giving them too much?
  2. Figure out how to collaborate with another class/teacher more smoothly.  I know this was made more difficult by the fact that I have the same group of students 1st and 2nd period, but the logistics of collaborating with a two religion classes was a real challenge.  Adam and I both had girls that were in both of our classes in different periods and therefore ended up hearing the same information and doing the same activity multiple times–not ideal.

…any ideas?

Student Submissions:

…thanks for your help Adam!

Redesigning Algebra

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The math department has been rethinking our math classes and
how we can personalize the learning for our students.  Inspired by what we’ve
learned from Jo Boaler’s How to Learn Math forTeachers online course, we’ve decided to remove tracking
in our Algebra classes.  We want to show
students that mathematics is creative and open ended.  It’s
about identifying problems and working toward finding a solution.  It’s
collaborative learning, real life problem solving, and sharing this information
with others.  We’ve had the opportunity to spend time in a small group discussing
what we want Algebra (with the intention of expanding out to other classes) to
look like next year.  Our work has just
begun but I thought it might be helpful to others to share our progress so far.
We really want to encourage our students to develop a growth
mindset.  We believe that all students
can do math and we want them to believe this about themselves as well.  
“When I asked over 800 teacher leaders in the
US recently which educational
practices develop and maintain fixed mindset
ideas in students the number one reason given was ability grouping. I agree, it
is hard to give a stronger fixed mindset message to students than by putting
them into groups and telling them they have a certain ability.  In a
recent study Romero (2013) found that significantly more students developed
growth mindsets after they were placed into high track groups. Students who
develop fixed mindsets will often do anything they can to maintain the idea
that they are ‘smart’ which can make them vulnerable to unproductive learning
behaviors and the avoidance of challenging work or higher-level math courses.
 It is extremely important that schools communicate growth mindset
messages to students, and don’t limit students’ achievement by giving fixed
mindset messages through grouping and other practices. This is important for
equity, it is important for students of all levels, and it may be the key to
unlocking the potential of millions of students in mathematics.”

We’ve decided that our Algebra classes
will all be of mixed abilities.  Instead
of having Algebra with Math Lab, Algebra, and Algebra Honors, we will mix in all
of the students together.  We will have 3
sections of Algebra during one period with 4 teachers team teaching these
students.  The curriculum will be broken
up into units.  We are moving away from
chapters and sections and instead we will be presenting the concepts in an
interconnected way so students see how one concept relates to another (we are
thinking of having the girls create their own concept maps and continue to fill
them in as we progress through the curriculum). 
Each unit students will self-pace through the following items:

  • Presentation of concept – this can be through a video or a discovery activity
  • Practice Problems – this is where students demonstrate mastery on the basic idea of the
    concept (can they solve an equation in one variable, for example)
  • Topic Challenge – these
    are application problems where students work collaboratively in small groups to
    apply the concept they learned.  Many
    problems will be presented as an open ended idea/problem and students will
    develop the plan to solve it.
  • Unit Challenge – this is
    a larger project that will connect all or most of the concepts from the
    unit.  This will also be completed
    collaboratively.  Some unit challenges
    will be worked on throughout the unit while others will be completed at the end
    of the unit.  Here is where students can
    really see how mathematics is used, how it connects big ideas, and how it
    applies across disciplines.  During the
    topic and unit challenges students will be working collaboratively to solve
    these challenges with the teacher acting as a coach to help guide them along
    the process.
  • Unit Assessment –
    Students demonstrate mastery on these short, application based
    assessments. 
  • Honors Challenge – If a
    student wishes to have the honors designation on their transcript at the end of
    the year then they need to complete this challenge.  These are what we like to call the     “wicked problems” where students really need to
    persist and make connections in the data and apply it to concepts they may not
    have initially been taught.      In addition,
    students will have a peer tutoring requirement and need to create instructional
    videos for their peers.     

We are envisioning using 3 classrooms where the students will move
fluidly among each one based on the needs of that class period (or a portion of
that class).  The 4 teachers will be
moving among each room too each day (which means we will need to touch base at
the end of every day to determine who will be facilitating each room the next
day).
Room 1: “Traditional,”
direct instruction
Topic challenges
Practice problems, collaboration
Room 2:  Teacher coaching/guiding, student-centered learning
Unit challenge
Honors wicked problem
Room 3:  Teacher monitoring, no direct instruction
Assessment
Practice problems for mastery
We have outlined the units for the Algebra curriculum and are now
beginning to determine which concepts will go under each unit and create the
assignments.  There is still a lot of
work left to do but we are excited to continue moving forward!

Quadrilateral Music Videos

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The Quadrilateral Unit of Geometry is usually my least favorite to teach.  While there are very cool properties of quadrilaterals and their interconnectedness, it is SO MUCH information to teach (not to mention so much information for the students to learn and remember).  Last year, I came up with the idea to have my students make a music video highlighting the various properties of quadrilaterals.  This was inspired by watching videos like these at home with my two young children.

This year, I took it one step further.  I made this Unit a Project-Based-Unit in which I led with the Music Video project and instead of teaching them all of the properties, I gave them large pictures of each quadrilateral.  With many tools available to them (rulers, patty paper, protractors, etc) they were instructed to figure out what was special about each of these shapes through direct measurement and investigation.  They had two weeks of free work periods to find what was special about the shapes, to make lyrics that incorporated what they found for a song of their choosing and to record their music video.

Only two of the six groups finished on time.  This continues to remind me that our students have a really hard time with self-pacing and open, free time.  Fortunately, the groups that didn’t finish on time were still able to produce something by the time the “Share” period was over:  Some had to do a live performance because of technical difficulties, and some had videos that showed amazing potential but that were ultimately disappointing due to time running out.  Two groups asked at the last minute (i.e. 15 minutes before the project was due) if they could have an extension.  To be fair there were no extensions, instead I told them they had to pull something together and “Make it Work!” to reference Tim Gunn.  The girls worked like crazy in those last moments and I could physically see their adrenaline pumping in the form of sweat, labored breaths and rosy cheeks!  While I certainly didn’t want the end of the project to go this way, I think it’s important for them to know that some deadlines are hard and to feel the urgency and teamwork needed to pull things together quickly.

I highlight here two videos that I think were nicely done for two different reasons.  The first is simply just fun to watch.  They had great visuals, choreography, and they clearly had fun with this project.  While their lyrics had some issues, the video matched the spirit I was hoping to see in this project.  The second video shows successful teamwork.  These girls split up the work nicely and put the effort in at home (or even in the car!) to complete the video on time.  Their lyrics were awesome.  While I wish they had a bit more passion and creativity, there is something about the raw nature of this video that just makes me smile.  Finally I share this video that clearly has technical issues.  This group created lyrics to a song by Macklemore and Kesha’s song, Good Old Days.  Their lyrics are awesome and I love that they chose this modern song which combines melody and rap.  If they could have executed this, it would have been amazing.

Finally, the students did have a fairly traditional test in which they could have their lyrics printed to use as a study guide.  Overall, this was a fun project and I’ll definitely do again, but perhaps with more check-ins so that the kids can finish and execute their amazing ideas on time.

FlipGrid – Take 1

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Last Monday, I used FlipGrid in class for the first
time.  I first heard of this app from the
English Department when they used it for the National Day on Writing.  As well, people in some of the ed chats I
follow on Twitter have recommended it as a way to engage students and to allow
them to have their voices heard.  So I
thought I would give it a try.
In my Algebra 1 with Math Lab class, we watched Carol Dweck’s
10-minute video “The Power of Believing You Can Improve” as a part of our
continuing discussion on growth mindsets. 
I asked the students to write down three take-aways as they
watched.  Then I gave them about 20
minutes to use FlipGrid to create a short video with their response.  I told them they could work with others, but
they had to stay on task.  OK, that didn’t
work.  While few of the videos were focused
and addressed the prompt, others were just down-right silly.  I realize now they needed more direction.
So I’m going to try it again this coming week as a review
tool for the Chapter 5 Test.  Here’s my
plan:  Students will work in pairs to show
and explain how to solve inequalities.  Each
pair will have four different questions to answer.  My hope is that by explaining how they
arrived at their solutions and by watching others do the same they will gain
confidence in their ability to solve these types of questions.  I also plan to review their videos before
making them visible to the class.
I’ll let you know how it turns out.  Stay tuned for my follow-up blog post “FlipGrid
– Take 2.”

A “Not Yet” Quiz

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I made a mistake today.  I had intended to give a quiz today and while I put it on our chapter schedule, I forgot to put it on Schoology.  It wouldn’t have been fair to give them the quiz, but I still wanted to make sure that they knew the material and have a way to assess that.

Inspired by Carol Dweck’s Ted Talk The Power of Yet, I decided to give a “Not Yet” quiz.  In this quiz everyone was required to get 100%.  Some students would get a 100% on the first try, others would need to retake the quiz (and continue to retake the quiz) again and again until they got a 100%.

The quiz I would have created had I not forgotten to schedule it would have looked like this:

1.  Quadratic function in vertex form:
     a.  Find the vertex
     b.  Sketch a graph by hand

2.  Quadratic function in standard form:
     a.  Find the vertex
     b.  Sketch a graph by hand

I partnered the students and instructed everyone to make a quiz for their partner and pass it to them.  They would take this very short quiz and check that they did it right by graphing it on their graphing calculator to verify their vertex was correct and that the pictures matched.  If they got it right, they were done.  If they made a mistake, they would work with their partner to find the error and they would ask their partner for a new quiz.  Partners were instructed to praise their peers when this happened with messages of “It’s great that you made a mistake!  Your brain just grew!  Let me make you another quiz.”

The kids did this and were so kind and supportive of each other.  One student, who took a few attempts to get her 100% at the end said, “I made it!  I feel so accomplished.”

Awesome.  Best Mistake Ever.

(Note:  I am aware that this will cause some grade inflation.  I won’t do this all the time but I do believe that the kids learned from this experience and got a big confidence boost.  As well as a lesson in the reward of grit and perseverance.)

Bring on the Challenge!

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This post is a bit of an update from a prior post of mine in which my attempt to foster a growth mindset through challenge majorly failed (to say the least).  Fast forward two chapters and my students and I have evolved.  After listening to their concerns, I changed my testing strategy.  I didn’t want to give up on the challenge, but I realized I had perhaps done too much too soon.  As a compromise I gave the students two options for the Chapter test:  [1] Take a standard test with fairly expected problems or [2] Complete an application-style test which consists of one large-scale real-life problem that would be different than anything they had seen but which would use the concepts taught in the chapter.  The carrot?  Option [2] would be completed in groups of 2-3 and have use of any and all resources (internet, books, notes, you name it).  Option [1] would be completed individually with limited resources (calculator and study guide).

Last chapter, when I first implemented this approach, 8 out of my 80+ Algebra 2 students went for the application style test.  Each of those students worked their tails off in the 45 minute period but all ultimately figured it out.  It was a small success!  This chapter, an amazing 23 students opted for the more challenging test, 11 being from my Period 4 class which had originally shown the most resistance to challenge and the idea that to truly grow one must struggle.  Again, it was a 45 minute period filled with sweat, hustle and adrenaline and again they all got to the end and correctly solved the problem.  This is remarkable considering that in this chapter, where we covered systems of equations and inequalities, I took my own risk and gave them a linear programming problem.  While related to systems of inequalities, I didn’t teach this method and these types of problems are complicated even when taught well.  It just goes to show that if we set our expectations high, our girls will rise to the challenge.

I feel like we’ve started a revolution in the Math Department and it’s so exciting.  When students make mistakes I celebrate and tell them, “your brain just grew!”  I regularly hear my students telling each other, “I believe in you!  You’ve got this!  Growth Mindset!”  While they are gently mocking me with these statements, I have to believe that some of these ideas are sinking in.  And now, I see them choosing challenge over predictability.  It’s awesome and exciting to be part of it.

Finally, I leave you with some encouraging feedback from a unit evaluation I gave immediately following today’s test.  Yes, I’m focusing on the positives (and no they weren’t all positive) but the balance of positive to negative comments is shifting and I’m riding that momentum:


I love doing challenging problem, like the performance task.

The group test really helped open my mind on the chapter and I get to view things differently.

Keep challenging us

I am starting to have a better growth mindset and open to challenges.

the performance task was kind of hard, but it made me work harder

Scatter Plots

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I was really excited to teach my Algebra Honors students about scatter plots and lines of best fit today.  I have been looking forward to this lesson because it allows me to teach statistics, which I love, to my students but also because we were going to use for the first time the HP Prime graphing calculator app on their iPads.  I knew I wanted to make this lesson interactive working with data they collected that would be interesting to them.  I found a PowerPoint that had 20 pictures of celebrities and students were supposed to guess their ages.  The students loved seeing pictures of Justin Bieber, Kim Kardashian, Nick Jonas, etc. and guessing their ages.  It definitely grabbed their interest!  I was feeling really confident about this lesson.  I then gave students the actual ages of the celebrities.  Students now had two rows of data and it was time to have some fun with the HP Prime.

Mistake #1:  I had thought that the HP Prime app was automatically downloaded on all freshmen iPads but I never checked with the students prior to class if this actually was the case.  Some students had it but many did not.  I thought this was easily fixed, the app was free, the students could just download it.  For the majority of students this worked but there were 3 students that were unable to download any apps. They unfortunately had to just watch on with a partner.

Mistake #2:  The HP Prime calculator is very different from the TI-84 calculator and there is a learning curve.  I had budgeted some time to play around with the basic functions of the calculator before we started the activity.  I was thinking of the first time I used the HP Prime at a conference last summer and how I was intimidated to push buttons randomly so I waited for the instructor to walk me through step by step initially.  What I didn’t anticipate was students feeling much more comfortable with new technology and have them start pushing all the buttons and not paying much attention to my tutorial.  This resulted in many students asking me how to do the same things over and over.

Mistake #3:  Once I got everyone back on track and stressed the importance of staying with me while we entered data we started creating our scatter plot.  Students did great entering their two columns of data but as soon as I showed them how to graph the points they immediately got excited about the touch screen of the graph and started going off in a bunch of different directions.  This resulted in a similar replay of mistake #2.  It was fine at first that they were playing around with zooming in on their window but when it became time to sketch a graph with their finger many instinctively hit OK which then saved their line – even if it was a line they didn’t feel represented their data well.  Rather than ask for help they continued to draw multiple lines which I had to walk around and delete.

While we did finish the lesson and I think I demonstrated to my students that the HP Prime calculator is superior to the TI-84, the lesson did not go at all how I envisioned or wanted.  In hindsight I think it might have worked out better if I gave students written step-by-step instructions for how to enter the data and do the functions working in their groups.  I then could have circulated around the room to see how things were going and troubleshoot if needed.  The girls are way more comfortable with technology than I gave them credit for and I think they could have figured out what I wanted them to do working together.  We then could have had a class discussion on the different lines of fit each student created and discussed the similarities of them.  Overall I would do things differently but I am glad that we approached scatter plots this way rather than just completing problems in the textbook in a more traditional sense.