Category Archives: Math

Algebra – Initial Thoughts

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Now that we have survived the first 5 weeks of our new Algebra program, I thought it would be helpful to write down some initial thoughts on how things are going.  It has been really exciting to see something that was just a crazy idea last year morph into a real program that we are already really proud of now.  There have been some adjustments already and there are some things we are still trying to figure out but overall the feedback from our students and the teachers involved has been positive.  

Here’s what has worked well:
  • We spent the first two and a half weeks establishing a group culture of the class.  We discussed growth mindset and had all of our students take the “How to Learn Math for Students” course.  We spent time discussing the messages from this course with our students. 
  • There are a lot of moving parts to this course (online learning through Carnegie Learning, collaborative activities, how Schoology is organized, GoFormative exit tickets, topic guides, etc.) which can be overwhelming if everything was explained at once.  Instead we used the first two and a half weeks to chunk things out and explain each one separately before putting it all together. 
  • We had all of our students take the Mathematic Diagnostic Testing Project High School Readiness test.  We will also test them again when they complete the Algebra program.  This will be one way we measure the success of this program.  
  • Students are truly moving at their own pace.  Those who are familiar with the concepts in the first topic have moved quickly through it and already taken their first assessment.  We also have some students who need the extra time to really master the content and are moving at the right pace for them.  This wouldn’t happen in a traditional classroom.  
  • The four teachers who are implementing this program (shout out to Mary Beth Dittrich, Kristina Levesque, and Christy Marin!) work well together.  We communicate often with one another.  We are all flexible with what we need to do each class period (even when it’s decided 10 minutes before class).  We do not always agree with one another but we are comfortable speaking our thoughts.  We are able to discuss things openly and honestly to come to a solution that is best for our students.  
  • Carnegie Learning is a good tool for our online component of the program.  The problems are rich and require students to be engaged.  The reporting section allows us as teachers to determine where they are struggling and what the students need extra support with.  Students are also able to go back into the program in review mode without losing their saved data which is a huge plus. 
  • The new furniture is amazing.  This program would not be what it is now with the old desks.  The new furniture allows students to work well in groups, using their smaller white boards when needed.  The individual desks work well for the online learning or assessments.  Plus the colors of the chairs brighten up the room and change the learning environment.  
This is just a short list of what has worked well.  There are so many more including hearing students comment that they really understood something or watching them work well with their groups on collaborative activities.  
We do have some challenges:
  • We have 4 teachers for over 94 students in period 3.  Sometimes we need an additional teacher (or two!) in the classroom to work with students, particularly those who are struggling.  In the past these students would have had 2 periods of math in a smaller class setting.  One requirement we’ve established in our program is that a teacher signs off on the collaborative activities.  When doing so we ask each student in the group to explain to us what they have just learned.  This is a great way for us to gauge student understanding and to see that everyone is responsible for their own learning.  However at times multiple groups need to be signed off and we just don’t have enough teachers to get around to everyone.  
  • Space is an issue.  The new furniture is great but having separate classrooms isn’t as ideal.  As teachers sometimes we are in one classroom but can’t necessarily see what is going on in other classrooms.  We try and pop in and out of the classrooms during the period but it’s not always possible.  If we had a larger, open space we would be able to  spread ourselves out to help the students more.  
  • The 45 minute class periods are tough.  It’s not quite enough time for students to transition to more than one activity once you account for taking attendance in the beginning, explaining which classrooms are for each activity, and ending a few minutes early for the exit ticket. The block periods work better for this program.  
  • As teachers we need to find more time to meet to discuss how things are working and what adjustments we need to make.  It’s hard finding that time even with a common 7th period prep.  Teachers have other classes they are teaching and need to use this prep for those classes at times.  We also have so much to discuss that even if we used every 7th period, there’s still always more to talk about.  This collaboration is so important though.  Yesterday we graded our first topic assessment together and this was necessary to make sure we are all grading the same way.  But it takes time which has become a precious commodity.  
Despite the challenges I am so excited for what we are doing.  I am loving watching our students engage in mathematics in a totally different way.  I’m inspired by the dedication and commitment from my colleagues.  I know that we will continue to reflect on what we are doing and revise it to continuously improve as we strive to meet the needs of all of our students.  
We would love you to come visit the organized chaos of our Algebra program.  We meet periods 3 and 4 in rooms 2, 5, 6, and MacLab.  Come by anytime as we would love to hear your feedback on what you observe and any ideas you may have to improve on this program.  

Fun Friday in Math

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One of our goals in the math department this year is to foster a culture of math that is collaborative, accessible and most importantly, fun!  Several times throughout the year we have planned Fun Fridays with today being our first one.  During these days, students from different classes work together in small groups on one challenging problem during their usual class period.  Students are purposefully placed in mixed groups to help encourage students to leverage their unique skill-sets and perspectives to come together to solve a problem they may otherwise not be able to solve on their own.  We hope students experience the value of teamwork and problem-solving on these Fun Fridays and that they realize that math is much broader than what they might currently be studying in their current math class.  We also hope this gives them a chance to practice grit, resilience, and resourcefulness.  Our whole department is working hard to promote the growth mindset with our students and this is one way we are modeling it.

Today was our first Fun Friday.  Each group received a bag with 27 cubes in total – 9 different colors, 3 of each color.  They were instructed to create a 3x3x3 cube with each of the 9 colors represented on each face, similar to a 3-D Sudoku puzzle. 

 

This was a problem that had multiple ways to solve or strategies to implement.  It was so fun as teachers to watch students work together to try and solve this.  They would think they had it solved only to realize their last cube wouldn’t work.  Many students tried and tried again with some finding success in the end and others not being able to solve it. 

 





































As students were working you could hear teachers telling students that it was OK if they didn’t solve it.  Their brains were growing as they learned from their mistakes!  Synapses were firing!  

 It was great to listen to their different strategies and ways of approaching the problem.  
 
Mr. Cushing and Mr. Schooler stopped by to try their hand at solving it.

   


They were modeling to our students the struggle of working on a problem that they couldn’t immediate see the solution to and also the joy that comes when you finally solve it.  
  
We would love to invite any faculty and staff members to come join us on our next Fun Friday October 5th.  It will take place in every period in rooms 2, 5, 6, 7, & 8.  Our first Fun Friday was a success and we are already excited about planning our next one!

Intersecting Bravery and Math

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It’s been another rocky start for me in Algebra 2 with Trigonometry.  [Algebra 2 is a predominantly Junior (some Sophomore and rare Freshman) level class.  These students had to meet a prerequisite to get into this class, so as you might imagine they are somewhat accustomed to receiving good grades and doing relatively well in Math.]  Just as with last year, I’m receiving fierce pushback against my style of teaching, which is more “guide on the side” than “sage on the stage.”  Students are  complaining that they aren’t being taught and that they are having to figure out too many things on their own.  Students are complaining that my assessments are too unpredictable and that they can’t handle the anxiety they feel at not knowing exactly on what they will be tested.  Students who had been really good at “doing school math class” up until now are suddenly finding that their usual strategies aren’t working.

There’s so much to unpack in this.  I am teaching them, just not quite as directly as they’d like.  But, perhaps more complicated is that we seem to have different definitions of what math is and what math should be.  I see math as an opportunity for problem solving and critical thinking, as a language and a set of tools that can be used to chip away at unsolvable problems.  My students, on the other hand, see math as a collection of formulas to be memorized and quantities to be calculated, and problems with only right or wrong answers.

To try to bridge this disconnect, and to open a dialogue with them about why I structure the course the way I do (it’s for their own good, I promise!), I planned an entire class period of no math but rather an opportunity for sharing.  I showed them Reshma Saujani’s 2016 Ted Talk, “Teach Girls Bravery, not Perfection.”  There’s so much I love about this talk.  I’ve probably watched it over twenty times and each time I watch it I get chills of excitement.  One of my favorite lines is:

We have to socialize our girls to be comfortable with imperfection, and we’ve got to do it now. …We have to teach them to be brave in schools and early in their careers, when it has the most potential to impact their lives and the lives of others, and we have to show them that they will be loved and accepted not for being perfect but for being courageous. 

I chose this video because I want them to know that they can achieve anything they imagine, but that, socially, they have been wired to back down from many of those dreams, out of caution and out of fear of failure.  In my classroom, in my small way, I want to give them opportunities, safe opportunities, to practice taking risks, practice failing so that it becomes common for them.  I want them to become so good at failing that they will bravely and courageously pursue any dream they have.

I then showed them a short video from Jo Boaler’s Youcubed.org about what math truly is and what it means to be successful in math.  A favorite line from this video is:

No matter how math is presented in school, you should know that math is not about memorization and it is not about calculations.  Math is a much broader subject about ideas, visualization, connections.  And don’t think test scores or grades define who you are or what you can do.  You can do anything.

I love looking around the room when this line is presented (it’s spoken by a fellow student in the video) and it’s such a powerful moment.  I think for many of our students it may be the first time they’ve heard the idea that math isn’t about calculations or formulas.  So many of our students have reduced math to this.

I gave the students surveys before and after watching these videos and we also had time for discussion (some periods more spirited than others) after each video.  This anecdotal feedback is so fascinating.  Below are some results and anecdotes that are particularly interesting,

The above graphic was from the Before Survey.  I find it fascinating how many students don’t see math as a vehicle for problem solving and critical thinking.  For many students, they don’t believe that math is a subject to be figured out.  They believe the only way to learn math is to be taught it by someone more knowledgeable than them.  I am working hard to show my students that much of math can be discovered through experimentation and deduced from very simple principles.

I was heartened to see so many students willing to change, adapt and try a new style of learning, especially after our discussion and the videos.  These comments are from the after survey:

At first, my initial response reflected on my irritation of not understanding the style of teaching. I have been taught that you need to understand the material the second you’re taught it. I’ve learned that it’s okay to not understand at first and feel lost because I will eventually know what is being taught.



I understand more why this class is the way it is, which is good but at first it is a bit of a shock to have things be so different. However, I think that this growth mindset concept is a good one to teach students, especially with all the research backing it up. It should be useful in the future.



I used to be primarily fixed mindset, but after watching these videos I feel as though I can learn to be better at math

I would say now that learning teamwork, critical thinking, and problem solving are things that I will value more during class time.



After learning about the research, I now realize that math is not all about math, but rather about how you look at the math material and your mindset going into it.


I know I will struggle somewhat in this class, but I think my mindset has grown a little more

I am more open to new topics and categories of math now that I haven’t learned before. I am motivated to learn new things and try harder on problems I think I can’t do rather than give up on them.

I really liked these videos and they helped me to realize that I have the ability to solve every math problem as long as I work hard and don’t give up. I like the way the course is going and I’m open to this new style of learning.

I think that this style of teaching will be more challenging in teh moment but more beneficial in the long run.

I used to be primarily fixed mindset, but after watching these videos I feel as though I can learn to be better at math

I appreciated the many students who brought up the very real burden of grades:

I want to be able to make mistakes and learn from them but I am too worried to fall back in grades. I think that it will be difficult to me to forget about the consequences of failing.

I think it’s good that we are learning to have a different mindset but it isn’t very realistic in today’s high school classes

I think that these people in the ted talks and math videos are making a valid point about having to fail to succeed but the way our school system works is that every mistake we make goes straight into the grade book, which makes making any mistake seem like the end of the world.

I think that struggling in math is essential to learning, and I think we will have the opportunity to do that in class. I think that this course may have some benefit for my particular learning style, but I am nervous about what will be graded in this class.

I like the idea of having the growth mindset, but its not always practical. In the video, they said that grades don’t define what you can do. This is sometimes true, but you were given a grade based on what you proved you could do. I think this is interesting, but it’s not realistic and it’s going to take a lot more than just videos to change the school and grading system from how it is now.

I like the idea of our minds becoming stronger, but I feel like the challenges involved cause a lot of anxiety because the grading system penalizes our mistakes.



I still feel hesitant. I believe it will help one-day, but our generation has not been raised me this, which for me, makes it harder for me to become minded about this new style of learning

Other than getting rid of grades altogether (is that a possibility for us?), I don’t have a good answer for this.  I do try to give my students lots of opportunity to fail without a huge grade impact (ability to remaster quizzes and group assessments are two of my favorites) but their concerns are real.

Finally, these two comments really hit me, as I think they are evidence of everything Reshma Saujani talks about:

After the first lesson I cried because I was so stressed about failing. I felt like you valued the benifets of failing but overlooked the grade aspect of it. I feel like this data is accurate though, and I relate. My mother is a perfectionist and stresses the importance of work being perfect. My father treats me like I’m so perfect and precious. Failure isn’t an option. Failing is hard for me.



As much as I’d like to believe in growth mindset I believe it will be difficult to change my view on math as a junior in high school.

Sometimes I think I’m crazy to try to change the way our students learn Math, especially for those who are about to walk out our doors.  I keep thinking it will be so much better in a few years once our Algebra program takes off and students walk into my classes already equipped with bravery and a growth mindset.  Other times I see it as my calling and I believe that it’s never too late to learn and practice courage and risk taking.

I’d love to hear how this is playing out in other departments.  My students tell me all the time that Math is just different:  you need to be taught math.  Critical reasoning is important in History or English but not in a Math classroom.  And the graphic above highlights this perception.  How do you see our girls being brave (or not) and how can I channel that bravery into my math classroom?

Individual Interviews vs. Typical Test

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What I did:

1.  I collaborated with my students to develop a rubric for their understanding of the Chapter 10 content.  The left side lists the four main skills from this chapter and the top shows the criteria necessary to earn each score.

2.  I created a schedule (in a shared Google Sheet) for students to sign up for their 10-minute appointment window.  Most were during class time, but some were before school or during a lunch/break/X-block.

3.  I created a sheet (for myself) with sample questions from each section so that I had questions ready to ask during the interview.  This also allowed me to ask different students different questions.  Students also used this sheet when they assessed themselves at the end because I broke the questions in to sections that matched the topics on the rubric.
4.  I interviewed each student for 7 minutes (with a 3 minute buffer).  During that time, I asked them questions about each topic (not just “Solve this…” but “Why did you do that step?”), students assessed themselves on the rubric, then I assessed them (as they were walking back to get the next student).
Why I did it: 

A few chapters ago, I asked students to make me a video (similar to a Khan Academy video) explaining the concepts we learned in that chapter (exponential expressions).  They did this in place of a typical written test.  The feedback was very positive and since then their ability to simplify exponential expressions has really impressed me.  Whereas a lot of math concepts tend to get forgotten after the chapter ends, these concepts seemed to stick.



PROS of the interview process: 

  1. I watched students hold each other to a higher standard when they were explaining and reviewing the concepts.
  2. The students said that they studied harder/better knowing they would have to explain themselves.
  3. Students were way more involved in the assessment process.
  4. I was finished grading as soon as the interviews were finished–there was nothing to take home!

CONS of the interview process:

  1. I had to limit myself to 7 minutes per student which limited the number of questions I could ask each student.
  2. The timing also limited how long we could spend on any given section.  If a student didn’t understand the concept, I would eventually have to move them along to the next section.
  3. I do not have the same amount of evidence to support my score of the student as I would have on a typical test.  Since the majority of the “work” for this assessment was a conversation (not every problem was worked out) I do not have a lot of evidence to base my score of the student on.  I wrote little notes and the students worked through some problems on paper (and they assessed themselves), but I definitely do not have written evidence to support every score.

Next steps:

I am going to continue with an “alternative assessment” for our last chapter, but my students and I are redesigning it with some changes.  Here are our ideas so far:
  1. Students pair up and assess each other.  This will allow students to have more time in their “interview” since multiple interviews will be going on at one time.
  2. Students will have the opportunity to re-explain a concept for a higher score on the rubric.

Do you have any ideas for us to consider?

What Happens When We (Teachers) Let Go?

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We recently covered Area in Geometry and instead of employing a usual teaching format (follow lessons in book with nightly homework in between) I tried a “blended classroom” approach.  I started the chapter with a project, a BIG project.  Students were told they had to design a house, make the blueprints, determine the total surface area of the house, and put it on the market (not really on the market, but they had to decide where they were going to build the house, come up with an address and based on “comps” in that neighborhood (i.e. houses with the same total area or square footage) set a price and create a realtor’s brochure).  The house had to have at least five of the shapes we had studied this year:  square, rectangle, triangle, parallelogram, rhombus, kite, trapezoid, any regular polygon, and circle.  It had to have at least two stories and they had 1 acre of land on which to build.

They had 2.5 weeks to get this done.  Class time, with the exception of the first 10 minutes or so was theirs to do what they wanted.  They could work anywhere on campus they wanted (I created a google form where they could tell me where they’d be) and in the course of those 2.5 weeks, they had to check in with me two times each week.   Homework was minimal.  Each night they had to derive, visually, a new area formula that I assigned.  We spent the first 10 minutes of each class sharing our derivations and after that the time was theirs to work.

Let me tell you a little about this class:  it’s a bear to teach.  I don’t know if it’s the mix of girls I have or the fact that it’s a seventh period class but the group as a whole would rather be pretty much anywhere else but my class, doing anything else but Geometry on a daily basis.  Fun, right?  Part of me thought it was a big risk to give these seemingly unmotivated girls so much freedom, and during seventh period no less!  The other part of me thought it really couldn’t get much worse, and we all needed a change of pace.

Once the project started, I was pleasantly surprised by a few things:  [1] I could always find the students when I would circulate the campus and, amazingly, they were usually excitedly working.  It seemed I had finally found something that they liked!  [2] Over and over again I heard, “I was working on this last night with my dad” or “My mom was helping me look up the area formulas.”  To me this is awesome.  I gave them something that they shared, voluntarily, with their families at home! [3] the nightly homework assignments in which they had to derive the area formulas on their own were amazing:  creative and thoughtful, they showed me that the students were really thinking and using their reasoning skills to figure them out.  Here are a few samples:

Look at all of that amazing thinking and reasoning?

When the project finished, I noticed a couple of other small observations that may or may not mean anything.  The project was due at the beginning of class on Wednesday.  That day no one was absent and everyone was excited, I might even say proud, to hand in their project.  No one needed an extension, and at first glance, they all looked amazing.

I gave an evaluation about how they liked this project and this way of learning a topic.  Here are some of the results:

 I found this next result particularly fascinating:  look how hard they worked!  I totally let go, I removed the usual scaffolding of a typical chapter and the micromanaged assignments, and they actually worked harder.

Finally, I share with you some anecdotes from the evaluation:

I enjoyed being able to work anywhere it gave me a different vibe then sitting in the classroom like I do in every other class. I liked how we needed to check in with you. I wish we could’ve done it as a group if we liked to, and needs a little bit more time maybe another 3 days since it’s a lot of small projects into one. It was a nice break from writing down on paper taking notes doing regular problems. It also opens me to think about how to find the are of a shapes I didn’t know and how I could use the area of shapes I already knew to solve for the shape. I think it would’ve been better to go over all the areas we might need first then give the project.

I really enjoyed this project, which is saying a lot because math is neither my strong suit nor my passion. This was a great opportunity for me to learn about what Area is and how it is solved while simultaneously incorporating my creativity. Instead of sticking to the routine of homework, quizzes, and tests, I was able to show the little skill I have in math. I also am better at learning when the material is applied to real-life situations. It gave me a lot of confidence as well.

I loved the project

I loved how this is something relevant that we can actually use in real life

It was actually fun making the floor plans and the brochure

it was a fun engaging project that made me look forward to this class.

There were some skeptics.  And, you can see from above there was a mixed review when it came to the question of whether they learned what they needed to know by doing this project.  What was so interesting to me were the comments by students who thought they didn’t learn what they needed to simply because I hadn’t traditionally taught them:

I didn’t really enjoy it because it didn’t help my understanding of area, it only tested my abilities to look up and follow the formulas. I didn’t really know what I was doing during this project and I would not feel confident or prepared if I took a quiz on area after completing this project

I feel like the project was a nice way to get immersed into the area section of geometry. However, I don’t think that the project was very efficient in teaching me the most about area. I also don’t think it prepared me very much for any tests/ quizzes we will have concerning area in the future.

I did learn how to do the area and stuff but I often found myself looking up the formula for specific shapes. I feel I might of retained the information better if we would of learned it from the book but I did like the project.

I really liked this project and i thought that it was the most enjoyable project. I think that learning about are and perimeter could have been accomplished in a different way. I think that i would have learned about area better if we did followed the book and did it like a normal class. (Normal meaning that the teacher teaches us the lesson and then the students do the problems in class and for homework and ask the teacher questions).

I would argue that having to look up and figure out formulas yourself makes you much more likely to remember them.  How can we convince our students that learning doesn’t need to happen with the teacher at the board and students passively receiving the information?  And, that perhaps this isn’t the best way to learn?  How do we teach our girls to be resourceful and to value that resourcefulness?  This has been a really awesome experiment for me and despite some mixed reviews, I truly believe that this is the way that real, sustaining, learning happens.  I look forward to continuing this free, blended style of project based learning and hope to see the culture shift where my students learn to trust this process and trust themselves and all that they’re capable of.

AP Statistics Research Day is Thursday, April 19th: Come Join Us!

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AP Statistics Research Day is right around the corner.  We are excited to share our research with you and we thank those of you planning to come to the Inner Court during Period 2 or during an x-block on this Thursday, April 19th.  To help you understand the amount of work these students have done, here’s how we completed this task:

August:  Right after learning about different types of variables
(quantitative vs. categorical), we made a very brief survey that we
posted on Schoology so that we could have some data to play with.
September:  Students got comfortable with Minitab software by creating
an analysis report of our brief survey. They learned how to create all of the
graphs and numerical summaries we had learned about in the first few chapters
of our book.
October:  After learning about study design, students submitted a proposal and
initial survey for their own research project.
November:  Students revised survey based on my feedback and consulted with an
“expert”. The expert was a teacher or family member who knew more about their
topic than I did. The expert reviewed their survey and gave feedback. Perhaps
some of you served as experts!
January:  Students selected their sample using the methods learned about
in this course. Most students conducted simple random samples or stratified
random samples. A small number of students conducted convenience samples
due to the nature of their research question.
February:  Students sent out surveys and monitored responses.  Some needed to
sample again due to low response.
March:  Students built a formal analysis report of their findings in four installments,
coinciding with the inferential statistics we were learning as part of our curriculum.
April: Students designed their posters.



I share these details because I hope that someday this symposium grows beyond AP Statistics:  Imagine “Carondelet Research Day” in which students from classes in many different disciplines showcase year-long research.  So, please do come join us on Thursday and when you do, think about how you might be able to join us next year, as a contributor and let’s continue to give our students opportunity to put what they are learning into action!  


Here is the list of topics and presenters:

2018 AP Statistics Research Day Presentations



April 19th, 2018


Inner Court

Carondelet High School

How Closely do Books Relate to Intelligence?  Will Buckley

How Many Students Take Classes Outside of the Normal Classroom?  Alyssa Dunn

Procrastination Station:  What causes most students to procrastinate on their work?  Caroline Edwards

Advantages of Playing Sports Leah Engle

Technology vs. Students  Jayla Fernandez

Is there a relationship between the number of colleges a student applies to and his or her GPA?
Makena Garrehy

Who Gets Injured the Most?  Briana Granados

How Do High School Students Form Their Political Views?  Courtney Lally

The Relationship between Sleep and Academic Performance  Ryan Leopold

What Factors Impact my Sleeping Habits?  Bryson Patterson

AP Classes and Their Effect on College and Future Jobs  Sarah Peterson

Exercise Habits of High School Students:  What are the workout and exercise habits of high school athletes and non-athletes?  Justin Pratt

How Much is Stress Really Affecting Teens?  Sarah Shaughnessy

Visual and Performing Arts in College:  Does being in a higher number of clubs or classes determine if you are going to continue with your VPA in college?  Cecily Sotomayor

What’s On Your Plate?  An observational study of the correlation between Carondelet Students and the Garaventa Cuisine  Sophia Veran Sanchez

Caffeine Consumption in Corporate Offices and High School Campuses  Adam Wong

Every x has a story

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Recently I attended training on a platform our Math Department has chosen as a tool in their Algebra curriculum next year.  Part of the training involved, unsurprisingly, doing some math problems. I would choose x and y as my variables as I wrote equations, but I noticed none of the math teachers did. They would choose s, b, or h.   “Don’t you guys use x anymore?”, I asked. “No”, I was told, “math can and should tell a story”. The choice of the variable can help tell the story, and so, when asked to determine how many books a certain number of student read in so many hours, s as the variable for student continues the story, as does h for hours and b for books. I was delighted because there is nothing I like more than storytelling. And I agree, the story and the problem make more sense when the variable name has some kinship to reality.

Then I learned the x and y axis can be called the independent and dependent variables.  Oh, my! What stories one can tell when you consider things that way.  Now, I know music and math are intertwined, and I had a breakthrough when I was listening to Malcolm Gladwell’s podcast Revisionist History on country music http://revisionisthistory.com/episodes/16-the-king-of-tears.  Country music songwriters tell some of the best stories. These songs have a dependent and independent variable! The amount of bad luck and heartbreak are variables dependent on the actions of the independent singers and their choices.
bl+hb =(s*c)/2.  Thank you Math Department. Every x does have a story.

Growth Mindset Isn’t Only About Math

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I am a bad writer.  I tell
myself this consciously or subconsciously every time I start to compose
something.  I will excitedly embrace the
most challenging math problem you can find, but staring at a blank Word
document paralyzes me.  In fact it has
taken me at least two weeks to get around to writing this blog post.  I reread an email multiple times before I hit
“send.”  Are my grammar and spelling
correct?  Did I get my message across clearly?  Do I sound like an idiot?  My Twitter posts are composed in my head long
before my fingers touch the screen.  Even
after posting I often question my words as I picture the broad audience they
reach.
I was thinking about this several months ago and came to the
conclusion that I’m probably not as bad of a writer as I think I am.  I’ve written a few blog posts.  I regularly communicate with parents,
students, and colleagues via email.  Maybe
people are just being polite, but I’ve never gotten any negative feedback about
what I’ve written.  I think it’s just
that I don’t like to write.  It doesn’t
come easy to me.  It takes me a lot of
time to come up with the ideas and even longer to figure out the best way to
express them.  It’s hard and painful work
for me.  It’s a struggle.  And I don’t think I’m very good at it.  (Right now I’m even questioning all the
contractions I’m using!)
Then it dawned on me. 
These are some of the exact phrases my Algebra 1 with Math Lab students
use when speaking about math.  “I’m not a
math person.”  “I’ve never done well in
math.”  “It’s too hard.”  All year I’ve been trying to nurture in them
a growth mindset: “Mistakes are good.”  “Your
brain is growing.”  And their favorite: “Synapses
are firing.”  I want to instill in them grit,
determination, positive thinking, and risk-taking.  I want them to see the beauty and creativity
of mathematics.  Well, of course, I
do.  I love math, and I want them to love
it, too.
I now realize what a hypocrite I have been.  Here I am encouraging a growth mindset in my
students with regard to math, while happily embracing (and nurturing) a fixed
mindset it when it comes to my own writing.
Recently I was “forced” to face my discomfort when one of my
husband’s colleagues asked me to write an article for a journal she edits.  Laura is one of four female scientists at
Lawrence Livermore National Lab who visited Carondelet four years ago at our
last Career Fair.  They came to talk with
our students about career opportunities for women in physics.  We had just embarked on our Physics 9 program
and I was excited to show them the steps we were taking to get girls into
science early.  More recently at a social
event, I was telling Laura about the flipped math classroom and some of the
changes we are planning for our Algebra 1 program.  It was after this that she invited me share
what was going on at CHS in an upcoming issue of the APS Forum on Physics and Society.
I was surprised, honored, and excited.  I’ve never been “published.”  At the same time, I was terrified.  Not only did I have to write, which is painful
enough in itself, but who knows who would be reading the article and what they
would think of me.  This is a journal for
scientists who “do” physics.  I just
teach math.  They’re so much smarter than
me.  What do they care about high
school?  I was overwhelmed with
self-doubt, but I knew that this was something I should do.
The first draft was bad – almost an embarrassment.  Too short, not enough detail, and a little
cheesy.  I spent two weeks psyching myself
up for the rewrite.  I set aside a whole
day.  I armed myself with a growth
mindset.  I told myself that I could do
this – that I could produce a quality product, knowing full well that it would
not be easy.  If the words weren’t
working, I took a short break, but knew that I was coming back to it.  Grit, determination, perseverance.  After about eight hours, it was done and I
felt pretty good about it.
So, what did I learn? 
I need to practice what I preach. 
If I expect my students to have a growth mindset about math, I need to
have a growth mindset about the things I find challenging.  I’ve come to realize that developing a growth
mindset is a process.  Just because I
wrote one article doesn’t mean I love writing. 
I need to keep working on it.  I
also know that I need to continue to be patient with my students.  To compassionately encourage them.  To stand beside them as they face their math
fears.  Because I, too, know what it is
like to work at something that is hard for you. 
But I also know the joy of completing the task and taking a few baby
steps of growth.
Synapses are firing!

Math Department Celebrates Pi Day

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The Math department celebrated Pi Day on 3/14 (you may remember that pi, an irrational number, is commonly rounded to 3.14).  We started our day together in a Math Department meeting where Mary Beth shared this beautiful Pi-related prayer:

The beauty of pi, in part, is that it puts infinity within reach. Even young children get this. The digits of pi never end and never show a pattern. They go on forever, seemingly at random—except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of pi.


We then departed and had many varied adventures with our students to celebrate pi.  Here are some highlights:

  • Amanda’s Geometry class was visited by Anne-Marie and four of her AP Calculus students.  These advanced students had recently used Calculus to find the volume of coca cola in a classic bottle.  The Geometry students were able to do the same task by using their knowledge of the volume of a cylinder (pi*r^2*height) to come up with a coarse estimate for the coca cola volume.  The Calculus students served as coaches throughout this project and at the end introduced the younger students to the idea of curve-fitting, area under a curve, and volumes of revolution to get a more sophisticated estimate.  It was a great activity to celebrate the many applications of pi in a collaborative way.  Both young and older students admitted they learned something new from each other.

  • Cathy’s Geometry classes derived pi empirically, by measuring the circumference and radius of various circles in the classroom and inner court and working backward using the circumference formula to derive pi.  Look how close they got!
  • Mary Beth not only treated the Faculty and Staff to many delicious pies, but she also encouraged the students to participate in a pi-tastic scavenger hunt.  See the tasks here.  
Until next year…

An Easy Way to “Flip” The Classroom Without Losing Student Discovery

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Flipped classrooms are very popular.  I see a lot of benefit in letting students have agency over their learning:  letting them learn when and where they want, and at a pace that works for them.  My main critique of the flipped classroom is that it removes most, if not all, student discovery.  Generally, videos are made that tell the students what they need to know.  Often, there’s little struggle or opportunity for students to figure things out. 

One strategy I’ve started doing is using a “flipped” approach in my classes where the “flip” is not when the learning happens but rather the roles the students and I play.  For example, today one of our class objectives was:  “I can simplify rational expressions.”  Instead of teaching the students how to do this, I put an example and the answer on the board and gave the students five minutes in their group to figure out how the answer was found.  They reverse engineered the method for simplifying rational expressions.  At the end of five minutes, the class had to teach me how to simplify.  I wrote down what they told me, synthesized their strategies and we moved on to the next skill.

This picture shows two skills done in this style.  The first is after they’ve taught me so you see the problem worked out and the strategies they came up with to do problems like these.  The second skill was the one they were working on when I took the picture.  They were working backwards to figure out how I got the answer I did. 

As I was doing this today, numerous students told me that “I like learning this way” and “This is fun” and “This is really making me think.”  I really like using this strategy when the skills can be deduced from prior knowledge.  There are of course times when they need me to introduce new concepts and there are problems that they simply cannot reverse engineer.  However, as much as we can we should require our students to discover, be resourceful and figure things out. 

I know that this works nicely in a Math class but I’d love to hear if this approach could be used in other departments and if any of us are already trying this type of “flipped” classroom.