Category Archives: Math

Exploring Patterns to Spark Curiosity

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My lofty (and somewhat nebulous) goal this year is to teach students to be curious.  This is particularly challenging in a subject that is often presented as “Memorize all of these patterns (formulas) that people figured out hundreds of years ago.  You will need them for a future test or class, but (probably) never again.”  With all of this memorization, what is there to be curious about?  And since that future test and future class are still looming realities, where is the time for curiosity?

Enter Algebra Readiness.  My Algebra Readiness class is designed to be a bridge for students between their middle school math experience and their high school math experience.  My goals are simple.  

  1. Understand that math is a creative subject.  

  2. Apply Growth Mindset strategies to shift your math experience.

  3. Explain foundational math concepts in a variety of ways (i.e. verbally, visually, numerically, etc.)

In an effort to show the inherent creativity in mathematics, I created a unit in which we are exploring patterns 一 lots and lots of patterns.  The first pattern we looked at was the Hailstone Sequence.  The Hailstone Sequence starts with any whole number and follows this pattern: If the number is even, divide it by 2.  If the number is odd, multiply it by 3 and add 1.  Keep applying these rules until the pattern appears to end.

 

For example: If you start with 7, the sequence looks like …

7 – 22 – 11 – 34 – 17 – 52 – 26 – 13 – 40 – 20 – 10 – 5 – 16 – 8 – 4 – 2 – 1 – 4 – 2 – 1 …

 

This is called a “Hailstone Sequence” because hailstones go up and down like this – they start in a cloud as drops of rainwater, then they are pushed higher in the atmosphere by wind where they freeze, sometimes several times, before eventually falling back to Earth.  These number sequences are called hailstone sequences because they go up and down like hailstones.  In 1937 a mathematician proposed his conjecture for these Hailstone Sequences, that for any number you pick, if you follow the procedure enough times you will eventually get to 1. Since then lots of mathematicians have been trying to prove or disprove it. So far every number that has been tried has followed his conjecture, and powerful computers have checked enormous numbers of numbers, but no one knows if there is a big number out there that might break the rule.   So this is classified as an unsolved problem in mathematics.

This, in and of itself, is pretty cool.  Since most problems we give students in math are problems that we (as teachers) already know the answer to, giving them an “unsolved problem” shifts the dynamic away from the cliche “sage on the stage”.  Students were simply asked to pick a starting number and run the sequence.  Then repeat this enough times until you are convinced of something.  I didn’t tell them where to start.  I didn’t tell them where to end.  I simply told them to figure it out.  As expected, some of them were super frustrated (as I would have been).  But they asked questions, they tried different starting points, they collaborated with each other and they all eventually came to the same conclusion as that mathematician did 80+ years ago.  But the coolest part was that after we concluded our time with this pattern I had multiple students ask me, “Ms. Levesque, are there any other unsolved problems in math that we can do?”  

So my question is this: In a world where students can get answers to most of their questions through a quick Google search, how do we teach them to be curious?


What Do We Teach?

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 I just read this article, Does America’s Math Curriculum Add Up? It says we should teach more “data analysis and problem solving; and linear equations.” This made me think of my freshman year in high school in 1961. The teacher told us we were only taking algebra to get into college. Most of us would never put algebra to any practical use, but all of us would benefit from going to college. I obviously remembered that conversation from nearly 60 years ago. Sounds like it is still true today. The article goes on to tell how difficult it is to get the public and many colleges to accept this new math reality. They want algebra and geometry, just like they had in high school.

This made me think about other subjects, like history. There are so many historical names, places, and dates that every students should know. Would any U.S. History course be complete without a unit on the great Wobblies movement, founded in 1905, or the Battle of Bunker Hill, Korean War, 1952. How embarrassing to not know these names and dates. 

There is so much historical information available now that it is mind boggling. Robert Caro spent ten years researching for his 3000 page magnum opus The Years of Lyndon Johnson. If students need to learn data analysis in math, they certainly need to learn research skills in their history classes. This boils down to data analysis. Math deals mostly with numbers. History deals mostly with events. But, when it comes down to it, both are trying to make sense of data. In both cases it is not the data that needs to be learned, it is the crunching of the data. 

I am no math expert, but I do know that it is impossible to cover the California history content standards in the time allotted, especially if we take seriously the need for research and analysis. We have only so much time available. This year, it seems like so much less time. How do we spend this time? How do we divide the covering of information with the deep dives that involve real research and analysis. How much time do we spend remembering, and how much time do we spend understanding, applying, analyzing, evaluating, and creating? This is such a basic question. Why is it so far from being resolved?


Cross-Curricular Projects: How??

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I always want to show the students how math is connected to other subjects and the real world.  Frankly, one of my big dreams for our math program is not to get more girls to calculus (but YAY! if that happens).  Rather my dream is that it starts to blow up the idea of “time” and “school day” enough that we can start to incorporate cross-curricular time in the day.  How cool would it be if instead of teaching dimensional analysis in physics and again in algebra, we taught it concurrently in the context of a bigger problem?  But how do we inch towards this?

This year I have tried two cross-curricular projects:

  1. Math (Financial Algebra) + College and Career: Ginger helped me design a unit about the realities of paying for college.  She taught the introductory lessons (on block periods) and popped in as I continued this mini unit through the following week.  The feedback was very positive and many thought that all of their junior classmates should have access to this unit too.
  2. Math (Algebra Readiness) + Religion: Adam and I got our classes together in the Innovation Center to explore examples of the Fibonacci Sequence in nature and discuss the implications.  Is this mathematical pattern proof of a common creator?  This was really fun, but the feedback that I got from my freshmen students was that it was awkward to work with a different class of students (in this case a mixed-gender class of seniors), especially for just one class period.  My personal feedback is that the lesson we designed should’ve been spread out over a week or more — it was really dense.
Moving forward I have questions:
  1. What is the most logistically efficient way to do a unit/project with another teacher/department?  Working with Ginger was easier than with Adam (no offense Adam!) simply because she did not have a classroom full of students that were expected to collaborate with mine.  I know my colleagues all have prep periods (which would eliminate the concern of having to join classes) but that’s a big ask and I’m just not there yet.
  2. Does a cross-curricular course make more sense than a cross-curricular lesson or project?  Yes, if the only concern is finding overlapping time and a similar student population.  No, because creating a new course feels like a huge barrier to cross-curricular work.  Also, if we keep increasing our course offerings do they eventually get watered down?  
  3. Anyone want to try another cross-curricular project/unit with me?

Bravery and The Growth Mindset: Why is the Pool so Different from School?

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Today marks the end of the second week of spring sports.  As the JV swim coach I have a love/exhausted relationship with this time of year, but when I step on deck and get to engage with our students in this different arena, I am reminded why I keep coming back to it.

Now in my third season, my JV roster has increased from about 30 swimmers (in each of the first two seasons) to 50 swimmers.  For a reason I can’t quite identify, there was a huge influx of new swimmers this year, particularly freshmen and sophomores.  And, to be clear, when I say “new” I mean, “no swimming experience whatsoever”.

When I share my shock with people, the common reaction is, “Well then cut some girls.”  There are about 15 – 20 girls this year that are swimming competitively for the very first time and standard operating procedure would dictate that I cut them (suspending the reality that swimming is advertised as a “no cut” sport).  But I can’t do it.  I cannot bring myself to cut them.

First, we cannot minimize the vulnerable position a teenage girl is putting herself into by walking out in a swim suit in front of her peers and jumping in to try a sport that she has no idea how to do.  In a world that feels increasingly judgmental (especially of our young people), how and why are they able to muster this kind of bravery? … (and here is my connecting point) … how can I get them to be this brave in math class?  More often than not, when my students see a problem they do not know how to do, they shut down.  They say they don’t know how.  They say they were never taught.  They question my validity as a teacher for daring to put something in front of them that I did not explicitly teach them how to do.  And yet in the pool, they jump right in.

I am not exaggerating when I tell you, this group of swimmers knows next to nothing about how to swim.  Only a few know how to swim freestyle, a smaller few have attempted the other strokes before this season, and none of them know how to dive or turn.  THEY FAIL CONSTANTLY.  And yet, they keep trying.  They take every word I say to heart and I watch in awe as they try to incorporate my advice to the best of their ability.  I can actually see their brains churning as they try to figure out how the heck to do a flip turn and not come up in the adjacent lane.  They keep failing and keep trying again.  Over and over.  Belly flop after belly flop until finally they dive in clean.  It is everything I want to see in my math class.  How can I get them to apply to school the same growth mindset they have in swimming?

I am so proud of their bravery.  Cut them?  No way.

¿Cómo Se Dice, “Maths” en Español?: A Collaborative Vlog

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WATCH THIS VIDEO, YOUR WORLD WILL CHANGE….(probably not, but just watch because I put work into it) 🙂


So I don’t like math. 
I never have. I like being competitive and getting points on Alludo though,
which is why I ended up signing myself up for an online maths course for math
teachers. (Yes, I said “maths”). I really didn’t have any intention of getting anything out of this
course and I really did just take it to get more Alludo points, because who
needs sleep? I also wanted to know what my students go through on a regular
basis to see if I can adjust my curriculum according to their needs and how
they learn, so I gave maths a whirl.


 Much to my surprise, it wasn’t really a course about math,
(maths) in the videos that I watched (on double speed to save time), but rather
a philosophy on teaching and learning that can be applied to various realms and
curricula. While I watched the videos I noticed language pertaining to “fixed
mindset” and “growth mindset”, and the concept of “yet.”
 Students in these
videos stated “I’m not good at math” “I’m just not a math person” “This isn’t
how my brain works”, and I began to make some connections: I noticed many of my
own students in Spanish saying similar things “I’m just no good at languages” “My
parents weren’t good at language, so neither am I” “I had bad teachers in middle
school, so I’m not very good”. I started to create a correlation between
Spanish teaching and learning and math, and when I approached Lesley Schooler
about this connection, she agreed that there might be some similarities. Like
math, students in Spanish are afraid to make mistakes, they put an obtrusive
filter on producing and speaking the language because they’re afraid they will
make mistakes and not be precise, so they just don’t speak. I found the
neurological studies in the math online course through Stanford to be
fascinating with the connections that I could make with my students in Spanish
class. I realized that the material needs to be slower and more attainable for
students, and not penalize mistakes, but point out mistakes, and allow students
to correct them (this is where the brain grows) and they shouldn’t be marked
down for making mistakes, but they should fix them so that they enjoy the process
of learning. The videos present the idea of the journey and process in learning. Students try and think aloud and defend and explain their findings rather than simply right and wrong and they move on. I am inspired to incorporate more of this style into my classes. The conundrum that I’m having is, while this is a great way to encourage learning and brain growth, I want to know that I am preparing students for college, and upper level learning where there might still be an institutionalized, systematic fixed mindset that they also need to be able to navigate. Would I be doing my students a disservice if I don’t require precision as well? I’m not sure. 
I shared these thoughts with Lesley and we made other
insights and connections as well. I think this is a good course to take, even
if you don’t teach math because a lot of the principles can be applied in many
fields of study. Also I actually learned some math, and I don’t hate it as much
as before. Yay!

What Do You Mean: No More Worksheets???

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When this email promoting a webinar appeared in
my inbox, I was intrigued … and terrified. 
What do you mean “reconsider using worksheets”?  These are the life blood of math classes.  How else are students supposed to practice
basic skills?
Intrigued, I signed up for the webinar.  Terrified, I listened carefully.
The webinar “Why We Should Reconsider Using Worksheets (And
What We Should Be Doing Instead)” was offered by Robert Kaplinsky, a math
educator from Southern California.  I
follow him on Twitter (@robertkaplinsky) so I had a good sense of his approach to mathematics
education.
In his webinar he presented his concerns with the use of
worksheets:

Full disclosure – I have used worksheets as busy work.  I’ll admit it, it’s an easy sub plan.  But I also have experienced his other three
points.  When students are given a
worksheet, they go into “git ‘er done” mode often without deeply understanding
the concepts behind the problems.
While there can be a role for traditional worksheets in some
situations, Kaplinsky promotes as an alternative “open middle problems.”  Here’s an example:
Notice that there is not just one solution.  There are, in fact, many.  This type of problem encourages students to
uncover the mathematical concepts behind the problem.  These are best done in groups where students
can talk through the problems.  Then,
sharing among the groups reveals the variety of solutions and leads the students
to a deeper understanding of the concept.
Here are the benefits he sees to these types of problems:
No longer terrified, but fully intrigued, I decided to try this
with my Algebra 2 Trig classes.  On
September 25 my Period 5 sophomores would be taking the NWEA MAP test with Sara
Anderson’s sophomores while I would have a block period with our combined 34
juniors!  Instead of giving them an 80-minute
study hall (which I’m sure would not have resulted in much studying), I decided
to test out the open middle problems.

I projected these problems in front of the class while groups
armed with whiteboards went at it:

While these problems do not reflect Algebra 2
Trig content, I thought that something familiar would be a good way to introduce
this type of problem.
What I found most interesting was that the students very
quickly focused in on the concepts behind the problems.  For the inequality, they knew that the first
box had to be a negative number.  Instead
of having them practice row after row of “multiplying or dividing an inequality
by a negative number” worksheet problems, this open middle problem brought to
the surface and had them apply what they had learned in middle school: multiplication
or division of an inequality by a negative number reverses the direction of the
inequality sign.
I also enjoyed watching them talk about math.  So often students just want to write out the
steps without explaining their thinking.
I had found Kaplinsky’s “benefits” to be true.
Here’s a link to his website with tons of open
middle problems.
The following week I had planned to teach piecewise
functions – a challenging topic for many students as it requires them to graph
only a “piece” of a function.  And if it’s
a linear function where the domain does not included the y-intercept, well, you can just forget about that!
I did my usual flipped “video the night before” and “exercises
in class the next day” on Tuesday.  But on the block period, I gave them my own version of open middle piecewise functions problems:

What I saw was students referring back to their work from
the previous day, explaining the concepts to each other, asking questions, and
being creative.  Multiple concepts beyond piecewise functions were
reinforced: slope, different types of functions, what a function is, how to
restrict the domain, among others. 
Success!
While I will still continue to use worksheets as an optional
review for skills-based problems, I am looking for other opportunities to
integrate open middle problems into the curriculum.
I am also wondering if this sort of problem could be used in
disciplines other than math.  Any ideas?

Mathematical Mindsets Conference

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One of the exciting benefits of our new math program is the relationship we’ve developed with Jo Boaler and the staff at YouCubed.org.  It’s been wonderful to get their validation of what we are doing and to have the opportunity to present at their Mathematical Mindsets conferences.  Kristina Levesque and I were invited back this past September to share out at their leadership conference about the changes we are making in how we teach our girls math.

While it is always wonderful to connect with other math educators, the best part of presenting at this conference is being able to take advantage of this professional development without having to pay the $1,000 per person fee.  😁  Every time we’ve attended this conference we have been able to take away something else that we can implement in our program.

This past conference we had the opportunity to play with a math problem involving a series of four figures of different dots.

With all YouCubed problems we were asked what do we wonder about each figure shown and what conjectures can we make for the the number of dots in figure 100.  Kristina and I loved playing around with the dots and developing an equation to determine what figure 100 would look like (without drawing that figure!).  We knew this would fit well into Topic 9 of our Algebra curriculum.  We added it in as a Topic Challenge and love that it shows students a visual representation of a quadratic equation.  One of the main takeaways from this conference was that engaging with math in multiple ways is so important for learning.  When we can add in the visual representation it can become a status equalizer.  Students at different levels can contribute and engage with the problem which is so important.  Our goal is to incorporate more visual representations in our math program and one of our plans moving forward is to find a visual for as many units as possible.

Now it’s your turn.  How many dots do you think will be in figure 100?  What would the figure look like?

Twelve Dots, Infinite Possibilities

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I often joke that my job is 50% math teacher, 50% motivational speaker because most students come into my classroom with their minds made up … they are not a math person.  Don’t even get me started on the ridiculousness of that statement 😉  Other than the two periods I teach in our new math program, I teach Algebra Readiness to freshmen (and some select sophomores) that do not demonstrate a strong math foundation and I teach Financial Algebra to juniors that are not tracking towards Calculus (for whatever reason).  I tell you this to illustrate that I teach a particular cross-section of students that are VERY mathematically-adverse. Many of them have felt beaten down by this beautiful subject that I love … so how do I bridge that gap?


I intentionally do a lot of “low-floor, high-ceiling” number puzzles to engage my students at the beginning of class.  For example, “Use four 4s in an expression to equal any number between 1 and 20.” Everyone can engage in this type of problem, even if it is just with addition, but some will venture to include exponents, radicals and factorials.  I opened up my classes with puzzles like this everyday this week and last, but never were they more engaged than when I put the numbers down and replaced them with dots. Yes. Dots.


I projected this image for about 30 seconds and asked the students to figure out how many dots are in the picture without counting each one individually.  


Look at how many different ways my students saw this problem!



I used this activity to build culture and illustrate the things that I value in my math classroom:


  1. Everybody can “do math”.  When we restrict “math” to memorizing formulas and solving equations, it is boring and so challenging.  Remember the last time you had to memorize something that didn’t matter to you at all? It’s nearly impossible.  Math is about finding patterns! When we show examples of math as a creative and visual subject that is all about figuring out patterns, it opens up the content to students that closed the door on it a long time ago.
  2. Visual math is the best math.  When we teach students to see math visually they are using more pathways in their brain (think “right brain vs left brain”) and learning at a deeper level.  If you are interested in reading more about the importance of visuals in mathematics, this is a great article: Seeing As Understanding: The Importance of Visual Mathematics for Our Brain and Learning
  3. There is more than one way to solve a problem.  12 dots and almost that many different ways of seeing the arrangement — in one class!  When I showed the original dot diagram after we share strategies, the students enjoy trying to see it in the different ways their classmates saw it.  There’s a lot of “Oh yeah!”’s and “I see it now!”s. Many students believe there is only one way to solve a problem — the way the teacher did it. The problem with that is if the student sees it differently than the teacher, they immediately think they are wrong, 
  4. Listening to multiple perspectives helps everyone to have a better understanding of the problem.  Have you ever looked at an optical illusion and searched for a face when all you can see is a vase?  Sometimes it doesn’t matter how long you stare at something, be it an illusion or a math problem, you just.  Can’t. See it. That is often what happens in math. The teacher explains one way to solve a problem and you just can’t see it that way.  When we encourage students to share their thoughts and strategies, it opens up possibilities for understanding. I looked back at the 11 strategies on the board and reminded the students that if all we hear all year is 2 or 3 of these perspectives, we will be failing to make the learning accessible to everyone.


How did you see the dot diagram?

What activities or discussions do you do/have in your class to build culture at the beginning of the year?

Math Program version 2.0

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Last year was an exciting and exhausting year for the math department.  We implemented a huge change to how we teach Algebra and it resulted in moments of happiness and frustration.  While we are extremely proud of the first version of the program, we always knew it was only the first iteration.  We knew we would go back and reflect on what worked and add modifications for the next version.

On Wednesday we had the opportunity to share out about our program to faculty and staff who wanted to learn more.  It was wonderful to see almost every department represented and many staff members as well.  Here is a link to the presentation we shared if you were unable to attend our session but would like to learn more. 

We initially started out with 3 goals for our program: 

  • de-track students
  • increase student agency
  • encourage collaboration and communication
Our program overall was successful in implementing these goals and we are continually refining what we’ve created.  We’ve created more opportunities for break out direct instruction every week.  We are tracking students’ progress more than ever through exit tickets, goal setting meetings, check ins with their lead teacher, and attendance at Math Power Hour.  In addition we’ve modified our Algebra Challenge Exam for incoming freshmen to make it mastery based.  Freshmen also had the opportunity to come in over the summer and get a head start on the Algebra curriculum and over 50 took advantage of this.  As a result we have over half of the freshmen already into the Algebra curriculum which will increase the likelihood of them beginning Geometry this year or give them the opportunity to slow down and focus on depth if needed.  
There are a lot of misconceptions about our program and I think they can be summarized here.  
It was wonderful to have the opportunity to share out with our community something we are really proud of and I hope other departments will do the same.  I would love to have the chance to learn more in depth about some of our other classes and programs.  

What if we treated academics like athletics?

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Hear me out.
As long as I’ve been teaching French, the rhetoric has always been that learning a language should be taught similarly to how we teach sports to children: You aren’t going to be Michael Jordan the first time you pick up a ball, utter a French sentence, play an instrument, or try to solve a difficult problem. 
These skills require honing through lots and lots of practice, dedication, repetition, encouragement, and passion from the instructor/coach. I’ve always tried to instill this mantra in the minds of my students, but it never hurts to keep coming back to it.
So when I opened my browser while lounging over a cup of coffee this morning, I was delighted to see an NPR interview with former-NFL player-turned mathematics Ph.D. candidate John Urschel (From The Gridiron To Multigrid Algorithms In ‘Mind And Matter’) touting his new book Mind and Matter. Then I fell into a rabbit hole, reading all I could about how Urschel is trying to change the way we talk about teaching and learning in American education. Then I stumbled upon his NY Times Opinion column “Math Teachers Should Be More Like Football Coaches,” and despite not teaching math, I greatly identified with much of what he’s saying. I ventured even further into the rabbit hole by perusing his Twitter account and had to share his story with some of my math teacher friends.
I feel so empowered to keep doing what we’re doing with role models like Urschel in both our and our students’ lives. He is giving students of all different walks of life access to an educational outlook they might not have otherwise held. And these lines really spoke to me, reaffirming my own goals as an educator of both young men and women from so many different backgrounds, social milieus, religions, family structures, political beliefs, mother tongues, home countries, you name it:

I recognize that because I’m a mathematician at MIT and I play professional football, I’m in the spotlight. And I have a responsibility to use this platform to show people the beauty of mathematics. To show people playing in the NFL, this isn’t your way out. You can do something mathematics. You can do something in STEM, even if you don’t necessarily look like what the majority of people in that field look like. 

And I have to say, okay, if you look at the field of mathematics, if you look at elite American mathematicians, there’s almost no African Americans. There aren’t many of us in PhD programs, there’s not many of us as undergrads, and what you’re sort of left with is the sad realization that there are brilliant young minds being born into this country that are somehow being lost — either because of the household they’re born into, or their socioeconomic situations, or sort of the social culture in their community. And this isn’t just a disservice to them, this is a disservice to us as a country.

Even at the end of the school year when I feel completely out of sorts, exhausted, and at times even ineffective as an educator, I needed this bit of reaffirmation to help me refocus and guide the students into the final stretch, the fourth quarter, the bottom of the ninth, or any other applicable sports metaphor you’d like to insert here:

A growing body of research shows that students are affected by more than just the quality of a lesson plan. They also respond to the passion of their teachers and the engagement of their peers, and they seek a sense of purpose. They benefit from specific instructions, constant feedback and a culture of learning that encourages resilience in the face of failure — not unlike a football practice. There are many ways to be an effective teacher, just as there are many ways to be an effective coach. But all good teachers, like good coaches, communicate that they care about your goals.

This speaks to what so many of us have been doing all year. And I wanted to end the year on a note of appreciation and gratitude for being part of such an innovative and supportive community of passionate educators!