Category Archives: Algebra

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?


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.  

Growing brains in Algebra Honors

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My Algebra Honors students had a test last Friday on systems of equations in 2 variables.  We had done a lot of work on this chapter and I knew that I didn’t want to assign them more of the same types of problems on our review day Wednesday.  Instead I had the class work in groups of 3 and I gave each group a sheet of paper with two word problems written out.  I told them that they had 50 minutes to solve these two problems using any strategy they wanted.  They just had to justify their answer with math.  I told the class that they are expected to explain their problem to the class toward the end of the period.

What I didn’t tell the class was that the problems involved 3 variables (an Algebra 2 concept) and that we hadn’t learned how to solve systems of equations in 3 variables.  Instead I told them that I believed in them and I knew they would be able to solve them. 
The students got to work solving the problems.  As I walked around the groups I loved seeing all of the different strategies the students used.  One group solved their problems on the white boards and every time they got stuck or realized they made a mistake they would start solving it again without erasing their work.  They labeled each try as a “take” and would refer back to what they did before to help them figure out where they made mistakes.  Finally “take 6” was successful and they were so proud of themselves for figuring it out.  
They labeled their final, correct work as “Holy Ground” and they were so proud of themselves.  
As this group worked I would hear them say things like “we’ve never solved equations with 3 variables before…I’m not sure how to set this up but I know Mrs. Schooler wouldn’t give us a problem we couldn’t solve.”  I also heard a lot of comments about how their brain was growing from the mistakes they made!  The groups for the most part worked without my help.  At times if a group was really stuck I asked them some questions to get them thinking about the problem in a different way 
and that was usually sufficient to get them working again.
As students presented their answers it was fascinating to see how almost every group solved both problems but even better was that not one group solved them the same exact way.  

The students listened to each group present and they would exclaim that they hadn’t thought of solving it that way or telling another group they were impressed at how they approached the problem.  


I surveyed the students after class asking them what they thought of the two problems and what approaches they used when they got stuck.  Here were some of their responses:


“When we got stuck we would look back to see where we went wrong and we listened to each other’s advice.” 
“Whenever we got stuck we would try a new strategy but left the old strategy on the board in case it helped us.”

I also asked the students how they felt after class.  Here’s what they said:


“I felt accomplished and proud of myself because I got to figure out a hard problem without the teacher’s help.  I also felt proud of my group because we worked really hard together to solve it.”
“I felt very challenged but in a good way.”
“I felt like I understood the problems a lot more after hearing how each group solved it.” 

I was so proud of my students.  They were given challenging problems to solve and were successful in solving them.  They made connections to what they knew about systems of equations in 2 variables and applied it to 3 variables.  This is a reminder to me that I need to always remember that my students are capable of so much and that if I am to prepare them to be strong mathematics students I need to give them more problems like this so I can help their brains grow.

Debating Systems of Equations Methods

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My Algebra Honors students have been learning how to solve systems of equations.  We learned 3 methods:  graphing, substitution, and elimination.  While each method will always work, there usually is one method that is easier to use based on the two equations given.  

In groups of 3 I assigned my students one method to solve.  I then gave the class a system of equations.  I gave each group a few minutes to discuss why the method they were given was the best method to solve that system, even if it may not be their preferred method.  Each group then shared out with the class why their method was the best.  Groups could then engage in a friendly debate on which method was the best method to solve that specific system.  It was wonderful to hear the reasons students came up with for each method and how they tried to sell their method to the class.  The students got excited and were really advocating for their method.  After we debated which method was best students then had to solve the system of equations by all 3 methods, starting with their method first.  After everyone solved the system of equations we circled back to our debate and asked if anyone’s opinions were changed based on solving the systems.  We were able to repeat this process 4 times with students getting assigned a different method each time. 
This lesson reinforced that no matter what method you use to solve the system of equations you should end up with the same answer.  I hope it also got students to think about the benefits of each method so they can decide which method to use when presented with the equations.  I loved how it got students talking about math as well which is a skill we are working to develop in our department.  

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!