Category Archives: Mary Beth Dittrich

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?

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.  

Lace as a Symbol of Unifying Love

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I think we are all familiar with our story of lace.  How the first Sisters in Le Puy taught women
how to make lace so that they could sell a product instead of selling their
bodies.  Which brings me to my first
point – that which seems extravagant or superfluous was actually very
practical.  I can only image how
expensive hand-made lace of that era was. 
Definitely something only for the very wealthy and probably used as a
way of showing off their wealth.  But for
the Sisters this extravagance was very practical – it was their livelihood –
funding not only their ministry but also giving the women of Le Puy a viable trade
– a way to support themselves and their families.  That
which is extravagant to the world is practical to us.

Secondly, lace is not very fashionable now.  Back in the 80’s many loud, flowery dresses
had lace collars.  Just go back and look
at the pictures in our yearbooks from that era. 
But even though lace is not fashionable or in style right now, it is to
us.  Lace is a key part of our
story.  We tell the story to our
freshmen.  We have a piece of Le Puy lace
framed in the Garaventa Center.  It is
woven into our activities.  At Mass on
our staff retreat we blessed a lace altar cloth from Le Puy that will be used
at our liturgies. That which is out of
style to the world is foundational to us.
Lastly, did you ever watch someone make lace?  Or even notice the first stitches when
knitting or crocheting?  It looks like a
bunch of knots – it’s very messy.  And
for a novice like me, it’s easy to get confused and make a mistake.  But with time, with care, with patience, the
threads are delicately interwoven and interconnected, and the project turns
into something beautiful.  So, too, are
our lives interwoven and connected.  We
come from all different backgrounds and experiences and have been dumped
together here at Carondelet to make something beautiful – to make lace out of
knots.  That which looks knotted to the world is beautiful to us.
So what does this have to do with unifying
love?
Unifying love is very practical.  However it is not a central component of most
schools.  It can’t be.  For the context in which we use it, that is,
the gospel, is not permitted in most schools. 
For them it is superfluous.  For
us it is key.  Who are we if we do not
love our students?  We are called not
only to teach them, but also to love them. 
If we as a staff embody and model unifying love, that is, if we love one
another – treat each other as the precious creations of God that we each are,
then we cannot but convey that love to our students as well.  We want them to “see how they love one
another.”  We do our best work – we are
our best selves when we act out of love. 
That which is superfluous to the
world makes us who we are.
Unifying love is not very fashionable right now.  We need look no further than the halls of
Congress or our local governments.  It’s
fashionable to argue, resist, disrupt. 
But not for us.  For us unifying
love is foundational.  It is what drove
the first six Sisters to reach out to the women of Le Puy and teach them lace
making.  It is what brought the first six
Sisters to St. Louis to teach the deaf. 
It is what called six women to open Carondelet High School.  We are called to go against what the world
says is fashionable.  We are called to
reach out in love.  That which is folly to the world makes us who we are.
Unifying love can be messy – like a knot.  We don’t always agree or get along.  To be honest, we don’t always like each
other.  And we’ve all had at least one
student who drove us crazy.  But unifying
love sets aside our weakness – our desire to notice differences – and shifts
our focus to what we share – our common story, our common purpose, our common
love – grounded in Christ and rooted in our love of the dear neighbor without
distinction.  That which is messy to the world makes us who we are.
So what do we do with this unifying love?  How does unifying love call us to act?  I invite you to watch this video:
The passage from Colossians really resonates with me.  It challenges me to look at my life – not
only my actions, but also my attitudes toward others.  It calls me to find ways to live the unifying
love that Jesus modeled for us in the gospel. 
I invite you to reflect on what you can do this school year to make real
one aspect of unifying love.
Since God chose you to be the holy people he loves,
      you must clothe yourselves
with
                  tenderhearted
mercy,
                  kindness,
                  humility,
                  gentleness,
                  and patience.
Make allowance for each other’s faults,
      and forgive anyone who
offends you.
Remember, the Lord forgave you,
      so you must forgive others.
Above all, clothe yourselves with love,
      which binds us all together
in perfect harmony.
And let the peace that comes from Christ rule in your hearts.
For as members of one body you are called to live in peace.
And always be thankful.
Colossians 3:12-15 NLT

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…

Shooting for the Stars

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

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

FlipGrid – Take 1

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

Finding Slope

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So I thought it would be really clever to insert a picture of Nemo here (Finding Slope – Finding Nemo), but I doubt it’s in Creative Commons and didn’t want Joan upset with me!

Last week I realized that my Algebra 1 students weren’t understanding slope as “rise over run” and therefore couldn’t find the slope of a line graphed on the coordinate plane.  So, armed with 50 cent brightly colored rulers purchased at a dollar store, I sent them out to find slope around campus.

Their first instinct was to make the ruler the slanted/diagonal line.  So I had to explain (and they had to demonstrate!) that the vertical (up/down) is the rise and the horizontal (left/right) is the run.  I then had them measure and calculate the slope, and document it with a photo.  After about twenty minutes of wandering around campus, we came back to the classroom and shared photos.

Their calculation weren’t always correct, but that wasn’t the point.  I wanted them to be able to visualize and have a “hands-on” experience of slope.

Maybe they’ll look at the Inner Court stairs differently now!

A Week of Inspirational Math

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After almost six weeks of slogging through the first two
chapters of the textbook, I decided to take a break with my Algebra 1 with Math
Lab class and do a Week of Inspirational Math. 
I wanted to apply and share with my students some of what the Math Department
is learning in Jo Boaler’s How to Learn Math for Teachers class.  The resources are on her website www.youcubed.org.  Each day we watched a short video and followed
it up with a discussion.  Then we did a
hands-on activity to illustrate the video’s message.  The topics included:

  • how our brains grow and change
  • how we see numbers and patterns differently
  • how our brains grow when we make mistakes
  • the importance of having confidence in your ability
  • the importance of visualizing math
  • that speed is not important, but deep thinking is

For me, this was an important activity to do with this
particular class.  These are our students
who struggle with math.  It has never
come easy to them and often they don’t feel very smart.  At the end of the week, I asked for their
input.  I found their comments very
encouraging:

Through the week of
inspirational math I learned a lot about myself and that my brain is ever
growing and changing and I can always improve my brain, and it is important and
ok to make mistakes because it helps my brain to grow and change.
It made sense that
they said that making mistakes is a learning opportunity because it gives our
brain a chance to grow. I learned that I should always take my time and get the
better grade, instead of rushing & making careless mistakes and getting
points taken off. I feel now that when I make a mistake, it grows my brain
more, which makes me more happy.
I feel like now that
I’ve watched those videos that I can have a better attitude towards math and
try my best at doing it as well.
I learned that I like
to visualize or draw things, more than just doing all of the math in my head.
It helps me understand what’s happening.
I guess I didn’t know what to expect heading into the
week.  But now seeing the impact it has
made on several of the students, there will definitely be more Weeks of
Inspirational Math in our future!

The Power of the Panda

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My appreciation of and affection for stuffed animals has spilled over into my classroom … with an unexpected result.  It all started several years ago when my Algebra 1A/1B students gave me a Build-a-Bear at the end of our second year together.  The bear is wearing a scout uniform and Birkenstocks with socks and is carrying a planner, pencil, and book bag.  My students knew me rather well.  Since then the collection has grown to include two pandas and an alpaca … each of which has its own story.

Now instead of just collecting dust sitting on the sideboard, they are playing a crucial role in my students’ well-being … especially on quiz and test day.

Yes, we have comfort pandas in Mrs. Dittrich’s classroom!  One student even brought her own.  Imani is holding her stuffed wolf, Astro.  If this trend continues, it will be all out panda-monium in my classroom!

It may appear somewhat childish to clutch a stuffed animal in a stressful situation.  But there are so many demands placed on our students and they are growing up so fast, I believe that a little reversion to the simpler things is appropriate … and often necessary.