A Day of Desmos – Parabola Slalom

On the Thursday before Thanksgiving break I had the opportunity to try out the Desmos activity builder Parabola Slalom in Scott Davidson’s Math 2 class. This is how that happened:

Tuesday: Spent the day at a Desmos training and played Parabola Slalom for the first time. So much fun and potential for learning! If only I could do this with students. . .

Wednesday morning: I reached out to the teachers in my district who I knew where teaching quadratics. “Can I come do a lesson on Thursday or Friday with your kids?”

Wednesday afternoon: I heard from Scott Davidson, the Math 2 lead at a high school in my district.  “Sure, but time is a little tight.  Thursday Block we are reviewing and Friday students are taking a midterm on all things quadratic.”

From there the whirl-wind began. 15 hours until I’d be teaching a lesson with students I had never met.  And some of that time needed to be spent on eating and sleeping.

This first steps I took were getting to know exactly where the students were at and what was expected of them.  I looked through the list of topics they had been studying and at the midterm assessment they would be taking on Friday. About 7 of the 20 problems were things I thought I could address with Parabola Slalom. Their learning had focused on vertex form, and this was  reflected on the exam. Students, however, were expected to consider which forms of quadratic functions would be helpful in various situations. Completing the square, quadratic formula, and imaginary solutions did not easily come up in this activity. Factoring as a skill did not come up, but I could discuss why factored form of a function was helpful. I asked Scott if I could have the first half of his block period.  The second half he could do last minute review on topics I did not address. Scott is wonderful, trusting, and adventurous. “Sure,” he said. “Can we invite my PLC in to watch?” My dream question. “Yes, Yes, YES!”  A couple teachers had prep.  For the others we got 15 minutes of coverage one after the other. Fortunately Scott has a big room. The stage was set.

The next step is where you’ll begin if you are a teacher using a Desmos activity with your class. I went to the activity builder and played it again. You can try it here. On Tuesday at the training I had the opportunity to talk with other teachers about which slides would be good to stop and talk at, but now I had specific students at a specific spot in their learning, and I needed to rethink where the valuable conversations would be for this group. After playing it again, I printed the attached Teacher Guide. (see button in top right corner)

parabola slalom

Teacher guides are a great place to start with any Desmos activity.  They allow you to think through what you want to get out of the activity, the teaching moves you will make, which things you can skip and which things all students will engage with. This checklist is at the beginning of each teacher guide:

activity builder checklist

I had already done step one. Check! That felt good. On to step 2 – learning targets.  Desmos activities are very flexible, and you might chose to use them at a wide variety of spots in a learning progression. I’d love to try Parabola Slalom as an introduction to vertex form sometime and let students explore what the happens as you change different numbers. If I did that, my goals and planning would be very different. For this day I settled on these as my goals:

learning targets for parabola slalom

Next, according to my checklist, it was time to think about how I wanted to lead discussion with these kids I did not know.  When would I bring the class together using “Teacher Pacing” and “Pause Class”? What would I discuss on those screens? I felt a lot of pressure to make sure the lesson prepared the students for the next day’s exam, because Scott was trusting me with this time. I wanted to make sure I picked up the students that were really struggling and gave them one more opportunity to make sense of this equations of parabolas thing.  I decided that screen 3 was the first place where equation of parabola came up, so I would stop and question to see where kids were at. For students that were confused, I planned to make a connection to the y=mx+b as an equation of a line.I decided to look for these examples to discuss:

  1. I wanted to look for a sample of student work that used an upside down graph and asked what in the equation made that happen.  
  2. I wanted to find a student that had made a parabola with vertex other than (0,0) and discuss what about that equation caused the shift.
  3. I also knew I wanted them to realize that the square was important, so decided to watch for someone who entered an equation without the square and then pause to discuss the resulting graph.

After this discussion, I would pace them to work on slides 3 to 5. The Tip for Teachers provided by Desmos gave me some suggestions to help lead conversation on slide 5.

teacher tip parabola slalom

Having looked at their previous work, I anticipated that most students would give an answer in vertex form. I decided I would focus on those answers here with the whole group, and help those lowest students focus on what vertex form looked like. I decided I would NOT give away yet what effect changing the a, h, and k would have, even though technically this should have been review.  Instead  I would encourage them to play with the form, trying in different numbers and seeing if they could figure out what changing each of them did to the graph.

I decided slide 6 and 9 were other places I wanted to have discussions. Slide 6 would give me a chance to talk about the usefulness of factored form. I considered taking time to explain why x-intercepts are easy to solve for from this form, depending on student response. This was a secondary goal for me and I didn’t want to overwhelm with too much information.  Slide 9 would give me a chance to consolidate the learning, and I would chose to push kids all to 9 at some point to catch the struggling students up and clear misconceptions.  In the discussion about slide 9 I would listen to what they had discovered about a, h, and k in vertex form, and make sure all students were hitting the learning targets.  To support their arguments about why each answer was incorrect I might open a desmos calculator page, graph the given points, and put in the equations. With their groups, I would ask them to propose an equation that would work. From there I would open slides 3 – 11 let them work at their own pace, giving extra time to students who had misunderstanding I had uncovered.

The teacher guide encouraged me to plan something for students who finished quickly, but I felt like slide 10 and 11 would cover that.  I only had an hour and the note at the beginning said the activity could take more than one regular class period.  I decided I would have students log in so they could go back to the activity at home.  

Last steps – I printed a copy of my teacher guide with notes for the teachers that would be coming in and out to observe.  That way they could quickly see where we were in the lesson and what had happened so far.  Excited to work with actual students tomorrow, I set my alarm and dropped into bed.

Thursday morning at 7:15 I arrived at Scott’s classroom.  I connected my computer to the projector, logged in, and made a class code for the activity.  I jotted the code down on my teacher guide and copied it to the whiteboard. I paused the activity so early students couldn’t begin before I was ready. Then I helped Scott set out laptops and soon students begin to come in.

The activity was even more fun than I imagined. Being with kids was such a rush, and watching the ah-ha moments gave me all the warm fuzzies. Man, I love teaching.

A few things that came up during the activity/lesson:

  • Making the connection between forms of equationsended up being a big thing for a few students. There was more good discussion to be had there about how an equation in two variables defines/creates as set of points that are arranged in a certain shape. Something to come back to.

 

  • As we consolidated their understanding of vertex form, I wrote the equationvertex form exampleon the board and asked students to predict where the graph would be and what it would look like.  Students readily volunteered a correct description of the graph, which I sketched on the board. I wanted to emphasize the horizontal shift being opposite the sign, so I asked the class “Is there anything surprising to you about the graph that you wouldn’t have expected just looking at the equation?” A girl raised her hand. “The negative three.”  Excited she was bringing up what I was looking for, I said “ And what was surprising about the negative three?” Her reply was a shock.  “ Well usually when there is a negative the parabola is upside down.” Man, I had almost missed this misunderstanding and assumed we were on the same page! But with a poker face I said, “That’s true. Lots of times a negative means the parabola will be upside down.  Does anyone have an idea how you can tell when the negative makes it upside down?” Students were ready and able to clarify that, and I let them be the voice that shared that it was only a being negative that made the parabola upside down.  “So what happens if the negative is in a different place?” And I unlocked the screen and let them experiment. How many opportunities like this have I missed because I fail to ask a follow up question and really listen?

 

  • Several students brought up the fact that the squared term is what made the graph a parabola.  This was overgeneralized by students, so that if they didn’t see a superscripted 2 written in the problem, they thought that it could not be a parabola.  They did not expect factored formto have a graph that was parabolic. I had not anticipated that misconception. And actually, I was lucky to have discovered it, because the students had not shared that confusion out loud. One of the visiting teachers brought it to my attention near the end of the hour, after she spent some time talking to a student. Scott then addressed this misunderstand with the class after I left. 

One of the many great things about an activity builder is that it gives the teacher:

  1. an easy way to spot struggling students (the dashboard), and
  2. the time to actually have a conversation with that student about what they are thinking.

If we want students to think in our classrooms, we need to give them opportunity to be more than a consumer of information. We need to let them figure things out, propose hypotheses and hash out what is true. Desmos graphing calculator and well crafted Desmos Activity Builders are a wonderful way to open the door to that opportunity. Listen to what the students said about the lesson:

What thing do you know understand better after the activity?
how to find the parabola and the equation
how to graph and where the numbers go
the vertex form
how to make parabolas
parabolas
how to graph better
Which letters in the function of the vertex and stuff.
How to make parabolas.
I understand the form used to graph parabolas
I understood the vertex form much better.
graphing and how to make an equation for the graph
How to look at points on a graph and make an equation for the parabola.
how to graph and move around parabolas
vertex’s
all of it
vertex form
how to plot
I understand how to graph the parabola with the equation.
how to make parabolas out of equations
I now know how to solve for parabolas.
parabolas
HOW TO GRAPH PARABOLAS
how to make the parabola wider or narrower.
which numbers you put onto the graph
I understand how to graph the parabolas and how to put the values into an equation.
the way graphing equations work
I already understood everything. I just got to put what I knew into practice.
More about how squares and how each letter in the needed equation works
The thing that I understand better is how to find what equations would go into the parabola.
the vertex form and how to correctly make a parabola/solution

 

Any other things you would like to share about the activity?
it was fun
i liked the activity
it wasn’t confusing as much as regular teaching
It was a fun way to learn how to make parabolas.
It helped me a good amount with solving parabolas, vertex, zero property and such.
it really helped me a lot
It was hard, but beneficial.
It was very fun to work in groups and solve together
It was fun.
It was better to do something a little different instead of sitting around and doing the review
This activity really helped me get a better understand parabolas and I now feel like I have a better chance of passing the mid-term then before.
IT’S FUN AND EASY TO USE
It was actually pretty fun
This activity helped
It was fun although it was math.
Not really, other than it was informative and helpful to my test
The activities really help you get practice and help you get better at finding the equations for the graphs.
We should do this more often.The learning experience we were able to have as a class was very helpful. We are very into technology these days so it is easy to connect and study using a resource that we have access to.

 

Holiday Giveaway – the gift of professional growth

Here’s an idea that has been really fun this year. It begin with recruiting support from administrators who work with math teachers at each of our secondary sites. They helped finance buying the book prizes. The remainder were provided by my office in Curriculum and Instruction. I chose to focus on Tracy Zager’s book, Becoming the Math Teacher You Wish You’d Had, because that was a learning I was interested in spreading around the district. It is more suited to my intermediate teachers than high school teachers, but there is so much good discussion about what it means to teach mathematics that I feel it is worthwhile for all of us. Perhaps in your district you would pick something different. Whatever awesome book you choose, you are getting great professional learning materials into the hands of interested teachers, so it is all a win. In addition to the book, I decided to give away a “Day of Desmos”. This is producing a list of wonderful and willing teachers for me to work with in the next semester.

 

Once the prizes were established and gathered, I sent the following email to the math teachers at each site, signing my name and the name of their supporting administrator. I included a link to a google form, where they put their name in for each prize they were interested in.  Winners of prizes contributed by site administration will be chosen just from teachers on their site. Winners of district office prizes will be chosen from everyone in the district that signed up for them.  I have a near infinite supply of days of desmos, so woo hoo – everyone there is a winner!

Here is the email I sent out:

The gift of professional growth

Why: Because we are grateful for you and the work you do every day with the students in your classroom.

What: We want to give you something to enrich your teaching and demonstrate the respect and friendships we have developed working together.

When: During the last two weeks of December, we will be giving gifts to teachers, based on their interest. Below find a chance to check out the gifts and put your name in a drawing for those you are interested in.

Start making your wish list here:

Book: Becoming the Math Teacher You Wish You’d Had by Tracy Zager   https://www.stenhouse.com/content/becoming-math-teacher-you-wish-youd-had         Video description here          Companion website and discussion forum here  

Set of posters:  10 posters that call out habits of mind exhibited by mathematicians (based on the 10 chapters in Becoming the Math Teacher You Wish You’d Had by Tracy Zager)

zager posters

Day of Desmos: Leeanne will come in a teach/co-teach your classes with a desmos lesson that supports whatever topic your students are currently studying. It can introduce a topic, replace a lecture style lesson, or practice applying something you already taught – Your choice!

Surprise: smaller gifts for your classroom

Sign up to participate in this holiday giveaway here (here is where the link went)

Looking forward: New Year, new resolutions. The learning you receive can be your present to yourself, your students, your PLC, and your team in the coming year.

 

The teachers are excited and emailing how much fun this is.  The administrators are excited because that was pretty easy and painless. I am excited because I get to show my appreciation for them AND give them awesome resources to improve their craft.  Win, win, win!

Practical Thoughts on Differentiation in the Secondary Math Classroom

“There might be a student that doesn’t need modeling, but there is always one who does. So why wouldn’t I provide that first?”

I heard this recently on a favorite site. It sounded reasonable, and was spoken by a teacher acutely interested in her students’ success. She was talking about the “I do, we do, you do” strategy of gradual release, making sure that her students fully understood the expectations of the task she set before them. At times it may seem like the perfect fit, but far too often in secondary math classrooms we overuse this well intentioned, time tested lesson plan.

One powerful reason NOT to provide that first is that the modeling robs students that don’t need it of a chance to be creative, problem solve, and use strategic thinking – all higher depth of knowledge ways of processing their learning.  When we provide scaffolds for all it is easy to reduce everything to a dok level 1 – “Watch and do exactly as I do.” For more on depth of knowledge, see Robert Kaplinsky’s work here.

Differentiation is an important topic in education. At the high school level many people find it challenging to truly differentiate in a 55 minute period.  But here is a simple opportunity that actually creates time in your period – time that students can engage in those higher depth of knowledge types of thinking.

How can that work? Several models are possible, but all revolve around reducing or eliminating the whole class modeling and instead planning timely hints appropriate to various sticking points in the process.  Start with thinking about hints for your highest kids. What is the least you can say? What question could you ask to spark their thinking? The next set of hints could be more directed, potentially less open, and may encourage them toward a specific method. For the most helpful hints, basically just do your “I do” – either in writing or by giving then a link to a short video they watch on their phone. Label the hints 1A, 1B, 1C, 2A, 2B, 2C . ..etc. 1 represents the earliest sticking point – how to start.  A, B, and C represent the level of the hint. Students should be encouraged to take the least help possible. It is amazing how quickly your highest kids will adapt to that suggestion . . they are inspired by the challenge. But kids who need more help have it readily available.  

Double the effectiveness of this strategy by putting the students on vertical non-permanent surfaces. Definitely worthy of a blog post all it’s own, but simple enough to implement tomorrow. For a look at this strategy and the research behind it, check out Laura Wheeler’s blog here. A quick summary of the fuller body of work, “Building Thinking Classrooms” by Peter Liljedahl, can be found here.

Struggling with the mismatch between some hypothetical classroom ideal where every student is motivated just because they want to learn and grow, and the reality of grade-chasing, point-grubbing, homework-copying in your classroom?*  Long term, the research above can really make a difference, but to help in the meantime, some teachers use a pointing system for their hints.  All groups start with 10 points (or more if it is a pretty hard task with lots of sticking points).  C hints cost 1 point, B hints 2 points, A hints 3 points.  Points that aren’t used are extra credit on the assignment, and students can spend into debt – making their highest possible score 95/100 for example. Remind struggling students that 95, or 86, or whatever they work their way down to is so much better than the 50% for an incomplete assignment, or a 0 for not turning it in. And to begin to change that culture, be sure you celebrate successful completions equally – let students have joy in completing the challenge no matter how many scaffolds were used along the way.

Every day and lesson is different, and occasionally a bit of “I do, we do, you do” may be just the perfect amount of scaffolding/guided release for the specific task your students are working on.  Optimize those moments by listening to your students thinking along the way. But too often in a diverse regular classroom, we mis-serve both the top and bottom segments of our classes by doing all the higher level thinking for them. This model of differentiation is one way to support ALL learners in those classes.

 
*Ready to try something different with grading? Matt Vaudrey has a nice post about his experiments with a variation of Standards Based Grading  here.  And there are a thousand other great ideas out there. Look here for what came up when I googled: MTBOS on homework.

It truly is a wonderful time to be a math teacher.

 

Over-scaffolding: the loving art of loosening our grip

Parenting is an ever-changing job.

In the youngest years we are required to give our children instruction and advice on how to do every minute life task: how to angle their foot to put on a shoe, how to walk safely in a parking lot, how to put their pee in the potty, how to pick up the edge of their long t-shirt before putting pee in the potty. Those things are eventually mastered and we smile as we watch our little ones handle them all by themselves. Then we move on to new tasks: how to pick out a matching outfit, how to make a bed, how to clear dishes, how to put those dishes in the dishwasher,  how to put the laundry in the hamper, how to take the laundry out to the washer and run it through. For most of my kids-at-home parenting years, these tasks did not seem to stick as well. They were learned, but never seemed to be owned. But when my children moved out, and had to do it themselves, that training finally came to fruition, and now they manage all of those things with ease and grace. I never have to call my 26 year old and ask him, “Did you wash your work clothes? Are you wearing clean underwear?” But they’ve been through the stage where they had to decide it was important to them to have clean dishes to eat from and clean clothes to wear to work. Clean clothes for a college chem lecture did not seem to be equally motivating at first, but happily they eventually began to see the advantages of not stinking to high heaven when sitting by an attractive co-ed, and they reassessed their prioritizations.

 

Maybe those early tasks are mastered with independence earlier because we let them go. When a problem arises – like pee along the front hem of a t-shirt – we can step in and provide some additional instructions and advice. We do not stand and watch our nine year olds put pee in the potty and critique their technique.  We provide scaffolds as needed when problems arise. And then we step back.

 

Stepping back has been the hard part of parenting. All of my early on the job training was at the level of minutiae, and detailed instruction with constant reminders were essential. But gradually, if we let them, they grow to be able to figure a few things out for themselves.  Maybe if I had let it happen my kids would have learned earlier that underwear with skid marks were not a good idea to wear to gym class and they better get a load through. Maybe if they had to scrape yesterday morning’s oatmeal off their bowl on a semi-regular basis, or their ice cream got served up in their dirty spaghetti bowl, timely cleaning of dishes would have seemed more important.

 

Thank goodness, in spite of me, they are making their way to being amazing and wonderful young men who can problem solve as things arise and handle life. I wish I could say I was as good at learning the parameters of my new role. “Did you call your landlord about that broken garbage disposal?” “Did you check reviews on that car repair shop?” “If you hang your clothes right when they come out, you won’t have to iron those dress shirts so much.” “Is that sweater dry clean only??? You really should check washing instructions before you buy. You can’t afford dry cleaning every week.”

 

When they are young, we tell our children what to do because we are applying our priorities to our life. It was a priority for me to get them out of diapers for my life, not just because eventually at 5 they’d be glad to not wear diapers to kindergarten.  But when they are adults, their actions and decisions have to come from their priorities. Fortunately life has a wonderful way of providing timely feedback to help us all make decisions around adjusting priorities. As parents, it’s important for us to learn how to get out of the way and let that feedback get to them.

 

Adulting is hard. . . and then it’s not. It’s up to us to give them the chance to get there.

 

So lately, I have been thinking about how this plays out in the classroom.

 

The current buzz word is “over-scaffolding”, which translates to helping more than students  really need. If we insist on owning the responsibility for all the thinking, they will eventually stop trying to think.  Instead we need to carefully structure activities that help only as much as they need.

This is extra tricky because students grow at differing rates. I used to think that was about ability, but now I think it is mostly because of the access or lack thereof to appropriate feedback. The student who waits in his group for others to decide on a solution path is never trying out his own ideas and learning what works and what doesn’t, or why it’s important to, for example, isolate the radical before squaring both sides of the equation.

 

Imagine if class begin with two or three simple equations on the board
equation 1

and you ask your students to work individually to play around with and figure out the answer. Many would use their natural number sense to guess and check, but a few would use their equation skills they are comfortable with and reason about what they might add to the new situation. It’s likely at least a few will come up with “ Well, if that equals five, then before the square root it must have been twenty-five.”

However they approached it, encourage them, still all by themselves, to check their answer and see how they did. Guess-and-checkers started with this, because they are still trying to make sense of the numbers.  But for many whose heads are full of procedures, they have let go of this sensemaking move, and only do it as the last step in a procedure if that is what their teacher requires.  So here we see the first of the difficulties we often cause for our students – we teach compliance over thoughtfulness. I am beginning to think the most important thing we can teach them is to seek and use feedback -especially feedback they seek out and create from their own efforts.

 

Next, gather answers. Listen to methods and value all of them. Form hypotheses together about how this might work.

Some guess-and-checkers will be be persuaded to the “square both sides” method, but others will continue to trust their better skill – number sense. That’s okay. Children learn at different rates. We need to give them the opportunities to learn. So up the ante with the next question. Give the higher students a chance to explore the edges of their thinking, and the slower ones a chance to clearly see the limitations of their method. Again, gather answers, listen to methods and what has been learned. Adjust hypotheses.

 

You might have to keep raising the level of difficulty several times before the strong-number-sense/poor-equation-sense students decide that maybe there is something to their classmates methods.  Here are a few ideas:

equation 2

Finally, when they are mostly won over, together compile the methods they have figured out with your careful guidance, using their own developing number sense and feedback-seeking skills.

 

Scaling parenting to 35 kids in one classroom is hard. But continuing to tell 100% of them exactly what to do is preventing them from growing up and owning their own learning. I challenge you to build a thinking classroom, one that respects the strengths of all your students and gently prods them to the next step in their development. Encourage them to try something and see if their answer makes sense, and to think for themselves about how to approach a problem.

When problems arise, as needed, give just enough help and then get out of the way.

Be watchful, be positive, and even prepare in advance some scaffolds they might need, but give them as needed.

 

And I encourage you to be patient with yourself in the process. Learning, real learning, takes feedback and adjustments. It’s messy. So you won’t be perfect the first time, but you’ll learn. And guess what – learning can be exciting and fun! And helping the children in your class, the children I am sure you love,  grow to an adulthood that doesn’t require the scaffolds of childhood, can be the most rewarding part of your job. It is why teaching is such a rich and rewarding profession. Let it be that for you.