Thursday, April 28, 2016

Algebra Assessments

This semester I've been purposeful about rewriting my Algebra unit tests to be more inline with the way I teach. Our Algebra team has had a focus on engaging all students, regardless of their readiness, during our lessons this year.  I've seen a huge improvement of student engagement through low floor-high ceiling tasks and differentiated instruction.  I was noticing that while daily lessons were leading to growth mindset for the students, we were still giving fixed mindset assessments.  It's been a huge benefit to collaborate with Kelly McBride, Megan Thornton and Dave Sladkey on these new tests.  In the past it's been hard to talk about assessment with an online community because we don't want to compromise test security.  However, if we're going to change teaching practice, which the #mtbos is doing really well, we have to honor students' learning with the appropriate measures of understanding.

If you just want to see what kinds of questions we're asking, click on the links to tests below.  If you want to know about the process, rational and experiences with these tests, keep reading.

Exponential Functions Test (I left off the skills part of this test and am just showing the differentiated piece of the test)

In our classrooms, we differentiate by readiness and by choice.  Differentiating by readiness on a test is something that I'm still wresting with and trying to wrap my mind around.  We have had great success with differentiating by choice on tests.  Our first attempt at this was the Exponential Functions test.  Most our test was pretty traditional covering topics such as determining if a table represents a linear or exponential function, graphing exponentials, determining end behavior and an exponential vs. linear growth word problem.  Influenced by Jo Boaler, we've been having students analyze patterns during class. Because of this focus in class, we wanted to have a pattern question on the test.  By giving students a choice of which pattern they wanted to analyze, students had more confidence on the question. They weren't just responding to what I wanted to know, they were creating their own test.  It was neat to see students take advantage of choice.  Their choices weren't always what I expected and there was a large variety in the class of what they picked.
We have tried to bring choice into more of our assessments since the EF unit. Recently we had this example of comparing different functions on our Graphing Quadratics Quiz. Having choice helps students demonstrate their understanding by not penalizing them for getting stuck on one part.  

We use Desmos almost daily in our classrooms.  Because of this, we wanted to make Desmos available to students during the test. We're looking forward to Desmos coming out with a test mode which will turn off sharing and Desmos solver features. Since we didn't want to wait for test mode, we decided to build Desmos solver into the assessments, encouraging students to use it to check their work. For our Solving Quadratics Quiz, we asked students to solve each quadratic equation in three ways.  They were allowed to use Desmos solver twice on the whole quiz. The students engaged in deep thinking by simply choosing which methods to use on the different questions.

Test Length
Instead of testing every skill and type of question that we presented throughout a unit, we wanted to give fewer questions that helped us know which targets students did and didn't understand.  An example of this was our Solving Quadratics Test. Instead of having 9 questions where students have to demonstrate solving with a particular method, we created a test that had 2 solving quadratic equation questions where students showed each of the methods at least once. It was interesting to see how students thought through these problems. In the past, students wouldn't check their answers.  But now, because they had to use multiple methods per problem, they were forced to check their work. And knowing the correct solutions helped them demonstrate learning on the methods that they were less comfortable with.  

New Types of Questions
Old Error Analysis
As we focus on the Standards for Math Practice during class, it's important that we challenge students to use all of these practices during the tests as well.  Many years ago we made a department goal to include an error analysis question on every test.  I like that the students know they will see this type of question and they are expected to write and explain their understanding on their test. Here's an example of how we modified an error analysis question on our Solving Quadratics Test. Our previous error analysis question, for this unit, didn't require students to demonstrate understanding, instead they were expected to memorize a formula. On our new error analysis question, students are demonstrating MP3: Construct viable arguments and critique the reasoning of others.
New Error Analysis
Besides error analysis, we've added open-ended questions to our tests. This allows students to solve a problem in a way that makes sense to them. We honors students' differences during class and celebrate unique answers. I'd like to be able to celebrate students' creativity on their test as well.
We've just begun to scratch the surface of what's possible in assessment. These new questions aren't perfect, but they're a start. What kinds of questions are you asking on your tests? How do you differentiate assessments?

Thursday, April 14, 2016

Let's talk about Creativity

There were many great sessions at the NCSM conference this year.  Math education is changing for the better and it's fun to engage in the change with so many enthusiastic educators.  I appreciated hearing about topics such as growth mindset, number talks, tasks, standards for math practice and most importantly equity.  Back in class this morning I could feel myself slowing down to listen more to my students because of the presentations I heard earlier this week.  We spent 30 min. talking about a WODB problem with high engagement and it was so much fun.
As a personal reflection after the conference, I asked myself what was new to me, what surprised me, what I disagreed with and also, was there anything missing from the conference, if so what? I want to take a minute to reflect upon what I thought was missing.  I love that we have moved from teacher led instruction of skills to creating learning opportunities for students' understanding, problem solving and critical thinking skills.  These are such important topics to be talking about in order to refine and improve our teaching practice.  However, all of these things were about students doing our problems and not about creating their own meaningful work.
As teachers, coaches and leaders, we learned how to create better tasks, formative assessments, engaging number talks, etc.  We're doing a lot of creating for students to consume, do and apply.  This teacher work is important for students learning, but it shouldn't be our only focus.  Just as we create on a daily basis, our students need that opportunity as well.  I'm not saying that teachers aren't allowing this opportunity, but we're not talking about how important it is for students to produce their own meaningful work. Quality, open-ended math tasks should lead students to problem solve with creative thinking.  In the past, students were expected to solve a problem with the same process as the teacher.  I think every speaker I heard this week values individual, creative thinking from students.  I want to challenge us to not just expect students to creatively solve our problems, but also to draw out their unique creative talents in an educational system that is focused on consuming knowledge and achieving academic success.
Assigning projects where students create something new gives opportunity for differentiation, student voice and multiple entry points.  When students create, they have pride in what they produce.  Creation gives students ownership of their learning.  They get to display their learning in their own style.  I've found that students who aren't engaged on a day-to-day basis come alive when I assign projects.  So often as teachers we feel pressure to find the topics and examples that will peek our students interest.  By assigning projects, we give students the power to bring their interests into our classroom.  When students create, I get to know them better.
The jobs we are training our students for are creation jobs.  They will be creating new technology, medicine, transportation, business, etc.  We should be fostering their creativity in our classrooms today.
With technology it is easier for students to create something meaningful to share with a larger audience than the teacher and their parents.  Creativity isn't limited to a poster like it was 10 years ago. Now students can display their learning on a dynamic infographic that can be shared via social media.  Let's have students create something that is fun, impactful, and appreciated in the school or community.
We don't have to have the perfect project to begin fostering creativity in our classrooms.  Start small.  Have students come up with the questions that you'll spend the class time seeking answers to.  Let students explore and play with Desmos Marbleslides.  Add a writing or drawing component to an existing task. For example, I had students read an article, answer some questions pertaining to linear functions and then on the 3rd page had students creatively engage with the topic of the article (stolen cattle). When you're ready to have students create something bigger, invite them into the planning by asking them what they want to create.

Friday, April 8, 2016

Movement, Open-Ended Questions, & Error Analysis

In Algebra 1, I taught a lesson that allowed for movement, student creativity, open-ended answers and error analysis all in one activity. The target for the day was to compare quadratic functions to the parent function.  To begin the lesson, we talked about the a, h and k values of vertex form.  They had played with these values on the previous night's assignment.  Then, I gave students a description of a quadratic function:

  • Translated Left
  • Vertically Compressed
  • Translated Down
Students had to write a quadratic function in vertex form that satisfied these requirements. We graphed about 8 of the examples students came up with on Desmos. 

When I thought that most of the misconceptions about the a, h and k values were addressed, I had students write their own description of a quadratic function on their desk.  After this, students got up, went to a new desk and were challenged to write a function that would satisfy the requirements of the desk they were at. Knowing that comparing quadratics in vertex form was a new topic, I anticipated that the equations they wrote might have some errors. I wanted to do an error analysis without students feeling defeated about the answers they put on a peer's desk. So I had them write a second equation of a parabola that they knew didn't satisfy the given requirements.  After this I had students return to their seat.  Back at their desks they had to work with the wrong equation first.  I had students "describe and correct the error."  They wrote out complete sentences about what was wrong with the equation.  After students were comfortable describing errors of an equation they knew was wrong, I had students check the equation that was supposed to match the description.  We used Desmos as a visual check to verify that the equation matched the description.  Because this was the first day of comparing functions, I anticipated that there were mistakes around the room.  Since students didn't know who had been at their desk, it was easy to talk about common mistakes with anonymity.  Common mistakes included forgetting the exponent on the quantity, incorrect "a" values for stretch/compress and wrong signs for the translations.  
This was fun, students were engaged, and we accomplished the target.  This activity also allowed for me to get an informal understanding of what the class understands as a whole by taking pictures of their desks.  The work on the desk was the exit slip that helped me plan the next day's lesson. I definitely want to do this again.  Next time I will have students move to more desks throughout the classroom.  

Wednesday, March 23, 2016

Open-Ended Algebra 2 Project

Recently a colleague asked Dave Sladkey and I about the benefits we've seen for students by asking more open-ended questions in our classes this year. Here was my list:
-students are more willing to take risks and try questions b/c they know their thinking will be valued 
-students wanting to share a different method for solving a problem with the class and wanting to find different answers than their peers 
-students' willingness to spend more time on a problem b/c the question doesn't have a distinct end/answer 
-creativity in answers b/c they're not trying to mimic my ways of solving closed problems 
-weaker students have more confidence b/c success isn't defined by the right numerical value or having the correct steps all the time.
-more exclamations of "this is fun!" And "let's do more problems/activities like this!"

Here's what Dave added to the conversation:
"Everyone has a voice.  Especially the ones that think differently.  Everyone is willing to try because there isn't much failure in an observation/opinion/guess.  On a teacher note:   I really love doing this and seeing the creativity of my students blossom.  It's more fun now for me too.  I'M MORE ENGAGED."
Last week I assigned an open-ended project in Honors Algebra 2 that is the best project I've ever done in my classroom.  I saw Zach Herrmann present at ICTM in the fall and he shared this Public Health project that he assigned during a probability unit in Algebra 2.  His presentation can be found here.  Here are Zach's slides for the project.  

I assigned the project the same way that Zach presented it, and it worked great.  His suggestion was to assign the project to groups then, for a week, teach regular scheduled lessons for the 1st half of class and let students work on the project for the 2nd half of the class period.  As soon as I assigned the project, students were talking excitedly to their groups.  Students collaborated really well because there are so many layers for them to consider.  I didn't give students any specific requirements for their presentation.  Most groups made a Google Slides presentation and one group made a poster.  I set a 5 min. limit on presentations so that we could fit all groups into one class period.  I didn't focus too closely on the time limit because I wanted students to be able to effectively communicate their plan without worrying about presentation details.
We spent a class period presenting their ideas.  I was impressed about how well students listened to their peers.  They were engaged during other's presentations because every group presented a different plan.  Strategies ranged from testing people 7 times to make sure that the false positives wouldn't be treated with an expensive treatment plan to charging people in the community to be tested.  We opened up the floor for the class to ask the presenting group questions after each presentation.  Their questions were thoughtful and appropriate.  Their questions either led the group to clarify their thoughts and explain them in a better way, or they sparked debate.  I was proud of each group for defending their plan and justifying their thinking.  I loved that students had to "make mathematical and ethical assumptions."  With closed questions, we don't often get to see students's creativity and imagination in math class.  The assumptions they made were very thoughtful and clever.  They came up with ideas I didn't once consider.  I can't wait to do this project again next year!

Sunday, March 6, 2016

How do we make loving math contagious?

The DVC math conference reignited my enthusaism and excitement about teaching math. Eli Luberoff and Andrew Stadel both spoke in such a way that made me want to be a math student again, exploring, discovering and problem solving in an active and engaging environment. As a teacher, creating these opportunities for my students isn't so much of a challenge as it is an invitation. I am invited to create a learning environment where students are honored as thinkers and celebrated as unique individuals with talents and passions. How each teacher accepts this invitation varies based on their unique teaching gifts and personality.
The teacher to-do list is long. When inspired by great speakers and hearing positive success stories, questions about time, difficult students, and limited resources can all pose as boundaries to experiencing that success for ourselves in our own classroom.
I listened to a Ted Radio Hour podcast from NPR, called How Things Spread, that has me thinking about how we can continue to be excited about math education a day, week or month after an energizing conference. The podcast plays clips of related Ted Talks and the host, Guy Raz, interviews the people who gave those Ted talks. This episode was about the "mysteries behind the many things we spread: laughter and sadness, imagination, viruses and viral ideas." I want to spread an excitement for teaching math among teachers as well as a love for learning math among students.
Spreading a passion for teaching
Teachers who are doing innovative, unique things to engage their students have to talk about these lessons and ideas with other teachers, administration and parents. Tweet, blog, email and tell someone else about the great things, big or small, that are happening in your classroom.  When more teachers are sharing their classroom successes, others will want to experience an engaged class for themselves. Sharing our ideas makes our profession collaborative. There's a greater joy in what we're doing when the people around us are as excited as we are about what's going on. Guy Raz interviewed cognitive neuroscientist, Sophie Scott, about why laughter is contagious. In her Ted Talk, Sophie says "When we laugh with people, we are hardly ever laughing at jokes. You are laughing to show people that you understand them, that you agree with them, that you are part of the same group as them. You're laughing to show that you like them, you might even love them." We have to share our classroom successes in order to connect with each other as professionals who respect what is going on in each other's classrooms and as friends who are genuinely happy for each other's delight in our experiences with students. We have to share in order to spread an enthusiasm for teaching that is as contagious as laughter.
Spreading student passion for learning
In order to ignite an epidemic of math love among our students, we have to market our subject.  In the podcast Guy Raz interviewed author Seth Godin about the best ways to market a product. A few quotes from the podcast got me thinking about how we get an idea like "love for mathematics" to spread. "Whatever you're trying to spread, whether it's an idea or brand or whatever it is, those things spread faster when the people you know and like talk about them" -Guy Raz. If a few students enjoy an experience in math class, can their perspective rub off on their classmates?
"Uber isn't big because they ran a lot of adds, they're big because someone took out their iphone and said to their friend 'watch this' and pressed a button and a car pulled up." -Seth Godin 
Desmos has figured out how to do this by making their product something people are talking about. There have been so many times that I've come to the math office after a class and said "look what my students made on Desmos!"  The question is, how do I get students to say to their friends at lunch "look what we did in math class today!"  It sounds cheesy, but I believe it's possible.
As we market our subject, it's important to strategize about which students we will focus on to spread our math passion to a greater student population. Guy Raz interviewed social scientist, Nicholas Christakis, about how ideas spread through social networks. Guy's conclusion from Nicholas' work is that "There are so many things that you could spread by identifying the right people and right entry points." I've had a big focus on differentiation this year.  I believe it's important to see students as individuals and to meet their unique learning needs. However, maybe now is a good time in the school year to pause from looking at each student individually and see my class as a whole.  How do I move the whole class to an understanding of Algebra 1 by May?  Are there a few students in class, that if ignited with a new found enthusiasm could spread the math love to other students? How could I let go of control of trying to orchestrate each students' individual appreciation for my subject and let the joy of problem solving spread through my classroom like wildfire? Having just been at the DVC math conference, the obvious ideas for how to do this would be to assign a new Desmos challenge that would engage students creativity and to use estimation 180 to spark students' curiosity and wonder. I'm not totally sure how to see this goal come to fruition, but my purpose this week is to make loving math contagious.

Tuesday, February 2, 2016

Fishbowl in a Math Classroom

I was challenged by Janice Schwarze to use a fishbowl model to differentiate in my math class.  I tried it today in my Honors Algebra 2 class and I can't wait to do it again.

When I 1st considered doing a fishbowl lesson, I had no idea what it was supposed to look like or how to execute it well.  I went to one of our LSCs for help getting started.  She arranged for me to see fishbowl in action in a Social Studies classroom.  This made all of the difference because I saw that the success of the fishbowl was in how I introduced the activity.

I began by asking students: when have you done a fishbowl (inside-outside circle) in other classes and what did that look like?  I followed that up by asking: what would a mathematical discussion look like?

I posted my success criteria for today's activity on the SmartBoard.

-Mistakes grow the brain - This is a learning experience for you and your peers, no one is expected to have the right answer right away.  
-Use mathematical language
-MP1:  Make sense of problems and persevere in solving them.
-MP3:  Construct viable arguments and critique the reasoning of others.
-MP5:  Use appropriate tools strategically (Whiteboard, Calculator, Chromebook, Your choice)
-Make eye contact
-Everyone must speak
-Connect your thought/idea to the previous student's thought.  Don't go on a tangent that doesn't make sense.
-Don't move on to a new problem until everyone on the inside circle understands it.
-Outside circle is listening for understanding.  Specifically listen to your partner.  Did you understand their explanation?  Did they use mathematical language? Did they accomplish the 3 math practices?  

As we were going over these bullets, I made about big point about saying that "Getting the right answer" wasn't my biggest objective today.  There's more to understanding mathematics than finishing a problem.  Our goal was to grow in being able to construct arguments and critique each other.

My original lesson for today was 6 example problems that I was going to show the class.  It was originally a very teacher-directed lesson.  I put students in readiness groups based on an exit slip from the previous day.  I was purposeful in picking which questions I wanted each group to work on.  I gave each group 3 of the 6 questions to work out together.  I also gave each group a pattern to figure out.

There were 3 circles going at once.  I had 4-5 students on the inside and 4-5 students on the outside.  Each inside student had an outside partner who was specifically listening to their contributions to the discussion.  After the 1st group went, I had the outside partners give 1 praise and 1 critique (Math Practice Standard 3).  I think this was my favorite part of the activity.  It was fun to see students encouraging one another.  And even cooler to see that we have a classroom environment where students feel comfortable offering constructive feedback.

Afterwards I had students talk as a class about the benefits of this kind of activity in math.  They appreciated that time wasn't a factor today.  There was no point in racing to finish the problem.  Students felt like the had space to ask questions and approach a problem more slowly for better understanding.  A few students commented that they liked hearing the different approaches to solving problems from their peers.  If they didn't understand the problem when one student described it, someone else was there to offer a different perspective.  Students felt that they had a better understanding because they had to speak and explain.  There was also a lot of pride expressed that they were able to figure out the problems as a group, without any teacher help.

Next time I do a fishbowl activity in class, I want to give students the opportunity to made decisions and have choices in their discussion with activities like Would You Rather Math or Which One Doesn't Belong.  I'd like to see their personality come out more in the discussion by having to defend their opinion.  A big thanks to Steve Stack and Megan Plackett for helping me plan such an engaging lesson!

Wednesday, January 6, 2016

I'm curious about...

On the first day of the semester I like to do an engaging math activity related to current events.  I do the same lesson in both Algebra 1 and Honors Algebra 2.  It gets students speaking math and asking questions.  The conversations vary between the two levels, but the activity can be easily adapted as an introductory lesson for any high school math class.

I found this interactive at that allows the user to compare frequently used words on Reddit. As I was playing around with it, I was excited to see that many of the graphs model linear, quadratic and exponential functions. Because of some inappropriate content on Reddit, and potentially in this interactive, I decided to make this activity all paper and pencil.  I had a few students help me pick common words/phrases for my list after they finished their 1st semester final. This is the handout I gave students.

The first step was to pick 4 out of 12 word comparisons that they wanted to know about.  Then they made predictions:  Which word is used more (on Reddit) today: Brah vs Bruh?  As I was listening to groups make predictions and share opinions, I overheard a student say "I'm curious about Lol vs Haha!"  I love that this lesson sparked curiosity.  That was a huge win.

I had printed out the graphs of word comparisons and posted them around the classroom.  Students drew the graphs on their papers, decided what function modeled those words the best, decided which word increased more quickly between 2012 and 2014 and then predicted which word would be more popular in 2016.

Students had great conversations that were beyond answering the questions on the paper.  One student was wondering about all of the spikes on the iphone graph and his teammate concluded those must have been the release dates of new iphones.  I heard another group talking about the "average rate of change" as they were comparing two graphs.

As I listened to conversations, I reminded students what their variables represented.  I heard students disagreeing with the Facebook graph saying that Facebook is not more popular than Snapchat and Twitter.  We talked about how the graphs represent the frequency of the word on Reddit, not it's popularity among the general population.  So, some of the occurrences of Facebook might have been negative, against Facebook, not in support of Facebook.

I liked this activity because students were able to relate to the language that is used online today.  I appreciate that the graphs are jagged and imperfect.  Students had to interpret "real" graphs that didn't have "nice" numbers.  We were also able to debate which function best represents each graph.  It gave students a chance to justify their thinking.  One of my students asked "Are we going to have to do real math today?"  I followed up with a class discussion about how identifying an appropriate function, making predictions and analyzing graphs is real math. Every activity that I facilitate in class is purposeful with an emphasis on math more than anything else. I hope that more activities like this one can convince them that math isn't dry, but interesting and thought provoking.