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.