Think About a Person Who is Good at Math (Class Day 1)

On my first day of class, I wanted to set the tone that all students are capable of learning mathematics.  This can be a hard sell in a "Business Calculus class", especially with juniors and seniors that have been removed from a math class by 4 or 5 years.  To help students see that the characteristics of a person who is good in math are actually achieveable by everyone and within their control, I took a lesson from the playbook of Howie Hua.  He sent out a tweet a few days ago about this activity.

Step 1: Instruct students to think of someone they know who is good at math.  The person could be a family member, sibling, friend or classmate.  But it had to be someone they knew.

Step 2: Think about the characteristics that person has that helps them to be good at math.

Step 3: Have students work together to compile their list of 5 characteristics.

Step 4: Display the lists in the room and ask students what they notice about the lists.

The lists generated by one of my classes is posted below.

We had to unpack what "Calculated" meant in the first group.  They described it as being organized and thoughtful about how they approached problems, a "calculated approach".  We also discussed "thinking outside the box".  This is partly seeing things from multiple approaches and can be improved with practice.

In my second class, we put the responses on paper and looked at them under the document camera.  That class had words like focused, organized, hardworking, determined and diligent.  I pointed out that these are characteristics that can be developed by anyone and that means they were all capable of becoming better math students.

I also told the story about my first math quiz grade in college - it was a 50%.  And I mentioned that I went to speak to the professor right away about that grade and then worked hard to understand the material in the course, looking for patterns in the problems, doing extra problems and working with friends.  The lowest quiz grade in that class was dropped and that was my lowest grade. I ended the class with a B+, because I put in the extra effort and got help when I needed it.

Today I had a student find me to check his work and he told me that exercise in class made him realize he should start working early and it motivated him to come in for a little help.  I also had another student email me about coming in for help.  She admitted that math has been very challenging for her in the past, but she wants to work.

So, thank you Howie for a great opening day activity!  And if you don't follow Howie on twitter you really should.

Thoughts on my Last Lesson

Today marks my last day at Kent Place School in Summit, New Jersey.  In August, I will be teaching full time at Moravain College, a much shorter commute from my home in Easton, PA. This marks the end of my 26th year of teaching.  You may think that after teaching for that length of time that I had nothing to learn about teaching, about students or about myself.  However, I have continued to grow as a teacher and I have gained insights in each of these areas this year .

Thoughts on My Last Lesson

I'll known for a while now that teaching is more than just understanding a subject.  It is about the students in front of you at that precise moment. Yesterday was the last day of school.  It was a half-day.  No exams, no regular classes.  Just final good-byes and one last chance to have students think and perhaps learn something new. Faculty were asked to design an activity and students would sign up to attend the 30 minute session of their choice.  The one I led was "The Case of the Stolen Jewels".  Seven volunteers were actors in a mini play.  The players were the cook, the chauffeur, the maid, the butler, inspector Euler Toots, Lady Shmendrick and the narrator.  Jewels were stolen from the mansion and the thief dragged his feet through the snow to throw off the authorities.  The testimony did not fit what was on the map.  However, I asked the students if they could tell from the map alone who stole the jewels.  (Can you tell?  Hint: Think Euler path.)

Map for "The Case of the Stolen Jewels"

I did this session three times and each time the student interaction was different.  Not only the interaction of the students with each other, but my questioning of the students.  Usually when I teach a lesson multiple times, my last time is the best.  Reflecting on how the lesson went the previous time(s), I can anticipate student questions and the direction of the lesson better.  However, my first time of leading "The Case of the Stolen Jewels" was the best yesterday and I think it was a result of my questionning.  I believe there are three main questionning techniques that can either open up a lesson or shut it down.

Number 1: Type of question  

Consider the following two questions. "We said you could trace a path if you start and end at the odd vertices.  What other questions might Euler have investigated related to odd and even vertices?" versus "Do you think it is possible to trace if we have three odd vertices?" 

The first questions directs students to focus on the vertices, but doesn't suggest any specific changes to make.  Any answer to this question is open to exploration. This was the question I posed to my first group and we had a lively discussion related to changes they suggested - 3 odd vertices, all even verticies, all odd vertices, etc. They were thinking like mathematicians. 

The second question removes the mathematician agency from the students.  It is a yes/no question and leads students in a very specific direction.  And since it definitely has a right answer - it can be traced or not - students hesitate to answer because there is a chance of being wrong and in the eyes of many students, even on the last day of school, wrong is bad.  This second version of the question was asked to my third group of students and it definitely changed the atmosphere of learning.  I had to follow it up with multiple questions and the discussion overall fell flat.

Number 2: Wait Time

This year I became better at wait time.  When I would ask a question, I would often restate the question - either word for word or slightly revised.  This gave students who hadn't fully heard the question the first time to hear it and it gave some more time for students to think.  Also, I did not go with the first person to raise her hand as I would have in the past.  Doing this often rewards the fastest thinkers and leads other students to think "I don't need to think about this question, because she will call on Susan. Susan knows all the answers."  Instead, I would wait until many hands were raised and often call on students who participate less often.  And, even if those students say something that isn't fully correct, we unpack it as a class.  Early in the year, students learn that mistakes are a valuable part of the learning process. You can learn so much from doing something wrong.  In response to mathematical misunderstandings, you can often hear me say things like, "I hadn't thought of that. Thank you for sharing that idea." or "Let's think about that some more.  There is something we all can learn from what you just said." or "That is an interesting thought.  Let's see what happens when we do that."

Number 3: Share with Others First

Not all students are risk takers. Think back to your days as a student. How many of you would say "I loved to share my thoughts that could be wrong and incomplete in front of 15-20 of my peers for them and the teacher to critique."  Students prefer to participate when they think they are mostly right and will be validated for their correctness.  So, how do we get students willing to take risks and share?  When I ask a question and see that no one wants to take a risk to answer it, I say "share some thoughts about that question with your group for a minute."  Then, I walk around the room and listen in on the conversation.  Sometimes, I will even tell specific students that I'll be calling on them to share their idea.  More students are willing to become risk takers after that minute of sharing with others first.

This summer I will be leading/co-presenting at several different workshops or conferences. Often the pace of the workshops is frantic, but that generally doesn't lead to deep understanding.  Sharing, risk-taking and reflecting will be a large part of my workshops this summer, as it has been in my classroom this year. Be sure to visit my blog in the fall to hear about my reflections in the college classroom and if you are a teacher, be sure to take some time for yourself this summer to relax and reflect.

Even Tech Has Its Limits: Learning from Students

One of the things I absolutely love about my teaching situation this year is that I have students that are very curious. They ask "what if" questions, make connections between concepts and notice things that I, even after twenty-six years of teaching, haven't noticed.  This type of noticing happened in Advanced Algebra on Friday and reminded me that math tech can have its limits.

Here is what happened. Do students really need to know about end behavior or behavior at x-intercepts for polynomial functions?  They can just plug the function into Desmos and see what it looks like.  But because I let my students discuss ideas and they can share them freely without fear of "being wrong", one of my students commented the function f(x) = x3(x+3)(x-4)  (shown here) seemed to have many

x-intecepts around the origin.  Because we had talked about behavior of x-intercepts of polynomials based on their factored form, she knew that there should only be x-intercepts at 0, -3 and 4.  Yet Desmos was telling her something different.  So, we decided to zoom in to verify that the function really only had one x-intercept at the origin.  And this is what we saw, after zooming in and tracing along the graph. It looked as though my student was right! The graph showed that several of the values close to zero had y-coordinates of 0, but we knew from the factored form of the polynomial that this could not be.  Desmos was lying to us!!! How could we trust it to do our math correctly?  I could tell that several students confidence in Desmos was shook.  At this point, we decided that  Desmos had to round the output to show it on the graph.  And based on what rounding convention Desmos used, it rounded the result to zero.  The students' understanding of polynomial behavior helped us understand what was really happening near the origin.  Incidentally, f(-0.002) is not zero, but it's pretty darn close to it at 9.598 x 10-8. Letting students share what they notice and wonder can be risky.  You don't know what they will say all the time and sometimes they may surprise you or throw you a curve ball, where you don't have a satisfying answer.  But letting students see that you are on a learning journey with them can definitely have unintended positive consequences, where even you as a teacher can learn something new.

Bonus tech tip: If students want to use the Desmos app on their phones, tell them to download the  "Desmos Test Mode" app and use it when doing their math homework.  If the select "Start Test", they can work in a distraction free way - no notifications of any kind will pop up on their phones!

Teacher-Centered vs Learner-Centered (The Post I Never Finished 'til Now)

Note: I have not blogged in a while.  It's been a little over 3 months since my last blog.  When I went to blog today, I noticed I had a draft of a blog going.  It was a blog that was to set the tone for the beginning of the school year.  It will now set the tone for going back to school after Christmas Break. For what you will be reading below to make sense, you need to read my previous blog entry. 

Image from https://blog.stetson.edu/faculty-engagement/2016/09/4849/

Reviewing The Old Transcript: Teacher-Centered Learning

1) Only the students who raised their hand are engaged in the lesson.  Those who didn't raise their hand could be engaged, but I don't know that for certain.  I have no sense of what those students know or understand about the lesson.  It could be that over half the class is totally lost.

2) The questions only allowed for one right answer.  This leads students to think that math is only about getting right answers and perhaps explaining how you got your answer.  To be honest, I used to think that math was only about getting the right answer until I had taught for many, many years and came to realize that learning math is more about the journey - the multiple ways to solve the problem and the connections between the concepts - than the destination of  reaching "the answer".

3) I was the creator of the knowledge.  Students weren't constructing their own knowledge.  I was seen as the source of right answers.  Students were not expected to evaluate their own answers or thinking.  Self-reflection was only reserved for the occasional student who had an inate sense of self-reflection as a key to deeper learning.

Some Observations Relative to Becoming Learner-Centered

Notice that the heading says "becoming learner-centered". Moving from a teacher-centered classroom to a learner-centered classroom is not like turning on a light switch.  It takes time.  It is easier for me to have a learner-centered classroom in Advanced Algebra and AP Statistics, courses that I have taught for much of my career.  It is my first year teaching AP Calculus and I don't have the same sense of security in the content, in common misunderstandings, pacing of lessons, and in how the content is inter-related.  In AP Calculus, it is very easy to, and often unintentionally, turn back to a more comfortable teacher-centered approach. 

So, what am I doing to get my classroom to be more learner-centered?  I am allowing more time for students to talk.  I truly believe that the more students talk about a concept the more they are engaged in learning.  With increased engagement and teacher guidance, there are more opportunites for ALL students to understand what is being taught.  Plus, it becomes easier for me to identify which students are struggling.

I'll talk more about the strategies I have developed for creating a learner-centered classroom in future blog entries.  But if you are ready for some now, I recommend reading this blog by Sara VanDerWerf.  In it she discusses a strategy called "Stand and Talk", a valuable stragegy I have begun to use in my classroom.