Sunday, February 3, 2019

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!

Thursday, January 3, 2019

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

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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.

Saturday, September 22, 2018

New School Weeks 2 & 3: The More They Talk (Part 1)

Now that I am into weeks 2 and 3 at my new school I am in the grading avoidance mode.  Even first year teachers recognize this mode within a month or so of school.  You look at your bag with the pile of papers in it and think of other things you would rather do.  It's Saturday morning.  I have plenty of time to attack those papers.  Here's an idea. I'm going to blog instead!

          An Old Transcript: Teacher-Centered

Are you a teacher who is afraid to have your students talk?  Does your classroom dynamic go something like this?

Teacher: Who can tell me what is f(6) for this function?

A few hands go up and then a few more. Teacher calls on one of the students who raised a hand.

Student: Twenty-two.

Teacher: Good. What about this question.  What is x if f (x) is 12?

The teacher gives sufficient wait time for a few hands to go up, but notices that they are mostly the same students as before.  Teacher calls on another student.

Student: x is 8/3.

Teacher: That is correct.  Can you explain how you got your answer.

Student: I wrote 3x + 4 = 12.  Then subtracted 4 to get 3x = 8 and then divided by 3.

In my early years of teaching, I would have considered this a successful classroom discussion.  I wasn't telling students the answers.  I was asking questions and students were giving me correct answers.  I look at this transcript now and cringe.  This is my twenty-sixth year of teaching and even now my classroom looks different than it did five years ago.  What does this look like for me in my classroom this year?

          A New Transcript: Student-Centered

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: Let's say that I wanted a function to make the input of 4 become an output of 3.  We could do that by writing f(x) = x - 1.  I want you to work with your partner or trio to come up with a different function that would contain the point (4, 3).  Write your function somewhere on the board.

Students work together and talk in their groups for a minute or so with each group putting a function on the board.  Here are several of the functions that were created.

Teacher: What do you notice about the functions that are written here?

Many students are raising their hands. Teacher calls on one of the students.

Student: Several of the functions are the same.

Teacher: That's true.  Is there anything that all or most of the functions have in common?

Teacher gives time to think and then calls on a student who does not have her hand raised.

Student: They all had something multipled to the x and then either added or subtracted a number.

Teacher: Good observation.  Which ones had addition and which ones had subtraction.

Teacher calls on student with hand raised.

Student: When you multiply by a number bigger than 1, it made the result get larger. So you needed to subtract.  When you multiplied by 1/2, it made the result smaller and that meant that you needed to add a number.

Teacher sees some students nodding in agreement as she restates what the student just said.

Teacher: I would like each group to answer the following questions. (Points to the board.) What is f(6) and what value of x would make f(x) = 8?  Write your answers and work under the function you wrote on the board.

Students work together and show their work.  A few students finish quickly and the teacher asks those students to create a new function that is more complex, maybe involving exponents.  Those students add their function to the board.

Teacher: Talk in your trio or with your partner about the difference between how you answered these two quesions.

Students talk together for a minute or two.

Teacher: Who would like to share what you just discussed?

Teacher acknowledges a student near the windows.

Student: The first question gives us x and we plug it in to get the answer.  The second question gives us the answer and we have to use it to find x.

Teacher: That's true.  Would anyone like to add to what [student's name] just said.

Teacher acknowledges a student near the classroom door.

Student: The first question gives us an input, or x and we find the output, or y.  The second question is asking the opposite.  We are told an output and we find the input.

Teacher: That's a good observation.  I'd like everyone to take a minute to write that observation in their notes.

Teacher observes students writing and waits for almost all the students to be done.

Teacher: Does anyone else have observations to make about what we just did?

Student raises her hand and teacher calls on student.

Student: (Referencing the answers on the board.) Why don't we all have the same answer?

Teacher: That's a good question, [student's name]. Why don't we all have the same answer?  Can anyone clarify this.

A few students raise their hand and teacher calls on one of them.

Student: They all went through the point (4, 3), but that doesn't mean that they will all have the same answer for the other questions.  They are different equations and will go through different points.

Teacher: That's true.  You all created equations of lines, but they aren't the same line for everyone.  They have different slopes and different y-intercepts.  Think back to geoemtry.  If you have a single point, there can me many lines that go through it.

Teacher looks at student who originally asked the question to see if this makes sense to that student.  Student is nodding head in agreement.

          Compare and Contrast: Old vs New  

My blogs don't often invite comments, but for this one I definitely want to invite comments.  I can see the difference between the old transcript and the new transcript; can you?  Write what you notice in the comments below.  I'll be blogging my compare and contrast thoughts about this lesson in a week or so. I attack those papers, clean the bathrooms or mow the lawn?