I Notice, I Wonder (Part 2)

(I apologize for getting this up a bit late.  Almost two months after the initial post.  I'm working on National Board Certification renewal and life got in the way.  But what can you do.)

Given this diagram, what do you notice?  What do you wonder?  Annie Fetter (@MFAnnie) of the Math Forum tweeted that she wondered if segments BE and CD were parallel.  This was not the first thing my students wondered about.  They wondered if certain triangles were congruent and if certain triangles were isosceles.  This was not surprising, because we had just completed a unit on congruent triangles and isosceles triangles.

So, what did they notice and wonder?  I gave students about 2 or 3 minutes to create a list individually and then I put them in groups of 3-4 to compile their lists.  Then, we went around the classroom with each group contributing one item and me creating a master list.  This is something like brainstorming.  All ideas were welcome.  Once or twice we did have to move something from the "I Notice" column to the "I Wonder" column.  For example, students said "I notice AGF is isosceles." However, since we were not given any segment lengths or angle measures, we had to move this to the "I Wonder" column.  Most of the time students are given diagrams with angles and segment lengths given and the fact that this diagram did not have any measurements was slightly disconcerting to some of them.

Here is a sample of what they "Noticed and Wondered".

Next, I gave them some additional information about angle measurements. They were told: Angle 1 was congruent to Angle 2. Angle 3 was congruent to Angle 4 and Angle 5 was congruent to Angle 6.

This allowed us to move some of the ideas from "I Wonder" to "I Notice", and this lead to discussing why we could make the moves from "I Wonder" to "I Notice".  Do you see the beginnings of a formal proof???

Class finished by having students work in their groups to write a formal proof focusing on congruent triangles.  The class discussion was rich and the geometry ideas plentiful!  Thanks, Annie Fetter for helping me and my students to "Notice and Wonder"!

I Notice, I Wonder (Part 1)

Over the past month or so my Honors Geometry students have begun to have issues with my class. Not issues with me personally, but issues nonetheless. The bane of their recent mathematical experiences?  The thing that gives some of them nightmares of broken pencils and worn out erasers?  Yes, the very thing that I personally loved about Geometry in high school has become the new four-letter word in Geometry.  OK...so it has five letters.  

Why don't students like proofs??  I think after 21 years of teaching I may have found the answer!  Well, at least part of the answer from attending a talk at ATMOPAV (Association of Teachers of Mathematics of Philadelphia and Vicinity) last Saturday.  The reason students don't like proofs...(insert drumroll here)...is because they don't understand them!  Thanks to Annie Fetter (@MFAnnie) from The Math Forum for helping me re-think this based on ideas from her talk "Ever Wonder What They'd Notice (If Only Someone Would Ask)?"

Gee - students don't like proofs, because they don't understand them.  Not really an earth-shattering piece of information there, Leigh.  I agree, but do we ask students to understand what is in front of them before they see the given information or what it is that they are to prove?  Do we require them to take some time to notice and wonder what is happening in a given diagram?  I know I had never done that before with my students. 

So on Sunday night, I found a problem from our textbook that I would turn into a 2 day "Notice & Wonder" problem.  The image shown here is what my students got from me.  Notice that it doesn't have any information about angles or segments.  There is no "Given" information.

First, my students worked individually for a few minutes, completing what they noticed and wondered about the diagram.  Then, they were put into randomly assigned groups (Did I mention that I love playing cards for random assignment?) to discuss what they noticed and wondered.  Last, we came together as a class to compile a master list.

Before I share their lists with you, what do you notice and wonder?  Be sure to check back to see their lists and how we moved some items from the wonder column to the notice column.

What I Like About Twitter and PLC's for Math

One of my favorite (buzzword altert!!!) PLC's is all the Twitter Math & Teaching peeps that I follow. It was only last spring that I realized how invaluable Twitter can be for math teachers. (A shout out and thanks to @mgolding for getting me hooked on Twitter and interested in/involved with The Global Math Department.)

Here is what I really love about Twitter: Within 2 minutes of looking at math tweets, I have found either something I can use in my classroom or something that someone in my department can use in their classroom.  Really - that is no lie!

Example #1: Finding alternative assessments that give each student something different to do is not always easy.  When looking through Twitter in April of last year, I found an idea that I shared with the PreCalc teacher at my school.  It was called the Birthday Polynomial.  She used the idea and made it own.  Plus we had tons of student created posters to decorate the math hallway.  I think I got it from here: Birthday Polynomial

Example #2: (From about 5 minutes ago) I am going to be renewing my National Board Certification this year and I need to find more ways to get students to reflect on what they are learning and how they think class is going.  While searching one of the suggested math blogs http://made4math.blogspot.com/, I found something that I can use!

You might be saying, "Sure - it's easy for you to find useful items, but I don't want to wade through tweets that don't apply to me."  You don't need to! I organize my Twitter feed in TweetDeck.  Here is a screenshot of my twitter feed as organized in Tweet Deck. This pic shows only 4 of my the twenty columns I have set up in Tweet Deck.  

If you want to learn how to TweetDeck, here is a link to a youtube video and here is a link to a wikihow on How to Use TweetDeck.

Mrs. Nataro's Postulates of Learning

This year I have set a goal to post to my blog twice a month.  Thanks to the blog prompts at Exploring the MathTwitterBlogosphere I may meet this goal for the next 2 months.  For this blog, I have decided to address the following:

What is one thing that happens in your classroom that makes it distinctly yours? 

Here is what makes my classroom unique to me:

About 10 years ago, I was writing my opening letter to parents and students and came up with the idea of Mrs. Nataro's Postulates of Learning.  I wanted students and parents to know what I personally believed about learning mathematics in my classroom.  These postulates are also prominently displayed in my classroom.  Here are my five postulates:

1)  All students are capable of learning math.

2)  The use of prior skills and knowledge is required to build new skills and knowledge.

3)   Learning is a collaborative effort.  Helping others learn will help you to learn.

4)  Asking thoughtful questions is a primary way to open your mind to learning.

5)   Choosing not to do an assignment will rob you of the opportunity to learn.

These postulates really do guide my teaching and what happens in my classroom.  First, I really do believe that all students are capable of learning math.  Once the student believes this of himself or herself progress can be made more easily.  The second postulate may seem obvious, but many times students start to have difficulty in math because of a misconception or shaky foundational knowledge.

The third and fourth postulates can be seen in action on a daily basis in my classroom.  Students quickly come to learn that I want them to work together on assignments.  Sometimes they are randomly assigned peers for a specific project and other times I have them compare work with their neighbors.  After I help a student with a problem, I have them explain the solution to one of their peers.  At first students think this is a little strange.  However, they soon see that they really do understand better after explaining an idea to someone else.  In addition to working together, students are not afraid to ask questions.  I encourage questions with phrases like "I am really glad you asked that question." or "That is something I hadn't thought about.  What an interesting question."  I often find myself sharing the questions that my first period class had with my second and sixth period classes.  The questions are that good and I want all my students to benefit from them.

For the fifth postulate, students quickly see that my assignments aren't "busy work" and that they really are an opportunity to learn more.  If a student doesn't have an assignment completely done on the day it is due, he or she still does the assignment to show to me, even when a point or two is deducted for the assignment being late.

As we review my postulates in class on the first day of school, I point out that these postulates are about learning in general.  And I encourage my students to apply them to any subject they are studying.

Math from Multiple Perspectives (Part 2)

In my previous blog post, I talked about a problem that I showed to my students on the first day of school.  This problem was to get them to think about math from multiple perspectives and it was taken directly from the Stanford University MOOC that I took over the summer called "How to Learn Math".

How many squares do you see along the border?  Don't count them one by one.

First, I told students to think about an answer and put their thumb up in front of them when they thought they had it. (A great idea I learned from the MOOC.)  Students didn't shout out answers in excitement and ruin it for others!  And those who normally would shoot their arm up in rapid fire speed didn't intimidate others in the classroom!  Next I called on students to explain how they found the answer.  The five different solutions that are shown here were obtained by by 3rd period class. 

I could tell by the expressions on student faces that several were surprised that there was more than one way to think about this problem.  Next, I asked some follow-up questions with some unanticipated results. 1)  Which method would you use if you had a 20 x 20 grid?  Most students thought the third way was easier to use.  And I always thought this method was easier, too.  After all, who would want to do 18 squared in their head?? Not me.

But then it dawned on me! (It only took the third time for me to teach this lesson to see it.)
20² - 18² wasn't that bad if you thought of it as (20 + 18)(20 - 18), a difference of squares!  My surprise was genuine and the fact that I didn't notice this right away wasn't so bad.  My third period students had a chance to see me think aloud through an idea, and they had a chance to see that I still get excited about math when I see something "new" that I hadn't seen before!

2)  How do we know these solutions will always give the same answer?  Students easily explained how to generalize each solution based on an n x n grid, where n was the number of sides on a square.  (This can be seen on the left.) When I asked how we might show the expressions were equivalent, I was thinking that my students might simplify each expression to show they were equivalent. However, in each of my three classes, they suggested setting 2 expressions equal and solving for n. (You can see the circled expressions that we set equal in this particular class.) In each case, an identity resulted and they could explain what that meant, that the expressions always yielded the same result for any value of n.

3)  How can graphing equations show these are equivalent expressions?  This one stumped my students and unfortunately, we ran a bit short on time for me to make sure they really got this idea.  However, in two of my classes I showed Desmos as a quick graphing tool.  [I became more acquainted with Desmos at NCTM in Denver in 2012. Thanks Desmos folks for the cool trivia prizes! I love my mint green Desmos t-shirt!]  We graphed y = 4x - 4 and y = x² - (x - 2)². A screenshot of the graph is seen here.

You might be wondering why only the first equation is graphed.  That is because I kept turning the second graph off and on in order for students to see that these two equations really produced the same graph. 

Could I have incorporated other ideas, such as domain or the meaning of slope, into this lesson?  Yes.  However, my goal was to have my students begin to see math from multiple perspectives.  That goal was met. 

Math from Multiple Perspectives (Part 1)

Over the summer I took a MOOC through Stanford University called "How To Learn Math". It was led by mathematics education professor Jo Boaler. Although I prefer face-to-face discussions over discussions via online discussion forums (especially when over 6,000 people are taking the course), the course was chock full of interesting and relevant research related to teaching mathematics. More importantly being engaged in this course reminded me of what it really means to teach mathematics. Math isn't about right answers and algorithmic processes. It isn't about memorization and speed.  What math is really about is creativity, connections, deep thinking and attacking problems from more than one angle.  In fact, these ideas are necessary to forge a lifetime of learning in any discipline!

In addition, making mistakes and asking questions are two of the best ways to learn.  Students know it is ok to make mistakes as they are learning a challenging piece of music, but for some reason they don't feel that way about learning math.  Not all student answers are perfect, but we can learn from all student answers.

Here is an interesting quote from Harvard College Office of Admissions relative to what students need to know about mathematics for higher education.  "You should acquire the habit of puzzling over mathematical relationships.  When you are given a formula, ask yourself why the definition was made that way.  It is the habit of questioning that will lead you to understand mathematics rather than merely to remember it, and it is this understanding that your college courses require."

So, this year I will be focusing on two main goals for improving my teaching: 1) encourage a culture of questioning as a means to learn and 2) designing lessons/activities to help students see math from multiple perspectives.

At the end of "How to Learn Math", I created the following poster. I shared it with my students on the first day of class and we even looked at a problem that reinforced this foundational idea. Here is the poster:

And this is the problem we worked on during the first day of class.

(adapted from the course "How to Learn Math"):

How many squares do you see along the border?  Don't count them one by one.

This is a very accessible problem and all of my students were able to figure out a way to solve it.  After a student gave the first solution, I asked more questions related to the task at hand.  What is another way to solve this problem?  Will your method always work for different size grids?  Which method would you prefer to use for a 20 by 20 grid and why?  How do we know a particular method will always work?  What evidence would convince you that two methods are equivalent?

Stay tuned for Part 2 of this blog for answers to the following questions: What were the five different ways my students solved this problem?  How did I incorporate Desmos into this lesson?  What new insight did I see on the third time this lesson was taught?