Friday, November 7, 2014

Student Curiosity

I have spent the last six weeks in my Geometry class developing definitions for rigid transformations (reflection, rotation, translation).  Some might call me crazy for devoting that much time, but I would argue that in addition to learning the mathematical content, students have learned mathematical habits of mind that will carry them through the rest of the year, and hopefully, the rest of their lives.  I'm a "big picture" kinda gal, and I want my students to learn skills that are transferrable to their experiences outside my classroom.

Toward the end of the unit, I gave my students a task that was written by Jade White (High Tech High) in order to sort of "bring it all together".

As you can probably guess, the last problem has students graph a letter "H", but it has been rotated 90 degrees and is therefore not oriented correctly.  The task I gave my students was:

Identify a single transformation that will orient this letter correctly and send it to Quadrant 1.

I was impressed to see that no students were trying translations.  I even heard them saying to each other, "We can't use a translation because that would not change the orientation."  Pretty awesome stuff!

Students grabbed patty paper and were testing out their ideas.  They were rotating, trying to figure out where the center of rotation should be.  They were reflecting, trying to find a line of reflection that would work.  I was impressed to see their level of commitment and engagement in this problem.  In addition to that, I was excited to see that they were using their definitions of transformations to look for their, what we call, "key ingredients" (line of reflection, center and degree of rotation).

Once students had arrived at an answer, I asked them to share a transformation that satisfied our needs (proper orientation and located in Quadrant 1).  One student had identified a line of reflection of y = -x.  We tested it, and it worked.  We connected it back to our definition of reflection and students saw the connections.  Students presented other lines of reflection.

Another student had found a center of rotation at the point (-3, 3) and rotated it 90 degrees clockwise.  This was also a viable transformation and again, students connected the fact that this "worked" back to their definition of rotation.  Students presented other centers and degrees of rotation.

At this point in the discussion, we explored what constituted as "Quadrant 1".  Students shared that they found lines of reflection or centers and degrees of rotation that had segments or points from "H" mapped onto the x- or y-axis.  Is that allowed?  Then a student asked, "Well, what's the definition of Quadrant 1?"  After I picked my jaw up off the floor, I said that my understanding of Quadrant 1 was that the x-values and y-values had to be positive.  Students decided for themselves if they wanted to "count" the x- and y-axis as Quadrant 1, but the fact that they even asked was exciting for me.

The question that inspired me to write this post came from a student who is always thoughtful in her learning.  She does not take things at face-value, she wants to know and understand concepts deeply and as completely as possible.

"I noticed that the center of rotation of (-3, 3) is on the line of reflection y = -x, and both satisfied our requirement of sending H to Quadrant 1 with the correct orientation.  Is it always the case that the center of rotation will be on a line of reflection that works?"

I didn't know the answer.  I gave students some room to explore, which I guess was a way for me to buy myself some time, but honestly, not all of them were engaged or willing to persevere through the question, so we moved on.  I have since explored this question using Geogebra and I have some ideas surrounding this question, but I still don't know the answer.

What are your thoughts?  What if it's not an "H" we're talking about?  Is this idea generalizable?

For me, this is exactly what I want to see in my classroom: students being curious about the mathematics they are learning and feeling safe enough to ask about it.  This is the kind of magic that happens when we give students the time and space to explore.

Wednesday, November 5, 2014

Mathematical Modeling: Counting Trees

Estimation is something that my students have always struggled with.  In my experience, some reasons for this are that a.) students are rarely asked to do it, b.) students are conditioned to not take risks in math class, and c.) there is a genuine fear of being wrong that students grapple with when engaging in mathematics.  But take a second to consider how often, in the course of a single day, do we make estimates?  Whether it be at the grocery store, driving in your car, or cooking a meal, I would argue that it is a skill that we use on a regular basis.

So, how do we get students to develop this skill?  I heard Dan Meyer ( speak a few months ago and as we were working through a series of problems, he asked us to write down an answer that we knew was too high and one we knew to be too low.  I had never done this before, but this particular strategy helped lower my fear of being wrong because, in a way, he was asking me to be exactly that...wrong.  Calling it out, and recognizing that this particular answer that I'm giving you is incorrect, provided the safe space to take risks and explore the solution more authentically.

I have made it a priority to provide my students with opportunities to engage in mathematical modeling tasks this year.  I wrote about our first modeling experience back in August (What Do You Mean, There's No Right Answer?!?) and since then, I have noticed that students are more willing to take risks, try a variety of methods, and overall be less comfortable (that's when I know they're learning).  Last week, I gave them a task called "Counting Trees" from the Mathematics Assessment Project website.  This particular task necessitates the use of estimation and communicating reasoning; two skills that I know my students need to work on.

When I handed this task to my students, there was a general feeling of annoyance and irritation.  "Do we really have to count all of these trees?  That's going to take forever!"  I told my students that I couldn't agree more, how annoying would it be to have to count all of these trees?  But then I directed them to the wording of the task, "Think of a method Tom could use to estimate the number of trees of each type."  This seemed to help with the grumbling and students felt a little less anxiety about having to actually count the trees.

When I worked on this problem for myself, I approached my estimation as a sampling of the entire collection of trees.  I broke it into smaller, more manageable pieces and then scaled up.  I have done this task with teachers and this is a very common approach, so I anticipated seeing it a lot with my students, but that was not the case.  More about that later.

In my experience, a modeling or open-ended task can be extremely meaningful for students, but to get the most "bang for my buck", I have learned that it is essential for me to have a game plan.  When I first started doing these kinds of tasks, I would just give them to my students and see what happened.  There were exciting moments, but I felt like the experience as a whole lacked coherence and a sense of culmination.  I hadn't really answered that "So what?" question.  For the Counting Trees task, I tried taking notes in a systematic way so that I could structure the conversation/debrief in a way that made sense as well as answered the "So what?" question.

The first picture is the document that I created for each class period.  I was curious what methods I would see my students using, but I wanted to remember who used that particular method, and then I wanted to record how many old and new trees their method got them.  The second picture is from all three of my classes.  I wanted to get a record of what numbers student were coming up with.  I then wrote the number of old and number of new trees on the board in order from least to greatest number of old (along with its corresponding number of new trees).  See below.

These records of student data (above) helped drive the conversation about how many old and new trees Tom could estimate to be in the plantation.  I began by asking students to look at the data and give an argument as to why they disagreed with a particular data set, and why.  For example, in the first picture, students were not comfortable with the 89, 37 numbers because they knew that there were way more than 126 trees in the picture.  In the second picture, students were not comfortable with the 330, 110 numbers because that was a total of 440 trees and they knew that there was enough space for 2500 trees, which would have accounted for only 18% of the plantation.  Looking at the picture, they knew that there needed to be more trees.  Similarly, the 1500, 750 data was, for some students, too high of an estimate.  They did not think that 90% of the plantation was covered with trees.  There was not consensus on this point, however.  Some students argued that this was reasonable.

As the discussion continued, I asked student to come to the document camera and explain the method they used to estimate the number of new and old trees in the plantation.  I started with the most common approach, which was to count the number of new and old trees in a single row and the number of new and old trees in a single column and multiply the two numbers together.  Some students described this as being similar to a multiplication chart, others thought of it as an area model.

Please note that all of these student work samples come from different classes and are a sample of each approach to the problem.

Row Times Column:

I then built on this idea, but showed a slightly different approach where students didn't look at a single row and column, but instead at the perimeter of the plantation.


Students discussed similarities and differences between the two methods, which they thought would be more accurate, and even how it was similar or different to their own method.

I then extended the conversation to include student work samples that showed using multiple rows in their sample (as opposed to a single row and column).  Students found that this may have been slightly more accurate because it was a larger sample, and still manageable, so they didn't have to try and count all of the trees in the plantation.

Using 4-5 Rows of Trees:

The next set of student work shows the estimation approach of breaking up the plantation in a variety of ways.  As I mentioned above, this was the most common approach I saw teachers use, but this was not true for students.  The single row and column approach (above) was the most common for students.

Breaking it into Smaller Regions:

After looking at these approaches, several students thought that these would be the most accurate.  It was interesting to talk about why different ways of breaking up the plantation may have yielded different estimations.  A lot of students said they preferred this method to others they saw or used themselves.

Something that came up in our discussion was and interesting use of division.  Some students calculated the number of total trees and then divided by two.  I put this back on the students: "What are they assuming by dividing the total number of trees by two?"  They were spot on with their response: "They're assuming that there are and equal number of new and old trees."  We talked about whether or not that was reasonable.  Most students felt that there were not an equal number and that there were more old than new.

Equal Numbers:

Another interesting idea that I saw students using was counting the number of blank spaces and using that to guide their estimate.  Essentially, there are three things to consider: number of new trees, number of old trees, and number of blank spaces.  Most students only focused on the number of trees, while others thought it was important to also include the number of blanks in their calculations.

Using the Blank Spaces:

Lastly, the approach that I saw that rounded out the conversation dealt with ratios of trees.  As you can see, some students were not able to clearly articulate how they intended to use ratios, but I was impressed to see that students were able to make a practical connection to the use of ratios to help guide their estimate.  As I said, this was the last approach I showed because for most students, it was the most sophisticated. 


In reflecting back on this experience, I think that I could improve it by making stronger connections to the similarities between the various methods.  I did my best to bring it back to the mathematics that the students were using, but also, what were the assumptions that they made?  Students are learning that just because not everyone arrives at the same answer, they're more or less correct.  They are learning that as long as you state your assumptions and use sound mathematics to answer the question, your solution will make sense.  

I feel that this was a great task to use for students in developing their writing skills.  I would like to place more emphasis on the precision of language next time because I feel like students were a little too vague in their descriptions of their approaches.  However, overall, I feel that they were successful and it was an authentic, meaningful experience.

And I know what you're all thinking...Did anyone just count the number of trees?  

Yes...yes, they did. :)

Saturday, September 27, 2014

Building Definitions, Bongard Style

In Geometry, I always find it difficult to get students to communicate the "Why?" or the "How do you know?" part of their solution.  At a very basic level, I believe that this is why traditional two-column proofs are so challenging for students; they are rarely asked to defend their reasoning in a math class. As I mentioned in a previous post, I've been doing a lot of things surrounding vocabulary and definitions this year, and I have been pleased with the results.  Students are starting to defend their solutions with statements like, "By the definition of midpoint, I know that segment AB is congruent to segment BC."  I believe that this transition is greatly attributed to the fact that they were given the opportunity to build on their own understanding and definition of a term, I just helped solidify it and give it a more academic spin.

What is a Widget?

My colleague and friend Amy Zimmer ( shared an image with me at the beginning of the year that comes from the Discovering Geometry textbook (see below).  The goal is for students to develop a definition of a Widget based on the pictures provided.

I displayed the picture on the board and asked students to quietly observe what they thought a Widget might be.  I asked them to be mindful of the characteristics that they were looking for to establish their definition.  After a minute of solo think time, I had students turn and talk to a partner about what they were noticing and what their initial ideas were for what it meant to be a Widget.  I called on a few students to share the characteristics they were considering, not to share their definition just yet (see list below).

Once the class agreed that we had listed all of the characteristics that were important, or even ones that didn't end up being relevant, we started to use those to refine our definition.  I had students turn and talk to their partner again to formalize their definition and I asked them to share it out with the rest of the class.  I wrote notes on the board and our definition was born.

Once we had our definition, we tested it on the "Widgets" and "Not Widgets" from the picture to make sure it held true and that we had not overlooked anything.  The last step of the process was to see which of the four images (A, B, C, and D) were Widgets.  Without having to ask, students were able to say Yes or No and provide a reason why.  I asked, "Is A a Widget?" and students responded with, "Yes, by our definition, A is a Widget."  When I asked, "Is B a Widget?" students said, "No, because it does not have exactly one eye and one claw."  It was obvious that they actually appreciated the concrete set of rules that they had developed in defending their answer.

In two of my classes, the rule about the eye and claw was not immediately as precise as "Exactly one eye and one claw".  Students had said something about they had to have eyes and claws.  When we looked at picture B, it required us to go back and revise our definition, which was so wonderful to show them that our first draft did not need to be our final draft.

It is my guess that for most students, this kind of abstract reasoning is not happening very often.  When students think about definitions, they probably think about their English class and having to look up words in a dictionary.  I wanted to show them that using definitions is an acceptable way to build an argument and that those definitions and arguments could be even stronger if they were given the chance to build them based on their own understanding.

Bongard Problems

Dr. Patrick Callahan (Co-Director of the California Math Project) had shown me these problems a while ago and I was recently re-inspired when I saw him share them with a group of teachers in Fresno during a professional development session.  Here is a slide from Dr. Callahan's Power Point:

A Bongard Problem is a collection of 12 images where the 6 on the left hand side all share a characteristic (or characteristics), and that characteristic is not present in any of the 6 images on the right hand side.  The task for the reader (or student in my case) is to determine a rule that works for all images on the left, and none on the right.  Here is an example (also taken from Dr. Callahan's PPT):

Can you figure out the rule?  This one is fairly easy.  Spoiler Alert!!!  All images on the left hand side are triangles and none of the figures on the right hand side are triangles.  Try this one:

Here's what I love about this particular problem (it's #4 on the website) -- students might not know the words concave or convex, but what so many of them pick up on is that the images on the right hand side are all "sunken in", or "have a bend in them", or even that they have a "part missing".  They notice that all of the images on the left hand side are "pushed out" or have "nothing missing".  This is exciting because it's an opportunity to build on what they already know and can observe about a figure to then introduce the more formal term for these ideas (concave and convex).

When I first introduced these to my students, I shared these two, along with some others that were more challenging.  What I found interesting was that there wasn't a sense that I was the one who held the answer.  Students had already picked up on the fact that they could check to see if their rule held true just by looking at the images on the left and the images on the right.  I know that some teachers struggle with the idea of not being the authority in the classroom in terms of knowledge, or having the answer, but this happened so naturally that I actually addressed it with my students.  I wanted them to acknowledge that they had the skills to be able to answer the question "Does that work?" or "Is that right?"  They learned very quickly that the responsibility of coming up with a rule fell on them, not on me.

For homework that night, I had students create their very own Bongard Problem.  I wanted to see what they would come up with and how creative they would be.  These are some of my favorites.  Which ones can you figure out the rule for?

And Now What?

I wanted to continue with this work because I feel that the use of definitions is something that we can rely on throughout the school year.  But, I wanted to formalize it and see if I could help students organize their thoughts with a template or structure that allowed them to articulate their thoughts more clearly.  This is what I came up with:

When I introduced the template, it was clear that students were a little apprehensive to write anything down because it seemed so final.  But with some practice, they realized that it was a work in progress and that they could take the risk of not having it perfectly stated the first time.

I asked a random student to pick a number between 1 and 100 (the number of problems that Bongard wrote) and in my first period class, the chosen student selected 94.  So, we looked at Bongard Problem #94:

This ended up being a great one to address precision of language.  Students realized that a subtle difference in wording could actually mean something that they did not intend to say.  With the template, students were given individual think time to develop their own rule and write ideas down, then they could turn and talk to a partner to refine their rule and perhaps see things they had not initially observed.  After about 2 minutes, I selected a group at random to share their rule with the class.  I gave them one minute to come to consensus as to what I would write on the board.  Once I had their first draft of a rule written on the board, it was the job of the rest of the class to brainstorm feedback or clarifying questions for this particular group.  It was not an opportunity for a new group to share the rule that they come up with for us to compare.  I said that we were committed to helping this particular group come up with the best version of their rule as we could.

Students provided feedback on things that surrounded the precision of language.  They referenced the actual picture to defend why they thought their feedback was important.  If the first draft of the rule said "The black dot is not at the end of the string of dots", students might ask for more specifics of the black dot's location in regards to the white dots.  Or, if the first draft said "The black dot is between 2 white dots", students might ask about the image on the left hand side that is on the right in the middle row.  This elicited more precision of language such as "between at least white dots", etc. 

Here are some student work samples of how they worked through the process of writing a rule:

We did five of these Bongard Problems as a class over the course of about 3 weeks.  I would start class with them, almost like a warm-up.  I especially like how this practice went along with the first unit in our textbook, which is very heavy with new terminology and vocabulary.  It was nice to have an example separate from the textbook to use that showed the importance of developing a definition to build an argument.  I am excited to see where this takes us in our next unit on transformations!

Lastly, I wanted to share some pictures from the last Bongard Problem that we did as a class.  These come from the three sections of Geometry that I teach, and I wanted to capture the original rule and the result of giving feedback and modeling revision that I used with my students.

The Bongard Problem was #58:

And here is how students developed their rule:

I would encourage teachers to use Bongard Problems in their classes to help students build a foundation for using definitions and understanding the importance of precision of language.  There are some Bongard Problems that are quite challenging, but for the most part, I have found that students enjoy the puzzle element of finding the rule and are willing to take the risk of being wrong the first time.  I have seen growth in their reasoning skills and I am encouraged by their perseverance and overall willingness to try and work hard.