Alexandra Lifka-Reselman5/15/2015Math 45

Reflecting on Your Teaching

I chose to have two classes of remedial Algebra I, tenth grade students, do an activity

exploring reflections of points over the x and y axis. Both of the classes have large amounts of

exceptional students so the teacher suggested that I split each period into two groups of about ten

students each. Each group worked for 15-20 minutes before returning to class. The first class

period had an associate who translated for a deaf student and an associate who translated for the

ESL students in the class. The second class period had an associate who did not offer support to

the students. I downloaded Geogebra onto a cluster of computers close together in a computer

lab as well as a teacher computer that was connected to the projector and had Geogebra running

when the students entered the room.

I implemented to two periods very differently, so I will address each section of this reflection

twice. The first class period implementation as well as worksheet failed so badly that I rewrote

the entire activity in the time I had between the two periods. I had hoped that because the

mathematical goal was of a very low level, the students would not be limited by their

exceptionalities and still be able to explore. This quickly proved to be incorrect and that the

students did not have enough experience with open ended questions to be able to make

conjectures and test them. I replaced the open ended questions with fill in the blanks and

significantly more specific wording for the second class period.

For the first group of the first class period, I started the lesson by having the students

explore Geogebra for a few minutes on their own. I then passed out the worksheets and gave

them very little direction except to explain their guess as in depth as possible and to clarify that

they remembered the difference between the x axis and the y axis. I do not think this lowered the

level of the task because it was providing necessary background knowledge. As I circulated I

noticed that many of the students put their initial point on an axis which would prevent the point

from being reflected. I could have let them keep the point there and create a guess that they later

disproved by dragging the point around, but I felt that they needed to see what the reflection

looked like immediately. I asked these students to select a new starting point, which lowered the

level of the task by preventing them from taking a different path than that of their peers.

For the second group of the first class period, I also started the lesson by having the

students explore Geogebra for a few minutes on their own. I once again reminded them about the

difference between the x and y axis, but I added an introduction about what a reflection is since

many students in the last group had not understood what the word meant. This may have been

more of a reading problem than a math problem. Explaining reflections lowered the level of the

task because it gave students a hint of what their conjecture should be. I requested from the

beginning that they find a point that is not on an axis. I also lead them through the set-up of their

first point and reflecting it over the x-axis to make sure they were using Geogebra correctly. This

may have limited exploration and pointed them in the direction that I wanted their conjectures to

go, but due to time constraints and the student’s limited experience with Geogebra I felt that it

was necessary.

For the second class period, I was even more rigid in my introduction. Even with my

modified activity and accompanying packet, I realized these students needed a lot of support.

Partially due to learning problems, partially due to lack of experience with technology, but

mostly due to inexperience with open ended questions, the students really struggled with making

guesses, checking their guesses, finding errors, and correcting their guesses. I added to my

introduction that the goal of the activity was not to come up with a correct answer, but explore

why they thought what was true, was true or not. I think this maintained the level of the task

because it put the focus on reassessing statements over and over, instead of trying to be correct

on their initial conjecture. I lead them through reflecting over the first axis as well as the other

introductions that I did with the first period which once again lowered the level of the task by

giving them too much leading information.

For both class periods I lowered the level of the task a lot during implementation. Very

few of the students identified places where their guess did not work when dragging the point and

its reflection around the graph. Most students only tested it horizontal to the x-axis(or vertical

with the y-axis) which supported most of their initial statements, or when they crossed the x-axis

they forgot which had been the initial point and which had been the reflection. For most students

I had to point out where their initial statement was no longer true. Instead of relying on Geogebra

for feedback, they relied on me to tell them whether they were right or wrong. This did not

surprise me since that is what has happened for most of their math education. They are not used

to identifying flaws in their work on their own.

Changing the written task helped students make more connections on their own. With the

original packet, students wrote specific points as their predictions and then became very

confused when they were prompted to write a conjecture that worked for all points. Once I

prompted them that their conjectures were incorrect, they were able to identify where they did

not work. I could see that they were making connections because while they could not create

rules that worked for all situations, they were able to identify that different areas of the graph

changed how the reflection corresponded with the initial point. A lot of the students were able to

verbalize that sometimes the point became positive and sometimes the point because negative,

but could not put it together into a concise statement. It was interesting that the first class period

referred to the points as a single concept (the point flipped, the point moved), but the second

class period referred to the x and y coordinates as separate situations (the x stays the same and

the y flips, the x stays the same and the y increases).

I think that my implementation of the task lowered the level a lot because I ended up

guiding almost over student to the path to the mathematical goal, or at least away from the very

wrong path. Even with this assistance, most of the students could not come up with a concise

conjecture that worked for a reflection over the x-axis. I assisted very little with the portion with

the y-axis and of the students that made it that far, few of them were clearly able to transfer their

connections from part one to part two. These students are used to being given what the right

answer is and working from there. While I guided them towards the correct answer, I would not

outright give it to them and they found it very frustrating. I think that they are too used to being

given the answer first and then working from it. The open-endedness of the activity just confused

them and very few of them made connections

Changing my written task for the second class period helped that students focus on the

mathematical goal, but it also lowered the level of the task. I wish I had tried the activity with a

non-remedial class as well so I could see if it was the task or students that were the source of the

confusion. I thought that the open-endedness of the task would encourage higher level thinking,

but instead the students put down exact ordered pairs as their guesses. This did count as a guess,

but it was not particularly useful. For the second written attempt, I removed that possibility by

having the students complete sentences that would not make sense with an ordered pair. I also

separated out the x and the y value so they were encouraged to think about how both aspects

changed, not just the point as a whole.

The first version had students identify where their guess was correct and use that to decide

how to progress. Many students did not read the directions carefully so they were confused on

what to do. They did not understand that they did not have to rewrite their guess if it had proven

true, or they did not realize they should rewrite their guess. This was especially confusing with

students who had only used a coordinate as their guess. Making the written task more specific

limited how much I had to intervene to encourage students down a mathematically useful path. It

also slowed down how much of the activity the students completed. Very few of the students

made it past part one.

The technology was very useful in that it sped up how quickly and correctly the students

were able to reflect a point. Almost all of them needed to be taught how to do so, even though

the directions specifically explained how. I am very happy that I chose to have them only reflect

a point as opposed to a shape like I had originally planned. When I worked with these classes in

the past, I saw them repeated incorrectly graph points. Geogebra was helpful in remedying this.

Once the students became used to Geogebra it was helpful, but the multiple tabs for each

button caused a lot of problems. Even though I had included images of each button they needed

to use, the students would select the wrong one without noticing and not understand why the

activity was not working. The students with behavior problems were distracted by the various

options in Geogebra and needed multiple prompts to only use tools that assisted with the

described task. Because I kept the task very simple there were not any technology problems that

I could not answer.

Instead of doing plan-teach-reflect, I did plan-teach-reflect-adjust-teach-reflect. Having

multiple opportunities to implement the task and several hours between the two class periods was

pivotal to the improvement of my task. Once I realized that my original written task was not

appropriate for the group of students I was working with, I had to change it. Having time to

observe and read through the first group of student’s answers helped me identify the failures of

the task. I tried to adjust the implementation with the second group and even that did not improve

the students’ confusion. It quickly became clear the whole task would need to be re-written.

Even though my rewrite did not have optimal results, it was still much better than the results of

the first class period. If I had a chance to try again I would either give the students much more

time, or have them only focus on one axis.