Asilomar Mathematical Modeling Presentation

This past weekend I had the pleasure of attending the annual CMC North, Asilomar Math Conference.  I have attended this conference before, but this was the first year that I was going to be a presenter at the conference.  The title of my presentation was "Embedding Authentic Modeling Problems in MS/HS Curriculum" and we featured two of Dan Meyer's 3-Act Video problems (http://blog.mrmeyer.com).  The audience had a choice to work on one of the two problems that we included and it seemed to go very well.  I've included the presentation below.  Please enjoy!

Re-Engagment Lesson

This lesson actually took place at the beginning of the year, but I am just getting around to it now...

           Through the last eight years that I have been a classroom teacher, one common thread that stands out to me is the amount of professional development that I have participated in.  I have learned about everything from strategies to engage English Language Learners, to how to incorporate technology into my classroom, to how to mentor a first-year beginning teacher.  With all of the different themes and directions that my professional growth has taken on, none of them have been as meaningful to me as those centered on Common Core and the Standards for Mathematical Practices.

            When I was asked to participate in the CPEC grant, I honestly had no idea what to expect, nor did I really know anyone who was involved.  I was anxious and excited and could not wait to see what was happening in my region with regards to Common Core.  The impact that this two-year grant has had on my teaching practice is invaluable and perhaps even immeasurable.  I continue to take risks and try new things in my classroom, as well as persevere when I am not successful the first or second, or even third time.  One of my favorite risks that I have taken recently was a re-engagement lesson in my Geometry classes.

            I have found in my eight years of teaching that the first chapter test is often the hardest for students.  I’m not sure if this has more to do with brushing off the dust from summer vacation, or the fact that Geometry is quite different than Algebra.  Nevertheless, my current Geometry students played into my theory and were not overly successful on their Chapter One test.  There were errors in notation, reasoning, and even setting up a correct equation to model a Geometric principle; all of which are areas that need to be addressed and remedied in order for a student to be successful in later chapters.

            After grading the Chapter One tests, it was obvious that I was faced with the age-old problem of the “haves” and the “have-nots” in the sense that some students "have" the understanding and others don't.  There was a large population of students who were struggling with basic concepts on the test, but there was a considerable amount of them who were successful and were ready to move on to new applications and ideas.  So, as their teacher, what do I do?  Do I move on and hope that the “have-nots” will catch on eventually?  Or do I start over and cause boredom and frustration amongst the “haves”?  My solution was to try a re-engagement lesson.

            I chose four areas/questions that I wanted my students to focus on.  The first three questions on the test related to Geometric notation and basic concepts such as where two planes intersect (many said a point).  The second area of concern for me was on an error-analysis question.  The majority of students were able to articulate which student in the scenario was correct and/or incorrect, however their explanations and justifications were weak and unconvincing.  The third and forth questions that I chose for my re-engagement both had Algebraic applications that stemmed from Geometry.  First, an Angle Addition Postulate problem that involved four adjacent angles (two of which were congruent) that formed a straight angle, and asked to find the value of the variable.  Second, an angle bisector scenario where students were given the measure of one of the smaller angles and the larger angle, both in the forms of an Algebraic expression, and asked to find the measure of the smaller angle.

            My current class schedule is laid out such that I teach three, 90-minute periods of Geometry every day.  Every other day I see the same group of students, so I see six periods over the span of two days.  The way that this particular test panned out was that I graded my first three classes during the first night and then graded my second set the next night.  However, once I graded the first set of tests, I was familiar with common errors and misconceptions, so I did not grade my second set of tests right away.  Instead, I went to the photocopy machine and made copies of student work from the second set of tests.  This allowed me to copy student work that was ungraded, so the students were unaware of the accuracy of the responses.  I selected student work that represented the common errors as well as a few correct solutions, some using a non-traditional approach. 

            The next time that I saw each group of students, this was also the day after their test, I told them that they would get their tests back at the end of the period, but they had some work to do first.  I knew that if they received their tests first, they would not buy in to the re-engagement activity.  On the white board I had written prompts that corresponded with each page of student work – there were four pages in total.

         
          For the first page, titled “Notation”, students were asked to find the error in each student’s notation and make appropriate corrections.  Because I photocopied the first three questions from the test and used that for each anonymous student representative, there were some pieces of each student work that were possibly correct.  If that was the case, students did not need to comment.

          The second page was called “True Statements About Angles” and required students to a. make sense of the student work sample, b. determine if the work was correct, c. if the student was correct, they needed to describe the approach, and d. if the work was incorrect, explain why.

           
       The directions for the third page, “Between vs. Midpoint”, were to first read the student explanation, and secondly write a question or a comment that could make the argument better.

            Finally, students worked on a page of sample work called, “Angle Bisectors”, which asked them to a. look at the student work and make sense of it, b. if the work was correct, state why, and c. if the work was not correct, determine where the student made their mistake and explain why it’s incorrect.

            As you can see, what was required of each student was a little different from page to page.  I did not want them to do the same thing for all, partly because the four pages dealt with four different mathematical ideas that I wanted to showcase.  My purpose was for them to become re-introduced to the problems from the test, and to use a critical eye to determine how they might make each piece of work better if it was incorrect, or expand their knowledge if it was an unfamiliar approach.

            As students began to engage in this activity, I was blown away by their interest and motivation to make sense of the mathematics.  Without asking them to do so, they started taking out their notes to use as a reference, they were communicating with their partner to see what they thought, and there were very few questions of me.  This was the polar opposite of what I normally see in my classroom.  It is a struggle for me to get them to interact with their notes and textbook, they usually turn right to me for the answers, and they don’t rely on their peers as resources or experts.

            As I observed my students working, I heard them discussing why a particular answer was true or false, they were critiquing the arguments that had been provided by the selected students, and they were working together to draw a conclusion.  I even heard comments like, “I’m going to be much more careful on my next test so that Ms. Balli doesn’t have to grade all of this crap!” or “Oh no, I don’t think this is my work, but I know that this is the way I did this problem, and now I see that it’s wrong.”

            There was one student work sample that I included because it was an example of an equation that had been set up incorrectly, but yielded the correct answer.  This was an interesting discussion starter because students asked me, “Would they get full credit?  They got the right answer!”  Eventually students began to realize that the importance did not rest in the correct answer, but rather the correct method.  It was obvious that, to them, math is all about getting the correct answer, but they were able to slowly move away from this because they saw evidence of getting the right answer through an invalid process.  I believe that this was the start of a cultural shift in my classroom – as their teacher, a correct answer is nice, but that is certainly not the only thing that I am looking for.

            When it came time to return the tests, I did a short debrief with them about the whole re-engagement experience.  Several students stated that they had never done anything like this before, and found it very useful.  The majority of my students said that they learned something from this lesson, and whether they scored an A or an F on the first chapter test, they felt strongly about these particular problems that they had examined and felt that they understood them better.  Regardless of his or her grade on the test, I did feel that each student was able to walk away from this experience with a deeper understanding of the mathematics.

            Just before handing back the tests, I shared with my students some of the research that has been done with relationship to feedback.  The three categories that were examined in this research study were teachers providing only a grade (no feedback), teachers providing both written feedback and a grade, and teachers providing only feedback (no grade).  The results showed that the greatest amount of student learning was from the population of students who only received feedback, no formal grade.  I had a good laugh with my students, because despite the findings from this research, I had provided them with both a formal grade and feedback.  I did this because I knew that I wanted to have my students participate in a re-engagement lesson prior to getting their tests back, so I felt that this would be a vehicle for student learning, regardless of me providing them with a grade.

            As the research predicted, once students received their tests with both written feedback and a formal grade, they appeared to be ready to move on to the next topic of discussion, as there was little interest in examining their mistakes.  However, I will say that they were able to connect mistakes that they made to mistakes that they saw from the student work samples.  The work samples may not have been a photocopy of their work, but the same mistake was made, and I do feel that there was something learned from making this connection.

            An observation that I made about my six Geometry classes was that, in some instances, there was not as much buy-in from my B-Day students (second day).  My B-Day classes have fewer Honors students, as the Honors classes (English, History, and Science) are offered more on B-Days, so I do find more Honors students in my A-Day classes.  However, I am not sure that the schedule of classes is the only culprit.  When I made photocopies of student work, I only made copies from my B-Day classes, so I am wondering if they felt discouraged because they recognized some of their work.  Are the students in these classes not as confident?  I am not sure what the reasoning is for the lack of connection, but I am interested to see that if I include work from both days, would there be an increase in buy-in?

            Finally, I want to take a moment to reflect on the two successful re-engagement lessons that I have undertaken.  The first re-engagement lesson that I taught successfully was using a MARS task that I had given my students as a formative assessment.  I was teaching a unit on surface area and volume and wanted to see how much they were able to apply to a composite three-dimensional figure.  The task involved finding the volume of a wine-shaped glass that was part cylinder, part hemisphere.  Students struggled with the composition of volume; forgetting to divide the volume of the sphere by two, or did not remember the four-thirds from the sphere formula.  There was a wide range of issues.

            When I did the re-engagement lesson, I provided students with six samples of student work, which they were asked to make sense of and then determine which student had the correct answer.  Without exception, every student in my three different classes was able to articulate why “Student B” was correct.  It was an amazing sight to see and I was confident that my students had really made a connection to the mathematics from the task. 

Fast-forward a week and I am now giving them their summative assessment.  The end of unit test included a problem asking them to find the volume of a hemisphere, and I was devastated to find that less than a quarter of my students were successful with this task.  This made me question the impact of the re-engagement lesson.  Was it not all that it was cracked up to be?  Was it really that effective?  Or in that moment during the re-engagement lesson did they actually connect to the mathematics and were unable to make the connection to this new problem?  Is the issue in their ability to see how these concepts work together?

I am finding that this year I am eager to assess my students, simply so that I can take them through the process of re-engagement.  I am also excited to explore the applications of re-engagement outside of assessment, as well as outside of mathematics.  Would a re-engagement lesson be applicable or effective in Science or Spanish?  What about English or History?  As I gain confidence in my teaching, I am open to sharing my thoughts and experiences with others, because I know that there are so many things that I have yet to learn.  Not only have I learned from professional development seminars, which are lead and populated by adults, but even more so I have learned from the people that are inside the walls of my classroom everyday; my students.