I have noticed an increased conceptual understanding almost across the board and this is reflected in the differing solutions on summative assessments. It is also nice to see their marks grow on these unit tests. I do not believe that paper-and-pencil tests are the best venues for displaying conceptual understanding, but it is awesome when the two intertwine.
My unit structure
I plan my courses in units of study, and attempt to build "pods" of conceptual understanding. I am fully aware that the topics will influence one another down the road, but I prefer to have those conversations when the students make the connections.
A unit begins with a strong focus on exploration with group problem solving. Students are randomly grouped and assigned tasks to complete with large whiteboards. These problems are designed to create peer discourse and stumble upon some key features to be later formalized into the "official" mathematics. Sometimes these problems contain a foreseeable problem to be overcome (i.e. graph a function with an asymptote they don't see coming) and others require the following of a pattern (i.e. giving an example of a linear relation and asking them to find a entry out of counting range).
These classes serve as anchor moments that can be referred back to during later times in the unit. They also provide time for students to conjecture, test, analyze, and critique. These activities shift the typical role students usually possess in a math class.
During the mid-stages of a unit I purposely plan more guided discovery lessons. Often times these include a handout with prescribed exemplars that will lead them to narrowed conjectures. This narrows their thoughts, but continues to include them in the "invention" of the math. In this stage, students begin to become more exact with their solutions and notations. They often check papers with others. Here, student explanation of abstraction is key. I nominate students to explain patterns and write rules. They are well prepared for this task because of the skills built with the more open problems.
The final stage looks much like a "traditional" class as we move toward a summative assessment. Here, we polish up methods and notation after the time has been spent to understand where they come from. This stage is dominated by lecture, seat work, and a quiz or two.
Shift in student disposition
The focus on group problem solving through large white boards and collection of formative data using digitized exit slips has changed the ways that my students interact in class. They carry an inquisitive approach into the last stages of the unit.
I was skeptical at first that I could cultivate an active disposition in students long term. I have had instances or lessons of great participation, but haven't tried such a large-scale change thus far in my career. Slowly, little victories convinced me it was possible. It is intimidating to give up control, and even the smallest changes take monumental efforts. Nothing about this post or blog in general is aimed at making change seem simple.
Three examples of active student disposition:
1) In-class conversations
Students have started looking past me as the sole provider of correct solutions. Many times they engage each other in conversations about differing methods. The discussions are initially centred around right and wrong answers, but quickly become arguments around efficiency.
For instance, when given the function "2y - 6x + 12 = 0" students debated whether it was quicker to isolate the "2y" on the right hand side rather than on the left (as was the normal practice in class). In previous years, students would have just copied down the example and told themselves that it must always be on the left hand side. The opportunities for them to exercise mathematical intuition create the climate for these moments of intellectual courage.
2) Formative data from exit slips
After group problem solving classes, I have students fill out their exit slips with things that they noticed or wondered. These snippets of formative data really help shape my next moves in class. They also provide the chance for every student to participate mathematically with me. After an introductory lesson on linear relations, students responded in the following ways:
Question: What did you notice / wonder from today's lesson?
"I noticed that numbers stand for specific things like corners and sides"
"I noticed there are multiple ways to figure out the constant and variables in the equation"
"I noticed that there are different ways of solving problems"
"I noticed that we got multiple different patterns from the same question. Depends on how you see it"
"I wonder if the inside of the square comes into it later"
"I noticed that some people subtract four corners and other add them in later"
The responses show the degree to which students were interacting with each other. The environment of student discourse allowed them to notice, debate, and wonder about varying strategies used by classmates. Having a private place to notice and wonder gives a chance for each student to discuss their individual thoughts.
3) Creative responses on examinations
Students that have the chance to build mathematics and discuss their rationale become more comfortable "doing" and not "reproducing" mathematics. As the culture shifted, I noticed more students becoming playfully astute on exams. This is a good sign; it means they are making conjectures, verifying their validity, and taking the risk despite the pressure of percentage grades.
On a recent exam on sequences and series, I asked the following question:
Build a sequence where every term is negative except the first term.
I got the following solutions:
"a=0 and d= any negative number. I know zero is not positive, but it is still not negative so it fits"
Here you can see the student teetering on the brink of mathematical precision. They creeped as close as possible to a non-solution; this is such a mathematically cheeky move.
"a=1,000,000 and d=-1,000,001."
I love this solution. It shows very clearly that the student knows that it won't matter how large the first term is as long as the common difference can cover it. Shows mathematical generalization, without notation or conjecture.
"Any a > 0 and d = -a + x."
This solution is not correct for all values of "x", but still shows the active nature of the student thought. They are willing to abstract the mathematics and take the risk well beyond the question's requirements. This solution served as a great class conversation the following class.
On some days my classroom looks drastically different than the norm. Students divide into groups and work on problems with one another. They migrate from group to group looking for verification. The class is loud and active. Even when I bring everyone back together, they still talk over me and debate their solutions with each other. Even desk work is often interrupted as pods form to discuss methods for certain problems.
On most days if you observed my classroom, you wouldn't find anything drastically different that the average classroom. Students are expected to follow examples and ask questions when confused. Students take notes and do seat work, but you would find students that question what is going on and have a tendency to break into conversations with one another. You would find students asking for me to go back and do examples in different ways. These mathematical activities are a direct result of building an environment of conjecturing, discussion, debate, verification, and sharing. Students' active disposition is thanks--in full--to the discourse effect.