So, as some of you know, I’m working on something like a high-stakes state test. Sort of like a much more complicated and elaborate and difficult version of the SAT, with a lot, lot, lot more writing. Part of it is having to write daily reflections about the lesson that I’m doing my PACT on (PACT is the name of the test). So I just wrote about five pages of reflections about the last two days of my first period Algebra class, and thought I’d share it with you all. I’d be curious to hear what you thought. I checked it a couple of times to make sure there are no specific names referenced, but let me know if I missed one.
Here is all is:
Daily Relfections
Day One:
1) What is working? What is not? For whom? Why? (Consider teaching and student learning with respect to both content and academic language development.)
2) How does this information inform what you plan to do next lesson?
Today’s lesson illuminated a flaw in my approach to building conceptual understanding and procedural fluency so far. Many of the students in the class can competently switch quadratic expressions from factored form to standard form, explain their process and the rationale for each step in that process, and they can explain the relationship between the terms in standard form and the factors. What I have not taught them how to do, is to recognize when factoring is an appropriate and useful tool for a situation. So students did not easily recognize how their knowledge of factoring applied to finding perfect squares.
About three-quarters of the way through the period, I discovered that groups were sorting all thirty-two cards by factoring every single one of them. I thought that the sheer number of cards would be all the prompting they needed to look for “shortcuts”. I expected them to “naturally” notice commonalities between the perfect squares they were sorting. It did not occur to me that students would use the “sledgehammer” of factoring to sort every card. As I walked from group to group, I realized that they were looking at each card individually and they never examined the piles they were making. A few students began to notice patterns anyway, and I think that, given more time, and many more cards to sort, the characteristics of perfect squares in each form would become apparent to them. However, in the context of a forty minute class and their finite patience for factoring trinomials, their efforts should be better focused to bring them to those realizations sooner. When I realized that the groups were ignoring their sorted piles and dealing with each card in a vacuum, I started prompting them to look for patterns. This had positive results, and students quickly began noticing the desired characteristics of perfect squares.
While thinking about this problem, it suddenly occurred to me that their strengths and weaknesses are probably indicative of its cause. Why would they have such a strong grasp of the connections between the various characteristics of quadratic expressions, but not know how to use them? How could they sit there and factor quadratic after quadratic without noticing a pattern? At least part of the problem I think is in the nature of the tasks I have given them this semester – today’s lesson included. Day after day, I have asked them to explain the relationships between things. I now wonder if this is a subtle but important mistake. Maybe instead of telling students the relationships I want them to explain, I think that perhaps I should be telling them the behaviors I want them to do. So instead of asking, “how can you tell that a polynomial in standard form is a perfect square,” I should have written something like “Every time you add a perfect square to the pile, compare it to the perfect squares already sorted. Every time you add a card to the pile that are not perfect squares, compare it to the cards already there. Compare the two piles to each other. Come up with a way to predict which cards will be perfect squares without factoring them. Test your prediction on the next card. Explain your process and how you made your predictions.” The activity I have planned for tomorrow is structured in this way.
While most of the class can competently switch between factored form and standard form, none of the students knew what a “perfect square” means in the context of polynomials. Since the period is only forty minutes including passing period, I quickly explained that perfect squares are expressions that can be written as the square of another expression. This was enough for most groups to begin working and making progress on their own, but there was still widespread uncertainty until, a couple of minutes later Kira asked, “is a perfect square any quadratic function where the factors are the same?” I asked her to explain her thinking to the class, and there was a chorus of “oh’s!” when she did a pair of examples on the board. From that point, many groups were able to complete the assignment with only gentle guiding questions.
I was generally pleased with my interactions with each group and my use of whole-class discussion, but think that I spent too much time at each table and as a result did not get the broad picture of each students’ engagement with the task that I was hoping for. I have two ideas about how to address this issue. First, I will keep my interactions to one question/answer per table visit. This will give me more chances to see each table, and will help keep me from over-directing groups. Second, instead of reteaching groups that have deep conceptual gaps, I will send them to work with another table for one of my circuits around the class. Since students learn better from each other, and talking more also improves learning, this frees up my time while also addressing each student’s learning needs. It also gives both groups a clear deadline since they know that they will be split up when I come back around to them and they can watch my progress, it will help them focus on getting the information they need and getting back to the task at hand.
I would say that today’s lesson was very effective both academically and linguistically for at least ten of the eighteen students in the class. These ten students demonstrated competency talking about perfect squares and began to discover for themselves the identifying characteristics of perfect squares. I would say that the lesson was at least moderately effective for another seven students. They gained at least some concrete experience talking about perfect squares and developed their procedural competency and fluidity with the concepts of various representations of polynomials. One student was totally disengaged from the task when I checked on his table about a third of the way through the period. He had been drawing and admitted to being spaced out. As I mentioned earlier, the amount of time I spent at each table meant that I was unable to assess student engagement to the extent that I would have liked. So the ten who I claim the lesson worked well for were the students who very clearly demonstrated their sustained engagement in the lesson, while the seven others were students who appeared to be engaged most of the time with major conceptual misunderstandings, or were sporadically engaged with the task. It is possible that the student who appeared checked out while his partner did all of the work was involved at another time, or became involved later in the period and I did not notice.
Partly because of students’ initial confusion regarding the definition of perfect squares, partly because groups persisted in the tedious task of factoring all of the trinomials, and partly because of the unusually short period, students did not make the kind of observations I had hoped they would
Day Two:
1) What is working? What is not? For whom? Why? (Consider teaching and student learning with respect to both content and academic language development.)
2) How does this information inform what you plan to do next lesson?
Thanks to our laughably short periods this week, none of the groups completed the assignment. Since they have to develop their method for solving equations of the form (ax+b)2=c and try it on a few problems before solving more general quadratic equations, we cannot move on to the lesson I had planned for tomorrow. Instead I will have students continue to work on the second task from today’s lesson.
I was pleased to see that the first task was not challenging for most students, because it was designed to recapitulate their knowledge of quadratic polynomials. Several students mistakenly constructed “squares” that exploited the fact that the length that algebra tiles use for x is very close to the length that corresponds to 5 units. This was a simple visual-spatial error that was easily corrected. I also noticed that a couple of tables that formed a square whose area was 4×2, but mistakenly identified it as x4 or with side length x2. This is an error I’ve noticed they make exclusively with algebra tiles, which leads me to believe it results my introduction to the manipulatives being somehow confusing.
I also felt really good about my new approach to circulating around the room. Yesterday I made slightly more than one complete circuit, today I made more than two. I’d still like to move around the room faster, but I think that keeping my interactions to one question/answer per table is a good thing. I still feel like most groups got the support they needed, even though I spent less time with them each, I also think they were able to figure out the things that I would have guided them through had I stayed. I would like to circulate through the class even faster, so tomorrow I will try having my first circuit be primarily observation, and only intervene if I see something seriously awry. During my second circuit, I will seek out student questions. Only on the third circuit, will I start asking students questions of my own. I think this approach will address the problem that even though I spent only about a minute at each table, it was still ten minutes into the group work before I saw the last tables on my circuit.
Thinking about this has made me wonder if the quality of a lesson is inversely proportional to my importance in students’ learning process. By this metric, the best lesson would be one where students inform me of their learning after it has already happened. I think it is pretty clear that I am not there yet as a teacher. I think the best I can currently aspire to is being a catalyst for a primarily student-driven learning process, and that target seems like a lot to shoot for already.
I felt rushed at the beginning of class to provide some closure for our lesson yesterday, which students basically completed. I felt like the sorting activity was really strong, but without the closure provided by a really good discussion, the lesson fell flat. When I decided to “forge ahead” and do today’s lesson as planned, I convinced myself that it would be okay, that students’ individual learning experience during the sorting activity made a strong impression. I had felt uneasy about this decision, but have been really doubting myself because of how far behind the district syllabus we have fallen. The extra time I have been dedicating is not getting the payoff in student understanding and retention that I had hoped for, so I have begun really feeling pressured to move on despite lingering confusion. Nevertheless, I think that continuing on was a mistake in the sense that it did not serve the academic needs of most of my students. Given how often teachers have moved on in the face of widespread confusion, it is a testament to human intelligence that any of them has a clue what is going on.
I forgot to try sending a struggling group to work with a group that was performing well. I think this is partly because I did not start with an observation-only circuit so I did not have a global picture of how the class was doing. I have been going back and forth between two ways to implement this idea. In one version, I send both partners to the same table, while in the other version I send each partner to a different table. I think that sending them to different tables will keep the host group as the “majority”, which is good because I want them to be able to keep control of their time, I also like the idea that this way, each partner will learn different things or in different ways, which will enrich their process when they get back together. I will make a point of trying this tomorrow.
Finally, I have been experimenting a lot with different ways to design tasks that inspire meaningful collaborative group work, and I am still not entirely satisfied with the design. I like the rubric for today’s activity, and we have had short class discussions about rubrics similar to this one, I think that most students would benefit from having a really deep, meaty exploration of what each level of the rubric means. I wrote each level of each of the three parts with particular students in mind, but I do not think that they see themselves in the rubric, which was the point of giving it to them in advance. This is something I will need to address in the future if I want to use rubrics as a teaching tool. Perhaps, like learning journals it will be something that I set aside for now and come back to when I have worked out its implementation, and introduction more fully so it can be more thoroughly embedded into the class culture. To be clear, I am not as interested in rubrics as an assessment tool as I am in using them as a tool to build metacognitive awareness, and executive function in students while giving them more granular control over their grades. I have begun toying with the idea with creating a set of generic rubrics for each student.
In this graph, each pair of dots is one polling station (the place where people go to vote). The height of the red dots shows the percentage of votes at that polling station that went to the current president Vladimir Putin, while the height of the blue dots show the percentage of votes at that polling station that went to Putin’s opponent. The dots are positioned left to right based on the percentage of people who could have voted at that polling station who showed up to vote. So each polling station is represented by two dots, a red one and a blue one that are lined up vertically (you can draw an up and down line to connect them).
This graph shows the number of polling stations that had each level of turnout. What we should see is a smooth, bell-shaped curve with one peak. Instead we see a bell like curve that is much steeper on the left side than the right, and that spikes right at the end. This is very, very unlikely. Combined with our other graph, it’s a pretty sure thing that the results of the election were faked.
mrbgreen 