Looking back, looking forward. (Content Methods Reflection)

For this week’s blog I’d actually like to reflect on the first “creative teaching strategy” that I tried out-- way back at the end of September. At the time I considered it a total failure-- for reasons that I will explain shortly.  I return to the experience now because I’ve been thinking a lot about unit planning recently, and this early experience changed the way I use so-called ‘creative activities.

Ideally a unit plan should maximize content delivery, provide a lot of independent practice, attract and retain student interest, provide easily graded and logical assessment points... etc, etc, etc and the list goes on. The activities in that we’ve been practicing in our Content Methods class are a great way to get student interest and independent practice. However in general, they don’t for direct-instruction -- that is to say these activities are great for a 2nd day work period or a review lesson before a quiz but not always appropriate for the first instance where students are practicing a certain type of problem.

I learned this lesson the hard way. Our second full unit this year in Transitions was on polynomials and we started with a week of exponent rules practice. On wednesday of that week I could tell students were getting a bit restless with the notes and practice routine so I decided to shake it up and try the matching activity that Dani modeled for us in an early lesson. Her version had us sorting through stacks of colored cards creating piles that match a function table to its equation and graph. In mine, students had to match an exponent rules problem (eg. x^2 * x^3) to its expansion (x*x * x *x *x) and the relevant rule (multiply bases = add exponents).

Making the cards was quite a bit more time consuming than I expected-- mostly because of the cutting and assembling piles with the appropriate assortment of cards. I was happy with the product. In class however, everything that could go wrong did. To start with, my instructions were ineffective. I did have written instructions on the board that explained what each color card was and what each pile would look like in the end. The difficulty was that the kids didn’t know where to start and they quickly became rowdy and off-task.

In retrospect I think that the trouble was because 1) I didn’t adequately model the thinking process that they could use to assemble a pile and 2) they were not strong enough with the background skills (simplifying exponential expressions and writing the expansion of an exponential expression) to have the sort of recognitions that would help them see how different cards belonged together. I probably could have solved this had we done some straight forward simplifying practice before trying the activity. 

To return to the question of unit planning and the appropriate placement for ‘creative teaching strategies’ the struggle is to have time to provide a strong enough foundation and practice before  the activity. It’s as if for any given topic you need a day for instruction and then a whole additional day for the activity... but than I always feel pressured to simply move on to the next subject. I hope to continue to use creative activities but only after we have already learned and practiced the skills in a straightforward manner. 

Chapter 3 Reflection (Content Literacy)

Chapter 3 of our Literacy textbook is dedicated to an examination of the term “comprehension.” The text reveals the naiveté in thinking that ‘comprehension’ is a simple matter of ‘ just understanding what is written’ and unpacks the many layers and indicators of comprehension as they apply to content area instruction. In particular the text reveals comprehension as the combination of decoding literal meaning of words and sentences, identifying the context of information conveyed through the written medium and processing and applying this informations. In the book’s more succinct, though also more ambiguous words “ Comprehension of print and other forms of text is a   meaning-making and  meaning-using process” (RAND reading study group as quoted in text p 45)

I am reminded of a story I heard my first night in Mississippi. Some second year teachers were sharing inclusion teacher horror stories. One described sending a struggling resource kid to take a reading test in the inclusion room only to have the inclusion teacher return later in the day saying “I don’t know what the problem is because when I read the test to him M----- does fine!” At this point the gen. ed teachers all laugh a little helplessly, knowing as we do that yes, if our students could just read they would be fine-- both on reading tests and i suspect in content areas. 

What the inclusion teacher meant to explain, I think, is that she had identified the student’s problem as one of decoding and not of comprehension. The student could not make meaning out of the symbols on the page, however if given that meaning the student could use that meaning to apply and analyze information contained in and implied by the text.

As a content area teacher I think that it is important to be cognizant of this distinction but    I am also challenged to know what to do with this information. In this example, yes it is very important for the teacher to know that the student’s ability to analyze/summarize the text would earn him a passing grade... but at the end of the day if the only situations in life and school that the student would be required to perform such analysis also require that he read the passage in the first place... well, then, how do we grade the kid? Based on our privileged knowledge/believe in his partial abilities and in the complex factors which explain(though not excuse) his poor literacy...? or by the same standards that a job application would hold?

Luckily, as a math teacher I can minimize the impact of poor decoding skills on my students; performance on assessments. However, comprehension is still a hugely important and relevant skill. The textbook discusses a number of strategies for building textbook reading skills (eg. bridging text ideas, mapping strategies, charts, opinionaires). Since I rely very little on the textbook these strategies interest me less. However, the discussions about developing a language of process through justifying content statements and group strategies like reciprocal teaching were of great interest.

As I have mentioned in previous blogs I believe that the main forum for literacy strategies in my high school math classroom is in the way that I employ “steps” to teach problem solving strategies. For each new topic that I introduce, one of the early lessons will be devoted to taking notes on key vocabulary and steps that describe a systematic approach to tackling this particular problem. For more complicated subjects we may spend up to two weeks getting to know the same steps or adding on new steps to the same sequence. To conclude this post I would like to briefly re-examine the use of explicit written “steps” in light of the text’s structural analysis of comprehension-- namely that it may be divided into 4 dimensions: 1) Cognitive, 2) Textual, 3) Personal, 4) Social.

1. The Textual Dimension to Using Explicit Written Steps in Algebra Instruction.

This is the simple one. The “textual structure” of my steps are always the same: numerically ordered steps in the form of direct imperatives sometimes with sub-bullets that describe different cases or outcomes. We acknowledge this structure in practice by reading and checking off the steps as we work through a problem.

1. The Cognitive Dimension to Using Explicit Written Steps in Algebra Instruction.

One of the challenges from the teachers’ perspective in writing the steps is to find a balance between detail, clarity and brevity. If the steps are too long, then students are reluctant to use and internalize them. However, if the steps are really short there is more work to be done to ensure that students correct interpret the written reminder. Our textbook discusses the ‘cognitive’ dimentson of comprehension as drawing together the students skills and background knowledge. To increase the effectiveness of “short steps” we can rely on words that may not have the exact mathematical meaning, but which we know are familiar to the students and convey a memorable concept. (eg. in dividing fractions instead of saying to “exchange the numerator and denominator” we just say “flip.”

1. The Personal Dimension to Using Explicit Written Steps in Algebra Instruction.

A key piece to using steps is to have the student themselves reading and speaking the steps aloud- both using the language of the generic instruction ( eg. “next we have to flip the fraction”) and in application to a particular problem (eg. “that means instead of 2/3 write 3/2”). To make this process more personal, and perhaps as a way to tackle the challenge described in section 2 would be to have students use their own words to articulate the steps as they think of them.

1. The Social Dimension to Using Explicit Written Steps in Algebra Instruction.

Ultimately my goal in using steps is to get student to think of math as a problem solving process and not just a race to the answers. Thus the goal is for students to be able to help themselves and their peers by giving hints and instructions instead of just giving away the answer. This social process both reinforces their mathematical knowledge and is a huge motivator/ ego-boost.

Chapter 4 Reflection (Content Literacy)

As a math teacher I have struggled to find an appropriate integration of our literacy strategies in my classroom. It’s not that I disagree that improving literacy is my responsibility-- not at all-- rather I struggle to fit literacy in when a) there is so little time and so much math to begin with and b) for many of my students my “in” is the little wedge of confidence that they have in the one class where low reading skills don’t immediately single them out--- well not as much that is.

Coming from this perspective I appreciated how widely applicable most of this chapter is. “The goal of literacy assessment” the authors write  is to encourage students to become “reflective, active, and purposeful learners.” This could just as well be the goal of any other type of assessment or educational strategy.

In taking this ‘goal’ literally I am slowly developing a niche for reading/writing in my classroom routines: namely, I am trying to make a habit of asking students to read out and write out steps and justifications of their thinking process and problem solving strategies. Not only is this just a general good practice, but I believe it has the dual benefit of 1) bolstering students with strong verbal skills and 2) challenging students who may already know what to do mathematically but not how to explain their own knowledge.

I was particularly interested by this chapters’ discussion of strategies to involve students in the assessment process. I love the idea of having students give input into the creation of rubrics. However I disagree that this type of engagement relieves teacher workload-- if anything it seems like facilitating this type of ‘mature’ student involvement would take a lot a lot of advance thought and planning. As a teacher I really struggle to draw out reflection or self-criticism on the part of my students.

A related struggle that stretches across discipline with implications for literacy is the challenge of getting students to use their own notes effectively. Of course in order for this to be possible their notes must be ‘usable’ i.e. ‘legible’ in the first place. However even those students who copy clearly must be explicitly taught how to look back and find answers to their own questions. It is a habit of self-reliance that sometimes seems impossible to build up in students with low confidence and low motivation... a crucial life skill that should be the goal of education in any discipline, but which in some ways seems more elusive that the most baffling phonics or factoring. 

Points of Comparison (Content Methods Reflection)

My green-book planner was great fun in the first months of school. “Function wheels” is scribbled in along a margin in mid-September between “Functions Test” and “start Unit -III.” There was, theoretically a 30min gap to fill. Needless to say, by the time that day rolled around between make-up work and management battles and piles of paperwork “function wheels” didn’t actually happen. 

Mock-teaching this activity the following saturday for our Content Methods Class then was bittersweet. I hate it that the few times I’ve tried creative activities have become management disasters; but I am also invigorated by the reminder that there are creative ways to represent material and engage students’ abstract thinking... something to aspire to.

Basically the activity was to build a manipulative that was the math-equivalent of a function table. We cut concentric circles of colored card stock and attached them in the center with a paper clip. We wrote input values around the outside circle and cut windows in the top disc so that the window would show the corresponding output as we rotate the outer circle.

In the ‘protected’ setting management was fine. Clear directions were super important. I’m glad that I had pre-copied the circles and also that I had students do the cutting instead of spending that time myself. In a larger class I probably would have a harder time walking around to check that people were folding and cutting correctly but that might be helped if I found a way to get students helping each other or have a visual of different stages of the assembly process.

Two strengths of the activity I think were that the “wheel” draws out the distinction between dependent and independent variable, and the “output windows” helped me to clarify f(x)-notation ( during my ‘traditional’ lesson this was a major point of confusion... the idea that f(x) isn’t itself an expression but rather the ‘name’ for an expression and which simultaneously represents an output). At the same time, I wonder about the pay-off of spending a whole 20 minutes on what really amounts to a single function table’s worth of calculations. Still, I could envision extending the activity to have partners composing their functions which I think would be powerful conceptually. I look forward to modifying the activity to try when I introduce functions in Transitional Algebra.