Carl's Teaching Blog

A place to talk about teaching and learning

Category: Philosophy (Page 1 of 2)

Problem-Solving, Theory-Buliding and Collaboration: How I stopped sucking

“Let them do it.” This was the constant mantra that Joe a retired NYC math teacher turned math coach repeated often to me during the classroom management struggles of my third year. At the time I was hoping to write material relevant to my South Bronx classroom by pulling from a different textbooks and 2007-era Google searches. Joe followed a similar routine when he made material to teach his East Harlem population using whatever people used in the early 90s. He quickly noticed something off about my tasks that I wouldn’t realize until after a few months working together. My efforts to control student behavior had seeped into a lot of the work that I had them do. Students weren’t doing the work of solving real problems, or making real connections with their prior math knowledge.  Instead they were walking down these narrow pathways of my own thinking. That’s exactly what I thought would help them, but in reality, I was doing them harm.
My teaching moves were as constricting as my curriculum, but I thought that this was necessary. My student population posed a lot of behavior and academic challenges. Changing the focus from my thinking to their thinking allowed me to stop worrying about their math levels or 8th grade scores and instead on how to build off the last lesson so kids really learn. When kids made mistakes, I typically told them what to do as clear and fast as possible. Joe, as politely as he could, shut that down.  Instead he’d ask me to start a conversation up at the front of the class to dissect the student error. He once said: “MAKE SURE THAT YOU NEVER, NEVER, NEVER GIVE AN ANSWER AND DON’T COMMENT ON ANY WRONG ANSWERS. Students will learn more from wrong answers that they discover themselves than they will if you tell them.” This is a quote from one of his emails, with his emphasis by the way. He continues, “They have been trained to look to the teacher to verify their answers and we want them to start depending on their own evidence for confirmation.” It was weird realizing that I was playing into this pyramid scheme of answer-getting, with the kids’ help, and preventing them from building their own critical thinking skills in the process. In trying to keep order, and keep my administrator from seeing chaos spill into the hallway, I played the answer giver that kids seemed to want, not the math teacher they needed.
To get students to do the kind of work they needed to build their thinking skills, I needed new tasks and new practices. One task Joe gave me that I still use today is Tina’s quilt squares. This task was, open, visual and had exactly the kind of thinking my kids needed to do. The old Carl would have totally strangled the joy  out of this. Taking his advice, I “let the kids do it.” The class was as quiet as any of my lectures as they all worked to find pattern. Some kids approached the task by actually drawing out each stage of the pattern. Others probed the tables for patterns in search of a larger explanation. These two types of student thinking were so interesting that I tried to validate both of them at the necessary end of class discussion and it became very clear which people should be presenting, and what I should, and shouldn’t, say about their thinking.
I thought about Joe, and my early years of teaching when I read The Two Cultures of Mathematics that Michael Pershan passed along. The paper’s author describes a tension among mathematicians among people who align with one of two statements:
(i) The point of solving problems is to understand mathematics better.
(ii) The point of understanding mathematics is to become better able to solve problems.
This creates a useful distinction of two kinds of math doers, the theory-builders and the problem-solvers. My first years was a time where I was torn between needing my students to become problem-solvers and wanting them to be theory builders. I was so focused on helping the students solve problems like being on the test, that I didn’t have time to let them really build the theory, so I figured I could do it for them with my awesome worksheets. The two styles seemed to be avenues for the kinds of approaches that students could work on and the kind of thinking that students could show in their student work. Looking back on it now, it was clear that I really didn’t understand what it meant for students to be problem-solving, theory-building, or doing mathematics in the first place.
The theory-building and problem solving avenues are different flavors of “Doing mathematics.” According to the Task Analysis Rubric by Stein et al., tasks are considered “Doing mathematics” if they:
  1. Require complex and non-algorithmic thinking.
  2. Require students to explore and understand the nature of mathematical concepts, processes or relationships.
  3. Demand self-monitoring or self-regulation of one’s own cognitive processes.
  4. Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.
  5. Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.
  6. Require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required.
The kinds of tasks that can fit into “Doing Math” could be either theory-building or problem solving, and possibly both. Tina’s quilt square was my first example of such a task, and seeing it first hand changed the rest of my teaching career. Students were doing complex thinking, analyzing the result, using multiple solutions. The kind of tasks I was doing before Joe would fall under the task guides memorization and procedures without connections, with little focus on making connections or, well, anything in that “Doing Math” category.
Before Joe, I spent hours doing the theory-building, and the problem solving for my kids. I then packaged up my results in a way that would help students easily retrace my steps. They didn’t see how these steps are useful, as they didn’t approach a real problem. They didn’t see how these steps could be built from prior understanding, as we only talked about theory superficially. All my efforts were going towards weren’t going towards making good future mathematicians, but good retracers. It stayed that way because any thoughts about doing things differently would throw my pacing off. It wasn’t until Joe’s frank meetings, and All-Caps emails, that I realized my current style had to change.
The whole point in going to teach on the east coast was to change the world with my teaching. Helping kids learn math seemed like the best path towards that goal because it could improve their test scores and make them eligible for higher education opportunities. Yet, I was lying to myself when I thought that it was enough for my class to do well on tests. My failed first test of calculus was proof that kids who do well on tests can still struggle at the next level. The right goal would be to make kids mathematicians. To help them actually learn how to solve big problems and make the connections needed to build theories. The application of those skills lies beyond just the math classroom, but can be actually used to think about tackling larger social problems and skills. The only way to get to my larger goal would be to not just teach Tina’s Quilt Squares the way Joe did, but to change my own thinking about my profession so I could make my own tasks to give to my future mathematicians.
After Joe, I sought to approach the the task of teaching my classes as my kids approached the task of learning algebra. I approached it like a problem-solver, looking to try and solve lots of little problems while simultaneously looking to connect big ideas and make sense of things like a theory-builder. Just like my students began to learn from each other and work things out in conversations, I began to talk with Joe and later other educators which helped me develop my practice. This was before I found the #MTBoS or joined NCTM or MFA, so this took a lot of time. Luckily Joe, and my Principal allowed to approach the task of improving as a teacher without having to retracing some steps or memorize some procedures, as it is probably the reason I’m spending another fall preparing to teach.
* * * * *
In writing this piece, I read W.T. Gowers 2000 article about The Two Cultures of Mathematics, of which I understood about 65%. I do know that is not about K-12 schools and what goes on there, but is actually a bit of a call two action about two cultures that exist among people in the field of mathematics. The prominent theory-builders belong to a culture that studies the fashionable ideas which are at the center of that field, while problem-solvers work around the periphery. This distinction reminded me a lot of math education. It feels like there is a prominent culture of around that approaches things in the old ways, like in this recent pro-memorization NYT article. Meanwhile the world seems to ignore ideas that come from problem-solvers working around the periphery in classrooms like Joe working with me.
The problem-solvers Gowers describes, who were creating the field of combinatorics, could actually benefit the theory-builders with their unique ways to solve problems. Technological advances would allow for major advances in combinatorics and math as a whole as the two cultures learned to collaborate and move the field forward. Hopefully writing this story of what happened in a cramped South Bronx classroom, might describe a different approach to how to improve teaching and learning. And hopefully the larger conference that it is a part of could promote the kind of collaboration across the world of mathematics education that has been seen among the problem-solvers and theory-builders of mathematics.

Grading, Assessments, and Hot Pockets

So you’re at a PD, a really awesome one at that. Everybody is quietly thinking about the prompt “What is assessment?” Your neighbors are writing things like “Assessment is knowing where kids are, where they need to go, and what you should do next.” These poetic statements allude to many parts of a real-time data gathering and analyzing process . Diagnostic assessment, summative assessment, formative assessment are all critical pieces of information that end up letting the teacher know what they need to maximize student growth and learning. The information gained from assessment become the ingredients that “Chef Teacher” can use to create any number of delicious stews, or salads, or souffles.

The facilitator tells everyone to stop writing and to stand up and share with someone new. After 15 awkward seconds of trying to lock eyes with someone, you find a partner across the room. After shaking hands you read your poetic statement with a serious flourish. Your partner responds with the following:

Assessment is how you give kids grades.

You wonder for a second if their table was given the same task. This statement describes a calculation chore that happens at the middle and the end of each term. Grades are what you show to parents and administrators if they want to know how the kids are doing. Assessment is a process that ensures that you have the information at any given point to be able to make the grade, but also to do so much more. Assessment can help you make decisions in the moment, tweak tomorrow’s lesson, or even alter your unit structure. Your assessments can tease out which students understand what you taught today and which ones are relying on the trick they learned last year. Viewing assessment as only a tool for finding grades is like “Chef Teacher” going to the kitchen, by passing all the groceries, and microwaving a Hot Pocket.

You rack your brain for how to begin a conversation about Grading, Assessments,…and Hot Pockets, when your partner cracks a smile. Turns out he was messing with you. He didn’t really believe that Assessment is solely for producing grades, but lots of teachers out there do. How would you describe all the things that assessment could be to someone who thinks it is only for getting the numbers to put on the report card?

Explaining my lack of “help”-fulness

This year our school is talking about student work in mixed groups. We have been placed into 7 groups of teachers and social workers, each of whom are related to one student. After each session, the teacher bringing the work gets ideas for their teaching, and the group gains insights into the student and how our work affects them. These conversations have only involved essays so far, but this past Friday I was the presenter.

Due to realities of our schedule I provided a student’s partially finished math project for our descriptive inquiry group to look through. It was a project where the student had to create a set of equations that then help her solve a larger problem. The student make a mistake early on in the assignment and continued finishing the work, not being able to see that answers stopped making any sense. The discussion about this did not just allow for us to talk about the student. It allowed the members of the group a chance to step into a math teachers shoes and decide to how to respond to student misconception.

Talking about this student’s work flared up and we ended up having to scrap the rest of the inquiry protocol. The issue that broke our group apart happened after I explained the project and everyone gave their initial impressions. Someone noticed that the student made the a mistake. “The student should have multiplied these answers by x,” the teacher stated, referring to the column with numbers far to small to make sense in the situation, “so the teacher should show them what correct answer should be.” I began to feel a little uncomfortable. My instincts say the first thing to do would be to understand why the student made the mistake. I would need to ask a series of questions before I gave any kind of instruction. Thesequestions would intend to help the student to understand why the need to correct it, not to correct the multiplication, thus preventing the student from making sense of the problem.

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Reviewing #NCTMAnnual proposals and the ‘Access and Equity Question’

For a few weeks last June I was one of the lucky volunteers who were able to review proposals for San Antonio’s 2017 NCTM conference. It was a lot of reading and it is a great opportunity to hear what math educators from around the world think should be talked about at the conference. Reading the words from hundreds of speakers provided a glimpse of math education thinking from the minds of teachers and educators across countless numbers of different contexts. It was a rare kind of opportunity that was both a great honor and also genuinely fun.

The Access and Equity Question

After reading over two hundred proposals, I noticed that some of the prompts from the application were better addressed than others. Particularly, the responses to the Access and Equity question were at times brief, or sometimes didn’t seem to address the question that was being asked. This question is the last written section of the proposal application form and asks:

“How does your presentation align with NCTM’s dedication to equity and access?”

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My Administrative Philosophy as Told By Unifix Cubes

So I was playing with these unifix cubes and it made me think about my work as a teacher and an administrator.


I imagined that each cubes represented one unit of productivity. So maybe this block represented one worksheet that I created, or a game that kids can play. For now let’s just say that this is instructional productivity, and not other things like discipline our data analysis.

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