Graphing Understanding

One of the things I would like to do with my students is have them graph their understanding of different objectives and mathematical practices. This would help them visualize their progress at different points throughout the semester. It would also help me see where they know they need help. Both asking questions in class approaching a teacher after can be intimidating and a bit nerve-wracking for some, so I think this would be a very low-key and passive way for them to tell me how they think they’re doing. I imagine something like a y-axis of levels of understanding, including statements like “I do not understand this at all,” “I understand parts of this,” and “I fully understand this.” On the x-axis would be the different elements students have been exposed to.

While this would not be used to grade them, I would use it as an informal, formative assessment to help let me know what to reteach or which students needed intervention. I think another way this could help my students is to visually show them that they have, in fact, learned something over the course of the class. If they start with a graph at the beginning of the year with low levels of understanding and increase over the semester or year, they will be able to give a tangible answer to the question, “What have you learned this year?” It would also be useful to show parents during conferences as a way to illustrate (albeit self-reported) learning or during interventions to show which elements the student definitely struggles with. It would also be interesting to see if some students understand past elements better after applying them to newly learned material. There is no doubt that students need to learn certain information before moving on. But would the deliberate practice and application help increase that learning after they did move on? Also, I wonder if I were to measure their understanding immediately before and after breaks, would there be a drop in knowledge due to the time off?

This is an activity I would like to take class time to go over at least three times a year, though ideally it would be done once every month. I think it would be helpful to me as an informal, formative assessment and a good illustration of self-reported knowledge for students, parents, and administrators. Let me know what you think in the comments!

Brochure for Parents

I designed this for a class I’m taking, and was hoping for some feedback from others. (Technically, I’ve just submitted it, so I haven’t received any feedback yet.) It think this is a good introduction to parents, and a few of the MOOCs I’ve taken have mentioned writing to parents. Would this be a better replacement for or addition to a letter to my students’ parents/legal guardians?

Parents' BrochureParents' Brochure 2

Homework Plan

Hello, again!

I’ve been auditing a second MOOC, How to Teach Us, from the Coursera platform. This week, one of the readings was a chapter from Fires in the Mind entitled “Is Homework Deliberate Practice?” This chapter covered how many teachers approach assigning and assessing homework in secondary grades, and included quotes from interviewed students. While reading this chapter, I started cementing my ideas (which have previously been vague, but purposeful) of how I wanted to assign homework in my classroom.

One thing that stood out was the idea that, in order to be deliberate practice, each homework assignment should fall into one of four categories known as ‘The four Rs’:

  1. Readies students for new learning (more complex concept/step)
  2. Repetitive in knowledge and application of skills
  3. Reviews previously learned material (such as practicing instrument exercises to improve the playing of a musical piece)
  4. Revises work or understanding (such as editing a paper)

Another point made was giving students an opportunity to complete work at school. Some students don’t have time or help at home of they get stuck. I will make sure to have tutoring time available during my prep, lunch, or after school each day. I will also encourage my school to have a study time available throughout the week.

The final thing that guided my plan was how students believe their teachers see homework. Many see it as a one-size-fits-all assessment. I’ve tried to cater to that by basing the final homework assignment for the lesson on what was missed in the first unit homework assignment. I can do this by having sets of questions (or worksheets) created before the unit for areas in which I foresee struggling, or using software to help find problem areas en masse (The Learning Company computer games were excellent at catering to the child’s level of understanding). A second concern the interviewed students voiced was that they believed teachers didn’t use or really looked at the assignments that were assigned. I would make it obvious that I go over and make notes and consider altering lesson plans based on what I am reviewing.

My Plan

I’ve tried to weed out the issues as I went, and did an overall review to add specifics after I first wrote it all down. But if you see something you think I’ve missed, please comment! This plan is for my future high school math classroom, which may include more at-risk students than the average classroom.

Daily Assignments

  • Formal, formative
  • Graded on completion (based on a holistic rubric– partial credit available)
  • Expected work time: 20 min.
  • Time available during class

This will be a short set of questions (less than ten) that are directly related to the material covered that day. These will be pre-created, but may be altered if necessary. When assessing my students work, I will look over and mark which parts were wrong (and what the correct steps would be), then make note of concepts they are struggling with. I will note class-wide trends for review during class time. When completing these, students should:

  • Do it! Partial credit keeps their grades up, and seeing their work helps me find gaps in understanding or if they are ready for more.
  • Show their work. Having that window into their process gives me better insight into where things may be going wrong in my lessons.
  • Work out the problems they struggle with to the best of their abilities. I don’t know where students get stuck if they don’t attempt the question.
  • Make note at the end of the assignment of points where they notice they struggle. This would also be an opportunity to note when too little time was spent on a concept during the lesson.

Unit Homework

  • Formal, formative
  • Graded on correctness (based on an answer key)
  • Expected work time: 45 min.
  • No work time given in class

This will be a long set of problems assigned at the beginning of the unit. These problems will be a mix of elements from each lesson, as well as questions that ‘put it all together’. Students are to complete this as they feel competent in each area. This will be collected at the end of the unit, giving students time to seek help and find time in their schedules. This will be an opportunity for students to practice time-management and academic honesty. The motivation to complete this with their own work is intrinsic, as the final assignment will be based on what they need practice in and will rely on them understanding each concept fully, without outside help.

Final Homework Assignment

  • Formal, summative
  • Graded on correctness (based on an answer key or analytic rubric)
  • Expected work time: 30-60 min. (varies for each student/activity)
  • No work time given in class

This will be a follow-up assignment based on what each student needed from the Unit Homework. Assignment activities will vary. Students may be given a problem set, worksheet, or BINGO card (containing multiple projects that fulfill one or more concepts). Activity purposes will be determined by assigning each student to one of four categories:

  1. Students who understand everything will receive deeper questions or require deeper thinking in their projects/application.
  2. Students who misunderstand one or two things will receive problems/projects based on improving, practicing, and applying that specific element.
  3. Students who misunderstand multiple things will receive problems dealing with specific base concepts, then combined and applied concepts in questions/projects .
  4. Students who understand little will (hopefully) have been given intervention as the unit went on. This will be built upon in this assignment by giving them questions and explanations dealing in base knowledge and practice. Reviewing past lessons and subjects will occur if necessary. Altered assessment will be considered for final testing.

That’s it! Let me know what you think!

Inventing Problems

I’m sure many of you have been annoyed by the story problems in math classes. They are often boring, worded funny, and require a large suspension of disbelief. You may ask yourself what kind of idiot would write questions like those.

Well, I would.

In my Probability class, one of the projects assigned was to come up with unique questions that would need to be solved using the methods we were learning in class. We didn’t do that well. Now, I don’t mean to insult my classmates. Our questions were clever and fun to think about. However, they fell short when we were asked to solve each others. They became as mundane and silly as the work we had done our of textbooks in previous math classes.

In Jo Boaler’s class, she mentioned a different approach to assigning larger problems. In most American classroom, Mathematics is taught as a subject, rather than a tool. Teachers search for a circumstance applicable to their current curriculum, rather than providing problems and giving the students tools and resources to solve it how they see fit. The questions my class and I came up with were synthesized with answers and a process already chosen. We had worked backward to get the problem, which made them straightforward and fake.

Jo spoke of a school that taught Mathematics as a tool. Instead of giving their students a procedure out of thin air and asking cookie cutter question that didn’t really apply to life outside the classroom, teachers would assign a project, then make a number of tools available for their class. These tools would include what the teachers would think would be of use, such as protractors and straight edges, along with random ones that were unlikely to be necessary. Students would have to think critically about how to go about solving the problem. Some students would even use the ‘unnecessary’ items to problem solve in a new, unconventional way. The students had to choose the procedure, rather than being force-fed one to plug-and-chug throughout the class period. When groups came to the teachers with questions, they would be led in the right direction, or given a concept or procedure useful for how the group decided to overcome their current hurdle. Students would understand why and how the procedure they had just learned worked and how it could be applied to real-world situations. The groups would even share what they learned with others. (I mentioned in an earlier entry how explaining to peers solidifies understanding of the material.) Mathematics was no longer a thing to be memorized and regurgitated, but something to experience first-hand and apply.

Inventing problems with specific information in mind does have a place in the classroom, though. By creating our own, personal questions in my stats class, we were deepening our understanding of how the procedures worked as well as where we could use them in our own lives. It gave us a chance to be creative and invent the most complicated (yet solvable) problem we could.

Class Discussions

In both of the EdX courses I took from Stanford, the professors often mentioned how important working with others and discussing justifications can be. I believed it worked– for others. It was one of my “I am awesome and independent” moments. Sure, I would encourage my students to pair up and talk about the questions and answers that came up in class, but it was never something that I had intended to practice myself. This changed Monday, when I swore during my Intro to Probability class.

We were asked how many permutations there were in the stringing of a necklace, consisting of three different-colored beads. We easily came up with six. Next, we were told to discuss within our table groups how many possibilities there were without repeats (Since the necklace would connect, rotations would fall into the same permutation). I came up with two. My partner claimed that there was only one. We argued a bit, then I finally raised my voice and said, “No. See, there are three rotations of this one [where I gestured to my model], and three reversed rotations of that one, and unless you flip the necklace– Shit!” My partner and professor grinned and some of my classmates snickered. My prof then waited as I explained to the rest of the doubters how there was only one unique way to string the necklace.

By justifying my reasoning that there were two options aloud, I was able to hear the flaw in my logic that my partner had seen. I was also able to solidify my understanding of the problem by tailoring my justification to make sense to other students who I had previously agreed with.

Communication is something my prof frequently encourages during class. There is a very small portion of class time spent watching him write on the board or explaining concepts without us interrupting with questions and comments. When we bring something up, he immediately asks us what our interpretation of the issue is and to try to come up with an answer in our own way and at our own pace. Class discussions continue to play an important part in my learning. When I believe I understand a concept, I can turn and explain my thought process to my partner, who then peppers me with questions about every step I’ve made. This helps me find and understand my mistakes and gives me a chance to practice my own teaching skills for the future. When he grasps something first, he’ll explain step-by-step, then ask me to re-explain it to him in my own way, to make sure I ‘get it’. These activities have deepened my level of understanding for the subject and helped me to double-check my work and fix mistakes early on.

While class discussion may seem overrated to introverts and independents (like me), it is a key element in the learning process– particularly due to the requirement of putting thoughts into words and listening to what comes out. I highly encourage classrooms to become less listen-and-repeat and more discussion-based. I was very surprised at how much of a difference it has already made in my education.