The Practice Predicament

So as promised, I am going to share my new teacher predicament between practice and understanding.

Last year I taught with curriculum that was based off of the idea that practice could lead to understanding.  The philosophy was that following an algorithm for how to solve a problem repeatedly would somehow reveal to a student why the algorithm worked.  One thing I know is that this does not work!  It is not fair for us to expect students to gain understanding when all we show them is a process.

However, not all practice is bad.  Students who have developed understanding often need practice in order to develop the skills to use the mathematics independently.  This should occur alongside opportunities to strengthen understanding.

My struggle is finding the appropriate balance of practice for students.  I find that often I actually stray on the side of too little (just basing this off of what I see on assessments after I feel like I have helped students develop a strong understanding).  Here are some problems I have had with getting students the right amount of practice:

• If it is an out-of-class assignment, many students do not complete/start it
• If it is in-class, not all students need the same amount of practice, so major differentiating needs to occur.  I struggle with differentiating appropriately as at different times different students are at different levels.
• If it is in-class, independent practice leads to more push back from some students, causing me to have to address many behavior issues all day long.  Perhaps this is because I do not differentiate enough or because I have not stated my expectations clearly enough.  Many reasons for this, but whatever the cause, it is a challenge for me.
• It is valuable to practice recall of the mathematics, so all learning cannot occur at one time.  Students who are on top of the learning one day in class may come back the next day and feel totally clueless.  The idea of working through recalling learning is super important for students.  I feel it is harder to address this when all practice is done in class.

Thinking through all of this, I have decided that a good next step is to work on differentiating for in-class time to practice independently.  I will have to state my plan clearly to students ahead of time, so that they understand I am not doing this to select favorites, but to help all students learn.  Hopefully this can be helpful in finding an appropriate balance.  I will let you know how it goes.

Multiplying Polynomials

As part of our homework for the challenge, we were supposed to try to increase the rigor of an upcoming lesson using the practice standard we had selected as a district to work on implementing more effectively.  We had chosen practice standard #1:  "Make sense of problems and persevere in solving them."

When they gave us that assignment I said to the people at my table, "I'm multiplying polynomials next week, how am I going to do this?"  So I brainstormed the whole way home, and I had a pretty good idea, but I decided to create the UbD template for the standard.  (Any feedback on what I have come up with would be welcomed as I am very much still struggling with knowledge vs. understanding and other things necessary for breaking down the standards.)  The standard is:  Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.  Unfortunately, with they way that the common core has ramped up our curriculum, it is unlikely I will be able to dive into the "analogous system" part of the standard as much as I would like :( But maybe next year...

After working on what the standard says, I did an internet search (using websites I know of that have CC aligned assessment) for ideas for this standard, but I came up short, so I decided to run with my idea and see how it went.  I used this chart to guide students through a repeated process of individual, small group, class discussions on a variety of multiplication scenarios that slowly increased in difficulty.  After going through each pair as a class, I assigned each small group another similar problem.  They were to process as a group and present to the class.   It took the good part of a week to make it through the chart, and it is still a steep stair between a monomial multiplied by a binomial and a binomial multiplied by a binomial.  I had the students complete a survey of several internet sites in order to see three methods of representing the multiplication of polynomials (vertical, distributive, and grid).  

Overall, I believe the process was productive for them.  I was able to quickly address misconceptions that arose with the entire class (a common one was that (x)(x) = x rather than x^2 which lead to the phrase "ninja one" which was much more successful than I would have ever thought) and give them more opportunities to practice.  (I still gave practice problems for students to work on individually, but I post an answer key also so they can check as they go.  I have struggled a lot with finding an appropriate balance in beginning mathematics (Algebra) between practicing too little and too much, but I will save all those thoughts for another time.)  I also thought that it lead to some of the best mathematical conversations that the students had ever had in my classroom.  I heard them discussing exponents, coefficients, and terms! Oh my!

Day 3

We had Day 3 of this project and we got to present right at the beginning of the day (Well after a Kid President video--he's hard to follow!).  We shared the key points of our action plan...almost like what I posted previously, but not the "why?" part.  I figured that part would be better to share verbally if the conversation went that way.

But I have to admit I was totally wrong in my assumption that people wanted to know why.  Most teachers are totally on board with the plan.  They want to see it happen, they just need help figuring out the logistics.  What was best about presenting was all the questions that came from this.  Everyone is trying to figure out how to make it work and so not all of the questions were addressed to us, some of them were addressed to others, too.  And that helped us realize some of the things we still have to figure out.  What great collaboration!

For the rest of the morning we split into groups:  Math teachers, Special Education teachers, and AEA staff. As Math teachers we continued our work to unpack standards in Unit 1 and find appropriate assessments.  We were working with 1-2 other teachers on a standard of our choice.  I worked with a teacher I had never talked to before, and we picked a standard she had worked on previously.  I felt as though it was much more productive this time.  She and I communicated well together, and I was able to be a little bit more flexible about the whole process.  I tried to avoid the things that had stressed me out too much previously.  We made some progress and we will continue to work on Unit 1 and maybe Unit 0 next time.  I hope that we can get the first part of the year planned prior to the beginning of the school year!

In the afternoon we discussed standards based grading in a little bit more detail.  This is one area that I feel like I have a good start to because of our school's PD, but there are lots of flaws that we need to work through.  I felt like the conversation we had around this were good, but I feel pressure to iron out the flaws prior to next school year.  I have to put my grading plan in my syllabus, and I want to have something fair figured out the communicate to students and parents.  This is a big challenge that I discussed with some of the AEA people.  They understand and also want to work to find a solution for us, however not all participants are going to fully jump into this shift next year, so it is going to be challenging to meet the needs of all the schools/teachers represented.

To conclude our day we got to be students in the lesson DVR Dilemma from Yummy Math.  The purpose was to get us to think through the practice standard that we are working on improving.  We have been challenged to work to improve our teaching strategies to better address this for our lesson plans in the upcoming week.  I will be working on teaching students to multiply polynomials...it's really hard for me to think about how to use this to help students "Make sense of problems and persevere in solving them."  But I've been thinking about it a lot after that challenge...maybe I will post again about what I decide to do and how it goes.  Going to be a big risk!