Math – Winter Quarter Reflections

As I approach the beginning of my full-time student teaching, I am excited and a bit anxious.  I have taught more whole class math lessons than any other subject, so I should be most comfortable there, right?  In my 5^th grade class, we use the Envision curriculum and it moves quickly with each lesson following the gradual release model (OLM) with demonstration, guided practice and independent practice.  Reflecting back over our intermediate math methods class, I know that I need to start doing more to augment the curriculum.  The frightening truth is the quote from the Lotan article on Group Worthy tasks that "one dimensional tasks require the same skills and the result is uniform success for some students and uniform failure for others."  The curriculum (at least how I have been using it), feels one dimensional and I need to focus attention on how to add to that.

This quarter, we have talked a lot about different ways to be “good at math” and the necessity of having multiple entry points into a math topic for our diverse populations of learners.  We talked about the importance of leveling the playing field so that socioeconomic, language and ability levels are blurred.  I want to be able to randomly group students so that they benefit from working with a variety of different students.  I am constantly trying to remember who I can put together and who I can’t.  Random groupings make a lot of sense and will simplify the process.

Manipulatives, virtual and real played an important role in our readings, class discussions and video reflections.  We have some manipulatives with the Envision curriculum, but they are mostly used for “games” at the end of the units.  I want to start integrating their use throughout the lessons to help get more of our students engaged in the learning.  I also found out that there are virtual manipulatives embedded into the on-line textbook.  We are all so busy getting familiar with the new curriculum and “getting through” the material that we are not using all the different modes we have to connect with our students.

I tend to be a planner, but I need to continue to get comfortable with the reality that sometimes things do not go as planned.  I need to be more flexible in my lessons, more flexible in my approaches and more flexible in my math thinking.  Our cohort has shown me that we all have different ways to approach math problems and now my students are showing me the same thing.  I need to get comfortable in being able to go where they lead and then bring them back as needed.  That journey can be uncomfortable but is how students create their own meaning.

Finding the ZPD for our students is one of the most important challenges that we have as teachers.  That continuum from boredom to anxiety is one that I see in my students faces everyday.  My own experience over the last two quarters tells me that I can be in a different place along the continuum on different days depending on the lesson, my background knowledge and a variety of other factors.  I know that some days, I was in the ZPD and that is what we want for our own students.  It won’t happen for every student everyday, but we need to be close enough to know that if not today, tomorrow.   

The more I know, the more I realize what I don’t know and that can feel frustrating.  I am going to try and relax and enjoy my student teaching experience.  This is what I have been waiting for!

Complex Identities

OK, I admit it, I like 1970s worksheets.  I am comfortable with them anyway because it is what I grew up using in math classes.  It is good to come full circle and realize that our students will be at an advantage because we can give them a variety of options for learning and understanding geometry.  They can “play” with shapes on the computer or make drawings with a pencil on a piece of paper.  We can provide the definitions of the concepts first, or let them do some discovering before labeling the shapes.  We have more tools at our disposal and we need to get familiar with all of them and make them available to our students so there are as many entry points to math as possible.

The measuring activity was enjoyable too.  It was great to use a variety of everyday objects to “measure” the rectangle.  I appreciated hearing the different thoughts about why the graph of the measurements was linear.  It was a great way for me to deepen my own understanding of measurement, graphing, algebra and slope.  Nothing fancy required here, so it is very accessible.  This helps me as I have been concerned with the availability of some of the math software in our schools.

I really like the idea in the complex identities article where “helping our students become aware of their mathematical identifies can empower them to make that relationship more meaningful”.  My school is having a math week later this month and there was a staff planning meeting about it this week.  I proposed to my master teacher that our class have a math community building activity.  We would we talk about all sort of different mathematically related activities (the table in the article).  We then have an opportunity to have our students consider and talk about the activities that they are good at doing.

I will be reviewing some other math activities that we might include.  Some pictures of  the Fibonacci sequence and connections to nature are floating through my mind…..

ZPD and Engagement

It is good to have some practical examples of what the ZPD might look in a math class.  Boredom on one end when math is too easy and anxiety on the other end when math is too difficult.  The ZPD looks like a pretty narrow target and I believe it is going to take some practice to hit the bull’s-eye and engagement is critical!

Going to the computer lab was a good example of how we might engage some of our students in different ways.  Geometers sketch pad was a lot of fun and would allow some of our students a different entry point for geometry. I need to review my geometry vocabulary as I had my kite and rombus a bit confused.  Using DASL in conjunction with  Fathom 2 was a great combination for engagement for statistics and graphing.  We have already talked about how using real data can allow students to connect and create meaning in ways that “made up” data sets does not allow.  I think analyzing data is lot more important that being able to graph data and this is a brilliant way to enable that to be easy.

I did ask my high school senior and he has never used any of these tools.  He attends a pretty resource rich high school (Issaquah), so it does give me pause about the cost or feasibility of schools using these to engage students.  He does have the same TI graphing calculator that he has used since middle school.  Although he has never hooked it up to a probe to do motion analysis, he has done something similar with Loggerpro that he tells me is pretty cool.  My understanding Loggerpro allows a variety of sensors (thermometer, ph, etc)  to plug into hardware that connects to a computer for graphing.  So maybe it will eventually be at our fingertips, or smart phones.

He was good at math.....

I have been struggling to focus on math (and school) this week.  But today I realized that Tyler’s story is full of math connections.  Tyler struggled academically but was a passionate and successful lacrosse player.  His desire to play at the college level resulted in a switch to a small private school this year.  He had recently been accepted at a Division II school in California where he was going to play lacrosse.  Last week a car accident took his life and ended those dreams.

I remember the first time I met Tyler’s mom, it was at the parent meeting for lacrosse when our boys were high school freshmen.  We met in the coach’s classroom, he used to split his time teaching math and phys ed.  The coach was pretty intense and I remember he spit out a lot of statistics about his team.  Patty whispered to me that Tyler was not good at math and maybe the coach would be able to help him out.  I remember telling her to have Tyler check in with my son Aidan, he was a “math kid”.  That was before I knew all the ways you could be good at math.

Tyler had an amazing ability to see the entire lacrosse field and know where all his teammates and opponents were located.  He was a defender (like my son), but his talent for always being where the action was earned him a spot on the varsity as a freshman.  In addition, he was the guy to go to when you needed to restring your lacrosse head.  This is an art that requires patience, hand eye coordination and 3 dimensional visioning – a kind of puzzle.  This week, I learned that Tyler also knew all the starting NBA line-ups and their jersey numbers. 

He was a fabulous lacrosse player and he was good at math.  He will be missed.

Informal Geometry

Today's activities provided another opportunity for people to be "smart at math" in different ways.  The visual and tactile nature of folding paper into boxes and squares made geometry very concrete.  As I am starting to get used to, it may have been informal, but I felt less smart in math as than usual.  I am more familiar with drawing a geometric shape and learning all the rules about how angles and sides are related.  My stretching is continuing.

I was hoping we would discuss the use of creative writing in math class that was the subject of the Halpern article.  I thought that was a different type of example of how to make geometry (or any math) more accessible for students who have strong written expression skills.  As students get into Middle School, I often hear that students start being divided into those that are more humanities focus and those that are more math/science focused.  I read The Phantom Tollbooth last quarter and I felt it was a fiction book that would appeal to students who love words and/or numbers.  A bored boy travels in a toy car through a tollbooth to a place filled with adventures based on letters and numbers.  The Mathemagician rules the Kingdom of Digitopolis and a character called the Dodecahedron has 12 faces with different expressions.  It is another possibility for giving our strong writers an informal entry point into geometry.

Group Worthy Tasks

I loved the readings this week.  I got inspired while reading the article on Orchestrating Discussions, so I went online to join NCTM to be able to start reading the middle school journal regularly.  The bag of marbles problems is a marvelous and simple example of a group worthy task and gives me something to contemplate for the lesson assignment.  The article makes so much sense to me that I pulled out the five practices planning worksheets from last quarter so that I actually start using them in my main placement.  I like the idea of trying to anticipate all the possible ways that my students will solve the problem- especially the ways that they may solve incorrectly due to common misconceptions.  Monitoring, selecting and sequencing becomes a whole lot easier when I know what I am looking for.  As a new teacher, I am not as good "on the fly" as I would like to be so I want to plan as much as I can until my comfort level increases.  Making connections to the lessons goals and then extending and making generalizations are all things to strive for.  But it gives me a template and something to try and I like that.

The Lotan article on GroupWorthy Tasks is exciting and a bit daunting.  I know it is a lot of work up front, but the group work in the article really did help even the playing field for students from different racial, ethnic and social groups.  I loved the multiple entry points and opportunities for students to display "smarts" in different ways in response to complicated problems.  The frightening truth is the quote from the article that "one dimensional tasks require the same skills and the result is uniform success for some students and uniform failure for others."  We need to carefully craft group work to enhance participation and learning for all our students - it is worth the effort.  Our lesson plan will be an authentic opportunity to consider this goal.

Finally, I continue to struggle with manipulatives.  The Mira symmetry manipulative was a challenge to me.  I was slowly getting the hang of it, but I need remedial work.  I think I will need to make my way to Math and Things store sometime soon.  My experience with virtual manipulatives is even more challenging.  Just give me the algorithm!

Getting Kids to Do the Talking

I really enjoyed the practical article that was full of strategies and tactics to try in math class.  I think the teacher had a terrific personal goal of trying to change his own practice by 10% every year.  I think that is a manageable goal that ensures teachers are continually improving their own practice.  As a new teacher, I liked many of the ideas and plan to incorporate them into my initial classroom plan.  One of ideas that was brand new to me was how to handle students who say " I do not get this" when responding to a problem.  I think that "I don't get it" should be banned from classrooms.  I loved trying to get the student to ask a question when they are stuck so that others (peers, teacher) can help them get unstuck.

I am wondering how to implement some of the ideas in my class where math is very scripted.  We have direct instruction, guided practice and independent practice every day.  I know that there are opportunities for pair/share, group work and asking questions.  I just need to get comfortable with the curriculum so I can add to it in a way to help more students learn math well.

In my main placement (5th grade), we are just starting algebra.  I will be looking for manipulatives tomorrow!  I loved the tiles and think that they will be very helpful in my class.  The students were very confused last week with the introduction of variables.  Statements like n=3 were confusing and I think it is a big shift for students.  If we do not have manipulatives, I plan to take a look at the virtual manipulatives to see what I can find to supplement the book.

Math Thinking

In math class today, I learned that I am much better seeing patterns in numbers than I am seeing patterns in pictures.  It felt natural for me to write the numbers in a chart and figure out the general equation for the nth number.  However, it took a while for me to figure out how to manipulate the tiles to show the associated geometry picture for the garden equation.  It took me even longer to figure out how to show the picture for the snake equation.

I am trying to figure out how to be more flexible in my own math thinking so that I can understand and help all my students.  Heather had a great (and very fast way) to do the garden problem and it took me several times before I was on the same page and understanding her thinking.  I need to figure out several ways how to solve problems so that I become more flexible and get out of long held patterns.  Next time, I should start with the picture and not the numbers!

 Last quarter, Prof. Allison Hintz gave us some tools –including worksheets to help us plan how we might want to use students’ math thinking to show different problem solving strategies to our class.  Sometimes a particular sequence can help guide our students to move to a more efficient strategy.   As a new teacher, I would benefit from using the planning sheets or trying to solve problems in a number of ways to help me preview what I might expect to hear/see in class. It is powerful to share students’ math thinking, but I think it needs to be facilitated well by a flexible thinker.