Math Week 9: Transformations through Reflection

This quarter we have studied so many aspects of mathematical learning.  We have talked about using manipulatives in order to root abstract concepts in concrete ideas and understanding.  Also, we have been involved in the group work process and exploring what it means to have meaningful and effective group work.  We have also interacted with a variety of technological tools to use to assist in mathematical learning such as iPod Touches, computers, and graphing calculators.  Throughout all of this, we have found ways to reach all students by offering multiple entry points for learning math.

It was difficult learning about teaching without having time to see different classrooms.  However, the videos that we watched were beneficial glimpses of good teaching practices and I tried to incorporate various methods of math teaching into the classroom I'm in.  I find that the most difficult part of math is accurately assessment students' abilities.  Some students are very good at showing their work and arithmetic, while others are superior at problem-solving and logical, creative thought.  One test certainly doesn't seem to fit all.  What are other ways of assessing student's math knowledge so that we can cater our teaching to suit their needs?

I have greatly valued the time I have been able to spend with small groups of students and individuals in the classroom.  A couple of times so far, I have had students write on a slip of paper what they would like to work on.  Then, I sorted the students into groups suiting their interests.  Being able to work with these small groups on particular mathematical concepts seems to be greatly benefiting them.  I got one comment from a student who said, "why can't we do math like this all the time?"  I also think that the one-on-one mini-conferences I've had with students has been beneficial for them.  In these cases, I had students write a question to me and in the mini-conference, we went over the answer to that question and any others they had.  When I see the results of the unit assessment next week, I hope to find evidence that the students benefited greatly from both of these learning approaches.

I believe that the integration of math with other subjects is critical for the growth of intellect and morale.  I've seen so many math classrooms (particularly at the middle and high school levels) where math is an isolated subject.  Thus, students either develop the belief that they are good or bad at math; there seems to be no middle ground.  If math was more integrated, we would be able to draw upon our students' skills and interests and connect new learning with background knowledge.  Not only would this significantly increase learning, it would also revive math to be a magical and mysterious subject, rather than a mundane and tedious one.

Math Week 8: Math + Philosophy = X

Today in math class I learned about different ways to teach geometry, how to think about graphing backwards, putting emotion into math, and having the right to fail.  Woah, that's a lot.

The backwards graphing caught most of us off guard.  We're so used to graphing with units in mind.  We've been trained and been marked down on tests for not labeling the units on the axes.  So today, when we had to plot data points to show the linear relationship in measurements, we felt like the earth had switched its rotation direction.  What makes present-day math teaching and learning so linear?  Why aren't more teachers allowing their students to explore different facets of math and offer different access points for learning?  Is it fear of failure?

This society also is very set in the mindset that we cannot let students fail.  We can see this on innumerable levels and understand why this is so ingrained in our culture.  I like what we talked about in class in terms of the lessons that we learn from failing.  Often, we have made our most life-changing choices when faced with these situations.  This is juxtaposed by the philosophy that in order for us to keep desiring learning, we need to be successful.  There is a balance between these philosophies; the question is where?  I suppose the answer is that it depends on the individual. 

Math Week 7: Techie Transformations

Today, I learned how to use some of the basic features of Geometer's Sketchpad.  This is a great, simple program that allows you to draw shapes.  In the process, you learn how to informally prove that a shape is what it claims to be, as well as explore the concepts of definitions of shapes.  This is a tool that we can use in our classrooms, on the computer or Smartboard.  We can also setup example problems, like the one we did in class today, and have our students interact with the software, solve, and discuss concepts.  I wonder how this software could be used as a method of assessment also.  It reminds me of when I used Solidworks in a class I had and took tests by building models using the program.  I think this type of assessment would be yet another access point for different learners to demonstrate their knowledge.

We also learned about all the add-on gadgets for graphing calculators which can provide ways to collect data and analyze graphs.  I like how this interactive process is engaging for the whole class while also helping students turn abstract ideas into concrete understandings.  As we wondered in class, when do we think technology will transform cell phones into graphing calculators?  After all, why have a bunch of gadgets when they can all be combined into one?

Math Week 6: Many Ways to Learn

This week in math, I learned about how using visual tools and manipulatives (such as Tangrams) allow the students in our classes to see how different people have different skills.  Some people have a mind for visualization of concepts while others thrive using logical thought processes.  Ideally, I think, it would be most beneficial to be able to offer the students a variety of modes of learning so that each student can access the learning in his or her own way.  What's the balance between encouraging all students to try different methods and making sure they are satisfied with the learning experiences they are having?

Also, we talked about how journaling is a great way to assess student understanding in a pressure-free way.  In this high-tech age, we can use blogs instead of journals so that students can keep their physical journals with them for reference at all times.  Also, as teachers, we can subscribe to our student's blogs so that we know when they are updated.  As we continue talking about different ways to assess students, I think it's so important to provide a multitude of opportunities for formative assessments so that we can successfully help all of our students learn.

Many people are still flustered by the idea of integrating math with other subjects.  We've talked about this topic so many times that I'm thinking this concern is not going to be resolved until we start delving into teaching.  Right now, I think that integrating subjects really is going to be smoother than we think.  This is one of those theories that will probably pay-off in the long run (students will learn more) even though the evidence that it's working probably isn't clear up-front.  I guess it's our job to be both daring and logical by collecting that evidence which shows that students are learning more through integrated methods.  (Then, we can write one of those scholarly articles that will be read by all those in the profession!)

Math Week 5: Can you prove it?

"It's a square because it looks like one" suddenly became an unsatisfactory answer!  How awesome!  This is when we math teachers insert a tiny "not drawn to scale" under each diagram and thereby stretch our students' mathematical thinking and visual perception.  In math class this week, I learned about the developmental stages of geometry learning.  I find it intriguing how there is a correlation between these stages and the ability for us to perceive geometry by trusting our logic and creativity rather than our eyes.

I am still left wondering about the use of the paper folding to build a conceptual knowledge of proofs.  In order to verbally express your reasoning, you need to have a certain vocabulary.  For example, we used the terms "linear pair", "alternate interior angles", and "perpendicular bisector".  So, would a lesson like the paper folding fall sequentially after learning terms and before formal proofs?  Would this type of activity be an effective learning tool for understanding terminology?  I'm curious about the different modifications for this paper folding activity so that I can use this hands-on, engaging activity to assist in the learning of different geometry concepts.

Now that I know there are distinct stages of developing geometric reasoning, it has become a lot clearer to me why some students seem to grasp geometry concepts quickly and can manipulate them while others are seemingly confined by the strict parameters of what they see.   I will utilize this new knowledge to help my students sequentially build their understanding of geometry.

Math Week 4: Mira, Mira, on the Wall!

In math this week, I learned what miras are!  No matter how exciting these tools were, though, it was incredibly sad to be reminded of all the math tools are stuck in closets, not being used.  And thus, I made a mental note and vow to always search the school for stuff before purchasing it.  I guess you could call this my personal school budget plan.

I also learned today that purposeful group work is very different than just working with other people.  Group work needs to be engaging and be a beneficial learning experience for all the group members to make it worthwhile.  Group roles are an excellent way to accomplish this.  Yet, the thing to remember is that group roles need to be monitored and upheld by the teacher, otherwise the students won't adhere to them. 

Even though a specific structure to group work might allow for greater learning opportunities, some part of me wants to think that all group work has benefit to it. Not only does it help kids learn how to interact with another person or people, it also then teaches accountability (when you do activities such as partner sharing and being held accountable for your group's work).  So, do all group work activities truly need to meet the "group worthy" criteria?  Or can group work sometimes just be about teaching students how to interact and resolve disputes?

Math Week 2: Manipulating Questions ≠ Questions + Manipulatives

Today, I learned all about the art of questioning.  As math teachers, we should never give answers or tell students that they're right...they'll shut off their brains.  Make students work together to find solutions.  Give them the tools they need to problem-solve and check their own work so that they build confidence in math ability. 

What else did we explore?  Make use of manipulatives whenever possible!  We learn every other subject by starting with concrete information and examples and move to abstract thoughts and concepts.  Why do we teach math the other way around?  So, by using manipulatives as tangible, concrete examples, we can begin to shift math teaching to align with the way the human brain is accustomed to learning.

I'm still wondering about a few theories.  If students rely on the use of manipulatives on tests, then how will they solve the problems on standardized tests?  How do we build a community that doesn't judge a student because they still need to use manipulatives to solve problems?  Also, we can be very "sly" to use tricks like think-pair-share so that students are held accountable for their group members' success, but that still doesn't negate the social issues that middle school students are going through and the feeling that students are at differing abilities if they use different math problem-solving strategies.

Math Week 1: ...Say what?

I love hearing how different people solve math problems!  It always seems that no matter how many different ways you come up with to solve a problem, there are always other people who have figured it out using another method.  In today's class, I learned that we need to build a supportive, patient classroom community so that all of our students are comfortable sharing their mathematical thinking.

I am still questioning the use of cards/randomness to call on students to share their ideas.  I liked the way that this tool was used at the beginning of each activity, to break the ice, and then substituted with "does anyone have a different way of solving this?" so that we didn't see too many repeat answers.  Nevertheless, even though our classroom environment is very supportive, there still seems to be nervousness when it comes to sharing our mathematical thinking with our classmates.

The activities that we did today were extremely engaging and inspired exploration and learning.  This type of activity will be a great tool to use in our future math classrooms.  The bonus is that these activities truly cover all of the state standards!