I don’t know about all of you but I was exhausted at the end
of our marathon Friday evening -all day Saturday immersion into the deeper look
at algebra. Notice that although we were doing algebra we did very little paper
and pencil procedural skills or practice. That was by design. It is my belief
that as we enter an algebra course we discard the notion that algebra is about
manipulating symbols and embrace the fact that algebra is a generalization of
arithmetic and is a logical, sequential, representation of the data that
surrounds us.
If you think about how algebra was invented back in ninth
century by the Arabic mathematician Al-Khwarizmi (whose word al jabr,
describing a common process that we use in algebra, gave us the word algebra) do
you picture a man sitting around simplifying expressions and solving equations,
or do you perceive this gentleman experimenting and investigating the phenomena
he saw around him, a man who tried to make logical sense of those phenomena and
develop a structure that would always hold true? He most certainly “played” with the
mathematics as he worked out a systemic process by making conjectures, testing
them, revising them, and testing them again.
What I hope we accomplished this weekend is that need to
model the phenomena whether it was through the water vases or the slinkies. The
inclusion of Hooke’s Law was a more formal method for discovering a general
equation that will always work. The equation y = 0.07x which most of you
discovered illustrates the k constant
rate of change to be 0.07 which is a ratio, the slope, and as someone in
class stated the rise over run. The problem that we worked on with the texting
options formalized how an equation with a constant rate of change can look in a
table, graph, and equation. Then the big question arose…what do you do with the
data? How do you interpret the information to make the best decision about
which text plan is the best FOR YOUR NEEDS? There really was no right answer
until the question became more specific and asked which plan was the least expensive
plan for 55 text messages?
In order to be successful with integers, it is necessary to
understand how they operate. I really
like the algeblocks because they can be used for all for operations, for
solving equations (something to look forward to), modeling multiplication on
the quadrant grid and actually illustrating a binomial times a trinomial. The
“aha” moments students have when they first get that three dimensional object
is heart- warming. We will continue
working with the algeblocks next weekend as we work with solving equations and
develop an understanding of function.
I hope you found the class interesting and engaging.
Anne
Please post your comments here.
ReplyDeleteI totally agree that the weekend was exhausting, both physically and mentally. That said, I learned so much about both me and my teaching methods in a very short time. I teach in the advanced program and have been teaching Algebra 1 for 3 years now. I am embarrassed to say that until this week I had never once taken out the algebra tiles. I have always assumed (and we know what happens when we assume) that my students didn't need them. I have always felt it was more important to plug along and attempt to get through all 13 chapters so that my kids could move on to geometry. I have "the nerds", who love to work with algorithms. I assumed (again) that is all they needed. In just one (long) weekend, I have had an "aha" moment. All students can benefit from visuals and hands on models. I can no longer assume that words and numbers are enough to foster the deeper more thorough understanding that the Common Core is asking for. Although I may not be looking forward the exhaustion of the up coming weekend, I am looking forward to what "aha" moment I might have next
ReplyDeleteHi Dawn,
DeleteI firmly believe all students no matter how bright benefit from various models. In the real world these students might become architects, contractors, graphic designers and they will be modeling before they ever get to build so the more experience they have with multiple models and representations the better they will be suited to doing that in a paying job.
I have always needed to make sense of things mathematical in order to learn them. I thought I had a problem. But making sense of phenomena through "playing, experimenting, conjecturing, and revising" leads to real understanding and apprehension of the mathematics. It's exciting and fascinating, and I hope to bring this back to my classroom.
ReplyDeleteI answered your comments but forgot to publish. so here I go again.
DeleteI am so glad you appreciate the need to play with the mathematics. Too many adults think they need to go straight to the algorithm without ever thinking about what the math looks like or how it behaves with concrete materials. I enjoy your enthusiastic attitude towards what we are doing.
Anne
I have to say our first weekend of class my brain was fried. It was unfortunate timing that the Math MTEL was on Saturday during class. However, I’m very excited to participate in the classes to come.
ReplyDeleteI’ve always liked algebra. Throughout my years working in the business world I always used the analogy “a + b = c” when training my staff members when handling money. I worked in the golf business for 5 years and supervised many employees (mostly part-time workers in their late teens and early twenties). Throughout a normal day at a golf course, the golf shop operations is faced with many challenges. The point of sale system could shut down. The phones most likely will ring nonstop. There is always a good possibility that you’ll field some complaints and potentially have to process something that your boss has not told you how to do yet. There were always discrepancies on what account to ring something up in or this golf shirt doesn’t have a price tag on it, etc. To comfort any new hires I would simply say, “At the end of the day, just make sure a + b = c”. I used this analogy to encourage them to do the best job they could and to make sure their cash drawer is balanced at the end of the night.
The other component to this class I have enjoyed so far is the graphing functions part. I’ve always been a visual learner as well as someone who enjoys visual design. Graphing functions is a great way for kids to incorporate mathematical data into a visual representation of that data. It also teaches them to work in a structured platform and to take their time in creating a well-drawn graph.
I’m excited for this weekend’s class. Hopefully I’ll have a proper functioning brain this time!
Mike Santoro
I like answers. I like knowing if my solution is right or wrong. As a student I think this is why I was drawn to math and not English. In math, especially as I moved towards early algebra in middle school, I could check my answer and prove whether a solution was right or wrong. In a sense, you can walk away from a math exam and know exactly how you did. In English, you can write what you consider to be a well-thought-out, quality essay ... only to have it totally ripped apart and criticized by the reader. Often times the same English composition could receive a wide range of scores if graded by different teachers.
ReplyDeleteI also like answers as a teacher. Most questions have only one solution and a student is either right or they are wrong. It makes grading simple.
Last session's activities got me rethinking my need for answers. When some of the questions weren't specific I started to consider all possible answers for many different questions. In the search for one answer it's possible to lose some of the process and miss out on some deeper understanding.
My favorite part of the first week was working with the Algeblocks. I have used algebra tiles very minimally in the past - primarily to show students basic factoring and completing the square, but now feel like I have such a better understanding of how they work. I feel like using Algeblocks is such a great way to model so many of the key concepts of Algebra that students struggle with. I also really enjoyed using the blocks to perform operations with integers. I know that that this is a topic that the 6th grade teachers in my district were struggling with trying to find the best way to teach their students. I plan on showing the 6th grade teachers how to use the Algeblocks to perform operations with integers. We already have unit cubes that we can use and will just need to create the "mats."
ReplyDeleteI have always liked teaching geometry more than teaching algebra. I found it easy to engage students by using activities involving building and discovery, which is why I preferred it. Students really seemed to be understanding the material. I’m assuming it was from the fact that some of it was more “hands on” than they had experienced in algebra. I am guilty of teaching algebra in such a way that is more based on lectures, note-taking, and examples rather than activities that get students out of their seats. I am excited to be taking this class and learning ways in which to get students involved in more hands-on activities in my algebra classes. That’s the point, right? To get students engaged and excited about learning, therefore learning more than they had expected? I think it is, and I am looking forward to changing my approach to teaching algebra.
ReplyDeleteI really liked the cell phone plan problem. Though I teach seventh grade so I am not sure if it a problem suited for their grade level. But what I liked about it was that it was a real life scenario. Most of our kids have cell phones these days but they do not pay for it so they are not aware of the costs. Giving the three companies information as a graph, a table, and an equation is a great way to connect the different methods. I also think that a presumption would be that if one company is cheaper for one scenario then they will be cheapest for all of the scenarios. But when you make a table and graph you have visual proof that that is not the case. This problem I would most likely simplify and have neater numbers because my students do not see that equations, graphs, and tables are inter-related yet. So to be able to convey that message to them I would not want to overwhelm them with daunting numbers.
ReplyDeleteI came away from week one determined to create more lessons that model the concepts I am teaching in my high school math classes. For all those students who are successful at quickly learning concepts and manipulating symbols, there are twice as many who would benefit from modeling the material prior to formal assessments. What I have found in my classes this year is that when I break away from the paper and pencil procedural skills, everyone seems to become more engaged in the process of learning how to think more meaningfully. They become more efficient thinkers. Our first class made a great selling point for Algeblocks, especially when we illustrated multiplying a binomial by a trinomial - a task often confusing because it is not your typical foil problem! While our class covered a significant amount of material in one weekend, there was something for everyone to get excited about and take back to their schools.
ReplyDelete