Higher Level Thinking

Some of the readings this week really made me think about how math was/is/should be taught. In the 5 Practices chapter, the importance of "higher level" tasks was highlighted. These types of demands include focusing on multiple approaches to problem solving, representing problems in various ways (manipulatives, visuals, symbols etc.), and overall- asks students to engage cognitively to explore and understand mathematics. In contrast, lower level tasks include various methods of basic memorization. When I reflect on my own mathematical learning, I feel I spent a significant amount of time simply memorizing. However, I was fortunate enough to grasp the concepts easily-and thus I really didn't struggle with the process of understanding mathematics. However, for those students who would spend time "memorizing" for a math test and then fail a test- this makes sense. Memorizing doesn't teach you anything if you don't know the underlying concepts. How do you get to the underlying concepts? Through higher level tasks!

In my placement, I would classify the classroom structure as following more "higher level" thinking in regards to mathematics. However, students are not yet to a level of mathematical reasoning in which memorization of various facts etc is required. Manipulatives are used as support as well as visual representations. Students also spend considerable time with their peers working through their activities, rather than individually or simply listening to the teacher. These activities as a whole seem to me, to be classified as "high level" rather than "low level" tasks. What do you see as high vs low level tasks in relation to your field placement?

However, after reading the article Children's Understanding of Equality I puzzled over the idea of higher level thinking further. The whole "equal sign" issue really seems to come down the issue of high versus low level tasks. It is ingrained into students' mathematical thinking that an equals sign is "a symbol describing a relationship rather than as a “do it”sign" (Falkner, 1999). As discussed in the article, if teachers are to take the equals sign concept to a "higher level" by changing its position in problems and varying how it is used, the concept is not only more accurately understood, but students are also more prepared and ready to engage in higher level math concepts. It seems to me that many mathematical concepts are likely able to be engaged with at a higher level to yield similar results.

I wonder how the students in my class view an equals sign?  Most students are not yet working with addition, and those that are working with addition are at a beginning level and have not been explicitly taught- they've simply been allowed to work on different activities and games to engage with the content. Because the article details students as young as kindergarten having the same misunderstandings even prior to working with equals signs, I'm assuming the students in my placement are the same. This concept is very interesting to me!

Addition and Subtraction

Something that I found really interesting was on page 19 of our CGI book.  This chapter discusses addition and subtraction and the page that I am talking about is a chart that talks about various counting strategies.  As a future teacher I found this really helpful. I do not remember how I learned simple addition and subtraction but this chart gives 6 different strategies that students may use or that teachers may want to teach to students.  I had no idea that there were so many different ways and I think this will be extremely helpful because we all know that students all have different ways of learning and any one of these strategies may help them.  I also think that learning addition and subtraction and mastering these skills is essential in learning math which is why these strategies are very important.

As for my field placement, I am unfortunately not observing much math because my CT only teaches literacy and some social studies while math and science are taught in Chinese. I have however reviewed their math book which gave me some insight into what type of things they are learning. It looks to me like they are focusing a lot on finding and making patterns but they will soon be starting very simple addition and subtraction problems (I am in a kindergarten classroom).  However, I have seen a little math instruction in my classroom when my CT talks about the calendar/weather in the morning.  The students show their counting skills by saying the various dates during the month and they also have a chart where they have the number of days they have been in school.  My CT uses this to show counting, patterns and sequences.  The students really seem to understand how the numbers on that chart work and are constantly answering the teachers questions correctly (like what do these numbers have in common? (10, 20, 30, 40....).  However, since I do not see much math I'm curious if you two do in your placements and what  exactly you are observing/what grade level?