Clarifying Misconceptions and Facilitating Deeper Thinking

            After going over the readings for this week’s class, I was really intrigued by the Falkner and Kabiri articles.  In Children’s Understanding of Equality: A Foundation for Algebra, Karen Falkner, Linda Levi, and Thomas Carpenter discussed how many children are very closed minded when it comes to the meaning of the equal sign.  Many of the students from the article believed that in an algebraic equation, the equal sign only means the answer.  This is of course false in a problems like 8+4=_ +5.  In the article the authors state, “even kindergarten children, however, appear to have enduring misconceptions about the meaning of the equal sign” (Carpenter, Falkner, & Levi, 1999, p.233).  This shows that this misconception is rooted early on, so in grades as early as preschool and kindergarten, it becomes imperative to teach the right thing.  I believe this article ties in very well with the article by Mary Kabiri and Nancy Smith titled, Turning Tradition Textbook Problems into Open-Ended Problems.

            If we as teachers can create more open-ended problems not only will our students expand their thinking, but concepts like the equal sign, will have more than one meaning.  In the article the authors state, “Often, when teachers plan instruction for their student’s, they focus on the middle achievers…However, many problems can be made more open-ended and accessible to a wide variety of student abilities with minimum effort” (Kabiri & Smith, 2003, p.186-187).  They go on to explain that this can be done simple by taking traditional textbook answers and adding an “additional dimension” where more than one answer is possible.  So far in my placement, I have not seen many open-ended questions in math, however, I am excited to see if I can somehow incorporate these type of questions into my lesson plan to see how the students react to a new style of questioning. 

Caitlin Shanahan: Seeing Math in the Field

I am fortunate in that I see math most of the days I am in field, but because I am placed at a Montessori school the way math is integrated in the classroom is a bit different than what might be seen as most other schools.  There is no specific portion of time set aside for math each day, but instead the students are free the pick the work they do.  As you can imagine, in a Kindergarten class this can be a bit problematic because the students who don't feel comfortable working with the math materials tend to shy away from them.  There are a handful of students who come to school excited to do more math activities with the teacher, but there are also a handful of students who I have never seen working with a single math related activity.  In a class of 25, the teacher only has so much time to work one-on-one with each student (there are no group lessons in a Montessori school), so many of the students are falling behind with their numbers and simple math concepts because they are not given time with the teacher. Just how much some students are struggling in math has become very apparent since the last report card testing, so the teacher has been trying to make it more of a priority to include math in their daily activities.  This worries me because it is half way through the year and some of the students can't even count to five, while others are learning simple addition and subtraction problems.
I think a big part of math has to do with how confident a student feels in actually doing math, and if some of these students who are moving on to first grade are already starting to fall behind, I worry that they will develop negative feelings toward the subject.  In Denise Cantlon's article she mentioned multiple ways of authentically assessing her students.  In my field placement specifically, I think the students would really benefit from having those ongoing assessment strategies available to them.  Right now, testing is just done at the end of every marking period so my MT is just realizing how much some students are struggling and they are already half way through the year.  If she did ongoing assessments through observations, conversations, and games she may be able to help some students sooner rather than later.