Classroom+Learning

What does mathematics learning look like in the classroom?

Learning begins with teachers specifying to students the goals of a given period of time (daily, weekly, monthly, etc.). In order for a teacher to discover if any learning has taken place, an assessment must be given (either formative or summative). Formative assessments (among other uses) will help establish where students are at in their learning. As teaching, and presumably learning, takes place, further formative (and finally summative) assessments can compare student knowledge to their previous assessment results. Assuming all assessments given are valid, reliable, and consistent, the results will show whether learning has taken place. If it has, the learning in the classroom is **ANYTHING** that has helped cause this result to occur. (JE, 1/19/10, 8:23 pm)

If I'm reading this correctly, you're saying that "learning ... is anything that has helped cause [learning] to occur" (?) (RK, 01/25)

In regards to the question of what learning should look like... - Learning should look like an interconnected mental map in which new knowledge builds on and connects to prior knowledge. Maybe this is more of an outcome of learning than the process of learning. (Support for this idea might be found in cognitive theories, as I picked up on some of these ideas when doing the readings for week 3; Cobb (2007) also summarizes cognitive theories to involve a constant reorganization of ones knowledge structures. I view the process of building new knowledge on previous knowledge as a type of mental organization. - NF 1/25/10) - I was really struck by Rob's classification of learning as active intellectual engagement (because it makes me think about the ways in which I've changed in how I learn mathematics. In particular, I used to attend lectures, copying notes without doing much thinking, I believed that the thinking was what you were supposed to do outside of class. Since beginning my career at Western, I've learned from other graduate students in mathematics the importance of being actively engaged during a mathematics lecture. I've learned this skill and have found that it contributes to my learning of mathematics. NF 1/25/10). - I also believe that in the process of learning, some people verbalize their thinking in order to learn it better. Some people learn by being engaged in discussion or in explaining their thinking - NF 1/19/10. (I think part of the support for this notion might come from constructivism and the notion of engaging in ones community of practice or learning. Some support for this could be gleamed from Davis (1997) - NF 1/25/10).

"What does learning look like?" My response to this question would be that learning looks like WORK: students must work problems in order to learn mathematics and they must use that work both to ask and answer their own questions. All of this work takes time, and it evolves over time. (CZ, 1/14/10)

I feel that learning should involve active intellectual engagement. I don't feel that "active intellectual engagement" is black and white (you are or you aren't), easy to measure, or even //necessary// for learning. I do feel that students who are actively and intellectually engaged with classroom mathematics will learn //more// mathematics and that they are more likely develop the habits of mind that will allow them to become successful creators and consumers of mathematics. (RK, 01/20/10) - There is specific support for this notion of learning in Dubinsky (1991): "in order to construct a mathematical idea it is necessary to be mentally active" (p. 122) - NF 1/25/10. - As well as in Sfard and Linchevski (1994): "A real change for the better will not come until teachers find ways to boost students' willingness to struggle for meaning" (pp. 224 - 225) - (RK, 01/25)

What does learning look like for me? Reflecting on my experience as a student and as a teacher, I see learning as a productive struggle. As a student, I am constantly faced with new concepts (not just mathematical) and the struggle is in connecting these concepts to what I already know and understand. The process of learning is long and hard, and I have to backtrack if the path I follow (or the connections I made) does not lead me to understanding the concept. The end result is new knowledge and understanding. As a teacher, my struggle is in how to guide my students to struggle productively- oftentimes it is easier just to feed them the information, but did they really learn? (JH, 1/25/10)