In Students' perceived sociomathematical norms: The missing paradigm, Levenson, Tirosh and Tsamir conducted research in 2 Grade 5 classrooms in Israel, looking at the effect of 3 aspects of sociomathematical norms: teachers' endorsed norms, teachers' and students' enacted norms and students' perceived norms.
Both of the classrooms were taught by experienced teachers, one, Rose, had 31 years teaching experience and the other, Hailey, had 14 years. The researchers investigated the sociomathematical norms that were related to mathematically-based (MB) and practically-based (PB) explanations. MB explanations would be based on previously learned material or mathematical properties, whereas PB would include context examples and/or manipulatives.
In Rose's interview, she felt that personally, MB explanations were the most convincing. However, in her classroom, her endorsed norms were to use either a PB explanation or a combination of the two. She explained that by doing so, she was teaching a method the majority would understand. Rose and her students' enacted norms showed she often asked her class to participate, but usually only the high-achieving students would. Students did not comment on each others solutions to problems, nor did they have to comment at all if they chose not to. Also, students only gave MB explanations to Rose's questions, and she would respond by inserting a PB explanation. The students' perceived norms were that Rose used whatever explanation was best for the student, but that the PB explanation was perceived to be for the low-ability students, as it was easier to understand. They did all agree that they could use either explanation and that Rose would accept either as correct.
In Hailey's interview, she also preferred the MB explanation herself. She also felt that the MB explanation was the best one for her classroom and couldn't understand why the PB explanations would be helpful, referring to them several times as "cute". Hailey's endorsed norms support this opinion, as she says she would only use manipulatives to introduce a topic and then would quickly move on to MB explanations. She believes the MB explanations lead to higher-order thinking skills. Hailey believes MB explanations "are more mathematical" (p.181). She feels her job is to
to any student, regardless of ability. One student related PB explanations to students of low ability, as some in Rose's class. They did agree that Hailey would accept any type of explanation.
This study really bothered me. I have a lot of differences of opinion with both the teachers, but for very different reasons. Rose, at first reading, seems to be the teacher that is using a variety of strategies in her classroom and is doing her best to reach all her students. However, the way in which she uses the PB explanations in her classroom leads them to be associated with “low” students. She also has a classroom where only the high-achieving students participate. I teach Grade 6 and 7 and am constantly using PB explanations for ALL my students. I find that having a good grasp in pictorial representations of problems means that students can make the leap to abstraction, particularly algebra, much more easily. Indeed, I have given my class problems that are almost impossible to solve without pictorial representations or manipulatives! Also, Rose does not foster participation. Participation is not an assessed part of my classroom, but engagement certainly is. Students should not be punished (in marks) due to their introversion, nor should they be allowed not to engage in the activities. Giving students a chance to work on problems in groups, to explain their thinking in pairs, is a way around the “problem” of participation in whole-class learning.
I also think Hailey is doing a huge disservice to her students. Shunning PB explanations is robbing students of a useful thinking and problem solving tool. MB explanations are not always the only way to challenge your students. Again, there are numerous way to pose very challenging PB questions to encourage PB explanations. I certainly have a problem with PB explanations being called “cute”. Shouldn’t mathematics teachers be teaching a variety of tools so that different types of learners can choose the tool that is best for their style of learning? Certainly some students prefer MB explanations, but by ignoring PB explanations, you are not giving your students a “complete” mathematics toolkit.
The other part that worried me was the students’ perceived norms. Although in both the classrooms, students claimed either MB or PB explanations were acceptable, no student in either class ever offered a PB explanation for any of the questions posed by the teachers. This seems like quite a contradiction. This also makes me wonder if their perception may be that although the teachers would accept the PB explanation, they would prefer and value a MB explanation more.
Hailey strongly believes that MB explanations are best in grades past Grade 2. Do you agree? Do you think MB and PB explanations are mutually exclusive?
In order to respond to your question, I revisited Wolfram’s popular TED talk titled “Stop Teaching Calculating, Start Teaching Math”. What I realized is that goals for mathematics instruction vary widely depending on one's conceptualization of what it means to understand mathematics. I take mathematics as a science of pattern, in turn, I personally favor MB over PB. I totally agree with you that teacher should adapt the teaching style that he/she is most comfortable with. However, I need more time to think whether pictorial understanding will boost the leap to abstraction understanding. One possible drawback would be that all students will pick up the norm and start from PB, reduce or even skip the step of MB due the time constraint. I haven’t been very successful in witnessing the leap to autonomous abstraction in my teaching experience when I enacted a PB norm. I am not saying that students won’t progress with PB (in fact, many students actually make great progress on tests); instead, I am with the author’s notion that for some students (maybe more advanced students) MB offers more cognitive opportunities.
ReplyDeleteI believe the balance of MB and PB in math class should be important. As students’ proceed to the next grade, the time proportion of PB might be decreasing and the time of MB is increasing. But I do not think devoting no or little time on the contents of PB explanation in math class is a good idea. For instance, the PB explanations might be effective to engage students in the introduction part of the math class, and both PB and MB can be utilized for the application what the students have learned before.
ReplyDeleteI suppose the interpretation of MB explanation might be difference among teacher and students. Moreover, it is not easy to clarify how we can exactly distinguish MB or PB in students’ explanations. But it is interesting to me that this article says both teacher and students understand PB is inferior to evaluate, compared to MB. And I agree with you, this kind of common sense might lead/force students to have similar mind with devaluing students’ uniqueness.