Knowledge, Beliefs & Mathematical Identities

The information presented in this chapter was, to some degree, not surprising. It was evident from preceding chapters that the experiences and development of mathematical knowledge at Amber Hill was very different from those experiences of Phoenix Park students. From preceding chapters it was apparent where the real learning took place- Phoenix Park. This occurred despite the efforts put forth by the teachers at Amber Hill- methodologies which they perceived would ensure student success, particularly on GCSE’s.

Amber Hill’s traditional methods and Phoenix Park’s open-ended approaches were not on “separate ends of the spectrum,” rather they fostered a different development of mathematical knowledge. The two schools differed in many aspects:
· Amber Hill students lacked confidence in their mathematical abilities and relied heavily on teacher support and cueing. In comparison, students at Phoenix Park were considerably more confident in their mathematical abilities.
· Amber Hill students made no connection of maths taught in the classroom to “real world” situations. Students at Phoenix Park were continually applying the math they learned in school to out-of-school situations.
· The students at Amber Hill were perceived as what Belencky, Clinchy, Goldberger and Tarule refer to as received knowers- expecting to accept knowledge from authoritative sources without contributing to knowledge or making connections across areas. In comparison, students at Phoenix Park were very mathematically empowered- they possessed the power to develop and use mathematical knowledge in the school setting.
· Unlike their Phoenix Park counterparts, the teachers at Amber Hill were continually motivating students and enforcing the importance of the upcoming GCSE’s. The students at Phoenix Park were largely self motivated, and the importance of the GCSE’s was rarely discussed (with exception to the couple of weeks leading up to the GCSE’s).
· Students at Amber Hill viewed mathematics as isolated and fragmented concepts. Their lack of success on applied assessment can be attributed to this notion that mathematical concepts are isolated.

It is evident that both sets of teachers believed they were using methodologies that worked best for the students. However, there was no mention of teachers taking the time to reflect and determine student areas of both strength and weakness. I, as do many teachers, often get wrapped up in the notion that “I know what work’s best” for my students. However, as is apparent in Boaler’s study, this is not always true. It is important to reflect, determine what “works” for students and what approaches to teaching are not as successful as others. That being said, it is very important to remember that what students find helpful one year may not be applicable the next. Although I have always considered myself to be aware of what methodologies are successful for students, I am continually exploring approaches that may work better. I am both anxious and excited to make use of Phoenix Park teaching strategies when I return to teaching in the upcoming year.

Exploring The Differences

Chapter 7 takes a close examination of the differences which existed between the two groupings of students, of which there were many. These differences can largely be attributed to the environment in which the two groups of students “learned” math. As mentioned previously, the learning environment at Amber Hill was very traditional in that the teacher introduced a particular topic at the beginning of the lesson followed by the completion of text book exercises which reinforced the concept just “taught” by the teacher. It was alarming to discover the huge sense of false understanding this created for the students. Alan, a year 10 student at Amber Hill, stresses this point in his statement “It’s stupid really cause when you’re in the lesson, when you’re doing work-even when it’s hard- you get the odd one or two wrong, but most of them you get right and you think well when I go into the exam I’m gonna get most of them right, cause you got all your chapters right. But you don’t.”

I wonder how many of us are victims of creating/having this false sense of knowledge in our teaching and learning experiences. It is no wonder that the children thought they could do the math, why would they think any otherwise when each class was conducted in exactly the same manner as the last- the teacher spends the better part of an hour introducing an isolated concept and the steps that have to be followed when presented with this particular type of problem. The students are then drilled to the point where every question is the exact same as the previous (which is the exact same as the examples given by the instructor), with the exception of the numbers present. When students did stumble upon a “problem” which they did not understand, students immediately looked to the teacher for help rather than thinking about the question. Assessment situations were no different from that of the classroom. Examinations were comprised of questions that were replicas of those found in the textbooks. The fact that students experienced success on the examinations may be attributed to the examinations comprised of isolated concepts which the students had just learned. Once the assessment was written, the mathematical concepts “learned” were forgotten.

Chapter 7 in Boaler’s study stresses, yet again, the importance of moving away from the more traditional methods of teaching as they simply aren’t working for all students. In order for students to experience success with mathematics we (educators) need to encourage them to delve deep, or to a depth that’s of comfort to them, into the subject and creating an understanding and thus knowledge of the subject.