Learning and Teaching With Understanding focuses on the issues of teaching mathematics in such a way that would encourage student learning with a conceptual understanding of the material. Different theories of learning are discussed that connect and represent a framework for understanding. Students' internal framework of knowledge is questionable- whether it develops as an interconnecting web or as a hierarchy of knowledge. The main point of the article expands on how this framework (no matter the form it takes) ties into students' innate ability to make relationships and connections to new knowledge and the knowledge they already have. If students can make connections between models, real-life scenarios, personal experiences, pictures, and other representations to mathematical symbols and then explain their reasoning then a deeper understanding is said to have been developed.
Within the article, Hiebert and Carpenter expand on the idea of developing a conceptual understanding. Developing this deeper understanding involves students internalizing the information and integrating into their internal framework so that it cohesively connects to existing knowledge in different ways. Hiebert and Carpenter relate the strength of each connection and relationship to the level of understanding a student retains. So as students make more connections and as these connections become stronger and cohesive, students are able to expand on their knowledge and their knowledge grows.
Recognizing student understanding/ misunderstanding can be done through viewing different works a student produces. Students who have a conceptual understanding of mathematical material will be able to represent this material in different. An example was given in the reading that described a student applying the subtraction operation to numbers with three digits. Using base 10 blocks a student with a deep understanding of the material can represent 403 in terms of the blocks (4-"100"blocks and 3-"single" blocks) and subtract 156 by replacing a "100" block with 10 "tens" blocks, and representing the act of borrowing a number. Not only can the student demonstrate the act of borrowing on manipulatives, but the student would also be able to explain himself verbally and symbolically (using mathematical notation).
Having a conceptual understanding allows the student to prosper in other areas of academia. Students who generate their own ideas and generate their own connections within their internal frameworks can recall information a lot easier. As noted by Hiebert and Carpenter, when students make connections within their own networks of knowledge, they are less likely to lose key bits of information because "an entire network of knowledge is less likely to deteriorate than an isolated piece of information" (p.75). Also, because this higher level of understanding is dependent on a network of knowledge, less information will need to be remembered. Hiebert and Carpenter made the connection between operations of fractions (addition, multiplication, subtraction, division) all incorporated (somewhat) the procedure of finding similar fractions (and simplifying them). If students tie all their fraction operational "rules" to similar fractions, then remembering each "rule" would be a lot easier to recall.
One take-away lesson that I personally found important was the idea of connecting street math to math found in the classroom. All too often I notice teachers and professionals emphasizing the importance of learning math and placing the education system on a pedestal. I think other types of knowledge, knowledge found outside of the classroom, knowledge used on a daily basis and informally is just as important. I hope that students will be able to realize the importance of all knowledge and as a teacher, I hope I will be able to aid them in connecting both types of math (street and classroom math) in a way that would be most useful and efficient for my future students.
I agree that teaching every-day uses of math is really important in math classrooms. However, I might even think it is even more important. Part of the reason we teach math is so that students CAN use it in real life. If math remains abstract and unapplied to real scenarios, students lose their interest and don't benefit from why we teach them math in the first place. I think that is also one of the reasons why it is so important to tie in student's cultures into math; because those are the environments and scenarios that they will be encountering that will involve students to apply their math knowledge.
ReplyDeleteI also find it important to connect street math and and academic math. Students will not find knowledge meaningful or useful if it is not connected to something they are familiar with. Not only will they find it more meaningful, it contributes to their retention of the concepts.
ReplyDeleteGreat point Mariah, as a child I learned mathematics from my family business we owned push carts in the city of Boston. I was very proficient at mental math because for 10 hours a day I was adding multiplying and subtracting costs of hotdogs, drinks and chips. Giving exact change was something I did in my head no paper or pen or calculator needed. As Erin mentioned connecting students culture is vital. Had I recognized the value of what I was doing as part of my education I would of been much more confident in mathematics. How can we make the connection for our students so they value their home life and see the connection to what they are doing in the classroom?
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