Prove It! Engaging Teachers as Learners to Enhance Conceptual Understanding -- Journal Article

Gathering teachers for a staff development workshop is a great way to get teachers thinking about the way they teach their students. In this article, teachers are asked to prove their answers rather than just telling the answer to a fraction problem. Doing so makes teachers think about the way they ask students to present their answers in their own classroom. Teachers need to learn to teach for understanding and discovery so students can be more comfortable with mathematics.

The activity presented to the teachers asked them to find the answer to a simple fraction problem. When asked how they got the answer they said "I found common denominators and multiplied." The facilitator then asked them to prove how that works; this is where teachers found the problem to be difficult. This opened the activity to other activities all throughout the mathematic spectrum. With these activities teachers realize the most important part of a math problem is discovering how to do it, not the answer to the problem. Learning for understanding is important to teachers also, not just the students. Teachers also need to reflect on their teaching experiences that way they have a better understanding of what to teach and how to teach it. In the end, learning is not a one step process its an entire circuit of questions, answers, and discovery.

Sweetland, J. and Fogarty, M. (2008). Prove it! Engaging teachers as learners to enhance

conceptual understanding. Teaching Children Mathematics 15(2), 68-73.

Reasoning and Proof - The Process Standard

The reasoning and proof standard asks students to be able to do simple things we all really do every day. Asking children questions throughout their mathematical learning is important step in helping them fully understand the problem that is set in front of them. They have to explore deeper and analyze all the evidence. Systematic reasoning is a building block of mathematics and is used at every age. Students learn a lot about mathematics through discovery. Everyone uses informed guessing without even thinking about it and students put it to good use when they apply it to examples and previous knowledge. Teachers can also feed of this guessing by asking students to recall back to their earlier math learning and expand their knowledge to a whole new level.

Once students have figured out the answer to a problem they need to be taught to look even deeper into that problem. Instead of asking for the answer ask how the problem works the way it does and if it always works that way. Given students a reason for math will make it easier on them to understand instead of knowing something happens "just because." As children get older, their arguments and ideas should get stronger. Students should work cooperatively as a class to develop strong mathematical reasoning and a good classroom environment. Lastly, as students get older, they need to be introduced to many types of mathematical forms and situations. With all kinds of different ideas they will become more fluent in learning math and why specific things happen the way they do.

Lessons on Variables - Video Blog

The purpose of the activity in the video was to get students comfortable with using variables. Students had to take variables that they use everyday, their names, and give them a number value. This helps students have a better understanding of what variables do in math and why we use them. The activity also served as a way to keep students asking questions and understanding why we do things rather than just knowing an answer and not being able to explain it. The teacher asked many questions that kept the students guessing and interested in getting the lowest/highest value for their names.

Questions
#1 - Describe how the teacher’s questioning, and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment.
#1 The teachers questioning and handling of responses were great factors in keeping a positive learning environment. When the teacher would ask a question related to the lesson, students would be constantly looking for an answer. She asked questions that were easy for students to understand and kept the lesson tied together. When listening to students answers she constantly gave constructive criticism or asked more questions to keep students moving and never getting bored with the lesson. She also got her students to ask questions and make predictions.

#2 Describe what the teacher does to support learning while students are working in groups.
#2 The teacher is always walking around and focusing on specific students to make sure they are focused and on task. She takes the role on on-looker and feeds on students questions. When one student gets confused she announces the misunderstanding to the entire class in case any other student was making the same mistake. The great thing about the groups was that the teacher almost turned it into a game, so the students were much more willing to participate in the activity.

#3 Describe how effectively the assessment of student understanding was aligned with the mathematical goals of the lesson.
#3 I feel that the assessment was right on track with the goals of the lesson. The best way to assess students in this situation is through communication and listening in on their groups conversations. Students are still learning because they are building off of each others ideas for the assignment. Also, students are comprehension can be easily seen by walking around the classroom.

Overall, I really enjoyed this lesson. I have never heard of learning variables like this and I think it is a great way to help students understand them. Having the variables tie to their name helps the students relate the lesson to their own life rather than thinking "we're never going to use this." I also enjoyed that they did this project mostly in groups but had to work individually as well. This gave students a reason to want to learn because it became a competition for them. Students seem to really enjoy this lesson and I would love to give it a shot in my own classroom someday.

Technology Rich Mathematics Instruction - Thach, K. J. and Norman, K. A

Thach and Norman reiterate the Principle of technology in Technology Rich Mathematics Instruction (2008). Society has greatly changed with the advances technology has given us. Instead of learning how to use technology, students are learning how to further develop their skills through technology. Through the use of computers, students are able to represent concepts and manipulate variables and objects that are difficult to do on paper. This has helped students become more interested in mathematics and ask their own questions rather than answering them out of a workbook.

In this article, Thach and Norman explain a technology based activity they use in their own classroom. To help upper elementary students better understand credit, they use algebra and number operations on electronic spreadsheets. Students are given a financial situation that they can relate to. They try to understand the affects of variables and learn the appropriate tools in solving these problems. Students use calculators to compute information and get to present their data using SMART boards. With an activity like this, students are more interested in learning and can have fun experiencing new technologies.

Thach, K. J. and Norman, K. A. (2008). Technology-rich mathematics instruction. Teaching

Children Mathematics 15 (3), 152-158.

Principle - Technology

Technology surrounds us everyday. Teachers need to take advantage of these technologies and utilize them in their classrooms. Mathematics is a vast subject that can be explored deeper and more fluently with the use of computers and state of the art calculators. Technology will only benefit our future in mathematics with its efficiency and varies ways of teaching. There are many examples of newer technology that can be used at any grade level. From graphing to working with simulations and manipulatives, math has become something that students can become more invloved in at an earlier age. With advances in technology, students are stretching their minds and creativity more than ever.

Teachers also have a big role in the use of technology. Technology never takes over, teachers still have to plan lessons in order for the technology to get used. Teachers are the deciding factor on when and how technology will be used in their classroom. They have also greatly benefited from technology. Now, teachers are able to look deeper into assessing students; rather than grading an answer they can see how students came to that conclusion. Thanks to technology, all parts of mathematics can be covered and in greater detail, not just the basics.