Video Blog #3

The videos on V-shape formations, beams and hair and nails had similarities. Each discussed patterns and formulas they could be created so that one could find the answer to a question with a high number. For example: If you went to the 100th V-pattern how many birds would be flying. Students can use the formula they came up with and tested with smaller numbers to determine this larger number. Connections to Standards of Mathematical Practice: The teacher introduced V-structure by inquiring about geese and their migration pattern in the sky. By relating it to this common site, children could easily use the manipulatives to form the shape with confidence. They also understood the real-life connection to the problem. (This also connects with the Process Standards of connections and representation). The progressive formalization structure was explained by a teacher within the video. Informal, pre-formal and formal stages are how students learn and participate. Informal is using pictures and manipulatives to solve, as in this example. Using these manipulatives, students could reason quantativley by counting and placing birds in a v-shape formation. During the lesson, students worked talked with eachother while constructing their formula, and it was evident that as soon as one student thought they had figured it out, they were defensive and tried to justify their answer so others understood it. This is also connected to the Process Standards of problem solving, reasoning and proof and communication. The teacher asked for participants to use the board manipulatives to model the v-shape and the addition of birds, so students could understand that well enough to then figure out the formula.
As all Process Standards and Standards of Mathematical Practice were apparent in the V-Shape lesson, they were in the beams and hair& nails lesson. In both, the teacher asked for prior knowledge or personal experiences about the topic (connecting and making sense of the problem--beams--asked if students noticed any building being built around town. H&N--asked student with short and student with long hair how long it took to grow etc). Both of these problems didn’t have manipulatives like the V-Shape, ones they could move around, but in beams they had the triangular pattern set-up they could add to and in the hair one they had a ruler to measure. I liked that both problems emphasized estimation because although it can make students feel uneasy that their answer isn’t correct, it comes into play when justifying an answer. Before they estimate they need to ask themselves if that number makes sense based on prior knowledge and other factors.
I saw the connection between the activity we did in our class on 3/30 with the excel spreadsheet and box dimensions. Technology can be introduced and used in the classroom when testing and using formulas to show the structure, accuracy and repetitive aspect of them.

Assessment Activity: Article on and Learning Logs

Combining different content areas into one subject or class can be challenging, but with learning logs, teachers have found that "students reflect on what they are learning and learn while they are reflecting what they are learning". This combination is a beautiful pairing because students are restating what was learned as they practice their writing skills and use of mathematics voabulary. Teachers feel confident about using learning logs for assessing student's knowledge along with assessing their teaching. Using learning logs is a guilt-free way to incorporate writing into the math classroom because the emphasis on math is still present. Using regularly scheduled writing in logs keeps a consistent importance of writing so students and teachers remember the necesity of wrtiting and keeps students in the habit of thinking about math. It is beneficial to see the teacher model reflection because it establishes value and effort. Effort is a large part of these enteries because there isn't necessarily a right or wrong answer, but justification and explaination is vital. Learning logs can be short or longer, reflect a specific assingment or lesson or a longer unit or project. They do not need to take much time and can be looked at brifly or more in depth depending on prompt. They can be prompted or self-reflective. Teachers can respond individually or as a whole and writing or verbally but feedback is important.

Draper, R. J, & McIntosh, M.E. (2001). Using learning logs in mathematics: writing
to learn. Mathematics Teacherq, 94(7), Retrieved from
http://www.pbs.org/teacherline/courses/rdla230/docs/session_3_mcintosh.pdf

March Articles

Professional Development Delivered Right to your Door
Teaching Children Mathematics

The authors of this article, Lynn Breyfogle and Barbara Spotts, write for an audience of pre-service and veteran teachers. For pre-service teachers, these tips and suggestions about professional development will become part of the routine, and for veteran teachers, the authors point out easy ways to incorporate it into an existing routine and emphasizes the importance of becoming a better teacher through some of the following things. They write that regular reflection on lessons, units, and assessments improves a teacher’s awareness of their strengths and weaknesses; they mention that collaborating with other teachers provides a team atmosphere with the common goal of teaching all students and creating the mentality of holding each other accountable for the success of their students. The creation of the “team” eliminates competition, and enhances balance between teacher’s strengths and weakness creating more stability for students among classes. Other avenues of professional development that one could do independently or as a team are reviewing professional articles, using teacher guides, create gallery walks with student’s work exhibited, conduct critiques, communicate and share with teachers a grade above and below yours and set one large and small goal for the year.
Professional Development is important at any stage of a teaching career and for every type of teacher. I am self-motivated and enjoy learning and through the strategies and activities presented in this article, I feel like I have the tools to effectively become a stronger teacher and help others become stronger too. Criticism is easy to dish out, but learning how to make it constructive so that it stays positive is really important. All of this information pertains to teachers of any content area, including math. Examples within the article describe teachers reflecting after lessons in math. They came to the conclusion that they need to leave more time for the students to come up with the answers after saying the problem, and how to ask interactive questions on a higher order of thinking and comprehension, instead of just yes or no.

Breyfogle, L, & Spotts, B. (2011, March). Professional development delivered right to your door. Teaching Children Mathematics, 17(7), Retrieved from http://nctm.org/eresources/view_media.asp?article_id=9648


Taiwanese Arithmetic and Algebra
Mathematics Teaching in the Middle School

The two female authors, Jane-Jane Lo and Feng-Chiu Tsai, dig into the culture of math academics in Taiwan. The information highlighted is valid because of the high success rate of students going through the Taiwan math curriculum. The main point of the article is the transition between arithmetic and algebra and three strategies students use when using arithmetic and algebra in problems. Taiwan middle school-aged students develop problem solving and reasoning abilities, deepen number and symbol sense and promote meaningful connections between arithmetic and algebraic reasoning. Their success comes from reading the problem carefully while thinking about different paths to solve it, and evaluating multiple solution paths of a given item, applying good number and symbol sense, before carrying out the computational steps. “By solving problems both arithmetically and algebraically, students not only develop in-depth understandings of quantitative relationships, but also discover the similarities and differences between arithmetic and algebraic approaches”. In summary, this article reviews the importance of connecting algebra with arithmetic to help students work through problems.
I think we can learn a lot from other cultures and their curriculum, especially from ones that have such a high success rate. If there are ways to use prior knowledge to assist in learning new knowledge there is no reason not to build upon it. I agree with this aspect of their curriculum but I don’t know if I agree with the huge pressure of the Basic Competency Test that evaluates their knowledge and places them in high school. As we have learned, standardized testing is not always an accurate way of assessing knowledge. Saying that, I think it is impressive that students seem to do so well on them, and I think that is directly proportional to their curriculum setup which we could borrow a few ideas from.

Lo, J, & Tsai, F. (2011, March). Taiwanese arithmetic and algebra. Mathematics in the Middle School, 16(7), Retrieved from http://www.nctm.org/eresources/view_media.asp?article_id=9621

Video Blog #2

Connecting this video to the CCSSI Math Practices I saw the following things. To understand the problem at hand, the teacher talked about terminology that would be used as well as a recent review of how to calculate surface area and volume. As a class they went over where to find the base, height and width on the object was so they could make appropriate measurements and calculations. The teacher modeled on an object where these items were so students could visually be on the same page (CCSSI practice--#4).The students were giving a word problem that put them in a certain role with a specific task (process standard--problem solving). They had to create a container that could house 24 blocks. They worked in groups (process standard--communication) to come up with surface area and other measurements. The teacher, at the beginning of the lesson, engaged prior knowledge asking students to remember back to the lesson wrapping boxes with netting (process standard--connections). As students started attempting the problem, individuals among the group had ideas on how to start or what to do and justified them with why they thought it would work, some listened more than others, and some lead with confidence (CCSSI math practice--#3 and process standard--reasoning and proof). As students continued working, the teacher checked on them to keep them structured and supported. Through some of the teachers interviews, it was made clear that through this Connected Mathematics Program, students learn a high level of math in each lesson, but because of how they are learning it (hands-on/inquiry-based) the content sticks with them better and longer, from lesson to lesson and they continue to apply prior knowledge to future discoveries. Students discussed ways in which they got to the surface area and answer and started noticing patterns among each others numbers and the pattern and correlation between surface area, volume and number of blocks (CCSSI practices #7,8).