Manipulative Blog
When using manipulatives, students have the opportunity to see the problem in different ways. They can analyze the problem and see the different ways of solving it. They can create a more relatable situation out of the problem using manipulatives. Using objects such as pattern blocks show students that the surface area of one shape is the same as a number of the others by comparing the shapes. There is a level of synthesis where students take their existing math knowledge and combine it with the manipulatives to create a solution. While the answers to many of the math questions (in this case) are the same, the process of creating the solution using objects can be an opportunity for creativity. Manipulatives help students justify their answer because they can see a physical object change is some way. Students gain a deeper understanding while using manipulatives because they are thinking in an analytical, synthetic and evaluative way. This knowledge is seen by the teacher through observation, interviewing and prompting questions. As we used manipulatives in class, Dr. Grant walked around the room looking at how we used the objects. She placed the manipulatives in certain ways and prompted us with high order of thinking questions to see if we could apply our knowledge. In one word I would say manipulatives are a way for students to communicate.
I believe that all types of learners benefit from using manipulatives and this is why. Kinesthetic learners can recall how they physically manipulated the objects. By creating the object, sequence or equation with objects, they can recreate it on paper or with available objects. Visual learners will remember using the objects, colors and shapes may help recall the math skill. Auditory learners may recall conversations about justification and problem solving while using them. These strategies will become practiced skills that can be recalled in any problem. Students could perform a task with the manipulatives in front of them and then perform it again without them in front of them to see if they would recall the strategies.
As mentioned before observation, interviewing (during manipulative exercises) and prompting questions/responses can give the teacher a solid idea of the students understanding. The other way to assess if manipulatives are beneficial to a student is to have an assessment with and then without manipulatives (both showing work) and compare work and scores to see the strengths and weaknesses. Students can also draw pictures or write justifications/processes down so that the teacher can see how the students were using them.
To touch on accountability among members in a partnership or team, a good example of shared work is what we did in class, changing recorders. By having each member responsible to write at some point, steers students towards a focus on the work needed to be done along with the engagement of the manipulatives. In a longer problem set, different parts could be delegated to different students, with a regrouping stage at the end. This would put responsibility on each student to work towards finding the solution.
Going off the idea above, if each student were responsible for their own part of the larger problem they could be assessed upon their individual work and the group work in general. Another way to assess students individually is if each student creates a problem with the manipulatives (first part of assessment). Second part of the assessment would be to work on perfecting their activity. The third part would be to solve another student’s activity. The three parts of the assignment would show the depth of understanding because it asks the students to use the manipulatives in at least two different ways. It is one thing to use them to solve a problem but it shows a different understanding if students can use them to create.
By incorporating manipulatives into the curriculum students are looking at problem solving as concepts and relationships and different sections instead of as one confusing problem. They can break down the problem into steps applying rules and strategies to each stage instead of making mistakes throughout.
Sunday, April 24, 2011
Monday, April 18, 2011
Technology Blog
Technology Blog
Technology is used in many aspects of the normal day-to-day activities but using it to benefit students instead of just using it to technology was used is sometimes a challenge. During the entire semester technology was incorporated through assignments, discussions and ended up becoming a necessary device without being instructed to use it. Of course computers and internet were used, but the sources like NCTM and CCSSI provided a variety of resources such as journal articles, standards, objectives, search engines etc. The option for hard copy journal articles was available, but through investigating the site we were comfortable to use it in other projects and classes that it wasn’t required, but useful, for. I think this is a smart way to educate students on appropriate and accurate sources and showing the importance for an immediate, large database of information.
Like in this case, we used an online blogging site to do our reflections on. The only difference of this assignment to any other reflection assignment was the online aspect. It was different, engaging and allowed all students to see each other’s instantly. It does not require any paper to accomplish this task, important to some for environmental and financial reasons. I personally enjoyed creating an individual page and the skills I learned from exploring blogger.com gives me yet another communicative source I can use elsewhere in life.
We talked about two technology-based tools specific to math; Geometer’s Sketchpad and a calculator. It was important to take class time to go over these tools because as teachers we will continue to grow as we use them with our students but we need a basic understanding so that accurate information and training is given to the students in a smooth way. The way we learned about these two tools is a good model as to how I would like to teach it to my students. We were allowed to explore both tools before doing an assignment with them. The assignment was to teach others something we found. For Geometer’s Sketchpad going up to the front and adding something new, showed a wide range of abilities of the program. With the calculator writing a problem and showing how to find it using a calculator let each of my classmates the opportunity to practice the operation and problem solve through struggles. All of the struggles we voiced during the process are bound to arise in any classroom of any aged students. I plan to incorporate technology in my classroom, and work with the materials provided to me by the school to allow students to use it as much as possible.
Technology is used in many aspects of the normal day-to-day activities but using it to benefit students instead of just using it to technology was used is sometimes a challenge. During the entire semester technology was incorporated through assignments, discussions and ended up becoming a necessary device without being instructed to use it. Of course computers and internet were used, but the sources like NCTM and CCSSI provided a variety of resources such as journal articles, standards, objectives, search engines etc. The option for hard copy journal articles was available, but through investigating the site we were comfortable to use it in other projects and classes that it wasn’t required, but useful, for. I think this is a smart way to educate students on appropriate and accurate sources and showing the importance for an immediate, large database of information.
Like in this case, we used an online blogging site to do our reflections on. The only difference of this assignment to any other reflection assignment was the online aspect. It was different, engaging and allowed all students to see each other’s instantly. It does not require any paper to accomplish this task, important to some for environmental and financial reasons. I personally enjoyed creating an individual page and the skills I learned from exploring blogger.com gives me yet another communicative source I can use elsewhere in life.
We talked about two technology-based tools specific to math; Geometer’s Sketchpad and a calculator. It was important to take class time to go over these tools because as teachers we will continue to grow as we use them with our students but we need a basic understanding so that accurate information and training is given to the students in a smooth way. The way we learned about these two tools is a good model as to how I would like to teach it to my students. We were allowed to explore both tools before doing an assignment with them. The assignment was to teach others something we found. For Geometer’s Sketchpad going up to the front and adding something new, showed a wide range of abilities of the program. With the calculator writing a problem and showing how to find it using a calculator let each of my classmates the opportunity to practice the operation and problem solve through struggles. All of the struggles we voiced during the process are bound to arise in any classroom of any aged students. I plan to incorporate technology in my classroom, and work with the materials provided to me by the school to allow students to use it as much as possible.
Friday, April 15, 2011
Error Problems
During the first two class periods when we did errors, I didn't understant the benefit I would get from it. As we went on and discussed within our groups and as a class, I realized taking notes and remembering strategies would benefit me and my students. I struggled with math and alot of the errors made and shown were similiar to what I used to do. I never had a teacher, until my tutor in high school, who could teach it to me in a differerent way. I didn't think it was possible really, and it also scared me to teach it in a different way for fear of what parents, administration of other kids would think/say. I saw the importance of learning/understanding the concepts versus just memorizing the rules, because without proper use, they mean nothing. I would assume that the errors made are common among students and through the different approaches we talked about, I believe they would help. One of the most beneficial parts of this extended activity was working through a problem as if I were the student. I got to see how they approached the problem and how confused but hard they were trying to get the answer. Effort was not lacking in these problems and usually the large concept wasn't lacking either, it was commonly just apply rules correctly that threw them off. The second thing that benfited me was practicing the strategy as a class or individually so we could really understand how to perform/teach it and know when to use it. After these discussions and this activity in total I am that much more confident to teach math and teach it in a way that kids understand for grades to come.
Sunday, April 10, 2011
April Journals
Math Club starting in Kindergarten
This article is about the implementation and reasons behind hosting a math club for students from grades kindergarten through eighth. The purpose of starting so young is to "enrich the classroom mathematics curriculum with hands-on activities and to have members participate in age-appropriate contests" (Perry, 2011). By engaging students with a variety of activities and opportunities to work with students, younger students practice math while getting attention from older students and older students practice math while sharing leadership and mathematic skills. Younger students are paired with older students during at least three out of the four meetings per semester. Those three meetings are after school. The fourth is a parent-involved night meeting. This promotes collaboration among parents and students as well as with the school and education in general. After describing the math club author Ann M. Perry talks about her experience setting up the club. The children are asked to provide supplies needed for the activity along with a snack for afterschool hunger. She found that encouraging children to bring a snack helped them focus throughout the hour. She also mentioned the importance of getting permission from administration and support from teachers so that the program is seen in a positive and empowering effort.
Through my own participation in educational extra-curricular activities and through learning about the benefits, I think starting a math club at an early age is a great idea. It is a way for students to have a different math experience then they do normally. I am not sure if it this article or just a culmination of this class but I am starting to become more confident in incorporating math into my lessons. If I can make it fun for me to incorporate it, hopefully it will be for the students too. There are many students who need a place to go after school and why not do something education four times a semester after school. The commitment for this is appropriate for the age range because they are balancing other activities and for some four time a semester is all the extra math they want to participate in. For students who want more activities on a regular basis I would provide some they could do at home, independently or with parents. I have also experienced and observed pairing younger students with older “study buddies” and I have always heard good things from both perspectives (younger and older) and I would like to find ways to incorporate this.
Perry, A. (2011, April). Math club starting in kindergarten. Teaching Children Mathematics, 17(8),
Retrieved from http://www.nctm.org/eresources/view_media.asp?article_id=9698
Addressing Cultural Bias
This article discusses the various ways to tackle the barriers ELL students face. Students who do not speak English as their first language or have not lived in the United States for a long period of time may be unaware of certain vocabulary specific to American culture. If used in word problems or educational materials, learning can become frustrated and stopped not because they do not understand how to do the math/work but because they do not understand the context. Twenty public school teachers were asked to find five examples from the educational materials they use that are culturally bias. They were asked why they picked those and how they could help students work through them. Teachers responded that they picked those examples because there were areas of potential confusion such as measurement and money terminology, places and certain material objects. When asked how to help students through this frustration, these teachers said to focus on understanding the mathematical concept and defining vocabulary. These techniques were discussed: provide diagrams, pictures, manipulatives, help to revise or rewrite problem using more familiar words, turning everything into a teachable moment, promoting conversation and intentional dialogue. Intentional dialogue is speaking with those familiar with the same cultural context.
These strategies and awareness of cultural barriers is important to all teachers and especially me since I am not sure where I would like to teach yet. There is already a large prominence of culturally diverse students in classrooms and there will continue to be more. I think this goes along with the idea that a teacher cannot assume students are familiar with all contexts and words, culturally diverse or not. It is great to have an ELL program in schools, but if there is not one present I want the best strategies to make learner enjoyable and without extra/unneeded struggles for all children.
Marinak, B, Strickland, M, & Wilburne, J. (2011, April). Addressing cultural bias.
Mathematics in the Middle School, 16(8), Retrieved from http://nctm.org/eresources/view_media.asp?article_id=9683
This article is about the implementation and reasons behind hosting a math club for students from grades kindergarten through eighth. The purpose of starting so young is to "enrich the classroom mathematics curriculum with hands-on activities and to have members participate in age-appropriate contests" (Perry, 2011). By engaging students with a variety of activities and opportunities to work with students, younger students practice math while getting attention from older students and older students practice math while sharing leadership and mathematic skills. Younger students are paired with older students during at least three out of the four meetings per semester. Those three meetings are after school. The fourth is a parent-involved night meeting. This promotes collaboration among parents and students as well as with the school and education in general. After describing the math club author Ann M. Perry talks about her experience setting up the club. The children are asked to provide supplies needed for the activity along with a snack for afterschool hunger. She found that encouraging children to bring a snack helped them focus throughout the hour. She also mentioned the importance of getting permission from administration and support from teachers so that the program is seen in a positive and empowering effort.
Through my own participation in educational extra-curricular activities and through learning about the benefits, I think starting a math club at an early age is a great idea. It is a way for students to have a different math experience then they do normally. I am not sure if it this article or just a culmination of this class but I am starting to become more confident in incorporating math into my lessons. If I can make it fun for me to incorporate it, hopefully it will be for the students too. There are many students who need a place to go after school and why not do something education four times a semester after school. The commitment for this is appropriate for the age range because they are balancing other activities and for some four time a semester is all the extra math they want to participate in. For students who want more activities on a regular basis I would provide some they could do at home, independently or with parents. I have also experienced and observed pairing younger students with older “study buddies” and I have always heard good things from both perspectives (younger and older) and I would like to find ways to incorporate this.
Perry, A. (2011, April). Math club starting in kindergarten. Teaching Children Mathematics, 17(8),
Retrieved from http://www.nctm.org/eresources/view_media.asp?article_id=9698
Addressing Cultural Bias
This article discusses the various ways to tackle the barriers ELL students face. Students who do not speak English as their first language or have not lived in the United States for a long period of time may be unaware of certain vocabulary specific to American culture. If used in word problems or educational materials, learning can become frustrated and stopped not because they do not understand how to do the math/work but because they do not understand the context. Twenty public school teachers were asked to find five examples from the educational materials they use that are culturally bias. They were asked why they picked those and how they could help students work through them. Teachers responded that they picked those examples because there were areas of potential confusion such as measurement and money terminology, places and certain material objects. When asked how to help students through this frustration, these teachers said to focus on understanding the mathematical concept and defining vocabulary. These techniques were discussed: provide diagrams, pictures, manipulatives, help to revise or rewrite problem using more familiar words, turning everything into a teachable moment, promoting conversation and intentional dialogue. Intentional dialogue is speaking with those familiar with the same cultural context.
These strategies and awareness of cultural barriers is important to all teachers and especially me since I am not sure where I would like to teach yet. There is already a large prominence of culturally diverse students in classrooms and there will continue to be more. I think this goes along with the idea that a teacher cannot assume students are familiar with all contexts and words, culturally diverse or not. It is great to have an ELL program in schools, but if there is not one present I want the best strategies to make learner enjoyable and without extra/unneeded struggles for all children.
Marinak, B, Strickland, M, & Wilburne, J. (2011, April). Addressing cultural bias.
Mathematics in the Middle School, 16(8), Retrieved from http://nctm.org/eresources/view_media.asp?article_id=9683
Wednesday, March 30, 2011
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.
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.
Tuesday, March 22, 2011
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
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
Monday, March 7, 2011
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
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
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