Manipulative Blog

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.

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.

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.

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

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).

February Articles

A Global Look at Math Instruction
Looking at the strategies and techniques that countries with students achievement in math leading far ahead, is beneficial for the United States so that we can adapt and help our students to progress into efficient math students. Within this focus, the studies and research the authors based this article, on centered on the achievement and understanding of fractions which seems to trouble American students, and even adults. The first comparison among curriculum in Japan and Korea with the United States is when fractions should be taught which seems to be third grade for both Korea and the U.S, although with closer look at their Focal Points and our NCTM curriculum, the whole concept of fractions, decimals and multiplication of fractions are taught earlier in Korea then in the U.S. Multiple areas of the teaching instruction of math is compared between the three countries, in an unbiased yet informative voice. Textbooks, use and choice of manipulatives and examples, time introduced and time spent on, and independent versus dependent learning is compared, leaving the reader with a new understanding about a sliver of math teaching in three cultures.

I was first curious about this article because of the title and the use of the word ‘culture’, since that is something I am constantly interested in learning about. As I started to get into the article I became even more interested because I realized it was comparing the United States with two Asian countries, but then I was corrected because this article is not set out to compare and criticize, but to compare for the purpose of gaining insight on how to best teach the concept of fractions. I was expected to read all about the long school days and years the students in Korea and Japan endure, and the strict structure their school system has adapted but I was happily surprised by this break from the “normal stereotype towards education”. I was also interested to read that we do have some similarities in curriculum with Korea, but that they introduce the entire concept in a short span of time, then we do. The adaption of textbooks to fit curriculum, through the seven times it has been updated, was interesting too. I think we have a lot to learn from countries that excel in mathematics training/teaching, they are our competition and we need to make sure in the future American children have a fair shot for those jobs, industries and academics.

Son, J. (2011, February). A global look at math instruction. Teaching Children
Mathematics, Retrieved from http://nctm.org/eresources/view_media.asp?
article_id=9602


Virtual Data Collection--For Real Understanding

As the push for technology incorporation increases, teachers look for academically challenging, safe and reasonably priced equipment, that still allows for some inquiry-based learning. The various tools examined in this article written by S. Asli Ozgun-Koca and Thomas G. Edwards, become great resources for teachers as they prove beneficial for all grade levels, altering intensity. In an activity given to eighth graders, TI-Nspire (calculators), GeoGebra or Geometer’s Sketchpad (virtual graph paper) were used. Instructions were provided and followed easily but students as they manipulated, tested, and set different radii, circumference and area of circles. They saw how changing the radius affected the circumference and the comparison between the two while also incorporating slope and pi. The virtual graphing devices were not necessary for the activity, student could have drawn out the diagrams, but the objectives for the lesson were not seeing if the students were good artists, but how they interpreted the data, used equations and translated the representations. Using the technology is a fast, effective, visually enhanced way of looking at the work being done by the students while engaging them with hands-on programs that they can use throughout schooling.
I think these programs are great, cost is the one thing I would be concerned with but in an ideal world I would hope each student would have a chance to work with them. I would hope these would not completely take the place of students drawing diagrams themselves, because I think that is a skill in itself that is highly important, but it’s amazing that technology has come this far to produce something like this. It is great visually for students to see instantly the transformation a circle makes when the radius is changed, in contrast to the slower connection it would be if students had to spend time drawing the circle in between each observation.


Ozgun-Koca, S, & Edwards, T. (2011, February). Virtual data collection: for real
understanding. Mathematics Teaching in the Middle School, 16(06), Retrieved
from http://nctm.org/eresources/view_media.asp?article_id=9589

Math Applet #2 How Many Under the Shell

Information: How many under the shell http://illuminations.nctm.org/ActivityDetail.aspx?ID=198
Summary: The menu allows players/students to select how many bubbles the game will start with, or a random number can be chosen. Students then pick if they would like subtraction or addition problems, a mixture of both can be choosen as well. After the initial set-up, a bubble with a number in it appears and covered by the shell. Numbered bubbles are added or taken away from the shell, but are all still hidden by the shell. The equation is in the top right corner for students to refer to, but no manupultives to count. The game repeats itself like this.
Critique: I did not like this math applet as much as the "grouping and grazing" for a few reasons. There were not as many different oppurtunities for learning (only addition and subtraction) and the format was slightly confusing. Not being able to touch the screen and count the bubbles (having them hid under the shell) might be hard for Pre-K and K, although it might have been better for first and second grade, because they could more easily use the equation.

Math Applet #1 Grouping and Grazing

Information: Grouping and Grazing http://illuminations.nctm.org/ActivityDetail.aspx?ID=218
Summary: This virtual manipulative uses the image of cows to teach children counting, place value, subtraction, addition and grouping using single and double-digit numbers. The player has the option of grouping the cows in fives or tens or to add or subtract them. At the end of each round, there is a button that tells the student if they were right or if they need to keep trying. During adding and subtraction the student inputs they number they think is there. A red barn is added to the bottom of the screen each time a player gets the answer correct.
Critique: I played first without looking at the directions, acting as if I were a pre-k or kindergarten aged student who didnt know how to read. It was easy enough for me to figure out through a little trial and error, and although directions should be gone over prior, a child Pre-K through 2nd, could figure it out. I liked that there were four skills to work on through this game allowing practice for those struggling and a challenge for those who are ahead. The motivational and positive red barns after each correct answer was appropraiate as was the box that read "try again", if needed. It had activities that ranged well between Pre-K and 2nd! I am brining this up to my co-teachers at my Preschool for the kids to use during computer time!

Problem Based Learning Part 3

Hawaiian School Carnival: Sixth grade Students work with a budget and fundraising to plan a Hawaiian themed carnival that supports a technology center for the school. The problem the students face is how they will fundraise the rest of the money they need for the event. This problem incorporates different subject areas including math (numbers and operations, algebra, geometry, measurement, probability), language arts, social studies, and fine arts. An overview of the project was laid out along with questions that will help prompt the students and also for the teacher to use to assess and check progress of the kids along the way. There are lesson plans included that introduce and teach about each of the math sections mentioned above. As the students work through the problem, they will be assisted by the lesson taught. Adaptations for children with special needs and challenges for those who need something extra were stated and discussed. Students would be assessed by the provided materials.

Strengths: Good outline of what standards will be met, lessons that will be used to teach students the skills that go along with the problem and I appreciated the inventive solutions to challenges and adaptations. Tying in Hawaiian culture from textbooks, internet and discussions, creating a PR campaign, writing a business letter and being involved in their community through discussion and newspapers is a create way to incorporate language arts and social/emotional standards into this project. Adding art into the project will allow some students to be successful through that area.

Needs Improvement: problem not state entirely and does not clearly state what the goal or problem is for the students. Doesn’t make clear if students are planning, budgeting or fundraising.

Assessment: The rubric was very detailed and well thought out. There was a separate one for the portfolio part and the oral presentation. I think using a rubric is a good form of assessment because it allows students to be more successful. They can receive different points for different areas, so that if one part is bad, the entire grade doesn’t reflect that.

Building A Playground: Sixth grade students will build a playground which will require them to have knowledge and practice skills in the following areas; geometry, special reasoning, numeric thinking, statistical thinking and measurement. The community will be involved whether it is speakers coming in to talk about playgrounds and the town or students interviewing people to get more information.

Needs Improvement: The section on process standards could have been more specific to the problem at hand. There was no assessment created, just what could be done. I did not understand what the importance of the day five lesson was. It was an interesting idea but I thought it was slightly irrelevant and written poorly. At the same time day seven lesson seemed irrelevant as well. Each day was started with guiding questions, but I thought part of problem-based learning was for students to think of these questions when they stumbled upon them and thought of them themselves.

Strengths: The breakdown of each day of the lessons and problem solving time. The incorporations of other subjects in this problem were strong.
Assessment: I think looking at journals and logs is a good assessment but there was not formal way to assess them, where was the rubric or checklist. Without having a rubric, it does not provide clear expectations for the students, they won’t know how they can do well, or where they stand when they turn in their project.

Overall:I felt that the Hawaiian School Carnival was a more engaging problem than the playground for the grade level of kids. Both were relevant but I’m not sure how interested middle school aged students would be in playgrounds. Both problems allowed students to use prior knowledge but also gave them opportunities to learn new information through lesson, self-guided research and team work. In both problem sets, there were positive and challenging questions that would guide students to get to their solution. I thought overall it was in good form and structure of problem-based learning.

Problem Based Learning Part 2

Summary: A teacher saw the love for a book through her students interest in Harry Potter. To make a math lesson more engaging, she linked budgeting, checks and balances, and purchasing. The teacher created a list of materials students at Hogwarts needed or would want for school, along with their prices. She assigned each student a character in the book that had a budget. The problem set forth was to see how they could budget their money to get the most, get the best, get what they wanted for school.

Review: I though this was a great way to study the idea of budgeting. There are many fun ways to do this, grocery lists, allowances, shopping etc but this teacher saw an outstanding and already existing passion her students had and incorporated it. I appreciated that she did not assume each child had read the story before, and therefore created a language arts activity in the room as well, allowing each child to read aloud. Not all student had the same budget, the result of pairing each student with a Hogwarts student, which allowed them to work through their own budget. It was mentioned that students helped each other periodically, but each person still felt individuality this way. Two class periods were used to do this lesson, which I thought was appropriate. This gave enough time for a structured introduction and reflective debrief which created a full-circle experience for the students.

Citation:
Beaton, Tisha. (2004). Harry potter in the mathematics classroom. Mathematics in the Middle School, 10(1), Retrieved from http://www.nctm.org/eresources/view_media.asp?article_id=6672

Problem Based Learning Part 1

I am somewhat familiar with problem-based learning from earlier math education classes, and was involved in a few of them. I remember getting a problem and a back bone of instructions and structure and told to find a solution. At first it was a little overwhelming but as my group and I started to brainstorm we really got into it and almost a little competitive to make our solution better than the others. It was a lot of team work but also some individual work. This is what I walked away from the activity realizing and I am glad to see through reading some of the materials on Problem-based learning that I got the right information from that activity. We talked in class yesterday about challenging your students and letting them struggle a little bit so that they can problem solve and become confident in their abilities to do so. Through these real-life problems there is brainstorming, sketching, crumbling up and throwing away ideas, setbacks and achievements with group and their individual work. They learn to use outside resources, teachers, community, classmates etc. Teachers are involved, but minimally. They are there for support, facilitating and debriefing. One of the documents I looked at, broke down the process and definition of problem-based learning in a very clear, direct and structured way. That document was great for someone who wants to start incorporating problem-based learning to their classroom effectively.

The authors of this article made a blunt statement that there is something missing in today’s math classes; reasoning and proof. In the past this has only been introduced and studied in high school geometry, but it has been noticed that it should be incorporated in math starting in kindergarten. Of course the rationale, support and reason will vary between kindergarten, fifth grade and twelth grade, asking students to understand why they get certain answers in math or set up a graph in that certain way is beneficial. By posing these open ended questions, students must think about the problem at hand, the answer they came up with and why it all matters. This will not engage every student who has a mental block towards math, but it will only help by relating math to real-life and putting it in different terms. This is done in a verbal or written reflection, discussion, question-and-answer, instead of this number plus this number equals this number. As teachers we try to tie subjects together, cross-curriculum. English teachers sometimes struggle with incorporating math, and visa versa. There is a fabulous way now—students can write about, create and distribute a survey, or interview each other to incorporate both language arts and math. Talking more about proving now, the article set forth examples of math problems that show contrasting answers and limitations that students can use to proof why an answer is the way it is. Two terms were defined that are important with this topic. Examples-based justification “means that students justify their generalization by stating that it worked for all the cases they tested” and structurally-based justification “which guarantees a proof”. After students worked through an activity dealing with making boxes out of toothpicks in different configurations, they contrasted the two justifications when debriefing the activity and talking about what worked and why.
Reasoning and proof is so important and needs to be re-implented for two reasons. More specifically for math, if students learn how to analyze and break down the math problems they are working with now, it will come more naturally to them later, and more generally, it is important for students to be able to research, question, test and support ideas, opinions and topics in school and the real world.

Knuth, E. J., Choppin, J. M. and Bieda, K. N. (2009). Proof: Examples and beyond. Mathematics
teaching in the middle school 15(4), 206-211.

Process Standards, Reasoning and Proof

Being able to understand how and why an answer was formed can help a child understand math in a less frustrating way, but reasoning and proof is also more specific to math problems themselves. This skill, of asking how and why and then finding the proof through patterns, structure and regularities in their real-life surroundings, is beneficial in the classroom and for life. It is often taught in the math classroom because patterns, structures and consistency is a normal occurrence within number lines, multiplication facts and elsewhere. Although in math class, these proofs are logical, the skill of reasoning and proving can lead to more creative answers in other paths of life. It is a skill taught over time through any lesson that it can be tied into. Children will not being able to successfully execute this skill after just one lesson, no matter how in depth. Questions such as “what do you think will happen next” and “Is this consistent no matter what” are questions that trigger thoughts in children. Their thought process or investigation is an informed one, or a mathematical conjecture. When open ended questions are posed to students, they may not even realize they are using “reasoning and proof” when answering, sometimes it will come more naturally then other times when a real “investigation” is taking place. When forming their answers, they can disagree with the information stated, agree with it, relate it to real-life, other math, or other content areas. They also may still question things that are not as measurable in the classroom.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

Video Blog

Purpose:The purpose of this set of video was to show the process of a lesson in a classroom with children that resemble those pre-service teachers will be working with. The video links were categorized into the different sections of a lesson including large group time, transitions, small group work and the products. Through these videos, viewers saw the interaction between students and teacher, the benefits of certain strategies and the weaknesses of others. It was not staged so human error on the child and teacher’s side occurred which only created a better learning moment for us as pre-service teachers.

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.

I thought she did a great job of setting the students up for success rather than failure through her questions and discussion. She relayed a sense of support to the students before posing the question so that they felt confident volunteering and comfortable answering. She gave enough time between question and picking for students to formulate answers and rephrased questions if students seemed confused. There was a large amount of respect from the students, while the teacher was talking and they stayed quiet until she picked on someone to answer the questions. No one was ever made fun of for their answer, but rather redirected or encouraged by classmates or the teacher. The teacher, with the help of her students, created a very positive environment for group discussion which is hard, but so important in any classroom.

How do you determine if group work is appropriate and effective?

Before allowing the students to go off into small groups she asked volunteers to explain and repeat the instructions she had given. This was an opportunity for students to ask questions for clarification as well. By doing this, it created a structured transition into small group work and also give the students to start and do their work without having to stop for clarification. I think an obvious sign of group work being unsuccessful is if students are working alone, and not engaging in the group or if the entire group is off task. These things will happen but the work given at group time should be different enough from normal individual work that they stayed somewhat engaged. If repeating the instructions prior to engagement and students stay on task while engaged, this would show appropriateness of material and activity.

Propose a strategy for dealing with one or more of the student mistakes or misconceptions that arose during the lesson.

I think it is important to make sure a child feels comfortable being redirected and is comfortable with constructive criticism. Saying that I think some children would feel more comfortable being one-on-one with a teacher while getting assistance so that they could ask questions, where as some children might feel more comfortable in a group so that they do not feel singled out. There are most probably going to be more than one student who needs extra clarification or help but if a teacher knows their students they can assess the situation and see how to best group the students and help them. Also a student might just need to talk with another student to hear the information on a different level and that can be beneficial to both students.

Overall Reflection: During the videos on reflection I realized something new. When I think of reflecting, self-reflection, on the lesson I did with the class I thought more about how the students reacted, what went well, what could be improved, but I realized it is also important to reflect and make sure the key themes and curriculum ideas were hit. Debriefing with the students after a lesson is good also so that I know what they took away from it and how it can improve in the future. Overall I thought this set of videos was a great breakdown of the different aspects that goes into a lesson and the atmosphere. I liked the breakdown and reflection questions on misconceptions and responding because those are areas in which I could use help from good models.

Preschoolers' Number Sense; Article on Equity

Equity means to be equal, but with average class sizes being well over twenty, finding a group of students that come into the year with equal strengths, and capabilities would be hard. In this particular article the subjects were preschool aged. This means attention spans and abilities weren't comparable to a student in a higher grade. Traditional assessments would not show accurate data among four and five year olds because they wouldn't know how to take a test, or fill in a scantron. Modifying this traditional assessment to fit these needs resulted in what this article was written on. Students sat with an observer and handed the following materials; a bucket of bears and cards with dots or numbers. Goal of the activity; where there were many but for the child, it was their goal to match the number of bears with the number or gathering of dots represented on their cards. The observer guided students to communicated the number of bears they picked up, how many they had in total and how many they had compared to the second player. Students were observed on their abilities to count, compare, identify, use the strategy of quantification, and one-to-one correspondence, counting strategies and cardinality. As said earlier some students will be far ahead, some average and some below but by using this assessment consistently, children are can start out with the correct adaptions, accommodations and support to make it to the finish line. Not to mention this is an activity that disguises math in a fun game, re-constructing the connotation towards math.

Moomaw, S., Carr, V., Boat,M. and Barnett, D. (2010). Preschoolers' number sense. Teaching children mathematics 16 (6), 332-340

The Equity Principle in Mathematics

The term "equal" is a common term to students in a math classroom. Equity, "high expectations and strong support for all students" should and is becoming a common term among teachers. Although this concept should be applied to curriculum throughout, this speaks specifically towards mathematics. As I looked at the three points that stood out to me at the end of my readings, I saw a similarity among them. The three dealt with the treatment of children who are seen as "different". Having a soft spot for students/children who come in with uncontrollable setbacks, I constantly brainstorm how to support and help them. I didn't expect to find some answers in an article about Principle of Math. The article stated that students not only should, but must have the opportunity to study. This does not require all children to study the same way though, and that is important to note. The second point I highlighted related to the prior. Teachers need to provide an environment, materials and resources that can be used to accommodate students coming from different backgrounds, such as low-income families, non-English speakers, or students with disabilities. An interesting point was written about the stereotype towards those who have, for lack of better words, "math-brains"; those who people assume will succeed in math verses those whom it seems impossible. The authors compared this to our push for the importance of high literacy and verbal rates in the English Language. We set a high standard for that, why not for math? Are we setting our students, children up for failure because we don't give all students equal expectations and goals? Just as every child with a set-back is different, students even in a mainstream classroom are different. They learn, listen, think, discover, brainstorm and problem solve differently, and it is up to the teacher to learn and try to understand the strengths and weaknesses of their students to better adapt a subject that has such a negative pre-notion.