Gabriels Final Post

I wanted to focus my final post on the article that dealt with assessments. I thought it fit perfectly because assessments normally come at the end of units, and this being our final assessment on the blog post, I decided to end it with this. I really liked how the article actually included math problems/ assessments that they were describing. I was able to understand better the assessment because I was able to visually see the math. One main point that I thought was the most important was the fact that there are many different ways to solve a problem, which shows the different answers we could see as an instructor. There are many different methods in solving math problems, and that’s beneficial for students, espically when the teacher talks about the different methods. Allowing students to show their work and methods gives the teacher a better understand of how students are learning and understanding the material; and when the teacher better understands, they are better able to teach. I think the article was just trying to get across that students will have different ways of solving answers, and providing an assessment in which they can explain their methods and solutions is best. I have had math teachers that had math exams in scantron form; which was not beneficial for me as a student at all. This is clearly easier for the teacher, which is why they assess like this, but what does this say to the student. To me, it says that methods don’t matter, its only the solution that counts; which is 100% not true, and does not send students the right message. My CT does not spend a lot of time on math, which is a bit disappointing. She uses the calendar as math, which makes sense, but I don’t think its enough. I have yet to see students write down any math. Lauren and Gabe, do you see your CT providing assessments in math that allow students to explain and justify their answers and understanding? If you please explain,.

4-20

Reading over these articles reminded me a lot of what we worked on in class last week. The equation shown on the first page of Baroody and Bartel’s article “5-3=3-5” reminded me of the problem we worked on last week. There were discrepancies between 9+3=12+3=15. When doing fast work, I would think this is fine… but when teaching, we need to be very careful with how we represent this… because 9+3 does not equal 12+3…. But if you are simply adding several numbers together, it saves time and it is more convenient. I need to remember when teaching to do the full problems, with all the details, and not leave anything out because my future students could possibly become confused and learn these basics wrong.

“The degree of a student’s understanding is determined by the number, accuracy, and strength of connections.” (Hiebert and Carpenter 1992) This article discusses linking the aspects of knowledge that are accurate and strong to comprehension of the problem. Making connections is discussed as understandings between two pieces of information. Math doesn’t need to be simply equations applied to the problems on one worksheet, and applied or thought of in other circumstances.

Some suggestions they gave to help the students make connections with math so that they use it in other circumstances rather than just one time on a test is through concept maps. This helps them to see when the concept can be used and what it applies to, rather than simply knowing the equation, but not

Math in America is taught poorly in the sense that students have received equations and only use them during that time. What I mean is that students are usually not able to use their background knowledge on equations they have already learned, and other cases they can apply them too. They simply memorize the equation without understanding the complexity of it, and scenarios when it is used. They memorize the equation for that test or exam, and do not use it again. As a teacher we need to teach them equations and help them make a connection on which scenarios to apply it to in everyday life. Taking math outside the classroom….. Giving them math facts in lessons that incorporate when to use it, other than just in the book. I liked the concept map on fractions and decimals, and when to use it. It helped students choose when it is time to use each, and scenarios where is more beneficial for them to use either one. Students are not only memorizing math terms, but also learning when to use them. That way students will use mathematical terms/equations they have learned in the past and are able to still use them today.

I am surprised that this article only discussed concept maps as a tool to helping the student differentiate real life scenarios on which equation to use. I liked the concept, but I would like to see more examples of how I can incorporate this into my lesson plans to help my students understand the complexity of the equation, and not just memorizing what numbers to plug in, and where. Rose and Lauren, do you have any ideas on how we can help students to remember the equations to apply to everyday life?

Cohen 4-14

I found the Cohen article on groupwork to be very intersting, eye-opening, and true! I have personally experienced as a students and seen as a future teacher in my placements so many of the things Cohen listed as problems seen in groupwork. I found the studies on status order to be very interrelated to these problems in groupwork. For example. you almost never come across a group of four students that are each participating exactly 1/4 each. No matter what the age or subject area, there is always one student dominating and one or two students being totally excluded and forgotten about. The article points out that the dominate student more times than not is percieved by his/her peers and the teacher as a strong reader and/or good student. The article points out this is even the case for non-academic groupwork too, such as the game in the study Shoot the Moon. "It is clear that percieved academic or intellectual ability is, whether it is actually relevant to the task or not, has the power to affect both participation and influence in small groups of students." I would argue this is true for all ages in any profession or instance where people have group meetings. In addition to this academic status, peer status plays a huge role in classroom discussion and groupwork. Students develop peer status for one another as they interact everyday in and outside of the classroom walls. Among other things, peer status can depend on popularity, athletic competence, or attractiveness. Those who are ranked highest on the peer status spectrum are also more likely to dominate class discussions and groupwork. Moreover, societal status plays a large part too. For example high class people will dominate conversations more than low class people, whites will dominate over minorities in conversations, and males will dominate over females in conversations. This is not to say this always happens, however, it is a very likely trend we see in Western societies. So, how od we change this all? Here are some ideas, and Rose and Gabe I would love to hear what you think! I liked the multiple ability strategy which alters the expectations of students, so the teacher has created a mixed set of expectations for everyone. Students are all supposed to show their strengths and weaknesses that they bring to the new task. This helps bridge the gap of competence between low and high status students.

Helping students with disabilities

I decided to read the article helping students with disabilities understand what mathematics means by Miller and Hudson. The article goes in depth about “five evidence based guidelines for implementing math instruction designed to promote conceptual understanding.” The article set up the guidelines in a very organized manner, which made it easy for me to understand and really relate what I was reading to my classroom. The first guideline is to “use various modes of representation” what this basically means is for instructions to present the material in different forms and formats for different various learning styles to be able to comprehend. The second guideline is “consider appropriate structures for teaching specific concepts.” This guideline explains that teachers should carefully plan and think critically about the way they structure their lessons. The structure of the lesson really has a large affect on how the information is presented and internalized by the students. It talked about how even though pictures, graphs, and diagrams are beneficial for the classroom, it’s also important that teachers know how to properly use these tools. Guideline three is “consider the language of mathematics.” In math I never really related it to language; however, they brought up excellent points about how important language is in math, and simply saying an equation wrong, or not explain methods in a clear way can cause such major long lasting problems for students. The fourth guideline is “integrate real-world application” this is one that I really liked and value. I think it’s very important to relate not only math, but all subject to real world situations. I remember sitting in high school and thinking to myself why am I learning this, what the point. I don’t even want my students to have those thoughts in my classroom. The last guideline is “provide explicit instruction.” This guideline “requires carefully designed lessons with clear and explicit teacher instruction.” This guideline “emphasizes teacher demonstrations, maximizing the students engagement and participation.” I think all these guidelines are beneficial and we as future teachers need to take these into consideration while we plan and teach our lessons. I see a lot of planning from my CT and believe she really does try to incorporate these guidelines into her teaching. What about you guys? Do you see your CT’s use any of these guidelines?

TTLP

Sometimes students can be confused by a lesson that is a continuation of a subject that they have already learned. They may not make a connection because it is not meaningful to them. I liked the quote from Smith, Bill, and Hughes that stated: “When this occurs, students must apply previously learned rules and procedures with no connection to meaning or understanding, and the opportunities for thinking and reasoning are lost.” From this quote, I understand it as that students will solve a problem step by step from what they learned by the teacher, but they do not actually understand what they  are doing, and how to apply it to other problems. As future teachers, we need to find a way to connect the meaning and understanding.

I liked the example they used for fractions. They determined the number of red marbles, and then blue marbles, turning them into fractions. It helped them to show that they are taking a portion out of the whole. I believe the students will be able to add this and apply this to everyday life.

The ttlp is a strategy to use the students’ mathematical thinking critically. Connecting their understanding to critical thinking. It is divided into three sections, 1. Selecting and setting up a mathematical task. 2. Supporting students’ exploration of the task, and 3. Sharing and discussing the task. These three sections of the TTLP help to make the mathematical activities more meaningful so the students can apply it to anything they need, not just in a worksheet. Before giving your lesson, evaluate it. Ask yourself, what ways does the task I’m about to teach build on student’s previous knowledge, life experiences, and culture? What methods will the students be able to use outside of class… what errors may they make? … what misconceptions could they have? Asking yourself these questions before hand will help you to better prepare for this so you can be more accommodating when it does happen. When giving a lesson, you need to be flexible. Not every lesson will go how you had planned. It’s almost better if it doesn’t, because that means that the students have become more involved and engaged, and letting them take the lead is an important part of being a flexible teacher.

Part two of the ttlp is ensuring that your students are on task. How can you ensure they are, and continue guiding them. Plan ahead by making the activities meaningful and entertaining.

The last part of the ttlp is sharing and discussing the task. How will you assess their understandings and review the main events. Do their answers display that they made sense of the mathematical ideas you wanted them to learn, etc.

The ttlp to me is a task for teachers, with step by step ideas on how to make your lesson more effective, and keeping the students engaged. I wonder if my CT knows about this. Yesterday in her math lesson I noticed that the students were not engaged, and bored by her repetitive approaches. I wonder if I showed her this TTLP, if it would help her realize that maybe she needs to adjust her ways so that her students are more interested and therefore more engaged.

Lauren and Rose, when looking at the TTLP graph , does it appear that your CT teaches in this manner. Does she have the students’ interest as she teaches? Is it meaningful to them, or are the students bored?

More fractions....

"Fractions both underpin the development of proportional reasoning and are important for future mathematics study, including that of algebra and probability. However, it is clear that many teachers find fractions difficult to understand and teach and many students find them difficult to learn" (Clarke, Roche, and Mitchell p. 373). Rose and Gabe, correct me if I am wrong, but I think it is clear to all of us that fractions are a difficult concept to teach and learn. The thing we are not so familiar with, is how to use different tactics to teach fractions more effectively. I thought the article gave great tips on how to make fractions more readily understandable for students. I also really liked the student's homework example on page 374 when the task asked Darcy to draw or write about 3/4 in as many ways as you can. She was able to come up with nine different examples, which seems to be really great to me. Some of the tips I thought were most helpful were the following: "give a greater emphasis to meaning of fractions than on procedures for manipulating them, emphasize that fractions are numbers making extensive use of number lines in representing fractions and decimals, and lastly provide a variety of models to represent fractions" (Clarke, Roche, and Mitchell p. 375). Gabe and Rose, what things did you find most interesting out of these three articles?

Rational Numbers!

"One topic that is particularly difficult for students is fractions" (Bezuk and Bieck). I find this to be a very true statement in my own experience. Rose and Gabe do you agree? I really liked the three ideas these authors have on making fractions easier to learn for students. They are: "a. instruction should be meaning oriented rather than symbol oriented, b. instead of delivering knowledge in prepackaged form, instruction should encourage students to construct their own knowledge, and c. instruction should provide students with structured learning experiences to help them acquire essential conceptual and procedural knowledge" (Bezuk and Bieck). It is important to teach children that rational numbers are ratios. In order to teach rational numbers more effectively, I believe we as teachers need to rely less on the conventional abstract math symbols, and rely more on using concrete tools. For example, all the materials listed on page 124 in this article seem much more helpful than a simple fraction sign. Another thing I found interesting in this article is to use word names such as one half before using symbol names, which seems like common sense to me, but I did not think of this before. Rose and Gabe, what things did you find interesting in these articles? Have you seen any rational number instruction in your placement? I have not yet...

Week 6 Lesson Study

Lesson study helps teachers to form long term goals for student learning and development, plan, conduct and observe a research lesson an designed to bring these long term goals to life. The lesson studies help teachers to have opportunities to observe student learning, engagement and the behavior during the lesson. Balancing teaching the lesson while also assessing each student is a difficult task, but important. During the lesson study process, the teachers are given opportunities to reflect on the teaching process and also the student learning. teachers need to learn how to balance these two aspects of teaching in order to make their lesson beneficial and meaningful. Lesson study began in Japanese schools a very long time ago. It helped their teachers to make sense of the educational ideas within their practice, change their perspectives and help them adapt to teaching more accommodating to their students' learning skills, learn from children' perspective, and lastly, collaborate among other colleagues. My TE friends have definitely discussed staying in touch in the future to share our stories, ideas and help each other with lesson plans. It is definitely beneficial to help other teachers with issues or give ideas for their classroom, as well as receive them! I really appreciated this article, talking about lesson study, but it is also something that should be required for teachers. They should have to work with other teachers and brainstorm and bounce ideas off each other in order to have better lesson plans. Everyone knows two brains are better than one. Why wouldn't we practice this?! Lesson studying with other teachers and colleagues will benefit you because you will also learn new teaching forms, if not new, different. In order to create a lesson study group, you can make it informal, amongst teacher friends, but as long as you are discussing and learning and adapting new ideas for lesson plans, I think it will be beneficial for your students. It helps putting more thought into it, and the lessons will be more successful. I do not see any of this happen at my placement school. The teachers during lunch in the lounge talk and discuss students, and ways to help them, but I have never seen them critique eachothers' lesson plans. Rosa and Lauranda, do you notice this at your school?

Developing Measurements

I found this week's reading to correspond very well with last week’s reading. Both articles dealt with measurements in the classroom; however this week’s had more to do with developing measurements. Van De Walle talked a lot about different kinds of units that can be measured; both standard and nonstandard. While reading I began to wonder which type of measurement was more beneficial early on in a child’s development (standard or nonstandard). Part of me things it would be difficult to teach very young students what a pound is, or how much volume something has, whereas teaching them its about as heavy as a brick would be more concrete in their mind and would allow students to relate their learning to the real world. On the other hand, I think it’s always beneficial for students to learn vocabulary and terms from a young age. In my classroom I see more standard units of measurement when teaching math, however I’m not fully sure if all the students are understanding the concepts. I believe this comprehension should be assessed more by my CT.
Another area of measuring Van De Walle brought up was estimating. I thought this went really nicely with what we have been doing in class these past few weeks, and especially last week. I think estimating can be a very effective tool, but only when then initial concept has been mastered. I think it will do more harm teaching students how to estimate before they are 100% confident in their ability to solve the problem in full. Some students may use estimating as crutch, and never learn how to fully solve a problem and get the exact answer. I have yet to see any estimating strategies in my kindergarten class and think it is for the best that they have not learned it yet. Some students are beginning to learn very basic addition problems, and I think they will need to develop addition and subtraction skills before they learn how to estimate.

Navigating through Measurement

The article I decided to focus my post on this week is the Navigating through measurements because it particularly pertains to grades 3-5, which are the grades I hope to one day teach. The article started off by saying “Measurement is one of the most fundamental of all mathematical processes” and I couldn’t agree more. I see this a lot in my placement, being in a kindergarten classroom I see the majority of math time being spent on simple counting routines. Every day the go over the calendar and count how many days they have been in school. They represent this number using money (dollars and cents) along with fish (because they are the “fish room”) and with sticks that they count and separate out by hundreds, tens, and ones. One thing I found interesting while reading was “instruction during grades 3-5 places more emphasis on developing familiarity with standard units in both customary (English) and metric systems.” I know I am in kindergarten and perhaps that’s why I don’t see any instruction on the metric system, but I was wondering if Laruen or Gabrielle have seen in being taught in their placements, and if so how important or how much of a role does it have? I think we should start teaching students the metric system for an early age, perhaps even earlier then 3rd grade, what do you guys think? The article also mentioned “children in prekindergarten through grade 2 should have similar hands on experiences to lay a foundation for other measurements concepts.” I agree with this statement and see it in my placement. There is a center during math time that has a large scale and the students are able to make hypothesis on what they believe will be heavier. I thought this was a really center for students to do, because it is a more fun approach to learning about mass and weight without dealing with such large concepts. What evidence of this do you guys see in your placements?

Mistakes

In reading student mistakes, I found it very interesting that students start to work with place values in or before kindergarten and continue to work on this idea through third grade, and sometimes beyond. This is a complicated idea that many students have trouble with and it is important to help them understand this concept by moving from concrete to abstract examples. There are many students in my placement that are having much difficulty with place value. They are in second grade, and it is frustrating for both them and me to learn/teach them the essential ideas one needs to know about the importance of place value.
I also really enjoyed reading the Value of Mistakes article. The authors say that "mistakes are catalysts for learning, or springboards for learning." I really enjoy both of these sayings, because it is very true that mistakes are very imperative to learning. I liked how Mrs. Phillips changed her practices from warning her students about possible future mistakes to giving them a solved problem with mistakes and asked them what was incorrect and to solve the problem correctly. "Rather than warn students about common mistakes as she has in the past, her new approach allows students to value the work of correcting errors by accepting ownership of the common mistakes and the reasoning processes used to overcome them." "Teachers should allow students to work through erroneous thinking as they would in a problem solving situation, that is, with minimal learning guidance." I learned this last statement is very important, because if no mistakes are made then no problem solving is taking place. We need to never ridicule nor allow other students to make fun of their peers for making mistakes. We need to let them know their mistakes are imperative to everyone else's learning. Rose and Gabe, what do you think about making mistakes? What do you see in your placement's that either enforce or go against what we've read and learned about making mistakes?

Week 4 Math

Within the articles we read, they discussed accommodating all students in different mathematical activities, by giving problems with many different solutions and teach the many different approaches on how to solve a problem. This helps the less talented students participate in the same task with the high talented students. Teachers are able to support high achievers, and also accommodate students who are a bit behind. The lesson plan is incorporating all talents. Open ended answers are great examples because students can write their responses to the story problems on however they solved it. There are many different ways of solving a problem, and it’s their preference, and what they feel comfortable with. They will still have confidence with themselves if they were able to get the same answer as the other high achieving students in their math class. Maybe once they are more comfortable with the problem they will be able to learn and observe their peers’ solutions.

I enjoyed the actual examples of the different solutions different students came up with. They all came up with the same answer, but getting there was different amongst their preferences. If they are taking the problem and solving it in their own way, you know that they are grasping the concept since they made it their own way of solving it. I noticed a pair of students working on an algebra problem in a picture, and I believe partners in assignments with this approach of open ended questions is wonderful because they can learn off each others’ solutions.

I liked the open ended form example in the Kabiri & Smith article. It asked the student to draw two different congruent right triangles. There are many different triangle shapes that contain these traits, so allowing them to draw their own helps those to understand the concept better. Open ended question makes the problems more authentic and meaningful to the students.

In my placement, I do not see any of these open ended questions and discussions in the math portion. She gives a lecture, allows questions, but they are not working together solving it in different ways. There are yes and no questions, and only one way to solve the problem. They work on worksheets and do a lot of fill in the blanks. It seems that majority of the students do a good job answering these worksheets, but I can tell that many of the students are so bored. This is not good that they are already bored with the repetitive questions at such a young age! This could be changed by giving fun story problems (maybe some that even use their names), more open ended questions, and group discussions or math activities. I believe my CT does do a good job accommodating all of the students, but I am sure she would be interested in reading these articles on ideas on accommodating the students that are struggling.

Week 3 Math

The reading I found most beneficial was definitely the student interview jigsaw choice. Since we are interviewing our students this week, I took special note of how they went about it, and how they got the most out of the students’ answers.  Action research is defined as the process of asking a worthwhile research question, collecting credible evidence to answer the question, and using the evidence to guide further improvement in a school. At the elementary school, Jefferson, in Oregon, the teachers were having issues with problem solving in their math classrooms. They found it difficult to teach and noted that the children were having difficulties grasping the concept. This was because the teachers did not have any knowledge of the student’s individual mathematical learning and understanding. The classes were too large and each student was different. I fear for this because how am I supposed to find enough time in 1 day to incorporate each student’s learning strategy in 1 lesson plan, especially in a larger classroom. I know it will be a bit of a challenge, but in the article they found a solution when they interviewed each student on which mathematical style of learning best suited them, and making teachers more aware of what individual children knew and what tasks they could perform with their knowledge. With that information the teachers began to increase their focus on meeting the needs of the individual students. Teachers met each student’s level by identifying who was able to move on the next level, and who needed more time. The teachers created their own story problems that best included the student’s in the classroom. The problems were more meaningful to the students, and they had an easier time understanding it. The teachers also made the classroom more of a discussion rather than one lecture. This helped the students learn the many different ways of solving a problem when they shared with their peers how they solved it, and witnessed the different ways the other students found solutions. Working together helped the students regain their confidence in math, and it was more interesting to them once they started to see results. The one on one time with the teacher was very beneficial to the students because it is rare to get that special time with one student. The teachers received immediate and specific feedback from each student, making the teachers more aware of what they needed to incorporate in the lessons to better accommodate all students. In my future teaching class, I will definitely assess each student one on one to see which ways they learn best, and incorporate each learning method in my lesson, showing multiple ways to solve the problem, giving the student options to find which one they prefer and understand the most. I believe its also a good idea to know which students learn best in which ways so you can match them up in the same groups or even different groups, so they can learn other styles from each other. Depending on what your goal is for that lesson, knowing who your students are will help you in the long run. The students will get the most out of your lessons if you can direct it at each of their different styles at once. It’s a challenge, but as long as you know your students, it’s worth it!