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...