Math in the Community

This is a picture of Cloud Gate at Millennium Park in Chicago, Illinois. Cloud Gate is better known as “the bean.”

Mathematics related:

·         Determine the surface area of the bean

·         Determine how you would find the volume of the bean

·         The distance of the image in the reflection is distorted due to its shape

o   Find true distance

o   Explore why the reflection is distorted

·         Many math problems can be derived from tourism

o   Prices of tickets

o   How many tourists per hour, day, week, month, year, etc.

o   Compare and contrast to environmental conditions (i.e. weather)

This is the Jay Pritzker Pavilion in Millennium Park, located in Chicago, Illinois. This pavilion is the location of many music festivals throughout the summer including the Downtown Sound, Loops and Variations, and Made in Chicago: World Class Jazz concert series.

Mathematics related:

·         Scheduling

·         Measuring time of shows

·         Measuring dimensions of stage and the lawn where the audience resides

·         Audience

o   Price to park

o   Number of people (i.e. demographics, capacity, etc.)

·         Food brought to the concert

o   Price

o   Fraction of people splitting the food

o   Number of people who brought food (probability)

This is a picture of a Cubs game at Wrigley Field Stadium in Chicago, Illinois.   

Mathematics related:

·         Prices

o   Tickets, transportation, food, etc.

·         The Cubs haven’t won a world series in over 100 years

o   How many games have they played in that time

o   Compare stats across the years

·         Fans

o   How many fans attend games every year

o   How many people fit in the stadium

·         Baseball field

o   Geometry-related with baseball diamond

o   Dimensions of field

o   Measurement of distance around bases

o   Speed of baseball thrown

o   Distance for home run

A popular question in Chicago is how long is too long to wait for the Cubs to win another World Series? It’s been 108 years and William and Lawrence have been die-hard Cubs fans their entire lives without seeing a World Series win. William is 89 years old and Lawrence is 78 years old. They both vowed to stop supporting the Cubs after attending their 2000^th game without another World Series win. Lawrence goes to 3 more games a year than William. How many games a year does each fan go to and what age will each of them be when they attend 2000 games?

The students in a Chicago Public School have always wanted to go to a Cubs game but they have never had the funds to do so. The 5^th grade teachers thought it would be a great idea to do a fundraiser to help students raise money for an end of the year field trip to a Cubs game. The students would sell school spirit items to raise money.

The items available for sale are:

·         T-shirts - $12

·         Home good crafts -$8

·         Pins- $1

·         Baked goods -$3

·         Posters - $2

The tickets for the Cubs games cost:

·         Upper tier: $20

·         Middle tier: $50

·         Lower tier: 150

·         Bleachers: $35

Each student was responsible for selling their own items to contribute to the class collection. There are 25 students in the class. Which items and how many of them could the students sell in order to help the class raise money for tickets? The goal is for the students to sit in the middle tier. 

Real Life Tasks

In the picture of the bird in the persons hand I will explain the mathematics obviously when I write about the task I will create for the picture. In the picture of the squirrel I would want to use a task that would focus on measurement or area. In the photo students could look at the size of the squirrel compared to what they estimate the tree size is. I like this photo because not only can you focus on measurement but also area. Maybe I could create a task that asks how much area they estimate the squirrel is taking up compared to the whole area of woodchips in the picture. Finally the picture of my family’s car in my own community I would focus on using measurement and comparison again like the squirrel picture. Here I would ask tasks about comparing the height of the traffic light pole to the height of the car, etc. In order to make these open ended I think I would want to have the students choose what to measure and compare and have them decided HOW and with what TOOLS they are going to use to go about this task.

I am going to make two tasks based on the photo of the bird eating seeds from the person’s hand in the forest. For my first higher level task I would ask the students to “pick two things in the picture and measure /compare the length of those two things to each other using what ever tool they want.” I am hoping that even if they use non-standard units the students would be able to start comparing objects and the see the differences in length between different objects of different size. The second task I would create would ask the students to “consider counting how many trees there are in the background of this picture? How would you go about this? Count ALL the trees in the picture? Estimate? If you estimated how would you go about estimating the amount?” My hope is that students would try to only count one section of the trees and then apply that number to the rest of the sections of trees. This is the most common way to estimate so I feel that it would be a common strategy for the students.

Performace Assessments

I really enjoyed the readings because they gave me insight on how to asses students differently from the traditional test and quizzes that are often seen in the classroom. Back in grade school, I was never the student who scored 100 on math exams because I was not a good test taker. Most times I would understand the material that we were being tested on, however I may have forgotten a step or two or used a positive instead of a negative and it cost me the grade that I wanted. Standardize test are also one of those things I did not do well on, either because I was nervous, or didn’t feel it was important because it didn’t affect my grade in the class. Both standardize and traditional test I feel do not always display a students understanding of a skill. After reading the NCTM Assessment book, I feel that the assessment that I would most use are performance-based assessments. This is because students are allowed to display their learning in an authentic manner. Many students do not know that they are being tested when these types of assessments are being done, so I feel that many students may perform better because of that fact. Performance assessments can also give students the opportunity to display learning with out trying to think of the written vocabulary that best describes their learning. With written test or quizzes a correct numerical answer may not always demonstrate that a students understands the problem. In the Stylianou reading it showed that a student could come up with the correct numerical answer and use faulty reasoning and errors in the arithmetic, therefore these assessments are not always the best way to show a students understanding.

One thing that can be difficult for a teacher to asses with these type of assessments is that they have to pay attention to the one key component that they are assessing while so many other things are happening in the learning. With a written or traditional assessment the teacher can identify whether or not a student understands a concept immediately through their correct written response and arithmetic solution, with a performance assessment it does not seem as though students have many opportunities for revision.  For this reason a question that I would pose to my peers is, can a student have the opportunity to revise their thinking in a performance assessment?

When I think of performance assessments I think of group work or students individually demonstrating their knowledge through the use of manipulative. After speaking with my MT, she does not have these opportunities for her students; they are only confined to paper/pencil test. As for my lesson objective in the lesson study project, I feel that a good performance assessment would be for students to verbally explain why they use certain objects to measure with and demonstrate it with the manipulative provided and show why others would not work as well. Seeing that I am focusing on one idea, posing a question would help me to better understand the students learning and reasoning.

Interpreting Math Understanding

I think the main mathematical goal of the lesson in the video is trying to help students recognize and understand the difference between area and perimeter. Based on the problem given, the students are forced to choose the area based on two choices; one of the choices is correct while the other one is a misconception and actually the perimeter of the square. By giving students these two choices, this makes me believe that the teacher wants students to identify that the incorrect answer is actually the perimeter while the correct answer is the area. By having students use the picture to help them determine whether or not they agree with Robbie, she is also pushing the students to recognize what a square unit is and how it is used to find area. Helping students develop this understanding of a square unit will help them make a connection to what area means, furthering their understanding of the difference between area and perimeter.

The debriefing is similar to our group lesson study because the observers will also be looking out for ways in which students solve the task. The goal of the observers was to write down how students interacted with the task and student work. Our observers for lesson study are more focused on a specific research question, which helps the observers focus their attention on specific conversations that students will have while completing the task. Our research question involves identifying what types of teaching from the teacher or methods the students use that help them understand that rectangles can have the same set perimeter yet their areas do not need to be the same. Observers will be listening for how the students come to this understanding, rather than just how they complete the task.

Since I will be teaching the lesson study this week, I paid extra attention to some of the suggestions the Whitenack article suggested. According to the author, instructional programs should enable students to develop and evaluate math arguments and proofs of their work. By providing opportunities for students to develop arguments that support or counter other students’ thinking or my thinking, they will be able to better understand the purpose of the lesson and work towards reaching the learning goal. I hope to gives students instructional opportunity to discuss their ideas so that others can benefit and develop their own math ideas or expand on peers’ ideas. Questioning is a great way to do this because it will help students decide whether they agree or disagree and evaluate on students’ responses to help them understand mathematical concepts.

I think the most challenging part of the lesson study will be to generate questions that the students may ask. This is always a fear before any lesson plan that I’ve taught because it gives you the sense that you may not be prepared to adequately respond to students. By thinking of various answers, questions, or misconceptions that students have, it enables teacher to be more prepared to answer or challenge questions that you may not be prepared for. I am also afraid that I may not have a way to adequately explain my understandings to help the students understand through questions. I know that it is beneficial to challenge students’ thinking in order to develop their understanding, but I am afraid that either me or the student will become so frustrated that I may just begin providing reasoning for students instead of allowing them to complete their discovery.

For me, the most beneficial aspect of this lesson study was to discuss and collaborate with other teacher-education majors. This is important to me because I recognized how well we worked on deriving a lesson plan solely based on conversation. Having a casual discussion about mathematics enabled us to create the task for the lesson plan, obtain ideas for great classroom management ideas, and proved that group work can be extremely helpful to completing any type of task. Our group work shows that all of our members are able to work together to help students develop their mathematical learning. This is extremely beneficial to put on a resume because I feel as though schools look for teachers who can interact with one another to share ideas of beneficial lesson ideas. Being able to get along with co-workers is another aspect that many employers may find valuable. Overall, I have seen the benefit in having a collaborative opportunity to create a lesson plan; our ideas were all shared with one another to try and develop the most effective and motivational way to help students understand that perimeter and area do not directly correlate. 

Teaching Math to ELL Students

This week, I read the Murrey and Wiest articles, in addition to, the Five Practices reading. The Murrey article focused on differentiated instruction in teaching mathematics to English Language Learners. A lot of the methods involved teaching students to be more proficient in the English language, both written and spoken. Wiest also mentioned a lot about how teachers need to incorporate language skills in the classroom for ELL students to perform better. I found it odd for both of the articles to touch on language arts so much. I initially thought it would be less efficient for them to do this, until I realized how much language and vocabulary there was in the subject of mathematics. Never really being a fan of math, I hated thinking about the connection between the two, because I understood literacy. I didn't understand math. When I read these articles, I realized that they are closely connected and that they can both play a major role in an ELL's fluency skills.

    In Wiest's article, she focused on tasks that demonstrated the importance of language skills in mathematics. The first task was called "The Chickens and Pigs Problem", and it involved the teacher giving the students a word problem. What I really liked, was that she used two students' names from the class in the problem (one boy and one girl). With ELL students, it is important to use vocabulary that they understand in combination with unknown vocabulary. This helps them understand context. This relates directly to Murrey's article. Murrey found contextualized instruction to be a major factor in language acquisition. In addition, she believes it is better to teach and define new math vocabulary after students do the lesson or activity, because they are better-able to connect the term with its context.

   In Wiest's second math task, she mentioned that teachers should "have students engage and remain immersed in a single context during a problem-solving session. This way, they can acquire appropriate background knowledge about the context." This is a great variation of how teachers can work on contextualized instruction. Overall, teachers must find various ways of strengthening ELL students' involvement in the classrooms. Both of the articles provided great pedagogical examples of how this can be done.

Fractions & Measurement Connect!

Although we do not have to try to focus on making a connection between the reading we pick from this week and then last week, I wanted to try to make a connection as best I could. The first reading I wanted to focus on is from last week from Siebert & Gaskin’s “Creating, Naming, & Justifying Fractions”. Most of my concentration went into one of the figures in this article, figure 3. This figure stood out to me most for two reasons, connection with my field placement and connection with measurement. In my placement, my students are very stuck on trying to understand the concept of a “whole”. Several students don’t realize that when you have a fraction the denominator equals the number of total parts in the whole. I liked this figure 3 because I feel like this would be a picture or diagram I would bring into the classroom to try to reiterate that concept. I really liked how the authors showed 1/8 in two different ways as well as partitioning the fraction and iterating the fraction. The picture shows area model and set model allows students to see a fraction being used in different types of criteria. In my placement showing this picture would be so key because I think my students need to see how a fraction is not “what is the shaded part of the object OVER how much is left over” like a subtraction problem, they need to see fractions as a whole. This cannot only help them understand wholes but also measurement.

I believe this figure effectively displays the use of fractions in terms of measurement especially when they show the partitioning portion of HOW 1/8 fits into a whole. The figure for the area model displays the notches in the “whole” to show how many 1/8s are in that whole. I believe this can help students understand the concept of measurement and how you have different units of different sizes in different “wholes” or even the same whole. This would be a good figure to show struggling students who cannot understand the concept of measurement maybe due to lack of understanding the concept of a whole. In the set model example we see how you can group this whole into 1/8 groups. This would definitely help students recognize the significance of determining a certain measurement based on the whole quantity.

Next I want to focus on one of this weeks reading by Van DeWalle “Developing Measurement Concepts”. I believe this further establishes the need to make connections between fractions and measurements because it effectively supports students learning of both concepts. Although DeWalle was talking about the integration of science and mathematics curriculums I felt this quote was also relevant to fractions for math. Measurement should be connected with fractions because of “the need for increased precision leads to fractional parts of units.” From my understanding of this point, DeWalle is saying that the importance of measurement is not only important for that subject alone but also can be an educational advantage to incorporate measurement into other subjects like fractions so the students can further benefit from a more integrative lesson.

Math is a Struggle

I was really intrigued by the readings this week since I was a student who struggled with math throughout school. The readings helped verify that struggling is not necessarily a bad thing, it can help you progress even further in the long run. If I had realized this in my earlier years of learning, I may have enjoyed my math experience more.

From doing the readings, I feel as though TTLP is imperative for lesson planning in order to promote understanding from students. By preparing for various outcomes ahead of time, it may be easier to push lessons back in a direction that can help students understand the learning goal. You cannot expect that continuously pushing ideas on a student will help them understand. Instead, it is our job as teachers to take what they already know to further push the students' understanding.

In DMI Ch. 3, many teachers struggled with place value and the questions their students had in terms of providing responses to the questions. By implementing TTLP, teachers could have used what they know of their students' knowledge to anticipate the students' thinking in order to push their understanding. Having questions to help draw out responses that could lead students in the direction of understanding are important for teachers to form ahead of time. Although some teachers did ask great questions, most stories were kind of open-ended so it was difficult to verify if the questions helped the students understand the concept of place values and counting. I observed a similar situation in the 5th grade classroom this week when Mrs. C. was thrown off by the lack of understanding in adding fractions with different denominators. She attempted to use clocks to show students how to complete the task, but many students were extremely confused. Since she realized most of the students were just getting frustrated, Mrs. C. decided it would be beneficial to stop the math lesson for the day and try to come up with a more meaningful way to teach the concept the next day. It was beneficial to cut the lesson short that day and plan something to help the students understand the next day; however, if Mrs. C. had implemented TTLP, she could have continued on with the lesson and used the students' questions to help drive the lesson to help them understand.

My favorite article was the The Value of Mistakes because it shows that students truly learn concepts by making mistakes. If we take this out of the math context, this becomes pretty obvious. Take a relationship for example. All of your friends and family can tell you that someone is not right for you but it takes multiple mistakes and learning how to work through these mistakes to finally make you see that what everyone else was telling you was right. This is parallel to what I have seen in math as a student and observing in the classroom. Regardless of how many times a teacher tries to repeat their method of completing a math problem, some students cannot completely come to terms with understanding if they don't see how and why that concept works. In the 5th grade classroom, the students have been working on reciprocal fractions and lowest terms. The teacher has a saying, "Whatever you do to the top, you do to the bottom." This saying is supposed to help the students remember how to create reciprocal fractions. Most of the students know how to do this but it seems as if they are not aware of why they are doing it and how it has any value to them. Although I know Mrs. C. does have good intentions by trying to find a way for students to remember how to create reciprocal fractions, the students are not able to form complete understanding of the concept because they have not had to struggle through understanding how to do this, which could help them understand why reciprocal fractions are important.

Although it may be difficult to implement TTLP everyday in extensive lesson plans, as one of the teachers said in the article, it's important to have three questions in mind when generating lessons everyday: 1. What are students' misconceptions? 2. How am I going to organize the work? 3. What are my questions? Since I know it will be impossible to create long lesson plans every day, I can at least incorporate these questions when I think about how I'd like to introduce concepts to my future students.