Math in Real World Context

Family Road Trip

1. Your family is going on a road trip.   Your family’s vehicle gets 22 miles per gallon of gas.  Looking at the price of Unleaded fuel above, if you have budgeted $200 for gas, how many miles can you go before you run out of gas?  (You can round the nearest whole number)

a.) Look at a map and decide possible places for your destination.  Remember you can only go as far as $200 of gas will get you!  Draw a line from point A (our community) to point B (your destination) on your map [integrating map work].

 

2. You are traveling with four people including yourself and you need snacks for the ride.   You decide to buy sandwiches from QD for a snack, but there are only three sandwiches left (oh no!).  Decide how you can feed all four people the same amount of food with only three sandwiches.  Use a representation of your choice to solve the problem so no one goes hungry!

Math in everyday life

Problem #1. A horse owner has 300 feet of ElectroBraid fencing to make a new pasture for her horses. She wants to make 1 new pasture, what are some different options for the area of pasture that she can make? How can she maximize the area without having to buy more fencing?

Problem #2. A storm knocked out half of a 600 square foot pastures fence. What are some possible amounts of fencing that the owner will need to buy?

Mathematics in the Community

1. You and three friends want to split an order of cheesy bread from Marco's pizza.  The only problem is, the cook forgot to divide up the pieces.  How will you partition the bread to make sure all of you get the same amount?  Show at least three different ways you can cut the bread.




2. You have $12.50 to spend on dinner at Marco's Pizza.  List all of the different items you can afford.

Informal and Formal Assessments

In looking at the information in the NCTM and Stylianou readings for this week I gained a lot of good incite about assessments.  Overall, the NCTM reading summarized different ways of assessing student work including more formal tests and quizzes, as well as informal documentation of observations.  What I liked most about this article is that it gave me brief listings of when an informal or formal means of assessment might be useful, as well as ways those methods of assessment can be improved to maximize the benefits of the student.  It’s no secret that tests and quizzes aren’t student’s most popular choice of assessment, but that comes as part of the requirements of being a teacher.  I really like the suggestion to have students explain their answers in multiple choice or true or false tests so that teachers can have a better idea of what a student knows, rather than what they don’t know.

In Stylianou’s article, the importance of allowing students to explain their answers was also brought up.  The article pointed out that, “Students’ explanations reveal aspects of students’ mathematical understanding and misunderstanding hat are invisible when only the final answer is reported.”   This quote made me think immediately of students who are referred to special education because of their low test scores.  So often teachers who refer that student might think he/she is incapable of doing the work or isn’t smart enough to follow the general education curriculum, when in reality it may just be a misunderstanding with the wording of a problem.  If we allow students to explain their thinking we as teachers are being given an incite into the minds of the students so we know where misunderstandings exist and how we can help clear them up.  “Silver, Kilpatrick, and Schlesinger (1995) commented that by sharing their thoughts and solutions, students not only learn from one another but can also help one another refine their views and improve their own understanding.”  As teachers, we can use student explanations to help other students since we are able to see the type of prior knowledge that we could build from to make connections to a new topic.

            The type of assessments I have been used to in my own schooling are usually formal tests or quizzes at the end of a unit.  In third grade, my teacher often gave us what she called, “Practice Tests,” at the end of every unit.  We would each take a paper and pencil test but we didn’t put our names on the top.  Then, the teacher would collect the tests and pass them out randomly to other students.  We would each have to grade our peers tests and explain why we thought wrong answers were wrong, or why we thought right answers were right.  We put on our name on those tests and my teacher would read our responses to better understand topics that needed to be covered in more detail before the real test.  I had always disliked this as a student because it was so time consuming, but now as a teacher I can see how beneficial this type of assessment was for all of us as students.  

Lesson Study

I think the main mathematical goal of this lesson is helping students to distinguish the difference between area and perimeter through an example with a difference of opinion. The teacher and he helpers are very active in trying to get the students to get them to prove their answer with a picture. This would help students to sort through their thinking and prove what the correct answer is. The debriefing video was similar to what I expected in the respect that each person shared their ideas about what they observed in the class being taught. The difference that I noticed in the video was that the teacher who taught the lesson did not share very much/at all. I would hope that in my lesson study debrief, the students that are teaching would share their thoughts on how the lesson went and how the students reacted to the lesson from their perspective. The Whitenack article states that,  “A teacher might encourage students to explain and justify their ideas during class discussions for many reasons. All students can benefit from these discussions, including the student who is explaining and the others who are participating in the discussion. When asked to explain or justify their thinking, students can revisit their mathematical ideas.” (Whitenack 2002) To me, this means that the teacher really needs to encourage students to explain their thinking behind the decisions that they made when doing a problem. Not only do I think teachers need to understand students thinking, but it will help students realize if their answers are flawed through verbalizing what they did. I think the most challenging part of the lesson study will be to take notes on everything that is being said throughout the lesson. I think this will be hard because some students in a first grade classroom will be speaking very quietly and some will speak very fast. I think the debriefing after the lesson will be the most beneficial part of this lesson study because it will be very nice to be able to collaborate with group members about what went on in the lesson. I know that I have reflected on my lessons on my own, but to have someone else watch the same lesson and then get their feedback about it will be very helpful for my personal learning. 

Differentiation

            The articles I had to read for this weeks differentiation blog included one about gifted students and another that discussed making the switch from higher level suburban schools to lower ses schools.   Both of the articles provide insight as to why differentiation is so pivotal in the classroom.  The dilemma in the first article by Wilkins described how most gifted students are not challenged enough.  Wilkins stated, “The goal of framework is to make it easier for elementary teachers to provide challenging activities for students working above their grade level in mathematics without having to plan a separate every” (2006, p. 70).  I think this a great point because I think a lot of the time teachers don’t know what to do with gifted students, and they are usually just passed along to help out other students that are struggling.  There is also the issue that sometimes gifted students don’t know how explain their thinking because a lot of them time they do everything so fast in their head they can’t keep track of how they figured the problem out.  Another problem Wilkins posed was that gifted students might need extra motivation, so this is why teachers still need to be involved and not leave the gifted students alone (2006, p. 12).  Now in regards to the other article, the teacher in this piece talked about how her methods with the gifted students in her suburban teaching did not work well with her students in the lower ses schools.  Through her teaching, her students complained that they were mostly angry at her for making them feel incompetent, even though she felt like the assignments she was giving had worked so well in the past (Robert, 2002, p. 292).  She also realized that what these students needed the most is simply confidence.  One of the most important lesson I learned from this article is the key to successful lesson include structure, encouragement, and time.  Overall, I thought the discoveries of these two articles were extremely important.  In my placement, there is one student who I think should be considered gifted but she is not given the extra support and challenges that she needs.  However, it is tricky situation because she is usually gone for half of the school week for her competitive gymnastics team.  My MT does not have the time to give her extra support because when she is in the classroom the student has to be focused on catching up with the rest of the class.  I think this case pertains a lot to the parent and teacher connection.  If her parents were to communicate more often when the child is going to be out the class, then perhaps they could work something out that would benefit the student more.

Learning About Measurement

We talked in class today about the trend of poor scoring for US students when it comes to measurement.  In particular, as mentioned in the Thompson and Preston article, Measurement in the Middle Grades, student scores are lowest on constructed response questions that involve students to explain their reasoning.  When I read this I immediately thought of the task level page 116 from the 5 Practices Book that determines what constitutes a lower or higher level task.  If students are unable to explain their thinking, or actually understand the mathematical content behind the measurement concept being tested, does that mean schools are presenting them with more lower level tasks that lack connections or procedure building?  We have spent so much time discussing the benefits of providing students with higher level tasks, but this is actual proof of why it is so important for students to build those connections and come up with procedures on their own.  The article stated that, “Teachers focus on low-level knowledge and skills with little effort to help students develop conceptual understanding or engage in complex problem solving."  In thinking of my own schooling I can remember being given a ruler and a worksheet in which I had to use a specific tool to measure an object.  While that did teach me how to use a ruler, I am not sure what it actually taught me about measurement.
For our lesson study, my group is choosing to focus on Measurement in a Kindergarten class.  We had a hard time coming up with a higher level task when we were talking during today's workshop time, but this article gave me some good incite.  Instead of teaching the students how to use a ruler to measure if something is "shorter" or "longer" I could make the task higher level by giving them multiple tools to measure with and allowing them to choose the best-fit tool for the job.  That way, the focus is on measurement, not just learning to use a ruler.

Supporting Diversity in the Classroom (Actual Post by Carrie Eby)

The kindergarten classroom that I am placed in this year is much different from the type of school that I went to as a kid. The school that I was in was predominantly filled with white, middle class families. This is in contrast to the highly diverse group of students that I placed in now. There are students from many different ethnic groups and socioeconomic status brackets. The Rosebery article that we read this week opened by saying that classroom populations are becoming more diverse, but the population of new teachers is not changing; it is still predominately white, female, and middle class. I definitely see this in field now, most of the teachers in the school fit this demographic and overall there is a very diverse population of students.
Because of this new more diverse group of learners that is being seen in schools, we as teachers must keep in mind that not all students learn the same way. The Lester reading highlighted this in a very interesting way. Lester points out that as teachers we should from day one have different types of lessons planned for math. This will help us get to know our students as learners and as people. It will help us plan and carry out the rest of our lessons for the year. The Kabiri article from last week talked about how it is important to turn traditional textbook problems into open-ended problems to create higher level thinking. Because classrooms are becoming more diverse, that means that students come into school with different experiences and it is important for us as teachers to help give each and every student an opportunity to learn and grow as a math learner. The best way that we can do this is what Lester and Kabiri suggest.

Clarifying Misconceptions and Facilitating Deeper Thinking

            After going over the readings for this week’s class, I was really intrigued by the Falkner and Kabiri articles.  In Children’s Understanding of Equality: A Foundation for Algebra, Karen Falkner, Linda Levi, and Thomas Carpenter discussed how many children are very closed minded when it comes to the meaning of the equal sign.  Many of the students from the article believed that in an algebraic equation, the equal sign only means the answer.  This is of course false in a problems like 8+4=_ +5.  In the article the authors state, “even kindergarten children, however, appear to have enduring misconceptions about the meaning of the equal sign” (Carpenter, Falkner, & Levi, 1999, p.233).  This shows that this misconception is rooted early on, so in grades as early as preschool and kindergarten, it becomes imperative to teach the right thing.  I believe this article ties in very well with the article by Mary Kabiri and Nancy Smith titled, Turning Tradition Textbook Problems into Open-Ended Problems.

            If we as teachers can create more open-ended problems not only will our students expand their thinking, but concepts like the equal sign, will have more than one meaning.  In the article the authors state, “Often, when teachers plan instruction for their student’s, they focus on the middle achievers…However, many problems can be made more open-ended and accessible to a wide variety of student abilities with minimum effort” (Kabiri & Smith, 2003, p.186-187).  They go on to explain that this can be done simple by taking traditional textbook answers and adding an “additional dimension” where more than one answer is possible.  So far in my placement, I have not seen many open-ended questions in math, however, I am excited to see if I can somehow incorporate these type of questions into my lesson plan to see how the students react to a new style of questioning. 

Caitlin Shanahan: Seeing Math in the Field

I am fortunate in that I see math most of the days I am in field, but because I am placed at a Montessori school the way math is integrated in the classroom is a bit different than what might be seen as most other schools.  There is no specific portion of time set aside for math each day, but instead the students are free the pick the work they do.  As you can imagine, in a Kindergarten class this can be a bit problematic because the students who don't feel comfortable working with the math materials tend to shy away from them.  There are a handful of students who come to school excited to do more math activities with the teacher, but there are also a handful of students who I have never seen working with a single math related activity.  In a class of 25, the teacher only has so much time to work one-on-one with each student (there are no group lessons in a Montessori school), so many of the students are falling behind with their numbers and simple math concepts because they are not given time with the teacher. Just how much some students are struggling in math has become very apparent since the last report card testing, so the teacher has been trying to make it more of a priority to include math in their daily activities.  This worries me because it is half way through the year and some of the students can't even count to five, while others are learning simple addition and subtraction problems.
I think a big part of math has to do with how confident a student feels in actually doing math, and if some of these students who are moving on to first grade are already starting to fall behind, I worry that they will develop negative feelings toward the subject.  In Denise Cantlon's article she mentioned multiple ways of authentically assessing her students.  In my field placement specifically, I think the students would really benefit from having those ongoing assessment strategies available to them.  Right now, testing is just done at the end of every marking period so my MT is just realizing how much some students are struggling and they are already half way through the year.  If she did ongoing assessments through observations, conversations, and games she may be able to help some students sooner rather than later.