Curriculum Plan Reflection

Creating the curriculum plan was the most challenging project for me this semester in math methods. It required the most amount of time, effort, and thinking. Although challenging, I can truly say this has been the project that I have taken the most away from. Organizing the different domain areas to be covered in each quarter was not very difficult for my group and I, nor were writing the objectives for each quarter. The challenging part was creating unique, interactive modeling and exploration tasks for the students to participate in within each grade. Additionally, creating the end of the semester exploration activities was a difficult task for us. However, I believe that my group members and I did an excellent job of coming up with unique activities for each grade level. This curriculum plan turned out so well, I would be more than comfortable using it in my own K-2 mathematics classroom one day! I feel that I learned a lot throughout the process of creating the curriculum plan. I realized how important it is for material to build off each other in each of the grade levels. It is to the benefit of both teachers and students to create curriculum plans for series of grade levels. This ensures that there is little overlap, and that teachers are maximizing on using students' prior knowledge. Although curriculum plans take a great amount of planning and time, they really can be beneficial to implement in a school.

After viewing the curriculum plans for both 3rd-5th and 6th-8th grades, it was interesting to see the flow of content and domains between the bands of grade levels. K-5th grades are very similar in domains, with the content obviously differing by grade level. However, once students reach 6th grade, the domains are changed quite a bit and become much more difficult and higher-level. Additionally, the measurement and data domain is completely eliminated in the upper grades (6-8). My assumption is that measurement and data is simply combined with geometry in these grades levels, and does not necessarily need its own domain area. I did see some overlap within the grade levels and bands, but I do not believe that this overlap is unnecessary. Actually, I think that some overlap (especially of difficult concepts) can be very beneficial to students. It just means that students are covering similar material in multiple grade levels, which can serve as a very good review. I also thought it was interesting that while most of the content domains changed every few grade levels, one domain was present throughout all grade levels. This domain is geometry. Geometry is heavily emphasized in each of the K-8 standards. It is clear that this is a highly important domain, and that students should be well experienced in geometry by the time they reach the upper grade levels.

Standards and Classroom Changes to Deepen Math Learning Reflection

Education in the K-8 classrooms across the country has changed drastically since my days in school. Even though that was not long ago, reform in education has required teachers to make huge changes in their teaching and assessing across all subject areas, including mathematics. I feel fortunate, however, that I am pursuing my college education at this point in time. The changes have been made, and now it is a matter of implementing those changes. I feel that I am at an advantage, because my education courses and experiences at Bradley are preparing me for these changes. Many current teachers do not have this practice, and were thrown into the changes and expected to make things work.

One of the biggest changes made to education was the creation and adoption (for most states, Illinois included) of the Common Core Standards for Learning. At this point in time, standards have been created for language arts and mathematics. In mathematics, these standards are often referred to as the Standards of Mathematical Practice, or SMP's. We have worked extensively in breaking these standards down this semester, in order to make sense of them ourselves. The standards themselves highlight what students should be able to do to show that they are proficient in mathematics at each grade level. There are 8 SMP's, and each are important to implement in all activities involving mathematical instruction. The Common Core Standards also provides specific standards highlighting what students should be able to do in various content areas of mathematics at each grade level. These standards should be utilized and aligned with lesson plans that teacher implement in the classroom.

Another change that has been made in mathematics education is the utilization of the NCTM Process Standards. These 5 process standards are composed of different areas of mathematics that students should be participating in during mathematics instruction. These 5 process standards include problem solving, reasoning and proof, communication, connection, and representations. All of these aspects should be included in any mathematics work that students are participating in. This is something that I will have to greatly familarize myself with and plan to implement in all of my mathematics lessons and activities.

Student engagement and active learning are two other major changes that have been added to mathematics instruction in K-8 classrooms. The days of students sitting at their desks listening to the teacher lecture and then completing a worksheet about the material are long gone. There is a major call for teachers to keep their students engaged throughout mathematics lessons and participating in active learning. This active learning does not include sitting at a desk and filling out worksheets or memorizing formulas. This active learning is problem-based, and involves more creative and engaging lessons and activities.

Finally, one of the biggest changes in mathematics instruction in K-8 classrooms has been the transition from content-based mathematics to taking meaning from mathematics. In short, students must now find the correct solutions but also explain and justify their solutions with reasoning. Simply finding the right answer is no longer enough. Students must learn how to justify their answers with appropriate reasoning and representations. This is a complicated task, and takes many students years to practice and carry out successfully.

It is clear that mathematics education has changed drastically in the last few years. There is a lot more planning, preparation, and reflection on the teacher's part. However, I feel that my past and continued experiences in college are preparing me for these changes.

Manipulative Reflection

How do you know if students deepen their understanding while using manipulatives?
I believe that a teacher can see if a student is deepening their understanding through the use of manipulatives through observation and questioning. The teacher can watch students to see if they are correctly utilizing the manipulatives they are working with. If this is not happening, the teacher can intervene to help the student and suggest alternative methods. The teacher can also see if students are deepening their understanding while using manipulatives by posing questions on the students. The teacher could ask students to represent something specific, explain their representation, and justify their representation. This could help the teacher gain insight on whether or not the student is really taking away meaning from the utilization of manipulatives.

How do you know if the students can transfer their understanding from manipulatives to other situations?
One way to see if students have the ability to transfer their understanding of an idea from manipulatives to other situations would be to ask the students to think about the manipulatives in their minds, or draw them out on paper. This would eliminate the actual use of the manipulatives, but the students could still utilize ideas they learned from the use of them. An example of this would be giving students the option of drawing base 10 pieces while solving a math problem. Although physical manipulatives would not be used, drawings could still help the student to solve the problem.

How can you assess that understanding or growth?
A teacher could assess the understanding or growth a student takes from the utilization of manipulatives by asking them to verbally explain their use of manipulatives, and what their manipulatives stand for. In addition, the teacher could ask students to write their explanation of their use of manipulatives and write also what the manipulatives stand for. These are two options for assessing student understanding of manipulatives. Another option would be for the teacher to provide manipulations on the SmartBoard or overhead projector, and ask students to explain the manipulations and their meaning to a partner, or ask students to share their ideas with the entire class. This could gauge whether or not students have true understanding of their use of manipulatives.

When students work in groups, how do you hold each youngster accountable for learning?
First of all, I think that any group work in the classroom should be done in very small groups of students, maybe 2 to 3 children in each group. When group sizes become too large, there are almost always students left out of discussions or students who feel they can "get away with" not participating. This is not acceptable, and really takes away from the overall point of group work in the first place. Having small groups should make all members of the group members participate and share their ideas. If a teacher notices a group member slacking or not participating, the teacher should pull that student aside and explain that their participation is not optional, and that their group members need their help and input during group work. I am not a huge believer in punishments, but I do believe that privileges should be taken away from students who do not participate in group work. All of these modifications will help students to be accountable for their learning, and promote participation and effort during group work instances.

When students work in groups, how do you assess each youngster's depth of understanding?
Assessment of students working in groups can be a difficult task, but I think there are two things that a teacher can use to help with this. The first item that can be used are checklists. Checklists can be used for individual students to monitor their progress and understanding throughout an activity or project that has students working in groups. While students are working, the teacher can observe and monitor student progress, behavior, participation, and understanding. The teacher can monitor understanding by asking students questions and trying to make sense of their responses. This may gauge whether or not the student is understanding the topic at hand. Conversation is a very effective tool for assessment, and a teacher can make great use of it during group work assessment. Another thing teachers can do to assess students during group work would be, if applicable, to have every student complete a handout or product. This ensures that all of the students in a group are participating and completing the task, and the teacher has something to look at for each and every student.

How are you improving students' problem solving skills with the manipulatives?
Manipulatives allow students to be hands on and visualize the mathematics they are carrying out. Many students need visualizations in order to understand mathematics, so manipulatives can be very useful. In addition, when using manipulatives, students can find errors and then fix those errors more easily. This can help lessen student frustration in problem solving. Manipulatives also provide a more concrete way of problem solving for students.

Technology in the Math Class Reflection

A great amount of technology has been utilized during this summer semester of math methods. At the beginning of each class period, we were asked to sign in on the SmartBoard, using something other than the pen. This allowed students to work with this piece of technology and familiarize themselves with features. Additionally, each student was assigned to show the rest of the class a SmartBoard feature at some point during the semester. This was a really useful activity, because I learned a lot of new things about the SmartBoard and some of its features. This will help me in my novice and student teaching placements, as well as in my own classroom once I graduate. Throughout the semester, one of our assignments was to watch two videos online that showed a math lesson taking place in a real classroom. Although these videos were long, I really took a lot away from both and wrote deep reflections on each. These videos were very interesting to watch and then reflect on. This was an excellent use of technology, and I enjoyed watching these videos much more than if we were asked to read an article about them. Two other forms of technology used this semester were Prezi and Jing recording software. I have used both of these in the past, but never simultaneously. We were to create a Prezi highlighting some of the Common Core Standards for Mathematical Practice, and then narrate that Prezi using Jing recording software. Although this took some practice and repeated tries (for me anyway!), this was a useful activity. These are two pieces of technology that I would definitely like to implement in my classroom one day.

We also utilized the website Blogger.com to post blogs continuously throughout the semester about our work. I have blogged before for other classes, and have always found it enjoyable. It is a great way to reflect on work. I know many practicing teachers who utilize blogs to keep their students' parents and family members up to date on class work. I think this is something I will definitely do in my classroom. Another assignment I completed this semester was searching for various math applets and apps for my assigned grade level, K-2. This was another meaningful experience, because I discovered cool applets and apps to use with my future students. We did use calculators a bit this semester, mostly for calculation work during activities. I feel very comfortable at this point in my education with the use of calculators, due to my past math classes in both high school and my early years of college. The final piece of technology utilized this semester was the creation of videos discussing our curriculum project. I have created videos before for assignments, but none as long as this video. This was an interesting and challenging experience for my group members and I. A great amount of technology was used this semester, and I feel that it will really benefit me with my future in teaching.

Assessments in Math Reflection

Assessment in mathematics has been a rather large topic of discussion during math methods this summer. In any subject area, assessment is the key to gauging whether or not students have learned the concepts and objectives taught within a period of time. A variety of assessments have been discussed, including traditional and authentic assessments. While both have their perks and downfalls, many schools are pushing their teachers to use more authentic assessments with their students. Formative and summative assessments were also discussed this semester. Formative assessments are ongoing and constantly occurring during a lesson and/or unit. This might involve the teaching asking questions to students, having classroom discussions, or simply observing students during activities. Summative assessment occurs at the end of a lesson or unit, and is used as a way to look at students' overall learning or their take away of concepts. In addition to classroom discussions on assessment, I read several articles regarding assessment in the mathematics classroom. One article that really struck me talked about the use of conversation as an assessment tool. I think too many times, schools and teachers want all assessment to be formalized so that a written product is produced. However, this article talked about how the element of conversation can be a very effective tool to use for assessment. A teacher can learn a lot about his or her students through conversation, including what they know, what they are learning, and what they do not know yet. This kind of conversation can help the teacher to quickly change the path of a lesson, or even plan ahead for future lessons. It can also help a teacher to modify instruction for certain students. I am a big believer in conversation as a method of assessment because it is relatively informal, easy to use, and can reap major benefits.

Throughout all of the assignments during this semester, I did some assessing of my own on my personal work. While completing each assignment, I regularly utilized formative assessment by rereading and checking my work. Additionally, I checked specific rubrics throughout my work to verify that I was on track with the assignment. I utilized summative assessment when I completed any assignment before turning it in by rereading my work and consulting the rubric once again. I also had the opportunity to assess some of my peers' work during this semester. I critiqued problem situations that were created by my classmates. In these critiques, I was to comment on strengths and weaknesses, and provide suggestions for improvement. This was a new experience for me, because I do not have much practice with critiquing/assessing my peers' work. I also worked with assessment when going through the More Errors document. This was a collection of student samples of mathematics problems that had mistakes within. I was to find the mistake in each sample and then try to apply that mistake to other, similar problems. This was also a new experience for me, and I had some difficulty with finding the mistakes in many of the problems. Finally, I worked with assessments by reading through the feedback my instructor gave me on all of my assignments. I did this by reading through the given rubrics and reading comments that were listed. This helped me to get an idea of what my instructor thought about my final work, and also allowed me to reflect on the assignments.

Error Analysis Reflection

The work I participated in while reviewing the More Errors document was quite interesting to dive into. A series of various student samples of mathematical problems were posed, many of which involved errors in solving and/or incorrect answers. I was to go through each problem and determine how the sample student was going about solving the problem, and if mistakes were made, where those mistakes were found. Then, I had to try and solve a few additional, similar problems using the sample student's method (which was usually incorrect). This seemed like a simple task at first, but as I explored deeper into the problems, I realized this was more difficult than I first thought. Some of the mistakes I was able to easily pick up on, while others caused me quite a challenge. I think most of the difficulty came from trying to read and understand someone else's work. It is easy for me to reread and understand my own work and my process for finding a solution, because I can internatlize my thoughts. However, when it comes to trying to read and understand someone else's methods and processes for finding a solution, it can be much more difficult. I had this problem throughout the entire More Errors document. Some of the problems really caught me off guard, because I could not figure out the methods/strategies the sample student was using to find their written solution. In additon to having difficulty in finding the mistakes the sample students used, it became evident to me that a large multitude of mistakes can be used when solving mathematics problems. I saw mistakes in the problems that I read which would never even cross my mind to do while solving a mathematics problems. I think this really opened up my mind to the fact that just because an idea, method, or strategy in mathematics makes sense to me does NOT mean that the same applies to someone else. Reading and understanding my student's mathematical work is something that I am going to do every single day. After completing the More Errors assignment, I now realize that I need a lot of practice and experience in doing this.

Article Discussions on Assessment

Getting Started with Open-Ended Assessment

Found in Teaching Children Mathematics, April 2005

This article discussed the high points and low points of utilizing open-ended assessment in the classroom. Open-ended problems are those where multiple strategies can be used to solve, and multiple solutions may be correct or accurate. These types of problems allow students to better demonstrate their actual understanding of a topic in mathematics. Instead of just finding an answer and writing it down, students must additionally explain their thinking and reasoning using detail. Open-ended problems do not rely on content-based mathematics. Instead, they require students to not only find a correct solution but also explain their process and reasoning for finding that solution. It is important for open-ended problems to utilize meaningful mathematics, allow for multiple strategies/solutions, and have the perfect medium of too little and not too much information provided. It was noted by a featured teacher in this article that creating and implementing open-ended problems can be very time-consuming and difficult. However, they can be very beneficial to students and their overall learning in mathematics. They require students to do a lot of work in justifying their answers, which may be a new concept for students. This featured teacher also suggested to not grade these problems at first, which will allow students to be more comfortable completing them. Additionally, she found it beneficial to show student samples on an overhead (with names removed) and talk about the pros/cons of each sample response to a problem. She explained that after using open-ended problems for an extensive period of time, she saw gains in student confidence and willingness to share their answers to peers. She also noted that as the teacher, she felt this provided her with a more concrete method of assessment, which helped her explain student achievement to parents.

This was an interesting article to read. I have heard of open-ended problems before, but have never really given them a lot of thought. It is obvious that they can be quite difficult to create and implement, but the benefits do seem to be great. I believe that using open-ended problems helps students to really look into their thinking and make sense of their strategies and processes for finding answers to problems. Open-ended problem solving takes a lot of effort on both the student's and teacher's part, but it is a very useful tool that can help students to take more meaning from mathematics.

Discussion Questions:

1. What are some ways that you could potentially implement open-ended problems in your future math lessons?

2. Could you implement open-ended problems into other curricular areas?

A Smorgasbord of Assessment Options

From Teaching Children Mathematics - April 2010

As teachers, we are faced with the task of creating authentic assessments to gauge our students' learning. Creating these types of assessment can be very challenging. To add to the challenge, there are an array of assessment options to choose from in which we can utilize in our classrooms. This article talked about one type of assessment, called student-centered assessment. This format benefits the students because they are participating in meaningful mathematics, but it also benefits the teacher because it helps to gauge student understanding and their level of learning. Mathematics is a process, and students must heavily rely on building upon prior knowledge as they work through concepts. In this article, a class of students who were working with understanding geometric shapes and solids was discussed. In this class, students were at a variety of levels of understanding in regards to geometric shapes. Regardless of their individual level, students used both knowledge and reasoning to come to various conclusions about geometric shapes they were working with. A "mastery" indication of these concepts occurred when students were readily able to communicate their thinking and reasoning to their peers. It was noted in this article that in order for student-centered assessment to be effective, modes of both formative and summative assessment must be utilized. It was also suggested that they best way of creating student-centered assessment is to first identify the standard, then select a target achievement level, and finally create assessment options for students to demonstrate their understanding of the concept.

Student-centered assessment seems to be a very valuable tool for teachers to use in the classroom. It not only get students focused and motivated on a topic, but it has them thinking more deeply and communicating their thoughts and reasoning to their peers. I think this is a very important skill for students to be able to master. Once they communicate their thoughts and rationale, it is a clear indication that they truly understand a subject themselves. It is definitely evident that creating these authentic, student-centered assessment is very time consuming and not exactly "easy". It takes a lot of planning and thought on the teacher's part, and it takes a greater amount of time to implement in the classroom. However, the benefits outweigh the downfalls, and student-centered assessment is definitely something I would like to use in my classroom to help engage my students.

Discussion Questions:

1. Can you think of any specific student-centered assessments that could be utilized in a mathematics lesson?

2. Are there any other benefits you have seen through observations/field experience that display student-centered assessment?

Understanding Student to Open-Ended Tasks

From Mathematics Teaching in the Middle School - April 2000

This was another article in regards to the use of open-ended tasks in mathematics classrooms. Open-ended tasks allow for student choice in the strategy to use for solving, and also allows (usually) for multiple correct responses. This allows students flexibility in solving, and using strategies that they are comfortable with. While open-ended tasks sound to be beneficial to students, they often are hard for students to grasp and understand. Some students have little experience or practice in actually communicating their reasoning, which open-ended tasks require. This article described some sample responses from 6th graders working with an irregular area task. Several responses were shown, some being incorrect and others being correct. The teacher of these students seemed to be pretty hard on giving feedback for all of the responses. In addition, students she thought understood the concept of area extremely well performed poorly on the task. Alternatively, students that she thought were struggling easily grasped the problem and explained their reasoning adequately. It was noted that the students who made great use of words, pictures, and diagrams to aid in explaining their thinking performed better on the task. The teacher had been working with this group of students over the course of a year, and felt that her feedback was allowing her students to generate more detailed and supportive responses when completing tasks like the one posed in the article.

I thought it was interesting how the teacher's viewpoint of student understanding was drastically changed after giving her students this task. The students that she was not concerned about turned out to be the ones who did not comprehend this task or provide adequate reasoning for their responses. Alternatively, the students she was concerned about really pulled through on this task to provide a correct answer and use supportive reasoning. I think that this was a clear indication that the teacher really needed a way to look further into her student's thinking and mathematical processes. I like that it was also noted within the article that any teacher beginning to provide students open-ended tasks like this one need to expect reasonable results. Students are not going to adequately explain their thinking on the first try, and will need a lot of practice in order to respond to these types of tasks correctly.

Discussion Questions:

1. What do you presume were some of the benefits that students experienced when working with open-ended tasks like the one described in the article?

2. Do you think this teacher was too harsh on her student's responses?

Assessing Students' Understanding through Conversations

From Teaching Children Mathematics - December 2007/January 2008

Conversation as an assessment tool is often an underrated element in the classroom. However, conversation is used as a method of assessment much more frequently that any other form of assessment. Why is this? I believe that it is because conversation is informal, for the most part. It is easy to participate in, and it allows for a lot of information to be exchanged between teachers and students. Conversations over mathematics can truly gauge what students do and do not understand about the topic at hand. This article talked about three different situations in which conversation was critical to understanding student knowledge. One 3rd grade teacher used informal conversation to gauge the true understanding of one of her students working with place value. Another 5th grade teacher used conversation to understand the mistakes made by a group of students on a geometry quiz. Another teacher used conversation during a lesson on measurement to realize that students were struggling with the basic concepts of measurement. All of these examples showed how critical conversation was to the lesson, because the teacher was much better able to gauge student understanding. This article also talked about promoting conversation in the classroom required that students feel safe in the environment. Students should feel comfortable sharing their thoughts, and be respective of others doing the same.

Conversation is such a valuable assessment tool, in my opinion. It is something I plan to use very regularly in my classroom. The reason that I like conversation so much is because it takes little planning, but it can still yield so much information. I believe that conversations as a class can also yield student participation, which is crucial in a classroom. This article discussed the idea of participation, and also the utilization of wait time in order to obtain responses from all students. Conversations about mathematics can not only gauge understanding, but can help students to find errors in their problem solving or clear up any misconceptions they may have. Conversation as an assessment tool is very useful to any classroom teacher.

Discussion Questions:

1. Can you think of any times during observations/field experiences where you have seen conversation being a great assessment tool?

2. How will you work conversation as assessment into your future classroom and lessons?

An Experiment in Using Portfolios in the Middle School

From Mathematics Teaching in the Middle School - March 2008

This was a very interesting article to read about the implementation and use of portfolios in a middle school mathematics classroom. The teacher featured in this article wanted to display her student's knowledge in a different way than traditional quizzes and tests. Therefore, she turned to the use of student-made portfolios. Over the course of several weeks, the teacher had the students collect pieces of their work. Students were to have sections of their portfolios discussing their mathematical attitude, problem solving, mathematical growth, mathematical writing, and mathematical connections. Students worked to select specific pieces of work for each of these areas. The portfolios additionally required a lot of student reflection on their choices, and rationale for picking these items. Once completed, the portfolios were sent home for parents to view and answer some questions about. The teacher assessed these portfolios by organization and the overall learning and understanding of mathematics that the student received from creating the portfolio. She used a checklist to do this. As far as benefits of utilizing the portfolios, the teacher explained that she began writing her lessons so that they were able to be included in the portfolios. She did this by creating assignments that involved more writing, real-world connections, and problem solving. She also noted that by reviewing the portfolios, she was better able to see what students had learned and where they experienced the most growth in their learning.

I really like the idea of using portfolios as a form of alternative assessment. I have made numerous portfolios myself for college courses, and they really are more meaningful than taking a final exam. They allow a student to examine their progress and reflect upon their work in various forms. I think that the students in this featured class seemed to take a lot away from creating their portfolios. In addition, the teacher required the students to reflect on why they choice certain pieces. The teacher also discussed how she had students write in learning logs after completing each assignment. Students were to write what they did and did not understand. This helped students when creating their portfolios to select pieces of their work that really represented their knowledge and understanding. Portfolios are a great form of assessment, and I would really like to use them in my future classroom.

Discussion Questions:

1. Where have you seen portfolios being used in the classroom?

2. Why would having students create portfolios be beneficial to use during parent-teacher conferences?


References:

Bacon, K.A. (2010). A smorgasbord of assessment options. Teaching Children Mathematics. 16(8).

Leatham, K.R., Lawrence, K., Mewborn, D.S. (2005). Getting started with open-ended assessment. Teaching Children Mathematics. 11(8).

Maxwell, V.L., Lassak, M.B. (2008). An experiment in using portfolios in the middle school. Mathematics Teaching in the Middle School. 13(7).

Moskal, B.M. (2000). Understanding student to open-ended tasks. Teaching Mathematics in the Middle School. 5(8).

Vanderhye, C.M., Zmijewski Demers, C.M. (2007/2008). Assessing students' understanding through conversations. Teaching Children Mathematics. 14(5).

Journal Summaries #2

Linking LEGO and Algebra

by: S. Asli Ozgun-Koca, Thomas G. Edwards, & Kenneth R. Chelst

(Found in Mathematics Teaching in the Middle School - March 2015, Vol. 20, No. 7)

This was an interesting journal article about an authentic, real-world mathematics situation that was posed on various classes of both 6th and 7th grade students. Under the Common Core State Standards for Mathematics, students should be working frequently to solve and analyze real-world mathematics. By doing this, students should be able to think realistically and allow for flexibility in their solving. In many cases, it is up to teachers to create these authentic math situations for students to work with. It is suggested that authentic learning have real-world connections, use a large amount of time and a vast amount of knowledge, utilize multiple viewpoints of the problem, allow for extensive group work, and require students to find a solution but also analyze that solution.

The activity described in this article was called the LEGO Pets Activity. The 6th and 7th grade students were asked to create dogs and ducks out of a set amount of LEGO pieces. The first part of the activity began by asking students to do this using 6 large and 13 small LEGO bricks. Students worked on some trial and error, and recorded the various combinations they came up with. During this, class discussions were held regarding how the students could work to create the most efficient and profitable amount of animals, using the pieces provided. It was decided that the most profitable combination of LEGO animals, given the amount of materials, would be achieved by building two ducks and one dog. Using an equation, students found that this would surmount to $57 in profit for the LEGO company. Students were then asked what would happen if they were given one additional large or small LEGO brick. Discussion occurred regarding this idea, and most students believed this would not have very large of an effect on profits.

Part 2 of this activity had students working with a larger number of materials. This time, they were asked to create more ducks and dogs, instead using 16 large bricks and 31 small. Students quickly realized this was going to require more work, so took the option of using a spreadsheet over creating the animals by hand. They were introduced to the spreadsheet and some various functions, and learned about the formulas within. Students also worked again to figure out the most profitable combination of animals. It was found that with these materials, four ducks and three dogs profited the most for LEGO at $135. At the end of the article, it was noted that the use of technology in this activity really promoted student learning. In addition, the activity was both authentic and made use of modeling. It was a very real-world activity by having students work with LEGO's and think in terms of efficiency and profit.

I thought this was an interesting article overall. The idea is good with the use of LEGO's and having the students think in terms of being efficient and also generating the largest possible profit. I also liked the idea of using the spreadsheet. However, if students did not have practice with a spreadsheet or understand the purpose of one, I think it would be pointless to present. I do not know the background knowledge of the classes this activity was done with, but I would definitely want to make sure students understood how to use spreadsheets before having them work with one. I do not think that I would implement this problem in my classroom, because I will not be teaching middle school mathematics. This would be too advanced for an elementary school classroom. However, if done correctly, I think this would work well and allow for a lot of student inquiry in a middle school mathematics classroom.

STEM Gives Meaning to Mathematics

by: Lukas J. Hefty

(Found in Teaching Children Mathematics - March 2015, Vol. 21, No. 7)

Both the NCTM Process Standards and the Common Core Standards for Mathematical Practice work together to emphasize the importance of students actually taking meaning away from mathematics. Simply memorizing content is not the basis of mathematical knowledge anymore. The school featured in this journal article was called Douglas L. Jamerson Jr. Elementary School, and their focus was to promote the use of engineering activities to involve real-world mathematics in the classroom. The teachers at this school collectively worked for multiple years to develop activities for each grade level that combined math and science concepts into engineering projects. School wide, the overall process for each project involves planning, designing, checking, and sharing. 

One example of an engineering project described in the article was rubber-band powered vehicles made of K'Nex. Students would do a variety of mathematics during this project, in addition to constructing via blueprint plans, measuring, investigating and experimenting, recording and analyzing data, and participating in class discussions. The goal for this project was for students to create the most efficient car to go a certain distance. 

Teachers at this school noted that they saw little frustration in students while working on the engineering projects. They also explained that when doing these projects, they still have a daily period of mathematics instruction in each individual classroom. Students then utilize what they are learning in math while working on their engineering projects. In addition, the teachers look to build upon student's prior math knowledge from previous grades. The implementation of these engineering projects have made some remarkable changes in the achievements of the students at this school. Teachers noted that students were better able to problem solve, think more critically, and apply mathematics to the real-world and to their own lives. The school also noted that they have seen pretty drastic changes in their student's math and science scores on standardized tests. Some suggestions were additionally given for teachers or schools interested in starting a similar program. They suggested that a good starting point would be to investigate simple ways to integrate mathematics into other subject areas. Then, group collaboration is highly beneficial in order to implement this type of program.

The idea of involving engineering projects that utilize mathematics and science is a great idea. However, I feel that this would be extremely time consuming. Most schools that I have observed in are very caught up in standardized testing. Therefore, they focus mostly on content rather than application of hands-on activities. While I do believe that these types of projects are very interesting and beneficial to students, I do not know how realistic they are to implement in a regular school schedule. Adding another block of time each day for engineering projects may be very difficult to squeeze in. I am not saying that I am against this type of school work, because that is certainly not the case. I think that authentic, project-based learning is highly beneficial to students. They take away so much more meaning from these types of activities. I just believe that this type of curriculum would be challenging to implement, unless drastic, school or district wide changes in the curriculum were made to allow for engineering projects like the ones described in this article.

References:

Hefty, L. J. (2015). STEM gives meaning to mathematics. Teaching Children Mathematics, 21(7).

Ozgun-Koca, S. A., Edwards, T. G., Chelst, K. R. (2015). Linking LEGO and algebra. Mathematics Teaching in the Middle School. 20(7). 

Problem/Project Activity Reflection

The problem/project activity was a very meaningful experience for me to participate in. At the beginning of the task, I was very unsure of the whole process and was not looking forward to a seemingly difficult task. However, I learned that this project was actually a very insightful task and I feel that I have taken a lot away from it. It is extremely important for me, as a future teacher, to keep my students engaged in the classroom. In addition, I must make my activities and lessons relatable to my students. I believe that it can be very difficult for teachers to do this on a regular basis. We are told what to teach and, in some instances, how to teach it. There is little wiggle room for creativity or “extra” projects. In hindsight, these real-world projects may actually be the one thing that gets students engaged and focused, and actually working toward an end result.

My group and I brainstormed a lot about potential problems/projects. When we finally chose the one we focused the project on, we were all very excited to begin planning. The problem that we posed to our students was that the local park was having some troubles. Trash was everywhere, the garden was full of weeds and dead plants, and beehives were all over the playground. It needed to be cleaned up so that all of the community could enjoy a beautiful, safe park. To help with this community issue, students were to fundraise money to donate to the park district. To do so, they would create products in their small groups to sell. Throughout this weeklong project, students used a variety of mathematics and other interdisciplinary content. I felt that this problem/project was an excellent one to pose to students. First and foremost, they were working towards an important cause. By fundraising the money, they were helping to better their local park for the enjoyment of themselves and others. Having students work to price, create, and sell products gave them hands-on experiences and a very basic idea of entrepreneurship. I think combining both of these ideas was a very interesting and effective way of having students work on this problem/project.

As I said before, this was a very interesting and insightful experience for me. I now know that real-world problems and situations can and should be integrated in the classroom, and can have some great benefits to both students and teachers.

Reflection of NAEP Student Work

Working with the NAEP student samples was a very engaging experience and really opened up my eyes to the importance of using good, well-developed rubrics. My group members and I were tasked with reviewing a data analysis problem samples from a set of 4th grade responses. Additionally, we were provided with the rubric that was created to score this problem. Upon first glimpse, the rubric seemed to be detailed enough to cover all of the bases within grading. However, once we actually started going through each of the sample responses and comparing them to the provided rubric, we had a change of thought about that. It became clear that many of the student responses could, technically, fall in two or more of the provided grading levels. As a result, it was really up to the individual or group grading the response to determine which level the response fell under. This led my group members and I to understand why this problem was taken out of the testing pool in the first place. We managed to get through all of the student sample responses and assign each to a specific score. It was shocking to us how many we scored as off task or minimal, the two lowest scoring possibilities. In addition, we were not completely sure that we scored each and every one of the samples correctly, considering the rubric was not exactly clear. Like I stated earlier, this project really opened my eyes to the use of rubrics. While they are a very useful tool for grading, they also must be written correctly and with enough detail. In addition, the different levels of a rubric must be clearly distinguished without any overlap occurring. This overlap is what leads to inaccurate grading of student work, similar to the grading issues and discrepancies we came across in the NAEP student samples. Rubrics are a fantastic tool, but only if written and utilized correctly! When selecting the three samples to create activities for, my group and I decided to pick three samples to cover the range of scores. Therefore, we chose an extended, partial, and incorrect/off task sample. We felt this was the best way to represent everyone in a classroom. We created one, overarching activity for students to do in which they physically created a human bar graph which was then transferred onto the SmartBoard. Each of these focus students worked on various areas, depending on their individual needs. I thought that it was actually quite difficult to differentiate for these three sample learners. One was clearly advanced and required more challenging work, while the other two were lacking and required some additional assistance in problem solving. However, I felt that this was an important obstacle to tackle, because differentiation of learning is something I will have to do every day in my classroom.

After viewing my colleagues' work on their individual NAEP problems, it became evident that my group was not the only one to struggle with scoring some of the student samples. Additionally, the two other rubrics we looked at also had some major flaws and were difficult to comprehend. One of the rubrics was extremely cut and dry, which was different to ours. I felt that our rubric was more well-developed and the different scoring levels were described a little better than this rubric. The other rubric we examined was very wordy and hard to understand/decipher between some components of the different areas. The other groups explained that they, too, experienced difficulties with determining what scoring level some of the student samples should have been graded as. I think that this was a common trend among all of the questions and sample responses. I believe that a big take-away from this project and experience is that creating and utilizing clear and cohesive rubrics is absolutely the key to accurately and fairly scoring student responses on a mathematical problem like the ones provided.

Video 2 Analysis - Number Operation- Multiplication & Division

The video Number Operation- Multiplication & Division featured a mathematics lesson that took place in a 4th grade classroom with math coach/teacher Becca Sherman. No video of the teacher discussing pre-planning or the goals was included in this video series, and the clips began immediately with the teacher instructing the lesson. However, the teacher did have the phrase "A picture is worth 1,000 words" written on the board, which she explained to students was the focus of that day's math lesson. The lesson itself was divided into four major parts, which were titled "problems" in the video clips.

On the whiteboard, Ms. Sherman had the words "multiplication" and "division" written side by side. For Problem 1, students were asked to talk with their partners about what multiplication means. Students then gave their responses, which Ms. Sherman copied on the board. Some of the students explained that it's like (repeated) adding and groups are made. The teacher asked for students to provide examples, and the students dwelled on the 2's multiplication facts. Ms. Sherman then explained that they needed to try to create equal groups, and could use pictures to help them.

Once the list of things regarding multiplication was created, Ms. Sherman asked students to do the same thinking for division (Problem 2). Students talked with partners momentarily, and then shared responses. One of the students said it was like subtraction, but could not explain why. Other students said that division is like multiplication, and you can "switch them" around. Many were able to see the clear connection between multiplication and division. Ms. Sherman wrote the number 100 on the whiteboard and asked students how they could divide this into equal groups. One student said that they could do two equal groups of 50. She asked for other solutions, but the number seemed too big for students to grapple with. Then, she wrote the number 12 and asked students the different ways to divide this number into equal groups. Students provided several correct solutions.

Problem 3 occurred next, and this involved students doing mental math. Ms. Sherman had a multiplication problem written on butcher paper that read 26 x 4. She told students to think about it quietly and try to solve. Then, she asked for students to explain their methods and answers. Most of the students chose to use the more traditional multiplication method by setting up a vertical equation and "carrying". However, many who did this got incorrect answers (likely due to the mental aspect of this task). Ms. Sherman asked students if there was more than one correct answer, and surprisingly, some answered yes (which was not true). She asked for additional methods, and explained that creating a picture representing 4 groups of 26 would be a great way to solve this. One student had an alternate approach, which I found interesting in itself. He explained that he knew there were 2 tens and 6 ones. So, he counted off 20, 40, 60, 80 (20 four times). Then, he counted off 6, 12, 18, 24 (6 four times). Finally, he added 80 + 24 and got 104, the correct answer. Ultimately, this student used his knowledge of place value to find the correct solution, which I found very interesting.

The final problem for the lesson, Problem 4, had students working with word problems. Ms. Sherman posed a sentence on butcher paper about money. The sentence said, in short, that Maria had $24, which was three times the amount of money that Wayne had. At first, no questions were posed, but students picked up on the fact that they needed to find out how much money Wayne had. There was a surprising split in the answer. Almost half of the class immediately saw "times" and did the operation 24 x 3. Clearly, they did not understand that this was a division problem and 24 was actually the whole. Other students were able to find the correct answer of $8, but could not explain why they thought this was correct. Students were asked to draw pictures and have words that explained their answers. However, most of the student's pictures did not explain the word problem in any way. The majority of students who wrote that Wayne has $8 could not explain their reasoning whatsoever! Next, Ms. Sherman wrote a picture on the board that explained the problem. She asked students to look at the picture and see if they understood how it was setup. Then, she gave students another sheet with a sample problem on it. "Charlie" set up his problem using a bar model, and 24 was Maria's whole while an unknown amount was Wayne's whole. She instructed students to try and find the missing numbers. Again, many students jumped right to putting 24 in each of the part boxes on the bar model. Finally, she gave the students two questions regarding the problem. Ms. Sherman tried to explain why 72 was not the correct answer, but it was clear that students were still struggling. To conclude, she had students write a few sentences about what they learned from "Charlie's" model in relation to the problem. Students seemed to struggle with this.

At the faculty debriefing, Ms. Sherman noted that at the beginning of the lesson, it was hard for students to expand from their 2's multiplication facts. Additionally, she noted that the whole idea of creating equal groups was not there. Many students explained that addition was like multiplication and subtraction was like division, but could not back those statements up. She believed that this was because students had been told this at some time in their education, but did not know why this was. Ms. Sherman also said that she felt the drawings portion of the lesson kept students engaged, which made her happy. She was displeased that over half of the students continued to believe that Wayne had $72, even after her attempts at explanation. Some of the faculty members noted that some students had correct answers initially, but changed them once they saw that their group member's had a different answer. This made it evident that there was a copying problem in the classroom. Most of the students also had drawings, yet they could not make any sense of their drawings or explain what it had to do with the story problem. The biggest take away that the faculty members drew from the lesson was that these students really needed to work on making connections in multiplication and division and their actual meanings, instead of just reciting number facts.

A few things really struck me in this video that I would like to now comment on. First of all, I believe that Ms. Sherman is a decent teacher. However, while watching the video, I noticed quite a few things that I think she should have avoided doing. During Problem 1, she flip flopped back and forth quite a bit between various student examples of multiplication. This was not beneficial to students; it just confused them. In Problem 2, one student shared an example and she asked him to draw a picture on paper to explain his example. When the student said he was finished, she simply said that they would look at the picture another time. I did not think this was the right thing to do. She asked the child to take the time to draw the picture, which relates to the topic at hand. Why would she have him share it another time? Also in Problem 2, she asked students to break the number 100 into equal groups. This was much too big of a number to begin with. It was clear her students were struggling with the concept of equal groups, and she should have given them a much smaller number to work with. In Problem 4, she was again jumping around between different pictures and kept saying "we'll come back to that". This was not at all helpful, and really confused students. I liked this lesson, but I saw the students really losing focus toward the ladder part of it. I think Ms. Sherman had good intentions of getting a lot of material covered, but the reality was that students lost focus quickly. The last task was to write a few sentences about "Charlie's" method, and some of the students were so lost and gone by that point that there was just no sense in writing.

I took a lot of interest in watching this video, and I think that the lesson overall was a great concept. However, it was clear that this group of 4th graders needed quite a bit of work on understanding the meaning behind multiplication and division operations.

Math Applets/Apps Review

The following are reviews of applets/apps that I found useful for grades K-2 in the content area of measurement & data and analysis & geometry:

Applet #1: Time - Match Clocks

From the National Library of Virtual Manipulatives

URL: http://nlvm.usu.edu/en/nav/category_g_1_t_4.html

This applet is a great way for students to practice telling time using both a digital and analog clock. Many students struggle with telling time on a regular, analog clock, especially in today's day and age of electronics. The applet features a digital clock on one side of the screen and an analog clock on the other side. The analog clock is set to a specific time, and the player is asked to change the digital clock to the time that the analog clock reads. The players use hour and minute arrows, respectively, to change the time on the digital clock. If they are able to change it to the correct time that is displayed on the analog clock, they have a "correct!" notice appear. Then, they can click for a new problem. The applet is easy to use and young students would likely be able to pick up on it easily. Visually, the applet is not extremely appealing with its grey background and blue areas within. I believe that the applet is challenging for students who are beginning to work with telling time.

I believe that this applet could greatly assist students in learning the concepts of telling time. The applet actually requires players to think about the time that is displayed on the analog clock. This is often a difficult task for young students just learning to tell time, so I think that this applet would help struggling students. One way a teacher could assess his or her students on their utilization of this app would be to have students write down both the analog clock reading and the digital clock reading on paper for each of their rounds in the game. The teacher could provide students with clock templates for them to fill in. The teacher could then see if the students really are following the game correctly and correctly finding the time. Some strengths of this applet are that it is easy to use, it involves the use of minutes and not just exact hours for each problem, and players can skip questions if they cannot figure out the correct time. One large weakness of this applet is that players do not work with changing the hour and minute hands of the analog clock. They only read the clock and then report the time using the digital clock. I think students would benefit by both reading and manipulating the analog clock, which this applet does not offer. Another weakness in this applet is that it is not very visually appealing, which may lead to young students losing interest after a while. In addition, every new game starts with the same order of times. Some students who use this regularly may start memorizing the times and not take any meaning away from using the applet.

Applet #2: Geoboard

From the National Library of Visual Manipulatives

URL: http://nlvm.usu.edu/en/nav/category_g_1_t_4.html

This applet is a great way for students to practice creating geometric shapes. One of the Common Core State Standards for Mathematics in 2nd grade requires students to work with geometric shapes and to partition a rectangle into smaller rows, columns, and squares of equal size. This applet would be a great way for students to practice that standard. Students can use the geoboard to construct various shapes to their liking. Shapes can be small or large, depending on how the student stretches the bands. Irregular shapes can also be created. Students can choose to color code different shapes. In addition, students can find the perimeter and area of the shapes that they create on the geoboard. The applet is fairly easy to use, but the students may need to be shown how to create their shapes with the bands (this was a little tricky at first). Visually, the app is quite appealing, especially with the various colors that students can shade their shapes with. On a side bar within the applet, there were activity suggestions for students who might be stuck or not know what to do on the applet.

Students could learn a lot from this applet. First, students could work to create various geometric shapes that they are learning about in mathematics. After doing this, advanced students could use the measure tool to find the perimeter and area of the shape. This concept of perimeter and area may be too advanced for most 2nd graders, however. One way to make the idea of area easier for young students would be to have them create regular rectangles, and ask them to count each individual square within the large rectangle. This is area, but an easier way of explaining the concept. Teachers could assess their students using this applet by giving students specific shapes to construct on the geoboard. Then, students would have the count the individual boxes within the shape and report their findings. Students could even print their virtual geoboard for the teacher to look at. Some strengths of this applet are that it is not visually overwhelming for students, and it provides suggestions on the side for student activities. One weakness of this applet is that it may be somewhat difficult for students to initially figure out how to use. This could be dealt with by the teacher demonstrating how to use the applet beforehand. Additionally, the geoboard is not huge, so students can only construct a few small or one large shape on each geoboard without having to clear it.

App #1: 2nd Grade Splash Math Games

From the iTunes Store - free app

URL: https://itunes.apple.com/us/app/2nd-grade-splash-math-games./id463471155?mt=8

This is a free app that can be downloaded on iPhones or iPads. The app itself has tons of games and activities for various areas of 2nd grade mathematics. However, for this review, I will be focusing on the measurement, time, and geometry sections of the app. Within each section is a game relating to that mathematical topic. Students answer questions and then are rewarded with an exciting tone if they answer correctly. Then, they move onto another question. The questions are read aloud to the player, so headphones would be needed. Students can leave the game at any time and return to the home screen to select another math section to practice. The app itself is very easy to use and is extremely visually appealing. The colors are very bright and the sounds are cute, which may help students to stay interested and excited about using the app. I feel that this app is challenging for students, and may be too challenging at times. Some of the questions in the games may not have been covered yet, so students may not know everything. However, I think this is acceptable, because it will introduce them to concepts they will be learning in the near future.

This app would allow students to learn a lot about mathematics in 2nd grade. In addition to measurement, time, and geometry, multiple other sections of mathematics are available for students to practice. The questions asked in each section are clearly written for students to understand. Even if students did not know the answers, it is still good practice for them to experience. One of the strengths of this app is that it is very visually appealing and the sounds are exciting for students to hear. Another strength is that it keeps track of individual students that play, and even can send reports to an identified email address. The teacher could potentially use this feature as a mode of assessment for students using this app. He or she could have reports sent to their email about student progress to make sure that the students are actually utilizing the app correctly and making progression throughout it. If this did not work for assessment purposes, the teacher could have students log the areas they worked with on the app each time they used it. This way, the teacher could make sure students are using correct sections of the app. One weakness of this app is that it can only be downloaded on Apple devices. I tried finding it on my Samsung device, and had no such luck. Additionally, headphones would be needed when using this app because of loud sounds. It may also be difficult to keep track of student progress if multiple students are using the app on the same device.

Article Discussion #2

Thinking through a Lesson: Successfully Implementing High-Level Tasks
by Margaret Schwan Smith, Victoria Bill, & Elizabeth K. Hughes

In mathematics education today, the importance of creating higher-level, meaningful tasks for student use has become a topic of great discussion. Many teachers plan extensively to create high-level tasks that they believe students will greatly benefit from and take meaning away from. However, research has shown that once these tasks are actually implemented in the classroom, a lot of the higher-level thinking is lost. Once this occurs, no meaning is taken away from the task. It is no secret that planning for and actually carrying out higher-level mathematical tasks is difficult. So, how is it that teachers can work to do this successfully? This article discussed the TTLP, or the Thinking through a Lesson Protocol. In short, this is a framework for teachers to use when creating math lessons that utilize a high amount of student thinking in order to further promote understanding. In addition the TTLP calls on teachers to think much more deeply about their planning and the content that they are presenting to their students. There are three overall parts to the TTLP: devising the task, student exploration of the task, and sharing of results and discussion of the task. All three of these parts take a lot of planning time in order to be carried out successfully. The article recommended some suggestions for implementing the TTLP. First, working collaboratively with colleagues may be beneficial. A high-quality task should first be selected, and then an overarching goal should be devised. What are the students going to be taking away from the task, specifically? Next, potential student responses should be anticipated ahead of time. The group should think about all of the approaches to solving the task. Finally, the group should write questions that can both gauge student thinking and understanding of the task, and help students to take their results from the task further.

The TTLP is an interesting and effective way to create and implement mathematics lessons, from what I have learned about it. It is very evident that a lot of time and preparation goes into planning the lessons, but the benefits seem really great. The article described that many teachers reported that they felt much more prepared for their lessons by using this framework. Additionally, some teachers felt that the questions they asked their students throughout the lessons were much more meaningful and well structured, since they created them ahead of time and not on the spot. Many teachers also reported that they were able to take key elements of the TTLP and integrate them into other subjects that they taught. I found it interesting that one teacher said that after she had written quite a few lesson plans using the TTLP, she found herself actually internalizing the process and framework. This, in turn, saved her time in actually writing the lesson plans using the framework. I think the TTLP may definitely be a useful tool for me in the future when creating mathematics lessons that are meaningful to students, and can help me to better be prepared and gauge student understanding.

Discussion Questions:
- The article suggested to collaboratively use the TTLP for creating lessons. Why might this be?

- All three parts of the TTLP are critical, but which do you think is the most important? Why?

References:
Smith, M. S., Bill, V. Hughes, E.K. (n.d.) Thinking through a lesson: Successfully implementing high-level tasks. Designing and Enacting Rich Instructional Experiences. p. 11-18.

Article Discussion #1

A Model for Understanding Understanding in Mathematics by Edward J. Davis

This was an article about the process of understanding mathematical ideas and content in the subject of math. The definition of understanding can vary greatly from person to person. In addition, the act of understanding is often seen as a process. It is very possible for someone to just partially understand something, instead of not understanding at all or completely understanding. The whole idea of understanding in mathematics is a very important topic for teachers to be concerned with. We must constantly try to assess our students' understanding of the material that is being presented to them. This article talked a lot about "moves", or the way that a teacher actually teaches mathematics. Extensive research has shown that "moves" contribute greatly to student understanding. Moreover, it has been found that teachers behave differently when teaching different mathematical content. The article discussed that teachers tend to instruct students differently when teaching mathematical concepts, like place value, than when teaching mathematical procedures, like long division. Additionally, the article touched on several areas of understanding in mathematics. Within each area, two distinct levels exist. Level 1 are the more basic elements of understanding, whereas level 2 are the higher-order elements of understanding. These areas include understanding mathematical concepts, understanding mathematical generalization, understanding mathematical procedures, and understanding number facts.

Gauging student understanding is a critical component of assessment in the classroom, and this article really dived into the logistics of understanding. I never realized how detailed and specific understanding really is. It was interesting to read about the various areas of understanding and the individual levels underneath each area. The author made sure to note that teachers should not be under the impression that students must achieve all aspects of Level 1 before moving onto Level 2. This is not realistic, as some students may be achieving parts of both levels simultaneously. In addition, the article discussed some key suggestions for creating "moves" while teaching mathematics. First, for younger students, it is important to begin with physical representations, move to picture representations, and finally, utilize symbolic representations of mathematics. Teachers should also emphasize to students the importance of continually utilizing physical representations and pictures when explaining their thinking in math. Correct answers are no longer sufficient enough, and students need to descriptively explain their thinking.

Discussion Questions:
- Why is understanding such an important element to take into consideration when teaching mathematics?

- Understanding is described as being a process. Why might this be? Does a student either not understand something or completely understand something, or is there room in between?

References:
Davis, E.J. (2006). A model for understanding understanding in mathematics. Mathematics Teaching in the Middle School. 12(4). 190-197.

Reflection of Rich Activity & Lesson Plan

Creating rich, group-worthy activities to share with our math methods class was a very interesting experience. I have not yet truly thought about whether or not activities that I might implement in my future classroom are rich and group-worthy activities. In short, I believe that this means that the activities students are doing are both meaningful to students and utilize meaningful mathematics, not just utilizing simple calculations to find answers. Our rich activity was called kaboom money sticks. In this activity, students drew popsicle sticks with coins attached to them from a bin, and counted their coins to report a value to their group members. Then, their group members would either agree or disagree with that value. Additionally, students were keeping track of each of their turns by writing the value they found and drawing corresponding pictures of coins to represent that value. This activity was rich in content because it involved multiple modes of learning. Small group, whole group, and individual work were utilized throughout this activity. In addition, students were not just counting coins. They were getting feedback from their other group members, and also writing down their values and the corresponding coins to those values. At the end of the activity, students ordered their values from least to greatest and also identified their highest and lowest values. Students also talked about how they could have spent their money.

The two other groups also had great rich, group-worthy activities that were shared with the class! One group had a surface area tin-man activity. Small groups were give various objects, such as spheres, cones, or cylinders. Then, they were given the formula for the surface area of that object. Using rulers, students were to find the surface area of each object. Then, they would report their findings to the teachers and receive some tin foil. The task was to cover the objects in tin foil with no extra space or leftovers. I thought this was a really cool activity. It had students doing math operations, but also allowed students to be creative. Additionally, it gave students a visual in order to aid in their understanding of the topic. Surface area can be a tricky concept for students to grasp, and this activity was a great solution to that. The other group did an activity for middle schoolers involving ratios and rates. This was another very interesting and interactive activity. Basically, students were to measure different body parts of a Barbie doll or G.I. Joe Then, they were to create a scale drawing of the figure as if it were a human. We did these drawings on floor-to-ceiling whiteboards. This activity was super interactive and hands on, and I really enjoyed it myself. I think it would be very effective in a classroom, if students could be appropriate enough to respectfully complete the task. Both of these activities were very rich in content and group-worthy, in my eyes.