May Journal *from MTMS August 2009

Stephan, M. (2009). What are you worth? Mathematics Teaching in the Middle School (15)1, 16-23.

Teaching seventh grade students the concepts of integers and operations can sometimes be a challenge. Many different approaches have been used to teach integers, but the abstract concept of integers leaves students with an incomplete understanding of the concept. This concept is typically taught using a number line and adding inverses. It has been improved to use plastic tiles and a mat to group positive and negatives. For example, the class in this journal explored temperatures in Pennsylvania. When a temperature increased from a negative degree, students would place that specific number of degrees in tiles in the negative work area and repeat this process with the positive degree. The students would then cancel the opposites. However, this still leaves students with an inadequate understanding of the reasons behind why the steps are used to solve a specific integer operation. The teacher in this article demonstrated a new way that she taught her students this abstract concept of integers...by determining how much people are worth. Using Oprah Winfrey as an example, who had a net worth of $1.5 billion dollars, the concept of net worth was discussed with students. The concept of debt was then brought up during class. The teacher introduced the concept and asked the students if they think that she has to pay money on her house and other things such as her studio. The children then came up with other possible things that Oprah probably owed money for, and the teacher wrote the responses on the board. Once students understand key terms and how integers work in this setting, students will be able to determine how much net worth a person has. Students can explore and solve the problem in whatever way they like, but most students add the assets, add the debts, "and subtract one total from the other total." This leaves students with the understanding that debt takes away from the overall net worth someone has. Once students understand this, they can work into more complex problems. A dating game was used in the article where three bachelors were introduced. Each bachelor had a positive number for something, a negative number for a loan, a positive number for income, etc, and the students would have to use these integers to solve which bachelor had the greatest or least net worth. This is where students are introduced to symbols and can actually put meaning behind the work and operations. The teacher then has the students create their own net worth statement for a specific person. This can be assessed to see if the students understood the concepts.

When I was in school, we learned integers through symbols first, and then worked on a number line to determine subtraction/addition for integers with real numbers. However, this always confused me. In my head, I always thought of negative numbers as money. I think that by using an example like this for students, it helps them realize negative numbers are a part of everyones lives...even the rich and famous. This also can help students realize the thought process behind developing a good savings plan as well as a good budget plan which can benefit them in real life. Developing the concept in a concrete form and moving to the abstract idea of symbols seems to be more beneficial in this case. I think that if i had used this as a lesson in school in seventh grade, I would have understood integers and integer operations much better.

Manipulatives Blog

1. How do you hold every student accountable?

Using manipulatives to support mathematical teaching can be incredibly effective. To manage student accountability, I think that teachers should make sure that the different types of manipulatives are introduced. This way students have the opportunity to learn how to use the manipulatives correctly and apply the different manipulatives to their concept skills. Another way I think that students can be held accountable with manipulatives is through allowing students to freely explore and use the manipulatives during a free period or down time during a learning center. By allowing students free access to different manipulatives, students will feel more comfortable with the material during class time. Another way to manage manipulatives during class time would be to assign a group leader to the table to ‘manage’ the manipulative tools. Dividing roles in groups helps to give each individual child a role during instruction. By giving each child a role, they are held accountable to both their group and themselves.

Manipulatives are sometimes viewed negatively when they are viewed as ‘playing materials.’ In a sense, I completely disagree with playing being a negative concept with manipulatives. Children love to play and explore. In a journal article I read earlier this semester, a sand play table was used in a classroom. Children were able to go ‘play’ at the sand table and communicate with their friends (incorporation of process standards). Once the students became comfortable with the table and the communication at the table, the teacher would introduce new material to the table to go along with concepts he/she was teaching. For example, the teacher would put different sized beakers in the area with the sand. The teacher would allow the students to first explore the new objects through play and communication. Eventually, the teacher noticed that the students would compare the different amounts of sand that each type of beaker could hold. Students began to explore mathematical concepts on their own. As the students played at the table, they began to question certain math concepts. The teacher slowly but surely would go to the play table and pose questions to the students and listen to their reasoning (another process standard). Concepts were explored through a simple station where children were able to play and communicate. I believe manipulatives should be used this way. Students need the ability to play, explore, and communicate. Eventually, students begin to question which leads them to a more exciting role in the math learning process. Teachers can then guide the students with questions or comments regarding the students’ questions. Eventually concepts are constructed and developed over time. Process standards are incorporated naturally into this process.

2. Why not “hands-on”, why is it “hands-on, minds-on?”

When students become engaged in the manipulatives they are working with, they are stimulating their minds. Rather than just playing with blocks at a table, students are communicating and manipulating materials. Students naturally begin to question ideas on their own through this process. However, if good instruction is provided before the introduction of manipulatives, students can then use the manipulatives to enhance the instruction. If students have been learning about fractions in the classroom and then pattern blocks are used as an exploration manipulative, students will begin to apply their learning to the manipulatives. Students will start to question, “I wonder how many triangles will fit into a hexagon.” Students then work and communicate with each other and construct their concepts in a more concrete manor. Manipulatives are not going to save mathematics. Teaching still needs to be effective and teachers need to stimulate interest in the subject and show real-life application. When students have strong instruction, they can succeed in the use of manipulatives. Students will build onto the foundation through exploration. This is where students being to use “hands-on, minds-on.” When students use manipulatives in this way, they will only strengthen abstract math concepts into more concrete representations.

3. Process Standards

When students have choice and control in the manipulatives that they use, students can apply reasoning and proof to show how that specific manipulative was used to deepen the understanding of an abstract idea into a concrete representation. Manipulatives and representation work cohesively. Using concrete representations of abstract mathematical concepts and ideas helps students to see where the idea or abstract material becomes a concrete concept that they understand with meaning. Manipulatives shift students’ thinking from basic memorization of rules and facts and more into concept construction. When students get to use manipulatives, they deepen their understanding of specific concepts. The more representation methods that the children can apply using the manipulatives, the more constructed these concepts will be shown through the students. Students also get to use communication and problem solving as they work through using different manipulatives. As teachers, we need to give up some of the control with students and allow them to explore more freely. By doing this, students learn to think on their own and build onto prior knowledge to create a foundation for greater concepts. Instead of the teacher being viewed as the ‘experts,’ the students begin to think they are the experts. When they take responsibility for their own learning and become engaged through the use of manipulatives, educational concepts can be constructed at greater levels. Connections can also be made through the use of different subjects such as art. When students learn concepts in mathematics such as slides, flips, rotations, etc. students can create tessellations. Tessellations can be viewed as both artistic and mathematical. When we use manipulatives for content material such as slides, flips, rotations, turns, etc., students can visually see the concept as well and physically manipulate the object through exploration.

Errors Blog

Going through the math errors document was probably my favorite and most beneficial activity this semester. When working through numerous types of mathematics concepts, it is really interesting to see how some students make mistakes. I had never imagined teachers actually sitting and looking at an error to see where a student made a mistake; however, I found it to be fascinating to see the ways in which some students logically think and apply methods that do not work for specific problems. When teachers take the time to look at how the student made the error, they can begin to see where a specific student needs some additional guidance. Students often are taught the formulas or told ‘this is just how you do it,’ which does not enforce any core concepts. Going through the errors enforced why the process standards are so important. When the content standards are appropriately used in the classroom, students are able to see concepts in more depth. Instead of just working though problems, students understand the meaning behind the problems and are less likely to make errors such as the ones presented. Instead of the student focusing on the ‘rule’ or ‘formula,’ they can recall how to do the problem through reasoning and proof. Students then are less likely to use incorrect rules or short cuts to simply solve the problem.

Learning different methods for teaching subtraction, additions, multiplication, division, and using fractions is something that was specifically beneficial to me. Growing up I learned the formulas and did not learn meaning behind the formulas. When I try to recall when one fraction is bigger or smaller than another, I often get confused. Learning to visualize fractions as portions of a whole is crucial. Students in younger grades often have a hard time determining which fraction is bigger or smaller and how to add, subtract, multiply, and divide them. Learning the meaning behind these concepts and using models to help students visualize these concepts is definitely reinforcing.

Looking for patterns in student errors and putting meaning behind the errors can help teachers focus on areas in which students are having difficulties. When we realize the mistakes the students are making, we can help address the problem and put more emphasis to the meaning of the mathematics concept. Teachers can help students put an end to misunderstandings when they realize the pattern behind the error.

Technology Blog

Technology is becoming better and better is what seems to be exponential time. As a future teacher, it is important to look at how technology can positively and negatively affect a classroom.

Our generation today has so much experience with technology. Growing up with a computer, and learning how to play around on programs has been beneficial. When I look at the older generation, I can generally say that many adults feel less comfortable with technology than my generation. Most adults are less likely to spend time experimenting with learning new programs; although this is not always the case. This proves to be beneficial for the younger generation teachers who are entering school districts and know how to learn programs on their own. During my Novice teaching experience, my teacher had me spend an entire day teaching her how to use different technology programs. Knowing how to learn, adapt, and change with the times is important as a teacher. With technology expanding so quickly, it is important that we learn how to figure things out on our own or collaboratively.

With technology expanding so rapidly, it is hard to keep up with all of the new and fancy programs. Students in today's society are used to spending time on computers and using different programs. They are learning how to work the technologies we are using just as quickly. However, I believe that technology can prove to be a negative factor in education if used in the wrong way. Although technology is seen as the new 'fast pace way to life,' technology can actually waste a lot of valuable time for learning as well. When technologies fail, or computer programs do not sync, technology can prove to be a hinderance. Programs are changing and becoming better every day, which means formatting for each program is now different. From computer to computer, technology can prove to be a challenge. Also, when the internet fails, the computer freezes, or any other technical difficulty is faced, technology can waste a lot of time. It is important if we, as teachers, use technology in our classrooms, that we do so in ways that value the learning opportunities of our students. Two examples of valuable educational technologies would be the smart board and the geometer's sketchpad program.

The smart board is an incredible presenting tool that students typically get excited to use. There are many opportunities that the smart board provides that can enhance certain educational topics. My very favorite math technology is Geometer's Sketchpad. For inquiry learning, I believe this is the most valuable math technology program available. Students can explore different math concepts from beginning learning experiences to very advanced learning experiences. I have personally worked on basic geometry concepts in this program as well as advanced calculus ideas. I had a teacher at Bradley who provided guided learning worksheets that went step-by-step through directions on how to create a specific geometry idea or concept. Once we had used the direction to complete the activity, we had to experiment with the shape/design/measurements to figure out what the diagram represented and meant. I learned how to complete many geometry proofs using this program. This is very beneficial, because sometimes students focus more on the formulas rather than the concept. When students get to figure out the concept on their own while visually seeing representations of something they created, they take more value in their learning process. Overall, this is my favorite program. I think this provides many valuable learning opportunities for students and demonstrates one great way that a teacher can apply technology in the classroom.

Calculators are another technology that can be used in the classroom. They can also be great for students to use and check their work. Calculators now can visually show representations of ideas. Students can even explore concepts on calculators to prove how a certain formula or concept works.

I do not believe that technology can 'create the learning experience,' nor do I believe it can solve all the problems for education. I think that when used in the right opportunities, technology can prove to be very beneficial. Math applets provided by Illuminations and NCTM can help students work through inquiry learning to deepen their understanding of concepts. When teachers use technologies that are appropriate, educational experiences can be enhanced. Teachers really need to learn when and when not is appropriate to use technology, find a balance for how much technology should be incorporated, and relate to both the positive and negative aspects of technology in the classroom.

April Journal Article MTMS

Obara, S. (2010). Constructing spatial understanding. Mathematics Teaching in the Middle School (15)8, 472-478.

Students need spatial sense and understanding when using two-dimentional shapes and three-dimentional diagrams. Sometimes, students experience difficulties with spatial understanding becaues it is not intuitive. Students need practice visualizing two-dimentional nets and their transformations to three-dimentional figures. In order to help students understand how shapes are creating through folding, curling, etc. to a net, teachers often use geometry technology software. One teacher used a triangular pyramid and a cube and asked the students to create the nets to these figures. Using Geometer's Sketchpad, the teacher showed the students how to construct a net and modeled construction as well as use of the Sketchpad. Students were then able to explore and create their own net to try to figure out the net for the solids she presented earlier. Another approach the teacher took to developing spatial understanding was to pass out congruent three-dimentional pieces of a shape and had her students try to develop the net for a specific figure. Students often started solving this problem by using a square out of the pieces given to them in order to start building their net. Students were able to see that by creating two identical pieces from equilateral triangles into the form of a trapezoid, that they could construct a tetrahedron from these two 3-D figures.

Working with students to help them develop spatial understanding between 2-D and 3-D shapes will benefit them in their understanding of geometry. By having this type of understanding as teachers, we will better be able to use hands-on activities to teach our own students how to better comprehend spatial sense and how it applies to different forms of geometry. When students use inquiry based learning such as the type given in this lesson, students get to explore different nets and models as well as incorporate technology into the curriculum. This overall will help students to develop spatial sense.

April Journal Article TCM

Treahy, D.L. and Gurganus, S.P. (2010). Models for special needs students. Teaching Children Mathematics (16)8, 484-490.

Co-teaching is a specific type of teaching where a special education teacher and a classroom teacher work collaboratively in order to develop a more cohesive progam for planning, teaching, and assessing/evaluating. By using collaborative efforts such as co-teaching, the classroom becomes unified and represented as a whole unit. In one specific classroom, a classroom teacher and special education teacher taught mathematics together. The classroom teacher would teach her mathematic concepts or lesson while the special education teacher would ask her to clarify specific points or to repeat important concepts that may need to be heard again. By doing this, the two teachers incorporate a strategy similar to the reading strategy, think-alouds. These two teachers also take turns teaching important concepts and share the role of instructing in the classroom such as a team-teaching role would present. By doing this, students are gaining a deeper understanding of difficult concepts. By repeating difficult material and clarifying it in different ways, students get to see different aspects of the problem. The article lists a number of different models for teaching a classroom of integrated learners; these include:

Team: shared role between the two teachers

Alternative:

• one teacher works with small groups for re-teaching, enrichment, pre-teaching, etc. while the other teacher works with the rest of the class
• teachers use think-alouds to clarify and to help train special needs students to work through the problems rather than applying a random formula or idea that they have learned; (lack of metacognitive skills)
• great for hands-on learning
• great for working one-on-on or in small groups to clarify concepts for struggling learners

One teach, one assist:

• allows one teacher to teach while the other teacher walks around and observes and helps students who need it
• can beneift students with behavior issues
• helps to focus more on special needs students; gifted learners, special needs, ELL
• allows individual assistance for students
• great way to monitor progress

Stations:

• content is divided into different stations and each teacher teaches a specific station
• great way to incorporate the NCTM standards
• teachers can focus on an area of their strength to benefit deeper understanding for students
• stations often incorporate real-life integration
• allows for monitoring and reinforcement
• can help prevent math anxieties early on

Parallel:

• the teachers plan cohesively but then each teacher teaches a half of the entire class
• teachers must plan for all areas of learners
• teachers must think about how all students learn and how to relate to all students

As a teacher, co-teaching would be something that would be benefical in the classroom. It provides rich learning experiences for all areas of learners. During these types of instruction mentioned in the article, teachers use models and clarification so that all students have the ability to understand concepts. Co-teaching helps create a common instructional language that all areas of learners can understand. This helps to unify the classroom and help students become more engaged in the materia. C0-teaching provides a rich learning opportunity for a wide-variety of students. The talents and energies of two teachers can be effective in a class of different types and styled learners.

Observations with Checklists

Cole, K.A. (1999). Walking around: getting more from informal assessment. Mathematics Teaching in the Middle School (4)4, 224-227.

This article describes how informal assessments such as observations with checklists can be used to gain, organize, and document information about students' performances and academic growth in order to produce coherent stories of student progress. Teachers walk around the room to gain types of informal assessments on student learning and progress. By doing this, it helps teachers to receive immediate feedback of how students are thinking and working on problems. For example, a teacher described in the article walked around the room in order to view students' thoughts about a math problem they had been working on. When she walked around to each group and talked with them about their thoughts/ideas, she round that come children had confused meters and feet in their problem they had been working on. By observing students informally, teachers can gain a lot of information about how students are working and certain misunderstandings students might have. Teachers can also introduce new ideas and pose more in-depth/higher-order thinking questions to students. The teacher in this article continued to walk around the room during different days throughout the weeks and found that her questions she had posed to certain groups had been effective toward their learning. Students were able to use their journals to further explain their mathematical reasonings. Using checklists can help us to organize our observations as we walk around the room. Checklists can focus on mathematical tasks, students' communication, using math language, and their ability to work well with others in the classroom. Checklists help to observe students and find out how well students are using the mathematical concepts and standards in the classroom.