Applets Reviews

http://math.rice.edu/~lanius/fractions/index.html
Who Wants Pizza? A fun way to learn about fractions.
Grades 3-5

This applet is laid out in a very organized manor. Students can easily explore an introduction to fractions, different methods for using fractions, and practice fractions (addition, equal, subtraction, multiplication). The ease of use of this applet is fairly simple. The organization of the applet is laid out nicely and the practice problems show both visualizations of the parts in comparison to the whole. Children can easily practice their knowledge after reading about the type of fraction use they are interested in. The description for using the fractions for different methods is also very descriptive and shows visualizations to clarify word meaning. At the end of each online section there is a place where students can test their knowledge and submit their answers to see if they were correct. There is also a bonus question at the bottom that helps children think more critically. Students are able to view the correct answer after they have thought about the problem. Overall, this applet can be as challenging as the student or teacher wants to make it. Students can explore different avenues of fractions (addition, subtraction, multiplication, etc.) which gives them different options for practice and challenges.

I believe that students would find valuable learning in this math applet for numerous reasons. As I went through the math applet, it was refreshing to have the information laid out in both words and pictures. Through description using visual fraction bars with individual boxes colored in to represent part(s) of a whole unit, students can easily see how different parts relate to a whole unit. This idea can be challenging for students to understand, and I think that this applet gives a great understanding to the part-whole relationship. This applet also gives a valuable assessment for the students because it provides them with practice questions and an opportunity to submit their answers at the end of their practice. Students get to use the mouse to shade in different fractions that the applet asks for. Through use of the computer and visual aids, students get to investigate fractions and work through their learning on their own. I think that this specific applet provides students with a lot of benefits for learning. Students are able to explore the concepts of different avenues of fractions on their own. They can pace themselves and can work on different areas that they find interesting or a struggle. Students also get to practice their knowledge of fractions through the computer assessment provided. A weakness that this applet has is that students might get carried away with the different amount of options provided. If students have not yet worked with multiplication with fractions, they might want to click on it in advance and it may confuse them. However, this could also provide students with the opportunity to explore different avenues of fractions. The information provided would give them a good basis for beginning exploration on different fraction uses. I liked that the students could relate the fraction use to real-life food situations; which students would love. This applet also provides the teachers with a teacher's page which shows how students can use the strategies in the applets to build upon fraction concepts. The grids provided in the applet can be printed out and used in congruence with the activity as well as be used in part for an assessment. This teacher's page also describes the use of how this applet coincides with the NCTM standards for 3-5th grade students. Overall I found this applet very user-friendly and educational in the use of fractions.

http://standards.nctm.org/document/eexamples/chap6/6.5/index.htm
Pythagorean Theorem Math Applet
grades 6-8

This applet is very user friendly and is to the point and is clear and concise. Basically, students get to explore the Pythagorean Theorem through the use of an interactive figure with two little squares, a triangle, and a bigger square. Each of the two little squares are shaded a color. These little squares can be moved through the click of the mouse to fit into the bigger square. This shows how the different legs of the triangle end up creating the actual symbolic representation of the Pythagorean theorem a^2+b^2=c^2. Students get to use the visualization in order to examine and describe how the relationship in the diagram represents the actual Pythagorean Theorem. Overall this is a very basic applet provided by the NCTM which aligns with student objectives and shows a basic, clear description of the Pythagorean Theorem.

Student learning in this applet is a little less guided than the previous applet described above. This applet is for grades 6-8 and is a little more based on inquiry learning. Students have more of an opportunity in this applet for exploration and learning through their own exploration with minimal guidance. Guided questions are posed so that students can answer them as they use the actual applet. This will help guide them to their overall answer at the end of the exploration process. Students are presented with more inquiry based questions asking them to explore if the relationship found in this applet could possibly exist among any other shapes or figures. This applet builds on prior knowledge and then adds to prior knowledge through challenges presented involving higher-level thinking. The strengths of this applet is that it is clear, concise, and straight to the point. Some students may understand the visualization really quickly and can use the posed questions to explore further investigations, whereas other students can explore the Pythagorean Theorem through proof form rather than symbolic representation. This builds on students' ability to analyze different relationships presented through the subject of geometry and also helps build on their understanding of the actual proof for the Pythagorean relationship. The weaknesses of this applet are that it does not provide a whole lot of guidance. There is no real way for the students to assess themselves based on this applet to see if they are doing the activity right. Depending on what type of activity they are working on, open or close-ended, this could pose as a strength or a weakness. I think that another strength to this activity is that there are extensions at the end that the students can build upon through the use of this particular applet. I believe that this applet could provide more of a student type assessment through the website in order for students to explore their thoughts and then get feedback on their work. However, this can also be a strength based on how the applet is being used. The actual activity in this inquiry based applet could pose as a good assessment method for the teacher rather than self-assessment for the student.

Feb. Journal Article TCM

Kilic, Hulya, Cross, D.I., Ersoz, F.A., Mewborn, D.S., Swanagan, D., and Kim, J. (2010). Techniques for small-group discourse. Teaching Children Mathematics (16)6, 350-256.

The way that a teacher facilitates the students in the classroom is very important. Student learning is partially dependent on the way that the teacher teaches and the way that the classroom expectations are set up. When students learn to think critically, reason through difficult problems, and work together through communication, there are more opportunities to understand the basis for mathematic concepts. Through reflection, teachers can try different methods for facilitation in the classroom in order to meet individual classroom needs. Expectations in the classroom should be set up in a way where students really understand that investigating and exploring mathematics is beneficial to their learning process. When teachers use meaningful context and facilitate students in their learning, students gain a deeper understanding. Tips to teach critical thinking were given in the article. These tips included focusing on the students' thinking, listening to their responses while encouraging them to listen to each other, and providing appropriate feedback throughout the learning experience. Many other tips were given, but each teacher can create different methods of their own for meeting the needs of their particular classroom.

As teachers, it is our responsibility to get our students to think at higher levels, provide real-life context for learning, and facilitate meaningful communication. All of these relate to helping our students get the most out of their education. In order for us to maintain productive discussions in our classrooms, the article stated that we needed to model through our own organization and communication. We can use think aloud methods so that the children see how we as teachers think through mathematical problems as well. Also, by showing how listening to others' strategies is important, students can learn to find meaningful methods through others' thinking. By walking around the room and monitoring classroom discussions, we can ask questions that relate to individual needs. I liked the example in the article where the teacher asked why the student chose that walking around the circumference of the circle would be longer than walking the diameter of the circle twice. Instead of telling the student she was wrong, the teacher simply asked how she reached her answer and listened to her thought process. She guided further discussion through other group members by involving them in discussing their methods for solving the problem. Through basic facilitation and questions about reasoning, the student was able to realize her mistake through listening and reasoning. The teacher facilitated the discussion but did not intervene and give the student the answer. It is important to ask the right type of questions that do not lead students into a specific direction of thinking. Creating a classroom where student conversation is important and meaningful is key. To do this, we can make sure that each group is following the expectations that their conversations are explanatory as well as clear in their reasoning. We need to allow students time to think, investigate, and work! Provide students with feedback on their thinking so that they can learn the best ways to learn.

Feb. Journal Article MTMS

Cramer, K., Monson, D., Whitney S., Leavitt, S., and Wyberg, T. (2010). Dividing fractions and problem

solving. Mathematics Teaching in the Middle School 15(5), 339-346.

Many students learn symbolic meaning for concepts without actually learning what the concept actually means. This article show how fraction division learning can be accomplished through the use of problem solving and use of pictures and reasoning. Students learned the basics of fractions at the sixth grade level before moving onto complex fraction problem solving. The students in this article completed an activity with the use of different colored paper circles--- all were the same size. Each of the different colored unit circles were folded into different amounts of pieces to show representation to students that different fractions are still in relation to the one complete unit. Students were able to see visual representations of the fraction parts to the whole unit which made the concept easier to apply to problem solving questions. Once students were able to investigate problem solving questions through pictorial representations, they were able to work on describing their process through language. As the students learned how to describe their picture representations through words and eventually symbols, they were able to understand the use of rational numbers. Students built upon this knowledge and eventually were able to learn more complex ideas such as dividing fractions. Through problem solving groups, class discussions, and reflections, students were able to understand the flexibility of a unit. These sixth grade students showed the capacity to learn complex ideas on which more abstract concepts could be built.

I believe that having students investigate the different ways to represent fractions is crucial to their understanding of the basic concept. As a student, I learned to jump right into symbolic work and never learned the basis for my concepts. Upon reading this article and seeing different student representations of their use of fractions and fraction divisions, I was able to visualize and understand the part-whole concept. Having the students take the different colored circles--all the same size--and fold them into different fraction representations really shows the importance of understanding a unit and how different fractions make up a unit. As a future teacher, it is crucial that we let our students learn the importance behind the symbolic work that they eventually do. I liked the way that the teachers let the students learn through discussions and reflections. Learning how to problem solve through picture representations of fractions before learning symbolic work was something that I really admired. We should not underestimate our students and should give them the opportunity to figure out the concepts so that they can build upon that knowledge in order to use symbols as representations.

PBL Comparisons

#1 Summary of PBL--Lounging Around (grades 7-8)

In this PBL assignment, students in grades 7-8 are to decorate a lounge area given a set budget by the school board. The students in this PBL use different skills such as measurement, prior knowledge about geometry, and budgeting skills. This problem that was assigned to the students relates to real-life skills as well as applies to a topic area that may be of interest to most students. This learning takes place over a time span of 16 days and involves the use of communication in groups. During these 16 days, students learn about different math concepts through worksheets in order to understand the meaning of budgeting. Students also make 3-D representations, listen to a guest speaker, and present their budget plan for their lounge area. All areas of curriculum are presented in the PBL and student uses a variety of different mathematical concepts to complete the PBL.

# 2 and 4/Comments About Strengths and Weaknesses/Critiques & Examples/Changes I would make

There are many strengths to this PBL but I also believe there are many weaknesses. When reading over this PBL, I can clearly see how the principles and process standards are incorporated into the overall PBL. Students are presented with a difficult problem where they must develop a budget plan for a lounge. This lounge area would be something that students in 7th and 8th grade would find interesting and relatable to their real life. As I looked read through the overall document, I felt that the PBL concept was good but the organization and ideas behind the concept did not flow together smoothly. The PBL was student centered but did not give each student an individual part to be accountable for. By working together, the PBL was student-centered and collaborative, but lacked individual accountability. Overall, I did not feel like the problem at hand was the difficult part to solve, but more so the many different tasks that did not seem to flow together. By completing a variety of assignments and homework papers, I feel that the students would get lost in paper work and methods for trying to solve the problem. Even working together, the project seemed completely overwhelming without the established roles given to each specific student. After re-reading over what makes a good PBL and the steps that makes a good model for a PBL, I think that this version would be overwhelming and confusing. However, the concept behind the PBL started off in a positive way, but became too divergent and messy toward the middle of the project. Students did not even start working on their budget plan until approximately day 14. This did not give the students much time to complete their actual budget plan. If I were to change anything about this PBL, I would make it more organized. I think that by giving each student a role to complete, students would better be able to research a specific part to contribute to the group. I also believe that the guided questions could be more focused and less divergent. Students working together on this project might get lost in the overwhelming amount of worksheets and assessments. I think that by creating a more authentic assessment relating to journal work or a portfolio, students would be able to show their individual learning process while working together. I would make my mini-lessons more incorporative into the actual PBL assignment rather than just basic activities that teach concepts of proportions and fractions. I feel like the mini-lessons given in this PBL are examples of the way that most every day math teachers teach. Instead of exploring a concept, they are given a worksheet in order to find answers. These are just a few ways I would change this PBL assignment.

# 1 Continued

Summary of PBL--Operation "Redo the Zoo" (grades 5-6)

Students get the opportunity to take a field trip to the zoo and create their own design and budget plan for an exhibit they'd like to see at the zoo. There are four groups of students who each are given a specific role and task to contribute to their overall problem. These students will become an expert on their role and will research information and bring their thoughts back to the remaining group members who will also share their information. Over a 15 day period, students will design an exhibit and a budget plan for a specific zoo area. Students will keep math journals to document information and their learning process. Through mini-lessons, students will develop a portfolio. At the end of the PBL, students will present their design and budget plan to the rest of the class.

# 2 and 4/Comments About Strengths and Weaknesses as well as Critiques and examples

I believe that overall this is a strong PBL for grades 5-6. The overall problem is relative to what the children know and adds on to their prior knowledge through an interesting topic. The subject of 'the zoo' is always very interesting at any grade level and designing your own exhibit and budget plan would be something that not every student gets to do in real-life. Although this is not directly related to an actual activity they could do in real-life, the concepts behind the PBL directly relate to real-life concepts that the students could connect to on an every day basis. The PBL is very organized and presents a a difficult problem in which students are each given the role of some sort of 'expert.' The students then are able to decipher the problem, make a plan, and establish methods for solving the plan--all essential components of a model PBL. Students then are able to take their 'expert' part and gather information that relates to their individual role for their group. When students come back together as a group they use communication skills to decipher the best method for designing and budgeting their problem. Students build upon their research throughout the project and complete graphs, charts, pictures, and representations. The problem is messy and seems overwhelming at first, but when divided among 4 other group members over a period of 15 days, students can realize that a big problem can be minimized through communication and individual accountability. Higher level thinking is evolved throughout the entire PBL and students keeps logs and journals of their learning process. The teacher guides the overall PBL by giving different challenges that can help break the overall problem down a little bit while also enhancing the learning process. Students seem to always be working hands-on or interactively through the use of applets and technology. They also get to listen to a speaker from a zoo and go on an actual field trip to the zoo. Cross-curricular needs are met through many of the standards and goals presented in other subject content areas. All of the objectives presented are well-established and are grade-related. I really do not see any weaknesses throughout this PBL. If I had more experience with creating a PBL or seeing one in use I might be able to pin-point specific weaknesses, but from the knowledge I have gained about PBL's I think this is a great example that models all of the necessary components of a challenging PBL that is grade appropriate. I don't think I would change a whole lot to the overall organization of this PBL. I might add some assessment ideas to go along with the actual PBL presented. By developing a portfolio along the way, students might find the learning process more valuable. If during the mini-lessons students were able to add to their portfolio and obtain valuable information guiding them back to their problem at hand, I think they would be encouraged to keep working toward better methods. This is the only part that I would change. I loved how the group incorporated different units for other subjects as well as incorporated grade-appropriate standards and goals.

# 3 Compare and Contrasting

PBL: Lounging Around VS. PBL: Operation "Redo the Zoo"

The Lounging Around PBL seemed to be challenging but way too overwhelming. The Operation "Redo the Zoo" PBL was challenging but was guided and divided into a more reasonable problem for the grade level given. After reading the problem for the Zoo PBL it seemed overwhelming, but as the guided questions and division among research through 'expert members,' students would be able to work through the stress of the situation. I also believe that through working individually and together, it provides a better opportunity for the groups to discuss reasoning and proof behind their research as well as lots of communication skills. I believe that in the second PBL, the standards and objectives were more directly related to the problem compared to the Lounging Around PBL. Although both experiences related to real-life situations, the budget plans and the amount of student-centered learning was completely different. Students in the Lounging around PBL seemed to not have any rhyme or reason for completing their PBL. When reading the model steps for a good PBL, it seemed as if the Operation "Redo the Zoo" PBL was incredible. Not only did it make the problem student-centered, higher-level, and learner-centered, the PBL focused on a situation where students used guided questions and steps to create a plan to solve their problem. Overall, I believe the Zoo PBL was a much better representation of a model PBL than the Lounge PBL.

#5. Thoughts on how math is or is not the main focus of the work/level expected

Lounging Around PBL

I think that math is definitely a strong focus in this PBL but I believe the objectives and standards behind the actual PBL are not strong. For example, I see that students are able to use transformations and flips and turns in order to create a lounge, but I do not think the actual concept is directly related to creating and building a lounge. When students would build the lounge they would primarily be focused on measurements and budgeting. The math concepts seem to diverge into other subject areas as the PBL continues. The overall PBL did not seem organized with math concepts or ideas. Doing worksheets and being assessed via worksheets does not seem authentic. Although math was a focus in the PBL, I think it was too unorganized to really see all the math concepts involved in the entire process.

Operation "Redo the Zoo" PBL

I could easily follow this PBL and was better able to see the actual math that went into the steps and model of the PBL. I liked that the students were able to take a field trip to the zoo to first being the process of visualization for the process. By planning out their math ideas first and then building on their prior knowledge they have a stronger basis to build from. Dividing each student up into an expert on part of the problem also gives the student an area of mathematics to focus on. When students come together they can learn about other ways that their group members used the same math concepts to relate to their area of expertise. When students use math journals and create diagrams, charts, graphs, etc. it really shows the use of math concepts and standards for this specific grade level as well as the authenticity that the teacher will be able to assess the students on. By keeping math logs of budget plans, measurements, time periods, students use many aspects of mathematics to explore concepts that they are familiar with. Students also use higher-level thinking in order to build upon their prior knowledge during the steps of this PBL.

# 6. Assessment

Students participating in the Lounging Around PBL were assessed based upon checklists and rubrics. I think that the checklists provide a valuable assessment for the teacher to see how the students are doing during specific assignments. The overall rubric assessment incorporated every activity and did not really show how math content or processes were evaluated. Their explanation of how math concepts and processes were detailed and understandable, but when it came down to the actual evaluation and assessment I think that the rubric did not do justice. I do not believe there was enough criteria for assessment for it to be an authentic assessment. I also believe that more math concepts needed to be incorporated into the assessment. In the PBL described below, students were able to use journals and portfolios to show their math concepts learning as well as their processes for completing the PBL. I think that this is a more authentic form of assessment based on the standards, objectives, and goals.

Students in the Zoo PBL were assessed both formally and informally. I liked how the students were assessed daily through student journals so the teacher could specifically which students may need more guiding throughout the process. This is where adaptations for some students could be realized and the teacher could better guide the student. Formal assessment thought a portfolio also shows a different type of assessment and quizzes were also given based on new knowledge learned. By incorporating both formal and informal assessments the PBL became more authentic. Students were also graded on their presentation of their overall PBL. Students are able to show their reasoning and proof through these types of assessment as well as show their communication and understanding through their presentation. Work in the portfolio also gives an example of how the students built upon prior knowledge and made connections to other subject areas. Overall I feel that the content standards and the processes were clearly shown through assessment in this PBL.

PBL Journal Article Review

The teacher in this article created a PBL based on the book, Harry Potter. What I really liked about this PBL assignment was that it was cross curricular. The teacher took their literature reading, Harry Potter, and turned the book into a real-life budgeting problem. The students were fascinated with the book and were excited to be assigned a character with a specific budget. Budgeting was a topic that was not completely unfamiliar with the students. The teacher introduced the budgeting problem and explained the term budget and how it applied directly to their real-life. The children were given a character with a specific budget amount and had to create a budget for their entire year worth of school supplies for Hogwarts. Students were given a price sheet showing the amount for specific items. Some of the items were required during the school year, and others were luxuries that the students could purchase if their budget allowed. This shows a great use of PBL in a classroom. The teacher took a real life problem and related to the world of her students through Harry Potter. In order to address the budgeting problem for the school year, the students had to use prior mathematical knowledge in order to break down the problem and start to find methods to solve the problem. The teacher noted that many students used different methods for solving their specific problem. One student decided to save some of her money and put it in the bank so that she could have access to more money throughout the school year. Other students wanted to buy luxury items and based their budget on trial and error. Students helped each other with their differing budgets and their methods for working on their problems. The teacher was more the facilitator while the students lead their own learning experience. Once the students had spent two class periods working on the PBL, the students discussed their methods to their specific budget plan with the class. In the debriefing period of the PBL assignment, the teacher stated that her class was able to make higher level connections about budgeting based on their own experiences working with their ‘imaginary’ Hogwarts budget. One child said that his parents had to budget a member in his family going to college while another student said that a previous teacher budgeted for school supplies for the entire classroom.

I really liked the entire PBL assignment in this article for numerous reasons. The students were already reading the book, Harry Potter, for their literature class. The teacher saw that the students were very interested in the subject of Harry Potter and incorporated it into a real-life budgeting problem. By using a higher level topic such as budgeting, the teacher was able to help the students understand the topic by bringing it down to their level. Through specific characters in the story, the teacher was able to assign each child a budget based on their character. I really liked how the students were all given a different budget so that they had to use their own method to buy supplies for school although the concept method for all the students could be similar. Students were able to discuss with each other the different methods and approaches they were using to buy their school supplies. By having required items and luxury items, students had to learn to manage their money appropriately. Students went about the problem in many different ways which was very interesting. Some students started with required items while others tried the method of trial and error. Allowing the students to discuss their methods showed other students in the class better methods for buying their school supplies based on their budget. The teacher really focused on the PBL being student centered while she took a step back and became the facilitator. Students learned the learning process through trying different methods for buying required items/luxury items with their given budget. In order to make the problem more difficult, the teacher stated that you could use ‘wizard’ currency which could be valued different than our American currency. This could be applied at a higher grade level as well as made cross curricular when studying different cultures and money. I did not believe that there were many weaknesses to this PBL method. I think that the teacher focused on the main criteria for a PBL assignment and did a great job relating the topic of budgeting to her students. The only weakness that I could really see was that the students did not really relate much to the subject of budgeting yet. However, I do not believe it is a bad idea to relate this subject to younger aged children if it can be done through their world. I think the teacher did a great job of relating a bigger concept to her students’ real world.



Beaton, T. (2004). Harry potter in the mathematics classroom. MathematicsTeaching in the Middle School 10(1), 23-25.

PBL Readings

Problem-based learning is where students are more involved through the learning experience while the teacher poses as the faciliator during the experience. This instructional strategy is student-centered and allows the investigation of challenging real-life problems. Students are presented with a difficult problem where they must work together to come to a solution. In order to do this, students must use communication skills to figure our what the exact problem is, what they already know based on prior knowledge/experience, and how they are going to get to where they need to end up. In the classroom, a basic KWL chart can be used to find out these basic questions and needs. The teacher helps guide through asking questions as well as directing students toward resources for finding the answers. Students will then use resources to collect information that can guide them toward finding methods for solving the challenging problem that they have been presented. However, the teacher allows the students to take the lead role in solving the problem. Each student uses communication in group work but also is responsible for their own part in the learning. These types of problems that students are faced with do not have a simple 'correct answer.' The students must work together to find potential solutions to the problem and then conclude on the best method for solving the problem. Ideas are then presented and the students get to pick the solution that best works with their problem. I liked in the readings that the teacher was preseented as the 'coach' in the teaching process. The teacher is there to guide and encourage the students while the students must actively find the solution. When the students come up with the best solution to their problem, it is important that they reflect on their overall problem and the solution that they fit to the problem. I like that during this process students are actually learning the actual learning process. PBL's can be used in any type of situation that involves a problem. Any type of career can use a PBL for a problem that they are facing. We work on problem-based learning on a day to day basis. We may not be writing down exact steps to how we are solving problems, but when we encounter a basic real-life problem, we being this process. For example, if you have no water pressure in your house, you must find the root cause of the problem. In order to do this, you need to develop a plan. This plan includes the process of figuring out what you know, what you need to know, and how are you going to get to a solution. There potentially could be many solutions for how to fix the problem, but the best method or approach is needed. I may gather background knowledge to try and save some money and then involve the use of other people, such as friends or family to try and solve the problem. Eventually, when I come to the solution for the best method, I am essentially using this same process of PBL that we are using with our students in education. PBL's are important in day-to-day life which is why it is critical that our students be prepared to use this type of learning daily.