Analyzing Geometric Patterns

Friel S.N. & K.M. (2009). A Framework for analyzing geometric patterns tasks. Mathematics Teaching in the Middle School. 15(1), 24.

This article begins by impressing the importance of students thinking algebraically in the middle grades. It explains that thinking algebraically is essentially recognizing patterns and devising functions to express those patterns. In the middle grades, students work largely with patterns that grow, and using geometric figures is an effective tool for helping students recognize patterns of growth and develop their functions. The authors of this article developed a framework for helping students find patterns and functions. First, they discuss patterns and develop rules to match their patterns. Then they record their information in tables. Finally, they organize their information from the tables into graphical representations.

I always struggled with figural reasoning when I was a student of mathematics. I could figure how many smiley faces were in the 10th stage, the 43rd stage, the millionth stage - but when they asked for the nth stage they lost me. For this reason, I would use the information in my future classroom to help students understand what the nth stage really is - it is the result of some pattern they have found, translated to a numerical formula. I think that using geometric figures is an excellent way to teach and reinforce algebraic thinking because not only does it provide a visual for the students when seeking patterns, it helps them connect algebra to geometry. For these two reasons, I would definitely use this information in my future classroom.

Manipulative Activity

When using manipulatives in math education, it is important to hold all students accountable throughout the activity. I would manage this challenge by first posing a question in conjunction with the manipulative for the students to consider. Then I would circulate the classroom asking more questions as necessary to keep the students thinking mathematically. I might also have students record their thinking the way we did during our manipulative activity in ETE 339.

Another point to consider in math education is the concept of "hands-on, minds-on" learning. While hands-on activities successfully engages students and prevents them from disrupting the classroom flow, the hands-on activities should also engage the students cognitively. Not only should the students explore with manipulative materials, they should think about the mathematical processes involved with the manipulative.

Using manipulatives in math education can easily involve all five process standards. Students use manipulatives to create and explore a variety of representations of math concepts. Manipulatives enable students to connect math concepts between the five content areas, guiding them to understand that math is not a host of disjointed ideas, but a network of interconnecting ideas. Working with manipulative materials lends itself to students working in pairs or small groups, enabling communication about math concepts among the individuals in the group. Manipulatives serve as a tool for students to use when explaining math processes, and teachers can provide manipulatives for students to use when solving problems, as manipulatives often include generous opportunities for activating prior knowledge to form new knowledge.

Error Analysis

I thought that the error analysis activities were some of the most beneficial aspects of the math methods course. It had not occurred to me before to look for patterns in the students' errors. I also really enjoyed learning different ways to teach addition, subtraction, multiplication, and division. I learned that often, the reasons students make errors is because they do not understand the processes of their problems. They simply try to memorize the process.

In my future classroom, I anticipate teaching students through the use of manipulatives and multiple representations so that my students more completely understand the operations they perform. I also would like to apply the way Dr. Grant taught us to teach the algorithms because I think that way makes more sense than the way I was taught. Finally, I hope to remember to look for patterns in my students errors, as those patterns provide insight to the students' thinking and can provide direction for me as a teacher.

Technology

The most useful forms of technology that I used in Math Methods were the basic communication tools - Google Docs and email. Working with large groups of people, it would have been a nightmare to prepare our projects without Google Docs. There was a number of times when several of us were editing the Google Docs at once - something we could not do with the Wiki. We also used email and chat extensively to communicate about meeting times and questions regarding our projects. These two tools - Google Docs and email - were life-savers for this class.

Other technology sources that we used included the wiki page, smart board, Geometer's Sketchpad, calculators, Sakai forums, and NCTM resources. The wiki page was frustrating because only one person could edit at a time. Therefore, if one person had three or four hours of information to add to the wiki, the other group members were handicapped for that period of time unless they stole the lock from the other person. If they chose to steal the lock, the other person's information would be lost. Google Docs was far superior to the wiki. The smart board is a tool that I have already used in Novice Teaching, and I presume that I will be using it in my future occupation. I appreciated the opportunities in Math Methods to play around with smart board. Geometer's Sketchpad is an excellent tool for helping students grasp geometry concepts without having to create their own representations by hand. Learning about this tool was very beneficial, and I can see myself using it in my own classroom some day. The calculators that could figure fractions are a long-time awaited invention. Now students will be able to discover math instead of simply doing math, by experimenting with LCM, GCF, adding, subtracting, multiplying, and dividing fractions, and other information regarding fractions. The Sakai forums were a useful tool for claiming our articles, although it would have been just as easy to send and email to relate that information. Finally, the NCTM website was great, with its vast resources including applets and illuminations. I am excited to know of such a rich resource where I can find activities to supplement or teach math concepts.
Overall, the use of technology in Math Methods had given me tools to fulfill the following quote: "The art of teaching is the art of assisting discovery." I look forward to using technology in the future as a means to "assist discovery" in my students.

Supporting Language Learners

Supporting Language Learners from Teaching Children Mathematics

This article suggests two methods of assisting language learners in the mathematics classroom. One method involves helping the students through their English language acquisition. Some strategies include using advanced organizers, bi-lingual organizers, explaining mathematical terms as homonyms (some,sum), visual cues, and adjusting teacher talk (repeating, slowing down, simplifying, consistency, and avoiding idioms). The article discussed two tasks which assist language learners in developing their terminology. One involved a story with manipulatives relating to the story (fishing) and checked the students' understanding of "more than" and "less than." The other activity shows how teachers can change the wording of directions to a problem to be more consistent with language learners vocabulary. A second method of helping language learners in the classroom is to promote a low-anxiety classroom environment. Teachers should reassure the language learner that mistakes are common. They should teach their students how to applaud and praise each other rather than ridicule mistakes. Routines also help language learners feel comfortable in the classroom - knowing what will come next in the day. Finally, teachers should use signals and assign classroom buddies to ease tension that language learners may feel .

I have already had the opportunity to apply some of these strategies - not with language learners, but with a student with Asperger's syndrome who struggled processing words and dictating his thoughts. I noticed that assisting him with understanding words was always necessary, regardless of the subject. He was an intelligent child, but simply needed extra help with vocabulary. I also noticed that he performed much higher when he was at ease in the classroom. If he felt like his peers were watching him or making fun of him, he lost all ability to function. I am sure that language learners will pose different experiences than I had with this student, but I do believe that applying the strategies for making the classroom a comfortable place and helping them with their vocabulary will assist them greatly in mathematics.

Making Technology Work

Making Technology Work from Mathematics Teaching in the Middle School

This article encourages teachers to ensure that their use of technology follows this criteria: it allows students to do something they could not have done before, or it allows students to do something they could do before in a better way. It warns against using technology when students could perform a task more quickly without the technology; yet it promotes using technology to help students use higher-level thinking and to visualize math better. The article provides an example of a beneficial use of technology - The Weather Project. In this project, the students must create a pamphlet about the weather in five different regions of Croatia. The project not only connects to other subject areas, it allows students to work with real data and to use technology to interpret and represent the data. The students took ownership in the project because it was a real to life experience.

I appreciated that this article addressed the need to use technology wisely. Technology can be powerful, but it can also be debilitating if it does not meet the criteria for beneficial uses of technology. If I take anything from the article, it will be to ask myself as a future teacher whether my desire to use technology truly allows students to do something they would not be able to do otherwise. Does the technology promote higher-level thinking? Will it enable students to engage in real-world applications of mathematics? Will it engage students in exploration and discovery? Technology cannot replace the brain - and teachers must be careful that they use it in a way that enhances learning.

Assessing Understanding Through Reasoning Books

Sally K. Roberts and Carla Tayeh

March 2010, Volume 15, Issue 7, Page 406
http://www.nctm.org/eresources/view_media.asp?article_id=9178

In this article, teacher candidates are challenged to make a booklet of reasoning and proof which show that they know how to communicate mathematical concepts and ideas to a larger audience. Their instructors do not allow them to say, "You know what I mean" even if it is obvious that they do know what the student means. It is important for teacher candidates to be able to understand a problem at a deeper level than by simply giving the correct answer. Later in the article, middle school students were challenged to use reasoning and proof to solve a problem without a clear-cut answer. Some of the students found an incorrect answer using faulty reasoning. Some of the students came up with the correct answer using good reasoning, but without providing adequate defense or proof to show how they found the answer. Finally, some of the students used good reasoning and proof to find and convey their answers to a wide audience.

I think that reasoning and proof can be the most challenging, yet important, component in mathematics. Because communicating our reasoning takes time and practice, teachers should begin to help students communicate their thoughts at a young age. I hope that I can resist the temptation to nod at a student and say, "Yep. You have the right idea. I know what you are trying to say. Good job." Students need to be given time to sort through their reasoning - to justify their thinking. This is perhaps as important as coming up with the correct answer.

Sand and Water Table Play

Teaching Children Mathematics
March 2010, Volume 16, Issue 7, Page 394

In this article, the teacher emphasizes the value of play in mathematical learning of young children. Children left alone to play may not develop mathematically, but children playing with a teacher in close proximity prompting them to think while playing offers great possibilities. This teacher kept a table in her kindergarten classroom filled with sand. She placed geometric containers, beakers, and bottles in the sandbox. At first, the students were allowed to play and the teacher simply observed the students' discourses and mathematical play, and considered possible math interventions that would naturally flow from the students' play. Then, the teacher played with the students. She asked only a few simple questions to enrich their play - to get them thinking about their play. Finally, when she noticed a frustration with some mathematical concept within the students' playing, she developed lessons from those frustrations to meet the state standards for kindergarten mathematics.

I am always in favor of hands-on learning, as long as the play truly causes learning and is not simply play. This article provided a good example of how to use play to interest the children and to make their learning relevent to them. I think that I will attempt to use such student-centered learning in my future classroom of (hopefully) intermediate or middle school students. I may not provide as much time for free-play with older grades, but I hope to allow a measure of freedom so that I can learn the subjects that matter most to my students, thereby creating lesson that they can relate to.

Portfolio Assessment

Damiani, V.B. (2004). Portfolio assessment in the classroom. In Helping Children at Home and School II: Handouts for Families and Educators (S3, 129-131). Retrieved from http://www.nasponline.org/communications/spawareness/portfolioassess.pdf

Using portfolios is an authentic method for assessing student work, from primary grades through college levels. A well-done portfolio contains a collection of student work from a specific topic or theme. Portfolios are valuable in that they show what students can do - not just what they know.

The portfolio process involves several steps:
1. Decide on a purpose or theme for the portfolio
2. Consider what samples of student work might best illustrate the goal being assessed.
3. Determine how samples will be selected.
4. Decide whether to assess both the process and the product, or just the product.
5. Develop an appropriate scoring system.
6. Share the scoring system with the students.
7. Engage the student in discussion about the product.

Portfolio assessment has several advantages and challenges.

Advantages -
1. Assesses what students can do - not just what they know.
2. Engages students actively
3. Fosters student-teacher communication
4. Fosters depth of exploration and learning
5. Helps parents understand the educational process
6. Provides goals for student learning
7. Offers an alternative to traditional testing

Challenges -
1. Establishing scoring systems that are reliable over time and with different raters
2. The process is time consuming.
3. Portfolios portray depth of student learning, but not breadth - assessment is over a narrow theme.
4. Keeping the process fair, when some students may have advantages in their homes, such as parental support and access to technology.
5. It is difficult to compare the students' learning to students across the nation.
6. Students may not accurately self-assess their work if they know it is being assessed by the teacher.

Video analysis - graphing

The purpose of the activities in this video is for students to represent graphs in a variety of ways and to extract meaning from the graphs. The students first create a story about the graph, which helps them learn that the graphs depict a relationship between two quantities. Next they create tables as another representation of the graph, and finally they develop an equation to show a fourth representation. The content of the graph is therefore shown as a graph, a table, a story, and an equation – four representations!
Lesson Analysis 1: Describe the primary task in this lesson and identify the mathematical skills and concepts that this task is designed to develop.
The primary task in this lesson is for students to create a sequence for their classmates and teachers to figure out missing numbers. In order to meet that goal, students will develop the skill of representing a concept in several different ways and problem solving to find a generalized equation for that sequence.
Reflective Task 1: Describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught.
I think the primary task in this lesson was very appropriate because the students had to use various representations to display the meaning of the graph, communicate mathematically in groups, and problem solve to figure out a sequence that would be a challenge even for their teacher. Not only were the process skills used, but the students developed a better understanding of the relationship between two variables in a graph because they attached personal meaning to their graphs through stories.
Lesson Analysis 2: Describe what the teacher does to support learning while students are working in groups.
The teacher supports learning by asking guiding questions to assist the students’ thought processes. She did not simply tell them the answers or tell them how to think, she guided their learning through questions. She also encouraged the students to communicate mathematically with each other, telling them to “talk about it in your groups.” As she rotated between groups, she listened to their thinking patterns to understand where they were misinterpreting or forgetting some important aspect of their problem.

Once again, I benefited from watching the videos because I could actually see the teacher leading the children to understanding. It is helpful to witness someone carrying out the strategies we learn in school. Observing her decisions motivated me to one day guide students to learning through the process standards rather than teach them the way I was taught.

Matching Clocks

Time - Match Clocks: http://nlvm.usu.edu/en/nav/frames_asid_317_g_1_t_4.html?from=grade_g_1.html

In this applet, students change the time on a digital clock to match the time on an analog clock, and change the time on an analog clock to match the digital clock. The objectives are that students will recognize that both representations show the same time and that students will read and create the time correctly on both types of clocks. It is easy to use, although I question that PreK students would be able to meet the objectives. The visual presentation is helpful, in that the clocks sit side-by-side where students can easily see the relationship between them. Once students have plenty of practice reading one type of clock, they are challenged to switch gears and read the next type. This could be difficult for some students. However, the activity really does not present sufficient challenges for first and second grade gifted students.

The information provided for the students is sufficient, since they are not at an age where they read proficiently and would not appreciate wordy directions. However, there is no information for teachers. Since the activity is straitforward, teachers should be able to use the applet to reinforce the relationships between the two representations of time, though. It promotes growth in understanding time, but does not require higher order thinking. Instead, he students practice one concept over and over. This repeated practice may reinforce what students have already learned, but does not push them to higher levels.

Pattern Blocks

Pattern Blocks: http://nlvm.usu.edu/en/nav/frames_asid_169_g_1_t_3.html?open=activities&from=grade_g_1.html

This applet designed for PK-2 grade students deals with making and describing patterns. Students drag pattern block shapes into a working area, creating a design of their choice. After playing with the shapes, the teacher instructs the students to make a wall with squares, a star with diamonds, and other simple shapes. While the names of the shapes are reinforced through the teacher's instructions, the teacher gradually guides students to create patterns based on variable names, like AAA, ABA, ABCCABCC. It is easy to drag the shapes, but PK-kindergarten students may have trouble rotating the shapes to fit into small areas. The visual display is attractive and the applet could keep students engaged. The challenges presented may vary depending on age level. For PK-K, simply learning the names of the shapes could be a challenge. For first and second graders, creating patterns in response to a teacher's use of variables may be a challenge.

The information presented to the students in this applet is focused strictly on shapes. Without teacher instruction, the applet might be a fun activity for children, but of little mathematical value. A simple lesson plan is provided for teachers to guide students to using mathematical discourse. The activity could potentially help students gain understanding of using different representations for the same concept (using letters instead of shapes to convey a pattern). It also provides early introduction to variables - using letters in place of a concept, albeit not a number in this case. It does not have much potential for improving problem solving skills, so would quickly bore a gifted student. The teacher could adapt the lesson, but the nature of the activity does not lend itself to problem solving, reasoning, and proof.

Mathematical Discourse

"Techniques for Small Group Discourse" from Teaching Children Mathematics

The subject of this article is using mathematical discourse in classrooms, including discourse between teachers and students and between students and other students. Two case studies highlighted two different teachers' responses to their students, who had just worked on problem solving in small groups. Some of the teachers' comments were helpful, in that they facilitated mathematical discourse, critical thinking and responsibility for learning; others of their comments hindered these qualities. The helpful responses from the teachers included requiring explanations from the students to help them reason through the math processes on their own instead of relying on the teacher. They also listened to the students' justifications, but did not evaluate them. Instead, one of the teachers in particular, turned to another group and encouraged their input. This promoted more mathematical discourse, while helping the students to take ideas from each other to note their own errors. One unhealthy response given by one of the teachers was that she did not allow her student to struggle and struggle with her own thought process. Instead, she jumped in and redirected her thinking so that she then figured out the answer using the teacher's preferred method. Not only did this hinder deep understanding contrived from the student's own reasoning, it halted mathematical discourse that could have stretched the student's understanding.

I definitely agree with the purpose of the article - that mathematical discourse improves student understanding and that teachers should be facilitators. It is my hope that I will allow time for my future students to engage in conversation and exchange ideas when working through math problems. Teaching math in this manner may not unveil immediate leaps in testing, but in the long run, I am sure that student understanding and achievement will increase.

Dividing Fractions

Summary of "Dividing Fractions and Problem Solving" from Mathematics teaching in the Middle School

This article outlines how teachers helped a class of sixth graders comprehend the algorithm for dividing fractions (invert and multiply). In most textbooks, the article states, fraction division is explained with one or maybe two pictures showing why the problem turns out a larger answer than the whole (1/2 divided by 6 equals 12). Some students may gain a superficial understanding of dividing fractions from these pictures, but most of the time, the vague examples go over their heads as the teacher dives directly into teaching the algorithm. Unfortunately, many students are detached from any problems dealing with fractional division because it makes no sense to them - memorizing the algorithm becomes their crutch. In this lesson, however, students solve problems using their own pictorial representations. Not only do they solve one or two problems, they solve many problems and explain their answers verbally. After sharing together as a class, the students begin to see patterns in their math processes. They naturally find the common denominators between the numbers they are working with, and naturally note correlations between the written symbols and their pictorial representations.

At first, I wondered how students could really comprehend fractional division. It seems like such an abstract concept, and looking back on my own education, memorizing the algorithm worked great for me. I never struggled with dividing fractions. It was easy - just invert and multiply! But as I continued reading and computing the problems pictorially as the children in the article did, I began to notice patterns that I had never realized before. In addition, for the first time I felt like I understood why we invert and multiply. Working through the problems definitely assisted my own comprehension. I can only imagine how helpful it must be for a sixth grader to build understanding of fractional division through personally created representations. Indeed, I will keep this article and use it in my future classroom. Not only will I use the problems given to work on, I will use the method of teaching fractional division presented in this article. In so doing, I will be addressing many of the mathematical process standards: the students will create, use, and translate representations; they will communicate mathematically in both written and verbal contexts; they will problem solve by determining what the part and the whole is and what they are searching for; and they will reason and prove their answers with their representations and their words.

PBL Comparison

Lounging Around - Summary
In this PBL lesson, students will design a lounge for their school. This problem is quite broad, especially since the students must keep within their budget. The problem is open-ended, with no one right answer, and the students must first decide what "mini problems" are within the big problem. The lesson includes guiding questions that lead students to connect with prior knowledge. However, their prior knowledge is not sufficient to resolve the many details of the project. They must then inquire about their options, which leads to mini lessons. The mini lessons are student centered, in that they must research and collaborate to find information (like looking through the classified ads for a job). After connecting to prior knowledge, learning from a number of mini lessons, and collaborating with classmates, students do further research via technology and finally, synthesize their data and information to create their plan.

Lounging Around - Critique
I have learned that PBLs should be open, ill-structured problems with more than one possible solution. This problem reflects that characteristic, as well as the need to first define the problem and second, devise a plan of action. PBLs are also supposed to involve higher order thinking, which is encouraged through the provided teaching questions. One aspect of the lesson that I do not recall reading about in PBLs is having a large number of mini lessons. The mini lessons in this PBL are great lessons, but they really seem to guide the students down one certain path. In reality, I think it makes sense for the teacher's questions to guide the students to certain concepts. I just do not know if true PBLs allow the course to go in several directions.

Recycling At Our School - Summary
In this PBL lesson, 3rd, 4th, or 5th grade students will start a recycling program at their school. The students must decide what they will recycle and how much they can recycle. The teacher will ask guiding questions that lead them in the direction of finding volume for the storage containers. They also must learn how much they will earn from recycling and estimate how much can be recycled in a week's time. The students will compile all of their information into a report and present it to the class to finish their project.

Recycling At Our School - Critique
This lesson started out with a very open, broad and messy question - just like PBLs are meant to originate. However, the teachers provide more structure at the beginning than I remember reading about in articles about PBLs. For example, right from the beginning, the teacher dictated the process they should take, instead of asking them questions that lead them to come up with their own process. Also, the mini lessons were not as inquiry based as a PBL lesson would have. The volume lesson is teacher directed rather than student-derived. The lesson on fractions was better developed because the students helped discover the fractions instead of the teacher just telling them how to find the answer. They also experienced modeling, guided practice, and understanding checks. The independent practice did not seem to fit PBL because it was not relevant to the students' lives.

Comparison of the 2 PBLs

"Lounging Around" fit PBL criteria very closely. It was student centered, inductive, involved the students' prior knowledge, and collaborative. I would feel comfortable using this plan in my classroom. "Recycling At Our School" was not as student centered because the teacher instructed more and asked fewer guiding questions. The teacher was still "in charge" of the project, while the students took possession of "Lounging Around" and the teacher simply facilitated. Also, "Recycling At Our School" involved less of the students' prior knowledge during the mini lessons. The teacher told them the new information instead of guiding them to discovering that information through prior knowledge.

Changes to "Recycling At Our School"
I would have begun this PBL by making the topic more personal to the students. 3rd, 4th, and 5th graders only care about recycling because they know that their teachers want them to care. I would ask them questions like, "How much garbage do you think each of us makes each week? Each month? Each year?" "How much garbage would it take to fill this classroom? How long would it take?" When the students realize how quickly their classroom would fill up if all of the students' garbage was dumped in the room, they might start to get the picture of why recycling is so important. In the volume mini lesson, I would have led my students to access their prior knowledge - perhaps their knowledge of lengths or of area. I would have had tools available for the students to figure volume such as large, empty boxes, yard sticks, base ten blocks - anything that would help realize the concept of volume. I would not make them use the tools, but they would be available to the students.

Math as the focus
I feel that math is definitely the focus in "Lounging Around". Already on the first day, students are in groups thinking mathematically. The guiding questions direct their minds to graphing and money. On day two, they take their mathematical thinking a step deeper, applying prior knowledge and choosing the best representations for displaying their polling data. Each day, the students apply prior knowledge to seek solutions to their problems, and when they learn a new concept like creating excel charts, the students synthesize prior knowledge with new knowledge. The knowledge they access each day is always mathematical.
Math is not always the focu in "Recycling At Our School". The opening activity was a quiz testing students' prior knowledge about the recycling problem. They were indeed accessing prior knowledge, but that knowledge was not mathematical. Most of the daily activities involved math, however. On day 7, they have to use estimation to figure how much they can recycle each day.

Assessment
"Lounging Around" had a variety of assessment methods - checklists, teacher observations, and rubrics. The final rubric was thorough and clear, assessing mathematical concepts such as representation (excel chart and scale) and communication (expectation that students can express learning verbally).
"Recycling At Our School" assessed daily assignments for completion and accuracy, but not for mathematical reasoning. The final assessment focused largely on mathematical computations, like correct volumes and estimations. It did, however, assess representations, reflections, and reasoning. The assessment was clear and thorough.

Stoller PBL Article

In the article, "What Is Problem Based Learning?" Dr. De Gallow of The University of California outlines the outstanding points of PBLs, addresses critics' concerns, and explains how and why it is important to incorporate PBL in the classroom. PBL main points are similar to the Best Practices of Education. They include the following characteristics: student centered and experiential, inductive, builds on prior knowledge, context specific, complex and require metacognition, creates cognitive conflict, and collaborative. Although critics suggest that students do not know what is important for them to learn, and therefore may not reach their cognitive potential, Dr. De Gallow asserts that research shows learning occurring at a deeper level with PBL. He then provides specific tactics for teachers to use in their classes which will help them make PBL successful. For example, to create a student centered problem, teachers should create authentic problems that relate directly to the students' lives.

Reading the articles and documents about PBLs, I think that this method of "teaching as a facilitator" is the ideal way to teach. It is backed by research in the characteristics it promotes, especially the higher order thinking that must take place in order to solve the problem. Instruction could be differentiated easily so that all students can reach their personal potentials. It would be motivating to students as the problems relate more closely to their lives than traditional problems. Finally, it addresses the top complaints from employers about college graduates - poor verbal, problem solving, and collaborative skills. The only concern I would have is the practicality of implementing PBL in upper grades where the students are not used to that style of learning. I think it is important and should be used, but it would take a few years for students to adjust to the different approach. It would be much better if teachers would use PBL beginning in the earliest grades and throughout the upper grades.

De Gallow. (n.d.). What is problem based learning? Retrieved from http://www.pbl.uci.edu/whatispbl.html

Stoller: PBL - What is it?

Problem Based Learning (PBL) is a method of teaching math that involves a problem relevant and age appropriate to the students. The problem is truly a problem, in that it features more than one answer. Emphasis is placed on the process of solving the problem, and students work together with the teacher serving as a facilitator. The teacher asks questions to guide learning and initiates an ill-structured problem which the students claim as their own, personal problems. The students are encouraged to seek multiple answers, the final assessment being student reflections of themselves, their group, and their product.
PBLs require a few important details. First, the problem must be ill-structured (messy) so that students first need to define the problem based on what they know already and what they need to know. It forces them to organize and gather information before they can generate solutions and decide on the best solution.

Models For Initial Decimal Ideas

This article is well worth the time for any math teacher to read. In one article, a wealth of knowledge and research is shifted to the readers, examples supporting the text.

The Rational Number Project (RNP), a group focused on children's learning tendencies of fractions, decimals, proportions, and ratios, developed a curriculum using the Lesh Translation model, which depicts five areas where representations should be used: real world situations, manipulatives, pictures, spoken symbols, and written symbols. Students should be able to represent any rational number (e.g. 0.26) in each of these categories. They should also be able to translate a representation from any one category into another representation from a different category. Theoretically, the students will develop a deeper understanding of rational numbers as opposed to rote memorization of how to manipulate them (adding, subtracting, etc.)

The crux of the RNP curriculum was using the traditional 10 x 10 grid. Students connected their spoken language of rational numbers (e.g. twenty-sixth hundredths, not point twenty-six!) with the pictorial representation. They also wrote the numbers with written symbols and translated the grid to a number line - another picture representation. The group discovered through their research that the 10 x 10grid was the most effective representation for students to begin studying rational numbers with. They found that students' initial ideas about decimals or other forms of rational numbers was a crucial element in how deeply they came to understand them.

The more I read about representation, the more I can see the how America's traditional math curriculum and teaching approach hindered mathematical reasoning. I just hope that when I become a teacher I will not become so caught up in preparing students for high-stakes testing that I cannot truly teach math from the "bottom up" and the "inside out."

Cramer, K. A., Monson, D. S., Wyberg, T., Leavitte, S. and Whitney, S. B. (2009). Models for initial decimal ideas. Teaching children mathematics 16(2), 106-116.

Representations

Representations are used daily in math instruction, yet students and teachers sometimes hardly realize that those representations are indeed...representations. That is, they are symbols used in place of a complex process as well as a function. For example, the + sign, we are told, tells us that 2 + 2 equals 4. In reality, that symbol represents a complex process - a mathematical concept.

Representations are not in and of themselves the process, but they are fundamentally important in learning and understanding mathematics. If an addition sign would not be the established representation for adding, we would inevitably create our own symbol at some point in life to describe a process in short hand. When we as teachers hand the children those representations, they quickly embrace the symbol and its function before grasping the process. Teachers should encourage students to create their own representations - perhaps even before the established mathematical symbol is presented them. This would help students think about the process and approach a meaningful symbol to represent that process, and would probably help the students to remember math better.
In addition to letting the students devise their own representations, teachers should encourage the students to share their representations with other students to increase flexibility in different types of representations for the same concepts. The article suggests that students tend to miss the connection between a circle divided into three equal parts, a division bar, and a fraction bar. If they are presented with a rigid, one-symbol approach to math, they will not be able to make appropriate connections between mathematical concepts.
Finally, once students are able to create representations and translate them to a variety of contexts, they should be able to use them as models for real-life situations. A model, as pointed out in the article, is an ideal picture of an object or a phenomenon. They give people something to work towards - a desired end. Many occupations use models, many of them containing mathematical representations, to engineer their product or goal. It is clear, then, that representations should not be used as an object, but as building blocks to deeper, more creative thinkers.

NCTM (2000). Representations. Principles and Standards for School Mathematics (pp. 76-70). Reston, VA: NCTM.

Fourth Grade Video

The fourth grade video addresses variables - a difficult topic for young children to understand. They have not worked extensively with variables up to this point, so the newness of the topic might intimidate them and inhibit learning. To help the students build understanding of variables - a mathematical concept - the teacher incorporated a lesson devoted to creating a variable machine. The variable machine was simply two strips of paper - one with the alphabet written vertically on the lined paper, the other with the numbers 0 - 25. Placed side-by-side and attached to eachother, each number was paired with a letter, or variable.

Reflective Task 1 says, describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught. I thought this primary task was valuable because it focused on the concept of variables before delving into the function of variables. One math curriculum that I have had to use is Saxon (in tutoring). Saxon math spends far too little time on developing concepts, while drilling problem after problem of various algorithms. The students become confused as to when to subtract from what side, where and when to place the decimals, and so often, I have found myself helping the students survive the class rather than helping them understand. I certainly can see the value of spending ample time on the concept of variables prior to using variables on paper.

Reflective Task 1 under discourse says, describe how the teacher's questioning, and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment. First of all, I noticed that the teacher encouraged thinking by telling the class to "Talk to me about what you were thinking." Apparently, her students were comfortable with this manner of academic speaking and explaining, because immediately one of the students responded, explaining her group's thought process much more adeptly than many adults could do. I noticed that some of the teacher's questions were lower level thinking questions, but many of them required evaluation, synthesis, or analysis. For example, "How could you make your table's names have a higher value?" challenged the students to analyze what they had already done, and synthesize that information to create a new pattern.

Lesson Analysis 2 says, Give two examples of evidence that students are having difficulty understanding the mathematics being taught. One of the tables included a little girl who could not explain why her group changed the numbers so that the low numbers were at the beginning of the alphabet. This indicates that she does not understand the concept of variables - that the letters are simply representations of the numbers, and by assigning numbers to certain letters, the value of the letters will change. Another group decided to increase their scores by changing the length of their names. They thought that if they would include their middle name, the values would increase. This indicates that they were not comprehending the main objective of the lesson. Their goal was to get a higher score in any way possible. They did not realize that the only way the scores could change without lengthening their names would be to assign different numbers to different letters (variables).

Overall, I benefited from watching an experienced teacher teach mathematics in a constructivist environment. I have wondered many times how the principles of mathematics could really be applied in the classroom, but seeing it on the video gave me a glimmer of hope that it can be done. I learned that if I am to be an effective teacher, I will have to branch off of the textbook and worksheets. I will need to facilitate mathematical discussion and explanations, and I will need to provide a student centered environment in which understanding is discovered - not spoonfed.