Polygon Properties May Journal Article

Polygon Properties
Teaching Children Mathematics
By Rebecca R. Robichaux and Paulettte R. Rodrigues

http://www.nctm.org/eresources/view_media.asp?article_id=9290

Summary

Can third grade students do geometry? Why, in fact they can do very sophisticated geometry with the right teaching and activities. Students in Mts. Robichaux dove into the properties of geometric shapes in a sorting and classifying activity. According to the van Htele theory of geometric thinking there are three levels of geometric thinking. At the first level, level 0 students think about thing on a visual level, at this point appearance and familiarity with shapes dominates the reasoning about geometric objects. As students progress into the second level, level 1, they develop description; they are about to sort and develop language that allows them to distinguish one property from another. At the third level, level 2, the informal deduction level the students can distinguish properties and even make arguments about why shapes are similar or not similar based on the properties that they do or do not possess.


It is expected that as they enter high school geometry classes that the students have passed through level 0 and level 1 of the van Htelen scale. The students must be comfortable and functioning at level 2. The NCTM standards also require that students are actively using all of the process standards and that each content standard is frequently being revisited and redefined.


In this article the students went through two activities that, through inquiry and exploration, allowed them to define the properties of polygons, distinguish and classify the shapes based on predetermined characteristics and develop questions and riddles based on their understanding of polygons and their properties. In the first activity the students were given shape and sort bags that contained many different polygons, these examples covered every thing imaginable, there were concave and convex polygons, a variety of vertices, edges, angle sizes and varying complexities. The students sorted the shapes as many different ways possible. Each time the students sorted their shapes they identified the qualities that they were looking for and recorded their answers on a response sheet. This process required the students to develop a representation of how each sort was completed. As the students worked through this process they developed more language and terminology which helped them in their discussion. Next, the students worked with geo-boards and attempted to create shapes based on predetermined “riddles.” At this point in their activities the students explored the idea of impossible polygons and determined through trial and error which characteristics could not exist in the same shape. The students then created their own riddle as an assessment, allowing the students to “show off what they know” rather than berating students who still needed additional work with the material. At that point in the activities all of the students could create a riddle; however, some were considerably more sophisticated than others.




Application

This is certainly applicable in the classroom. Activities like these present teachers with a way of teaching material in a way that is real and alive. Polygons are interactive and changing in this activity. They have properties and they can fall into nonexistence when certain properties are combined. They students are working in a hands-on, minds-on activity that allows them the opportunity to make mistakes and discuss their mathematical ideas in a non-threatening and encouraging environment. I hope that I will have the opportunity to use this activity in the classroom one day.

Robichaux, R. & Rodrigues, P. (2010). Polygon Properties. Teaching Children Mathematics. 16(9) 524.

Manipulatives

1. How do you hold every student accountable?
Initially I think it is important to allow every student to play and handle the manipulatives. It is natural to want to play with new things. College students have trouble with this sort of thing. Thus, I strongly feel that it is unreasonable of teachers to put manipulatives in front of students and expect them to overcome the urge to play. Furthermore, I feel that it sends the wrong message to tell students to play. Lifelong learning comes from a desire to explore and understand. Yet, teachers are insistent on stomping these behaviors out of children.


After the students have been allowed a period of exploration and play I feel that the teacher should take some time to model and introduce the materials. This is an important step as it will start to show students they ways that the materials can be applied to the mathematics content. This is the point when the teacher can ask guiding questions and get students vocalizing their thoughts and ideas. Students will build their vocabulary and ability to talk about the manipulatives only through hearing the teacher speak about the manipulatives. I feel that it is unreasonable to expect the students to use communication effectively if the teacher has not first modeled this step.

I don’t know that I know all the ways that teachers can make students accountable and find an observable way to measure the students learning using manipulatives. However, these are some of the ideas I have. Hands-on activities and group work activities lend themselves very well to observation and checklists. Watching students work through a problem when they are required to use communication and teamwork is a perfect opportunity for the teacher to take an observation role. In the older grades it might even be appropriate for the teacher to “interview” each student and ask them to explain their reasoning in a formal manner.

Another technique that a teacher might consider would be to have the students produce some kind of product. The students might journal on their experiences. Older students might consider blogging about their problem solving and thought processes. Students could draw a picture, type their work into a word processing software, capture a screen shot of their work, or even produce a project, poster or brochure.

I also think that a series of steps, guiding questions or a guided handout might be useful for keeping students on-task and engaged with the material. I saw a very interesting program that had students experimenting with pattern blocks. The teacher had created a series of questions that corresponded with a power point. Each question was multiple choice and as the students used the pattern blocks to determine the answer they answered with automatic clickers. The clicker software immediately polled the class and showed the data as a bar graph. Then the teacher asked the students with the correct answer to explain how they reached their conclusion. I felt that this was one great example of how to make mathematics education everything it can be. The students were using the manipulatives in a systematic and directed fashion; they were using technology in a meaningful way. Students were accountable for their work and they were talking about their problem solving and reasoning.

2. I have already addressed “hands-on” and “hands-on, minds-on” in the previous question. When the students are playing and becoming oriented with the materials they are “hands-on.” However, at that stage there knowledge is undirected. It takes purpose and guidance for students to progress to the level when they are “hands-on, minds-on.” Again, it is the difference between good teaching and poor teaching. Manipulatives are not enough and without strong instruction they are all useless as worksheets. I would even go as far to say that a teacher’s classroom management makes a big difference with their ability to successfully employ manipulatives in the classroom. I can easily see a lesson going wrong simply because the teacher doesn’t have assertive discipline and clearly defined boundaries in the classroom. Finally, when working with manipulatives a teacher is going need to be especially sensitive to the special needs in the classroom as well as the gifted students.

3. Process Standards
Problem Solving- I feel that it is fairly evident how students are using problem solving as they work with math manipulatives. Manipulatives allows students to easily and quickly engage in guess and check. It is especially important in this day and age that students get feedback as they work. Students who play video games and use technology frequently are accustomed to offering input every ten to fifteen seconds and accustomed to some level of feedback every twenty to thirty seconds. Manipulatives offer this level of engagement and can easily supplement activities that would otherwise leave students unengaged and disinterested.
Reasoning & Proof- Reasoning and proof doesn’t just happen as students work with manipulatives. Rather, they emerge in the way that the teachers couple the manipulatives with other work. When students must defend their work and document their proof this process standard becomes a natural part of the activity.
Communication- Communication is another process standard that flows naturally with the activity. However, the danger is that the students don’t use the correct terms and vocabulary. Thus, it runs the risk that the students are able to work through the activity but cannot explain what they did, how they did it or why they did it. It takes the teachers active modeling for students to learn the correct habits. Teachers should not assume that students will learn this on their own. As a result, communication becomes one of the valuable indicators of whether or not the instruction is effective or not.
Connections- Students have a remarkable ability to make connections between mathematics and other experiences in their lives. However, I feel that the teacher must be the facilitator in helping the students to realize the connections within mathematics.
Representation- Representation and math manipulatives go hand in hand. However, I feel that students abilities to utilize representation is sometimes far more capable than my own. Student’s creativity and cleverness is a constant surprise to me. The ways that they think about mathematics and problem solving should be nurtured and encouraged through the careful and deliberate use of math manipulatives.

Technology

In an ever changing technological world there is a lot to be said about the technological tools that can be used in mathematics. The past generation was stuck in a very dangerous place, the students knew far more about the technology than the adults who were teaching them. This created a very strong technological push in education. Now, master teachers, like my Novice teacher, are much more comfortable with technology. However, the newest generation of teachers, entering the field, has the biggest advantage.

One of the most important aspects of working with technology is the comfort level. Personally, I am very comfortable with technology. I enjoy playing with programs and learning what they can do and how they can be used in the classroom. I can directly compare this will my parents generation, they are almost afraid of technology; afraid to break it, afraid to make mistakes, afraid to play. But because I was virtually raised with all the technology that I use now, I have a very different view of the tools.

Two of the most valuable technological tools that I have learned about in class are the smart board and the geometers sketchpad. Both are tools that are going to be very commonplace, especially in the upcoming years. I was very impressed with the wide ranging applications that both tools are equipped to handle. Its amazing how much the programs have improved in the last few years. Furthermore, it is exciting to think that these programs will continue to be improved and how much more they will be able to do in the future. The most important thing that I have taken away from exposure to these programs is that as a teacher I have to take the initiative to make myself comfortable with the programs. These technologies are not a gimmick. In the same way that math manipulatives cannot save mathematics, neither can technology. There is still the pressing need for good teachers and strong instruction. However, technology can be used in congruence with the existing curriculum to augment and supplement learning.

There have also been some wonderful programs that I have been exposed to outside of class. Just recently I attended a webinar on mathematics visualization using internet software. There are a number of programs such as moodles and thatquiz that can be used in the teaching and assessing mathematical concepts. I also feel that some of the illumination applets are extremely useful. However, as with all things these programs must be coupled with instruction, guidance and hands-on exploration.

There are times when the natural constraints of technology are a benefit. There are also times when the nature of technology programs and software limit students understanding and ability to explore the material. Therefore, it is the teacher’s know how that makes or breaks a classroom.

Errors

Working with student errors has been one of the more interesting, frustrating and enlightening components of the Math Methods classroom. Its it amazing to see how logical student mistakes are.

Literally two days after we finished the Errors in class I experienced a real life example that cemented my understanding of how important it is to understand student errors. When you understand the error it becomes infinitely easier to correct the problem. I was grading papers before class with a student, this student is very bright and often comes in early to socialize with people who are at his level, teachers, and he was looking at the paper I was grading and commented that it appeared that the student seemed to think that the fraction with the largest denominator was the largest fraction. I studied the paper for a moment. He was right. I called that student up for a few minutes at the beginning of the class and allowed her to make correction on her paper. The second time around she only made a few computational errors. I pulled the young man aside and complimented him that he realized something that they teach in college courses, something that I still struggle with, he was very proud of himself, deservingly so.

This experience made me appreciate how much frustration, both for the teacher and the student, can be avoided when the teacher is knowledgeable about the types of mistakes that students make. Students generally make mistakes that are very logical, they confuse a rule or even make up their own, but generally their misunderstandings can be correctly if they are understood. One of the greatest dangers is when teachers include “short cuts.” Instead of simplifying the problem teachers add a new layer of rules and procedures. It actually becomes more confusing to the student. Sadly, teachers think that shortcuts are useful because they assign worksheets that ask the students to perform the same computations over and over again, creating a need for a quick solution. If teachers assigned work that focused on quality over content this wouldn’t be such an issue.

All in all I feel that I’ve learned a lot from seeing the wrong way to approach mathematics.

A smorgasbord of assessment options

Summary- April

This article fits perfectly with the assessment assignment that the class recently finished. The author of this article, Kathy Bacon, reminds teachers that there are a plethora of assessment options and that student and teacher must remember that choosing the correct type of assessment is as critical as the information being taught. Bacon takes a moment to remind her audience what a student centered classroom should look like: it should be a place where the instruction is guided from the data gathered during assessment. This simply means that instruction and assessment must match the students thinking and way of communicating.

What is the best way to understand how to pair assignment and assessment; through the strategic use of excellent classroom examples. The students in Bacon’s classroom use every type of assessment. There is an example of a performance based assessment where a student created a three-dimensional duck and demonstrated her ability to identify shapes. Another student started to explore the properties of different solid shapes in a rudimentary proof. Students in the class keep learning logs and take pen and pencil exams as well. The author even provides an example of a test she gave to her students which contains true/false, short answer, multiple choice and essay questions. The key to success is that the teacher frequently uses formative and summative assessments. Both the teacher and the students are aware of the purpose of the assessments and are cognizant of the way that the data generated will be used.


Application

It is clear from the article that Kathy Bacon is an exemplary mathematics teacher. Her fifth grade students are producing very complex, sophisticated and thoughtful work. Their understanding of reasoning and proof is quite remarkable. I am currently novicing in a fourth grade classroom and based on my understanding of the students ability levels I feel certain that this work would be far too difficult for them in fifth grade. However, it seems evident that Kathy Bacon has been able to bring her students to a high achievement level. I am certain that this is due in part to her ability to assess and mold her lessons to the students needs. She stresses the fact that she often chooses assessment on the most effective way to get the information she needs with the smallest investment of time. She is fortunate that her school has a block schedule and that she has a ninety-minute block of time in which her students can be engaged in inquiry based activities.

I feel that as a teacher the most important thing is to take the time to do things well. Initially, Bacon probably spent a great deal of time designing her units and matching the assessment to the content being taught. But with repetition the process gets easier and less time consuming. It is an investment initially; however, the results in the long run are undeniable. Her students are high achieving and their knowledge is more than skin deep, they have an enduring understanding of the material due to the nature in which is has been taught.

Teaching Children Mathematics
Kathy A. Bacon
http://www.nctm.org/eresources/view_media.asp?article_id=9232


Bacon, K. (2010). A smorgasbord of assessment options. Teaching Children Mathematics. 16(8) 458.

Tech-Knowledgy and Diverse Learners

Summary
This article goes hand in hand with the topics that have been covered recently in class as well as what was talked about at the last BSEA meeting. Technology is the topic and its application in the classroom continues to baffle many and excite others. The response to technology ranges from extreme dislike to extreme enthusiasm. Jennifer Suh, the author of Tech Knowledgy & Diverse Learners, stresses the most important components of technology, the knowledge and skills that the teacher must possess when working with technology, the limitations of technology and the instances when she feels that technology is actually more appropriate than math manipulatives. Her article focuses on the application that technology has when working with diverse learners; however, I would argue that the same is true for all students.


Jennifer Suh begins the article by discussing the specific problems that ELL and special needs students struggle with. The practice is that when these students struggle that teachers use math manipulatives to make the abstract concepts more concrete; however, this can introduce a new problem. This can result in information overload. Suh proposes that the “built-in constraints” actually work to the students benefit. One of the points that I most agree with is that the computer software generally offers immediate feedback; thus, maximizing on the time spent engaged with the material. These programs are also generally very good at forcing the students to use the mathematical vocabulary, which may not be something they naturally do when working with the math manipulatives.


The article contains three specific examples of computer programs that fit all of the qualifications that Suh identifies as essential components of exceptional technology based tools. I was most impressed by the interactive line graph on the Healthy Forest site. The application shows not only the diagram and the simulation but also incorporates tiered lessons and math vocabulary. I feel that the more abstract examples are still relevant. However, they are not as interactive and user friendly.


The article finished with Suh final thoughts on the subject. Truly the technology is only as effectives as the teachers ability to recognize effective programs and software. The teacher must also be able to marry the technology with appropriate forms of assessment. Finally, the work must be combined with other forms of experimentation and opportunities for the students to use the information in other contexts.


Application
I feel that this article offers three specific and excellent uses of technology; however, I have a growing concern regarding teachers ability to assess this type of work. While discovery and exploration are necessary components of the learning process I don’t see an effective way to assess students as they work with and learn from these technology programs. Unless the teacher decides to award participation points or checklists, which I do not feel are an adequate assessment of this math content, it is virtually impossible to generate a grade or truly measure the student’s understanding. This just reaffirms the fact that technology, like manipulatives, cannot be the only method of instruction used in the classroom.

Mathematics Teaching in the Middle School
Jennifer M. Suh
http://my.nctm.org/eresources/view_media.asp?article_id=9197


Suh, J. (2010). Tech-knowledgy and diverse learners. Mathematics Teaching in the Middle School. 15(8) 440.

Walking Around: Getting More from Informal Assessment

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

In this article, Walking Around: Getting More From Informal Assessment, the author, Karen A. Cole, addresses walking around the classroom and using frequent informal observation as a component of her teaching approach. Her research was funded by the National Science Foundations in corporation with the Middle School Math through Applications Project. This program was targeted toward middle school mathematics classes and was a comprehensive curriculum that was project-based and student centered. The teacher described the two most important aspects of the process; observation and conference. Observation is fairly straightforward; the teacher must listen to the student’s conversation and stay attuned to potential misconceptions, clarifications and opportunities to ask more in depth and higher order thinking questions.

A possible weakness or problem with this strategy is that it can be very difficult to observe all students equally. Thus, the teacher must be very deliberate when it comes to this type of assessment. The teacher must allow every student the opportunity to talk, provide the students with regular excellent examples of proper discussion and explanation. Furthermore, the teacher can use one-on-one conferences as a time to catch up with students and offer them individualized instruction and attention. I feel that this article might have benefited from some specific suggestions that the author of Informal Assessment: A Story from the Classroom, used. I feel that this technique is more effective if the teacher selects specific students to observe and specific traits to observe.

Informal Assessment: A Story from the Classroom

Vincent, M.L. and Wilson, L. (2006). Informal Assessment: A story from the classroom. Mathematics Teacher 89 (2), 284-292.

In this article, Informal Assessment: A Story from the Classroom, Mary Lynn Vincent, a twenty year veteran mathematics teacher, worked in collaboration with Linda Wilson of the University of Delaware to revitalize her means of assessment in the classroom. Vincent experimented with various means of observation and documentation, in the form of rubrics and checklists, to determine a holistic and effective way to document not only the student’s procedural, factual and computations knowledge, as traditionally assessed through written assessments, but to assess their problem solving, communication and critical-thinking skills. In the end, she choose observation and checklist as the preferred means of recording and assessing these skills.

I felt that this article was most useful as the teacher offered excellent examples of how, when and why she choose the types of assessment that she used. Furthermore, she referenced the NCTM standards multiple times throughout the article, indicating that she is attuned to the best teaching practices and makes good pedagogy a daily part of her teaching repertoire. I also appreciated that the author provided examples and non-examples of how to use each strategy. The reader is able to learn through Mary Lynn Vincents mistakes and triumphs in the classroom and apply the information to his/her own subject and grade level.

Assessing Understanding through Reasoning Books

Assessing Understanding through Reasoning Books
Mathematics Teaching in Middle School.

Summary

This article starts in on a very theoretical note. As teachers do we sometimes avoid the difficult questions? Do we choose to opt for the easy answers? Are we doing a disservice to our students? Asking difficult questions gets overlooked because it’s an investment. Asking difficult questions requires time, effort, a deep understanding of the material and adequate exploration and communication.

This article focuses o the use of mathematical reasoning books as a tool which will help students develop vocabulary, reasoning and proof skills, communication skills and a forum in which to explore and answer the “difficult questions.” The introduction on page 408 can be used with students; it illustrates the purpose of the assignment and the format that is to be used. I also think that figure four and five on page 412 can be used to help students with the assignment. Figure 4 is a feedback checklist that students can use to evaluate if their answers and responses are through enough. Figure 5 is a self reflection rubric that will help the teacher and students reflect on the process.

The article is really a series of prompts and student responses. The authors then analyze the student’s responses for understanding. There are student samples at each level of achievement. Some students fell upon the belief that there was not enough information to reach a conclusion, which is typical of students who are struggling when it comes to transferring the information into a new context. Some students had the right idea but lacked the justification to support their answer. Other answers were very sophisticated and showed a true understanding of the question, answer and the terminology.

In conclusion, the mathematical reasoning book is a very useful tool that, when used properly and taught correctly, can be used to assess students reasoning and proof skill at the middle school level.


Application

I liked this article. Seeing the types of errors and understanding why the student made an error is an important step. Furthermore, I was surprised by the degree of difficulty of some of the problems and how well some students preformed with their math reasoning books. However, I am suspicious of the article as a whole because it is clearly a plug for the NCTM Reasoning and Sense Making book. While I liked the questions that were provided as an example I don’t know that I need a text book to teach me this strategy. It seems to me that this article did a fair job explaining the math reasoning book, which is really a glorified learning log. Personally, I have been using this technique in my classroom without an actual notebook. The students have been preparing for their ISAT exams and my novice teacher has been using the opportunity to review the mathematics extended response, which requires the students to defend their problem solving strategies in paragraph form. My teacher uses a t-chart to teach this concept. The students write the problem at the top and use the right column for the math- diagrams, equations, etc and explain what and why they did what they did in the left column. I like this approach. I feel that with the t-chart approach as prior knowledge that it would be fairly easy to teach the students to keep a mathematic reasoning book or leaning log.

Roberts, S. & Tayeh, C. (2010). Assessing understanding through reasoning books. Mathematics Teaching in Middle School. 15(7) 406.

Paint Bucket Polygons

Paint Bucket Polygons
Teaching Children Mathematics
http://my.nctm.org/eresources/view_media.asp?article_id=9163

Summary
This article is presented as the combined efforts of intermediate-level school teachers and college methods instructors. The group worked together to determine a series of lessons that would help the students develop a more sophisticated understanding of geometric concepts. In this particular lesson the fifth grade class was attempting to build an understanding of polygons and what the characteristic of a polygon are. The goal of the lesson: to use the pain bucket function of popular photo editing software, which is also located in the paint application of virtually every computer, to allow students to explore and build an understanding of polygons. The intent was the students would understand if a shape was not closed due to the fact that the paint would “spill out” and color not only the shape but also the background. Also if a shape had intersecting lines inside the shape only part of the image would be colored. As a result, the students would build an understanding of the closed and similar shapes.
The students gathered together and worked with several prototypes and non prototypes. In this instance popular prototypes are triangles, squares, rectangles and pentagons. Typical non prototypes are crescents and circles. While this can be a useful tool it can greatly limit students thinking and the intent of the lesson was to introduce the shapes and quickly graduate to more complex and less typical shapes. However, there was a more difficulty than expected. The students spent much more time than expected defining the word polygon and had considerably difficulty with distinguishing the prototypes and non prototypes.
Eventually the students were able to get to the main focus of the lesson, the use of the software to explore the attributes of polygons. The students were clearly able to understand the distinction of simple and closed using the software but there were limitations of the software and still areas which would need to be addressed using another medium.

Application
This article brings to light a problem that exists in many instances across Mathematics curriculum. Terms and definitions are used in student text books that are ambiguous and vague. Some texts use language that is clearly not student friendly or is all together too broad. When the authors of this article researched 80 different curricula they identified over 21 different definitions. This is clearly a point of confusion and dissension. If students are going to understand the concepts they need an understanding of the vocabulary and precise, student-friendly definitions. While, I like this lesson as a whole, I am surprised by the general tone of the lesson. The authors present the paint tool like a wonderful, unheard of, very creative approach. I consider paint to be old software and I am surprised by the notion that this is a new or novel idea. As soon as I get to school on Monday I want to see if the same tool can be used on the Smart Board, as I believe it can, as that could be another way to teach this lesson. Furthermore, with software like Geometers Sketchpad and other more student oriented applets and programs I feel that the approach is outdated. Like we’re discussed in class, manipulatives and software can not save the subject if it is to be held back by poorly written definitions, text books and lack of inventiveness with upcoming software and technology.

Edwards, M. & Harper, S. (2010). Paint bucket polygons. Teaching Children Mathematics. 16(7) 420.

Video #7 Lesson on Graphs

Purpose for the Activity
In this activity the teacher wanted the students to understand the relationship between a word problem or story problem, a data table that numerically shows the data and the information graphed on a coordinate plane. The students practiced generating ordered pairs. They also worked on identifying and assigning the x and y axes. Next, the students generated the algebraic equation. All in all, the teacher wanted the students to be able to show each problem as a graph, a table and an algebraic expression.
The teacher had the students talk about the graphs as a story. I liked the example she gave, comparing the relationship of the graph to an oven baking cookies. The oven’s temperature slowly climbs at a steady rate until its reached temperature. Once it has reached the temperature it has been set to it fluctuates around that number for the amount of time that it is used. Then as the oven is turned off it slowly looses heat until it has reached room temperature. Good pedagogy suggests that students understand mathematic relationships better when they can apply it to a story or in another context. The students took time to create a story that fix a graph which helped them to understand the foundation which the rest of the assignment built on. As the lesson progressed the students worked with generating tables, graphs and the algebraic equations.

Questions:

#1 Describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught.

I feel that the teacher did a thorough job with this topic. She set up a context for the students to explore the concept on their own and attack the problems from different angles; some of the groups started with the tables and progressed onto the graphs and algebraic equations, some of the groups started by making the graph and then generated the table. However, I do feel that the teacher’s initial instructions were a little vague. She spent a lot of time reiterating the directions; I feel that her initial instructions, given to the whole class, should have been more thorough. Furthermore, considering the time constraints, a ninety minute block period every-other day, I think that the teacher balanced the amount of time spend in whole class discussion and group work well.

#2 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?

The teacher discussed her questioning style at length in the interview section of the video. I was very pleased to see the degree to which she used errors to teach concepts and emphasize ideas and relationships. Furthermore, the teacher acknowledged that the students feel comfortable sharing mistakes and asking questions because of the positive classroom learning environment that she created. Clearly, she was sensitive to the student’s mistakes and very tactfully dealt with errors. Her sensitivity to the student’s mistakes was possible because she clearly understood the types of errors that the students made and why they made the mistakes they did. She quickly addressed misconceptions and mistakes in a very non-threatening manner. However, she never just gave students the answers, she always lead them to the answer through her line of questioning.

#3 Describe what the teacher does to support learning while students are working in groups.
The interview section of this video addressed the strategies that the teacher used to support the individual groups and help them reach the next level of understanding. She knew her students well enough that she knew when a simple question would be enough to redirect their thinking and lead them to the correct answer. However, she knew that some group needed more personal attention and stopped to discuss the problems with them at length. The teacher said that in this lesson they tried a different grouping arrangement, she allowed them to select their own groups. Personally, I would not do this; I feel that having done it once that the students would frequently ask to choose their own group and that it might become a point of contention. Perhaps, had she grouped the students more purposely they would have avoided some of the confusion that may have been caused by the students distraction caused by grouping situation.


Overall Use of the Video
Unlike the last video I felt that this video was less effective because it was broken into so many pieces. The former lesson was shorter, with less parts and the general flow of the lesson was more apparent. In this video series, I was very confused as to the sequence of events and the general progression of the lesson. The video quality was very poor and that detracts slightly from what I, as an observer, get from the lesson. However, the interview section of the video was more valuable in my opinion. Sand Allen provided very specific examples of techniques she used, errors that they students made, the objectives that she had for the lesson and what she believed the students got from the experience. Overall, she spoke about the lesson using specific examples and was less theoretical than Rosemary Klein, the teacher from the previous lesson.
There are some aspects of the video that are frustrating. I wish that I could see the overhead properly. Some of the audio is difficult to understand and the subtitles do not consistently make up for lost on inaudible speech. Being able to see the student work as the video progressed would have been helpful as well. However, overall I feel that the video is very useful. It has certainly given me some ideas about how to best teach this information in my own classroom.

Math Applet: How Many Under the Shell: Grade K-2

How Many Under the Shell
K-2
http://illuminations.ncthttp://illuminations.nctm.org/ActivityDetail.aspx?ID=73m.org/ActivityDetail.aspx?ID=198

Summary:
This is a very simple application that allows young students, from kindergarten to second grade, to practice with addition and subtraction. It is a very straightforward game that will help the students with their counting, addition and subtraction skills. The bubbles pop up, as each one appears it is counted, then the shell covers the bubbles and some of the bubbles are pulled away or added, and some remain under the shell. Then the octopus asks, “How many bubbles are under the shell?” and the equation appears. The students then answer the question using the number pad. If the students select the correct answer the applet dings that the answers is correct, if it is incorrect an angry buzz is heard and the student has another opportunity to answer the question. In general, it is a very simple applet, ideal for the young student just learning to use these types of tools.

Critique:
I feel that this is a very useful application that can be used to introduce young children how to use math applets. The interface is very friendly and the program is very intuitive. Again, I feel that the biggest problem is that the program does not record student responses in any way to show how many questions that a student has answered in a session. Unlike the applet for the third for fifth grade student this application does not have a “help” option, which I feel might be helpful for the students. Furthermore, the instructions are a very small font and are not appropriate for the kindergarten through second grade student. It is obvious that the parent or teacher would have to explain the directions if the students didn’t automatically get it and that the instructions are there for the adult audience. This applet is especially useful in the young grades as the teacher can have the student select a specific skill; the questions can have either, all addition, subtraction or a random assortment of both addition and subtraction. The program can also be test specific numbers, any number one through nine. This would be useful if you were working with a student who had a strong understanding of addition and subtraction with the numbers one through five but needed assistance with five through nine.

Math Applet: Concentration: Grades 3-5

Concentration
3-5
http://illuminations.nctm.org/ActivityDetail.aspx?ID=73

Summary:
The concentration math applet, available through the NCTM Illuminations, resources for teaching mathematics, is a game which requires students to use matching skills in a memory game. The program allows for a great deal of flexibility of the content being taught. The students can elect to play the game with simple number relations, numbers represented as blocks, dots, words or numbers. Then the students can graduate to more difficult numbers. You can set the program to test the understanding of geometrical shapes, multiplication, fractions and even percentages. At each skill level the students can play solo or with a partner. In the way that the program responds; beeps, shows correct answers and resets the game, it feels very much like a game. It is very user friendly and easy to understand. However, despite its ease of use, the applet is able to test students on a variety of skill levels.

Critique:
I feel that this applet is a very useful tool. Because it fees so much like a game I feel that students would enjoy using the program. Furthermore, because the program is interactive the students are motivated to continue to the more difficult levels. In a number of the other applets I noticed that they allowed the students to manipulate; however, without a focus I can see the students becoming bored and giving up. As a teacher, I appreciate that the program is self sufficient, with the other programs I would have to create a problem or purposeful context for the students were they to use the applet for any prolonged period of time. Having tested the program I feel that the fractions version of the game is the most challenging. It requires the students to match the numerical fraction to a visual representation. Students who struggle with this concept can use the “glass pained window” to make the process easier. It makes the task much easier. Then the students can really focus on identifying the correct fraction pair with less focus on the time constraint. The only thing I would change about this applet is that it does not record the student’s results. Were I to use this applet in the classroom, as a reward, during computer time or to get a feel for the student’s prior knowledge on the subject, it would impossible to know their results unless you sat and watched each student play the game.

100 Students by Riskowski, Obricht and Wilson Mathematics Teaching in the Middle School

Summary

This article is a great example of project based learning and its use in a middle school classroom. The learning goals were that students would learn to talk about and analyze statistical data, collect a representative sample of a population and analyze the results in proportion. The project was modeled after 100 people world, which was a project by The Miniature Earth that sought to analyze how the earth would look if only 100 people lived on it, a representative 100 people that proportionally represented the Earth’s population from the 2001 statistics. The students set out to discover what their school would look like if a representative 100 students were chosen according to proportion to represent the entire school. Students were really responsible for carrying out the activity. They came up with the research questions, examined the questions in depth to analyze for bias, administered the surveys, collected the results, entered the data, analyzed the data and made an informational video about their project. In the end the teacher reported that not only did the students learn about statistics and data but that they learned about respect for others. At the end of the project the students sat down and talked about things they would have done differently and it was clear from their conversation that they understood how to get more accurate results from the school had the changed their questions, their sample and how they interpreted their data.

Application
This type of activity is so wonderful. In high school, these are the types of activities that I loved and still remember. These students were actively involved in the project because it was about them, their school, their peers and their lives. The students were involved in complex tasks that required a lot of planning and reflective practice. I admire the way the teachers taught this lesson, they allowed the students a lot of freedom; yet, they were there to ask thought provoking questions and prompt the students to analyze their research methods. I feel that this project is multidisciplinary and incorporates many advanced tasks; for instance, the students edited their own video, I have worked with editing software and that is not small feat for middle school students. I also appreciate the way that this project required students to work with their peers in coorporative groups. By the end of the project the students reported that they felt they had a better attitude toward their peers and felt that they should be less quick to judge and kinder to their classmates. If for that reason alone I feel that it was a time worthy project and its amazing how much they learned as well.

Riskowski, J. Olbricht, G. and Wilson, J. (2010) 100 students. Mathematics Teaching in the Middle School. 15(6) p 320

Storyboards for meaningful patterns by Dubon and Shafer Teaching Children Mathematics

Summary

This article is about Dubon, a Novice teacher who works in a kindergarten classroom in a Title 1 school in northwestern Indiana. While the class frequently uses patterns and predictability, in classroom management as well as in following the daily routine, the students were unable to generate patterns on their own and were not able to discuss or talk about the patterns that they saw or had created. Dubon’s principal eventually paired her with a professor, Shafer, from the local college who arrived at the school armed with manipulatives and ideas on how to teach the children patterns. Initially, Shafer tested the student’s ability to generate a simple pattern, a topic that they had studied in class; few of the students were able to perform the task. So Shafer began her lesson, she used multiple examples using, people (boy-girl-boy-girl), sounds (snap, clap, snap, clap) and other patterns. She then introduced storyboards to the children. The storyboards require the students to start story problems and then introduce the snap cubes as a way to visually represent the objects in the story. The image shown in the article as cats and dogs in an alternating AB pattern; one of the students made up a story about the cat and dogs kissing and fighting, etc. Where all the other strategies had failed this one was successful. The students were able to take this model and extend it to many other situations and stories. Eventually the students were able to successfully use pattern blocks to present many things, assign meaning and tell a story. The key to the strategies success is that it allows the students to assign meaning to an otherwise meaningless activity.

Application
I feel that this article is a wonderful example of how a reflective teacher solves a problem in the classroom. Dubon’s students did not understand the material the way that she approached it, so she approached it a different way and sought the help and guidance that she needed. It can be difficult for a teacher to admit that she dose not know how to help her students and I feel that it is a mark of professionalism to admit when you need help. This article is also very helpful in that it reminds teachers that what we teach must always carry meaning in students lives. The best lessons appeal to student’s interests and common knowledge, are holistic and allow students to participate in a meaningful way. In this article the students spend a lot of time coming up with the problems that they then solved; which, in part was why they remembered the material.

Dubon, L. and Shafer, G. (2010). Storyboards for meaningful patterns. Teaching Children Mathematics. 16(6) 325.

PBL Student Work Analysis

1-1) Creating Candy- imagine that you are a candy company who is struggling to compete with another company who has just released their most successful candy ever! That is the problem the fifth grade students in my first selection were asked to do. The students needed to determine what kind of candy they were going to create, based on consumer feedback, how it should be packaged, and how much they can charge for their candy.

1-2) After School Special: A PBL Unit for Grades 7th- 8th-in this PBL a group of students is being asked to design a space in their town’s community center. The center is designating the space to be used as a teen center; however, they do not know what the space should become. They are eliciting the help of local teenagers to create a plan that is educational and yet recreational. The students need to budget, create a floor plan, determine what materials will be needed and be ready to defend why their plan was the best.

1) The two plans are very different. In the Creating Candy PBL one of the strengths was the range of mathematic objectives and extension objectives. Overall, the Candy group put a lot of detail and guidance in the project, which allows for the wide range of math concept covered. However, there are a couple of weaknesses in the project. The group’s guided questions were not very helpful. The guided questions are intended to prompt students when they are stuck or confused. The guided question in this project more or less reiterated the instructional goals of the assignment. Furthermore, the group left very little flexibility in the assignment, every day was meticulously planned out, perhaps so much that it began to detract from the meaning of the assignment. The second PBL, After School Special, was strong in that it allowed for a lot of creativity and independence though out the entire process. This group chose to incorporate journaling, as a means of assessing the students reasoning methods and justification for their decisions regarding the youth wing. Each day of their lesson outline included the activities and the possible guided questions that would be helpful to the students as they worked through the process. Overall I felt that the After School Session PBL is a cohesive project, I didn’t recognize any major flaws in the assignment.

2) The Creating Candy PBL is a very restrictive PBL assignment. However, After School Special allows the students a lot of independence and flexibility in their work. At this point I don’t know what level of flexibility is best for students; however, it is clear that these two groups choose two very different approaches to this assignment. When comparing the two PBL’s I noticed a substantial difference in the way each group incorporated mini lessons in the overall project. I felt that the Creating Candy mini lessons did not fit well with the flow of the project; whereas, the budgeting assignment in the After School Special PBL, was a logical assignment that helped the students complete the rest of the project.


3) In general I feel that both assignments are rather restrictive. My understanding of the Problem Based Learning assignment is that it should allow students flexibility and creativity. In the second project, the School Special PBL, the group used journaling to help make the students accountable for their work. I like this idea; however, I would change the assignment a bit, I would have the students keep a learning log, all their mathematic problems and scratch work would be done in their learning logs and the students would write their feelings, reactions and frustrations. This would also help the teacher have a log of the areas of the project that were problematic which might need revision in the future.

4) In the Creating Candy the students were clearly focused on incorporating as many mathematic concepts as possible. There are many content standards involved in the project; however, the group seemed to neglect the many process standards, they are clearly present in the work; however, the group did not call attention to them or specifically assess them. In the second project, After School Special, it’s clear that the students were attentive to both the content standards and the process standards; each was addressed numerous times in three separate rubrics that the students created to assess the PBL.


5) The Creating Candy PBL was very thorough in assessing all the mathematics objectives that were taught in the lesson. However, there are some specific problems with the rubric. For instance students who have earned an “excellent” in the rubric must have demonstrated “much evidence” supporting the mathematic topics covered. I question a teacher’s ability to measure “much evidence.” My experience is that rubrics should be much more explicit. However, in the second group no specific mathematic concepts are assessed, merely that the group was neat, organized and that their final product was effective.

How to but a car 101. Problem Based Learning in Action

1) This journal article was about a middle school class, who used Problem Based Learning to work on and solve a problem that would interest them; selecting a car to purchase. The students were presented with very specific restrictions, Mr. Jones, their ‘client’ needed to buy a car that would be affordable on his budget, have good gas mileage and be appropriate for Mr. Jones and his wife. The project was full of mathematic content. The students had to determine Mr. Jones’s monthly payments, which included the price of gas and take into account the interest rate. The project was involved, holistic and the students were engaged and excited about it. One of the most challenging things for the teacher is learning to let the students work on their own and not giving them the answers; rather, the teacher needs to act as a facilitator and resource.
2) The article definitely addressed the strengths and weakness of the project. Firstly, the teacher needs to have established an environment where the students are capable of working independently with others. Secondly, the teacher must learn restraint and allow the students to struggle and make decisions on their own. Finally, the project requires very specific assessment tools, if the teacher is not clear about the requirements initially the students will struggle. The teacher must be attentive to the point which the students struggle with and make alterations as the project develops which is a process which takes a lot of time and effort. As a reader, I appreciate all the visuals that the author included in the article. It defiantly helps me to visualize and understand the process and final products when I see examples.
3) APA Citation: Flores, C. (2006). How to buy a car 101. The National Council of Teachers of Mathematics. 12(3), 161-164.

Problem Based Learning

Problem based learning is really a holistic approach of teaching. I have personally experienced problem based learning in two of my teacher education mathematics classes. However, this is the first time that I have really had to look at the instructional objectives that an instructor would set when designing a problem based learning assignment (from here on referred to as a PBL).

Problem based learning exercises require the students to work on several different skills that are typically aimed at solving a problem outside of the classroom. In some examples I have seen the students were attempting to solve an actual problem that effected their daily lives, in some instances the students were attempting to solve a fictitious problem, and in some instances the students were competing to find the most efficient, most creative or most effective ways of solving the problem.

When a teacher grades the PBL work the assessment is very important. This is a project when the process is as important as the product and the teacher should be aware of how well the group members worked together, how effective their plan or solution was and how creative they were in accommodating the constraints.

Problem based learning can be used in any discipline; however, I especially like its application in mathematics. In science classes I have experiences units when I was completely immersed in the content. Yet, in math courses I rarely spend more than thirty minutes filling out the assigned work sheet. Imagine how much more engaged in mathematics students can become when you make it a project that requires teamwork, discussion and application.

One of my favorite examples of problem based learning was from an article in a science journal. One student was fed up with recess being canceled due to flooding on the playground. The teacher used the situation as an opportunity to have a PBL assignment. He divided the class into teams, assigned budgetary constraints, time constraints, material constraints, labor constraints and asked that the students come up with the most creative use for the diverted water. The students were sent out to collect the data; they measured the slope of the playground and collected all the measurements they needed. Each group proposed their idea and the teacher selected the winning plans. In the end, the students were able to get their playground back as well as have a small pond and water collection tanks for the school’s garden.

Another instance when I have encountered PBL’s is at the Science Olympiad competitions. Students compete in different areas; they build robots, solve simulated crimes, build model cities from the future (solving problem like recycling and exponential population growth), they have even taken the egg drop to the next level. The response that you see from students when you present them with this type of project proves that it is not the mathematics that they don’t like, it’s the way that it’s taught.

Problem based learning is important because it really incorporates all of the process skills. One of the most important things, in my opinion, is that it teaches students perseverance, dedication and when the students are finished there is a huge sense of accomplishment. It promotes that intrinsic motivation, which will help the students be successful even when they are not encourages by parents or by their peers.

Video #4: Lesson on Variables

Purpose for the Activity

The purpose of these videos is for education students to be able to see how an experienced math teacher would go about teaching a math lesson in the fourth grade. There are other videos that one can access through the website and the goal of these videos is for students to see the variation in instructional strategies and content across the grade levels. The teacher in the film used many helpful strategies that are worth emulating. I quickly notices the progression of the lesson and ways she allowed the students to practice with the material and experiment and reason through problems as well as the very thought provoking questions that she asked the students.
The students in the film were working on learning variables, in compliance with the state standards and NCTM guidelines. In a group with her peers the teacher addressed the issue that the students were familiar with using letters and symbols to represent numbers but were struggling to understand the concept of variables. The teacher knew this information because she is very cognoscente of her students abilities and their prior knowledge, this is a key element of the lesson and frequently becomes apparent as you see her ask different questions of different students and the degree to which she probes students to elicit a desired response. For some students this lesson is very challenging and when they are on the right track she offers praise. For the more advanced students she requires that they explain their answer more fully and even worded her questions in such a way that it made the students explain the reasoning and steps in their thought process.
I think it was really interesting to see the degree in which the teacher was able to connect the mathematic content with other subject matters. Obviously, the lesson involved a lot of letters and words, as that was the basis of the variable machine. However, she was also able to incorporate vocabulary when she asked the students about their prior knowledge with the word ‘variable’. She connected their prior knowledge with their social studies project, studying the weather in Belize and Guyana. Both cities have very constant temperatures which she contrasted with the variable weather in their home town. The teacher also made connections between student’s names and made the class pet part of the lesson. I really commend the way that she made this material attainable for the students by using so many elements that they were comfortable with.



Questions:

#1 Describe the primary task in this lesson and identify the mathematical skills and concepts that this task is designed to develop.

This lesson was about variables. The students needed to learn the vocabulary word itself as well as understand what a variable is, how it can be altered and how altering the value of the variable can affect the outcome of the problem. In order to teach the lesson the teacher had the students make a “variable machine” which could be used to assign the twenty six letters in the alphabet different variables. Initially the teacher provided examples and then allowed the students to use their own names to practice the concept. By the end of the lesson the students were assigned the word “bear” and attempted to find the lowest and highest possible score for the word. The students were really required to use a lot of trial and error initially but as the lesson progressed they began to recognize the patterns that arose when working with the variable machine.


#2 Identify an example of when the teacher responds to students by offering clarification, by explaining, by questioning, and by letting the student struggle.

At one point in the lesson all of the groups but one had identified the lowest possible score for the word ‘bear.” Instead of allowing the group to move on or allowing them to struggle and potentially become frustrated, a problem that may be been exacerbated by the presence of the cameras and newcomers in the classroom, the teacher asked a group who had gotten the answer to give some strategies to the group that was having trouble. This was a way of asking the groups to explain how they reached their answer that also is part of her classroom climate, an environment where the students help each other.

#3 Describe the student-teacher interactions during the task debriefing discussions and assess the effectiveness of these interactions.

An aspect of teaching that I have really come to value is the debriefing part of a lesson. Many teachers overlook this essential component of teaching. The teacher wrapped up the lesson by drawing attention to the main goals she wanted to accomplish. She wanted the students to have a firm grasp of the word ‘variable,’ she wanted them to recognize that the value of a variable can change, she wanted them to understand that changing the value of the variable effect the outcome and she wanted them to realize the patterns that arose when working with the variable machine. The teacher accomplished these objectives by asking very strategic questions. She asked the students to explain what a variable was and asked them would happen if you changed the value of the letter in a word. As a teacher she was able to informally assess if they understood the content and she boosted the student’s confidence in the material. One of the things she mentioned in her conversation with her fellow teachers was that some of the students needed to feel successful with the material, they needed the positive feedback and sense of accomplishment, likely because they had previously had negative experience with math and their lack of confidence was affecting their ability to absorb new information.

Overall Impression

Overall, I feel that the video was a very useful tool. However, having just studied the equity principal I was very aware of some of the teaching and conversational phrases that the teacher used in her instruction. When she used phrases like “big ticket item” and “hike your score” I was struck by the face that many ELL students, as well as some English speaking students may not be aware of the meaning of these particular phrases. Furthermore, when she asked students to contribute their ideas she would indicate that she wanted them to answer by saying “speak.” Personally, this is something I would not do, I found it disrespectful. Also her primary mode of instruction was verbal. In an attempt to reach visual learners, I might have also written or drawn more of the examples on the white board. However, I do feel that the teacher from the films was a very experienced mathematics teacher and her strategies and skills are defiantly something I would use in the classroom.

Reasoning and sense making. Reasoning and Proof Process Standards

W. Gary Martin & Lisa Kasmer
Reasoning and sense making

What is reasoning and sense making? Essentially, it is learning. Learning is an active process; data is imputed into the mental machine, processed and filed away with all the other data, according to its type. Students learn how to make sense of new data, new material by an active reasoning process. It is the process of breaking large pieces of information down into manageable pieces, reassembling the information, assessing the intended computation and working through the problem to reach an answer. Students need to challenge the question, they need to challenge the answer, and they need to challenge each other. It is essential that students learn how to talk about math. It is essential that students learn how to defend their ideas. Students are naturally curious and endless worksheets do not foster that curiosity. Students who are taught in this way are given the foundation for advanced problem solving and creativity, skills that are important not only in high school and college but in the world, as they enter the job force. A teacher can create this type of atmosphere by making students feel comfortable. Students have to feel that they can make mistakes. The teachers must learn the right questions to ask.

Consider a specific example: the teacher asked the student to work through a word problem on subtraction. The students were permitted to use any subtraction process they liked. One of the students decomposed the numbers in the problem before subtracting. The teacher and students asked questions about how and why the student worked through the problem and she, the student, was very comfortable and eloquent in defending her choice. This is wonderful example of a classroom where mathematical reasoning is used and questions are encouraged.

Martin, W. G. and Kasmer, L. (2010). Reasoning and sense making. Teaching children
mathematics 16(5), 284-291.

Reasoning and Proof Process Standard

This article focuses on four key concepts:

All students, from the pre-kindergarten classroom through senior year of high school should be able to:

1. Understand proof and reasoning methods as a ways of understanding, explaining and comprehending mathmatics.
2. Develop mathematical conjectures based on the problem given
3. Develop,analyze and represent mathematical reasoning
4. Understand when and how to utilize the various reasoning methods


Students need to be able talk about mathematics. From an early age children are able to justify and explain what they see in the world around them. One of the first things a child will recognize is a pattern and they can quickly learn to anticipate the next element in a pattern and explain why it is the correct choice. As students get older and enter school this is generally called reasoning or proof. When most students think of "proofs" they think about geometry; however, proofs and justification are a part of daily life and an essential component of mathematics and true mathematical understanding. As students grow older their level of reasoning will become more refined and sophisticated; however, the skills are quite the same.

Every time a teacher asks a student to justify his or her answer the teacher is asking for the reasoning, asking that the student demonstrate the proof or line of though that led the student to the answer. This is a important aspect of mathematics that can often become overlooked in worksheets and endless problems.
The second point revolves around conjecture. Conjecture, or guessing, is a part of mathematics, it is a part of learning. Students have to feel that they are able to guess, make mistakes, be wrong and learn from the experiences. Mathematics is a stressful subject for many students, due in part because they fear being wrong and this fear can stop them for interacting with the material. As defined in the process standard, students learn best when they are able to work cooperatively with their peers and reason through their mistakes.
As students learn how to go through the process of reasoning they need to also learn how to represent their work. In the elementary grades it may be most appropriate to have students draw and color their work. In the older grades, students should practice showing their work with sketches, numbers and even in paragraph form, not only as a two columned proof.
Finally, students need to be able to recognize when and how to utilize various reasoning methods. For instance, the nuances of an algebraic equation may vary considerably from a geometric problem. Essentially, as they progress through their schooling students should start to recognize how best to go about a problem and various strategies that they can use.

Is mathematics a universal language? by Tim Whiteford

This article dives into the realities of teaching in a diverse setting. The challenge is more than just communication but rather truly understanding the student’s prior knowledge regarding mathematics. Some students enter the classroom with a strong mathematics foundation; however, there are significant differences in the way that math is taught in other countries. Teachers struggle with the conceptual differences in the way math is taught and understood. Yet, other students may have virtually no prior knowledge of math at all. There are times when students have essentially the same mathematics background but have learned different procedural ways to solve problems. Whitford offers a wonderful example, Molita from Bosnia struggles because she has been taught the equal addition method of subtraction rather than the traditional American decomposition method, taught in schools today. I can strongly connect with Molita as when I was a child I was home-schooled by my mother who taught me the equal addition method. When I attended fourth grade my math teacher publically called me out on my work and penalized me on assignments because I did not know the “correct” way to subtract. Looking back on the experience, I can recall how embarrassed and defeated I felt. I never want my students to feel that way and I want to make myself aware of the cultural differences and sensitivities that children may have. As teachers we must learn about the various procedural methods in order to make ourselves aware of the multitude of ways that mathematics can be taught. I think that one of the key ways that these differences will be understood is by talking about mathematics and engaging our students in conversation.
The next hurdle in teaching mathematics to a highly diverse student body is the difference in the way that people think about mathematics. Teachers fail to realize how important the way they speak is. Subtle variations in the way we word questions can have a profound impact on how a student perceives the question. Students who have limited English proficiency can be perceived on having very poor math skills simply because they lack the ability to communicate their ideas. Teachers must seek out the proper support, be it an interpreter or an aid, in order to understand the students meaning. Teachers must also be sensitive to the challenges that the English language pose to an ELL student. Even the number system can be challenging to a student who isn’t fluent in English. Students may also struggle to understand the units of measurement, which vary considerably from culture to culture.


In conclusion, teachers have to invest the time to understand a student’s background, their prior knowledge and their comfort level with the English language, the thinking process, the units of measurement and the method of instruction. The bottom line is that teachers have to believe that all students can learn mathematics. They have to understand what support a student will require in order to understand the material. Furthermore, teachers should embrace the opportunity to learn about the cultural variation in how mathematics is taught and understood.


Whiteford, T. (2010). Is mathematics a universal language? Teaching children mathematics 16(5), 276-283.

The Equity Principal

The core idea of the equity principal of mathematics is that all students are entitled to mathematic education which is challenging, meaningful and purposeful. Teachers do not have the right to choose which students will be successful and unsuccessful in mathematics; rather, it is their responsibility to make every effort to see that each and every child is able to be taught and able to learn mathematics. Sometimes this means that special accommodations must be made to reach the specific needs of the child. One of the key dispositions that all teachers must have is high expectations for all their students, free of discrimination of all types. Technology is a tool that can be used to reach students who would otherwise struggle. New technologies can help overcome language barriers and assist students with learning disabilities. Furthermore, in the teaching profession, equity requires the support of the school administration. Teachers must be supplied with the technology, funds, time and personal support to make this concept a reality.

One of the notes that really resonated with me was the final comment that the author made to the teachers personal biases. It is so disappointing that in this day and age we still must confront prejudices and preconceived notions about a child’s ability to be successful in education based on color, sex or economic status (among many others). This is a challenge that as educators we must confront daily, to fight for the rights of children.