Motivating Students

Motivating students is tough. There is no explicit algorithm or formula teachers can use to get their students to show an interest in math. Despite this, Sobel (1985) suggests ten actions teachers can take to help make their math classrooms more interesting, and, quite fun. Although the article was written more than 30 years ago, many of the suggestions, or the derivatives of such, are still used today and are really effective.

Finding relavent articles in newspapers and magazines, videos on the web, and other current events can have a profund effect on students. It shows that math has a place in the real world, and it's not just a bunch of symbol manipulation.

I as a math teacher enjoy learning the origins of topics in math, even if the way they were learned and taught in the past differs so much from today. The example Sobel gives is called the rule of false position, an Egyptian method of solving equations. I believe it only works in multiplicative relationships, so it might be a cool thing for Algebra I students to explore but I wouldn't recommend teaching it. If the related history is presented then students might see the motivation behind the math.

We need not, should not, and certainly cannot show practical applications for all that we teach? (Sobel, 1975, p. 481). Why not? Does the author disagree with showing practical applications for everything, or is he simply stating that it cannot be realistically done? Granted, given the time allotted for topics in secondary school math, there is not a lot of freedom regarding exploring and investigating practical applications; but in my experience I have always seen textbooks present word problems involving real-world situations for every mathematical subject in the text. In fact, my philosophy of math education shows that there is always a practical application for any math concept. Else why would it exist?

Lab experiments are always fun for students, no matter the context. In my student teaching experience, I began the unit on Direct and Inverse Variation with two labs (one per day) in Algebra II. The direct variation is discovered through Hooke's Law, which states that the distance a string stretches is directly proportional to the weight that is applied. The inverse variation lab involved swinging a pendulum of different lengths and measuring the frequency. My students had a lot of fun collecting data, which in itself is a great use of the NCTM's Measurement standard. Another experiment I used in the same class was the visual representation of the difference of two squares. I was happy to read about Sobel's description of this because it was almost identical to the one I had used in class.

I was surprised to read the author's opinion that there is no clear indication that student discovery leads to increased learning (p. 483). Perhaps this article is too old to address the research on discovery and investigation, but if there's one thing I learned from my undergraduate and graduate mathematics education classes, it's that students learn best by figuring stuff out.

In 2004, Stevens, et al. published a study measuring affective attributes of students of different ethnicities. In two groups consisting of Hispanic and Caucasian high school students respectively, the researchers analyzed the effects of self-efficacy and motivational orientation on the students' performance and their plans to take future courses in math.

References

• Stevens, T., Olivarez, A., Jr., Lan, W. Y., & Tallent-Runnels, M. K. (2004). Role of mathematics self-efficacy and motivation in mathematics performance across ethnicity. Journal of Educational Research, 97(4), 208–221.
• Sobel, M. A. (1975). Junior high school mathematics: Motivation vs. monotony. Mathematics Teacher, 68(6), 479–485.

Mathematical Miseducation

This article provides commentary on the traditional mathematics education paradigm, its detrimental consequences, and evidence and reasons for shifting to current reform in primary and secondary education. Battista introduces controversial criticisms to current reform movements being made in mathematics education and how the majority of American schools still follow the traditional route when teaching students math. He explains that this route leads to failure of students to retain understanding, thus wasting enormous amounts of time and money to reteach students concepts they had been taught in previous years. After reviewing a short history of the mathematics education reform movement, Battista elaborates on the constructivist view that students must personally construct mathematical ideas as they make sense of a situation.

I think like any major change in society, the reform movement in mathematics education will take time to catch on. Scientific studies on this subject have only come into play within the past quarter-century, and in the world of scholarly research, this is relatively young. The thesis of Battista’s article is not to educate readers about mathematics education reform, but to educate readers about mathematics educators who are not educating themselves about mathematics education reform. In a way, he is “preaching to the choir.” By that I mean, his audience by definition will have already agreed with his point of view. The people he is trying to persuade are not likely aware of this article. However, I think that research about research can help, through recommendation and word of mouth. Readers of this article can recommend it to their uninformed colleagues, and every bit of awareness helps.

The more teachers that take Battista’s research into account and spread the news to fellow educators, administrators, and policymakers, the more the mathematics education community can be lead in the right direction. Soon the traditional mode of teaching will no longer be mainstream, and students will be accustomed to constructing their own understanding.

References

• Battista, M. T. (1999). The mathematical miseducation of America's youth: Ignoring research and scientific study in education. Phi Delta Kappan, 80(6), 424–432.

Teachers' Support

A three-minute video of the student-teacher dialog

Jacobs and Philipp (2010) explore the possibilities of support that teachers can provide to students during problem solving. They discussed four basic types of support:

1. support that focuses on the teacher’s mathematical thinking,
2. support that focuses on the child’s mathematical thinking,
3. support that focuses on the child’s affect, and
4. support that includes general teaching moves.

When working with students on an individual basis, support that focuses on that student’s thinking is best. Teachers must adjust their instructional behavior to fit every individual student’s needs. In the example of Rex, this teacher said she would use what she learned about Rex’s thinking on the first two problems to help him solve the third, more challenging, one. Rex liked to use his fingers instead of the manipulatives that were available, so teachers should not push for use of one manipulative over another. In addition, children often think about math differently than adults do, and many can offer new ideas to the table. Provided the opportunity, teachers can learn from their students all the time.

That is not to say that the other three types are not appropriate. In a class discussion or lecture-type class, support that focuses on the teacher’s thinking is more suitable, since the teacher is providing content and modeling processes for the students. Students’ affect should always be taken into account, especially for struggling students; and general teaching moves are useful ‘go-to’ tools in everyday instruction.

This article would have major effectiveness in a traditional classroom. In this type of classroom, all students follow along with the teacher at the same pace, and individual needs are rarely attended to. Usually, problem solving is nonexistent because there is only one or two ‘right’ ways to solve a problem, and these ways are presented by the teacher. The students’ job is to remember how the teacher solved it and perform the same procedure on a similar problem in assessment. Rather, problem solving itself includes the task of figuring out on one’s own the method of solving the problem, and then carrying out that method. Once the algorithm for completing a task is known, the problem is no longer a ‘problem’ even if the answer is still unknown (Harvey, 2010b, p. 3).

Teachers could, and will, benefit from concentrating on students’ mathematical thinking when instructing their classes. By calling for more student involvement and keeping their classes stimulating, engaging, and fun, teachers can get to know their students and how they think mathematically.

References

• Jacobs, V. R., & Philipp, R. A. (2010). Supporting children's problem solving. Teaching Children Mathematics, 17(2), 98–105.

High-Level Tasks

Henningsen and Stein (1997) begin with some background information about the (relatively) new theories about thinking and doing mathematics, which depends on educators' view of the nature of mathematics. The research shows that one theory that has been increasing in popularity is one that emphasizes exploration and has a dynamic stance of education, and holds that student learning is seen as the process in which students develop their own opinions, beliefs, and perspectives of mathematics.

Activities that promote having a mathematical disposition include the following (Henningsen & Stein, 1997, p. 525). These are the processes that students should take away from their mathematical studies, not necessarily the specific content that is covered in their classes.

• Looking for and exploring patterns
• Using available resources effectively and appropriately to solve problems
• Thinking and reasoning in flexible ways
• Conjecturing, generalizing, justifying, and communicating one's mathematical ideas
• Deciding on whether mathematical results are reasonable

In order to promote these activities, teachers must situate their classroom environments such that students engage in high-level tasks. The introduction of this article includes an in-depth discussion of the importance of high-level tasks, the difficulties that may arise from implementing such tasks, and the ways of supporting such implementations.

The authors report that tasks are essential in a mathematics lesson because they give students messages about what “doing mathematics” is. So tasks teachers pose have a major influence on how students think mathematically, and thus they have an altered perception of mathematics. This brings us to the point of developing tasks with a high level of cognitive demand. But implementing these tasks in a high-level manner is not so easy. Since students are comfortable with being told what to do, sometimes teachers lower the rating of a task by making it more simple or giving students a procedure.

To avoid implementing high-level tasks in a low-level manner, Henningsen and Stein (1997) offer some factors, as outlined in Henningsen (2000, p. 245). Teachers should focus on the meaning of the content, rather than the procedures involved in calculating it. Scaffolding is another useful tool. By using students' prior knowledge and building them up to a new concept, they will more easily connect ideas together creating a richer conceptual understanding of a topic. Other strategies mentioned are modeling and self-monitoring.

The remainder of the research in Henningsen and Stein's (1997) article is shown in their empirical study, in which they demonstrate factors that support high-level thinking, reasoning, and sense-making.

References

• Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematical Education, 28(5), 524–549.
• Henningsen, M. A. (2000). Triumph through adversity: Supporting high-level thinking. Mathematics Teaching in the Middle School, 6(4), 244–248.
• Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14(3), 132–138.

Alternative Assessment

Goetz (2005) believes cooperation plays an important role in learning, and that, we should assess what we value (p. 12). Ergo, his assessments in his precalculus classes contain cooperative activities in which students' grades are partially dependent on the interactivity and communication between the students. His tasks are posed as open-ended word problems with real-world applications. In one example, given a data set the students were expected to construct a mathematical equation that models that data. There was a diverse range of answers varying from polynomial to rational to exponential functions.

The students' grades were based on a grading rubric, which the students were aware of before the exam. One of the elements on the rubric took group participation into account, and the students were expected to explain the specific roles each member played in the cooperative activity. That way, each student would get the deserved amount of credit. Goetz believes in using assessment as a tool for learning, such that students will turn an exam into a learning experience. This corresponds with NCTM's view that assessment should enhance student learning.

Grading rubrics are essential for assessment. In my personal experience and from advice from my educators, I can report that students' scores on assessments without rubrics can become subjective and open to interpretation, and are a potential source of conflict between teacher and parents. In order to provide an explicit and objective grade, a grading rubric must be used. The higher resolution a rubric has, the more accurate students' score will be on that particular task. Sometimes, though, teachers create rubrics that are hard for students to understand, thus the students might score more poorly than if they had a clear indication of what is expected of them.

Brown-Herbst (1999) had her middle-school students construct their own grading rubric. Her class's rubric was based on a final draft submitted by teachers from twelve schools participating in a statewide project in Alaska. While constructing the rubric, her students had to interpret the language used by teachers to gain an understanding of the spectrums of performance. After three days of debate and discussion among the middle-schoolers, they finalized a rubric that was to be used on not only their end of year exam, but also on that project itself. In other words, the students were being assessed by their own criteria.

A project such as the one implemented by Brown-Herbst (1999) takes much time and planning, but the knowledge and skills gained by the students are worth it. Students reflected NCTM's (2000) Communication process standard: they translated mathematical teacher language into mathematical student language, and conveyed concepts and ideas to one another and refined them. Even previously implicit ideas have been made explicit by students who asked each other to clarify meaning; e.g., [a] seventh-grade girl spoke up: I need to know exactly what the math thing is (p. 453).

A final example as another form of assessment is offered by Bailey and Chen (2005). They introduced the graphing portfolio, in which students are expected to trace out a picture or graphic by using functions (either cartesian or a combination of cartesian and polar) to illustrate the lines in the graphic. This method is slightly related to Goetz's (2005) example, in that students are working backwards with functions: given a function's graph (or a curve of best fit), they need to find the equation. Graphing portfolios are useful for an artistic and creative touch in a mathematics course.

References

• Bailey, E. C., & Chen, F. (2005). Graphing portfolios in calculus: Reinforcing concepts and inviting creativity. Mathematics Teacher, 98(6), 404–407.
• Brown-Herbst, K. (1999). So math isn't just answers. Mathematics Teaching in the Middle School, 4(7), 448–455.
• Goetz, A. (2005). Using open-ended problems for assessment. Mathematics Teacher, 99(1), 12–17.

Activating Prior Knowledge

Stillman (2000) elaborates on the different classes of, and the importance of, prior knowledge in the way students approach a task. Prior knowledge of the academic variety consists of knowledge gained through outside academic experiences, such as content learned in another class or study. Encyclopaedic prior knowledge includes general facts and trivia of the world, and episodic prior knowledge is that which is gained by a learner from personal experiences outside of an academic setting.

Sloyer (2004) offers a strategy he calls the extension-reduction strategy. This is a pedagogical strategy in which a teacher will present a problem that requires his or her students to use their prior knowledge to construct new knowledge. The goal of the strategy is to get the students to reduce the new problem down to a simpler and better understood problem (e.g., finding the area of a polygon by subdividing it into triangles). The teacher's role is to help and guide the students in activating their prior knowledge. Even though the students may have this prior knowledge, they may not always know how to use it productively. The example that Sloyer gives is a problem in which students try to find the volume of a segment of a cone. The prior knowledge here was that of finding a part of a whole. The value of the desired part is found by taking the value of the whole (whether that be a region's area, a finite series, a solid's volume, etc.) and subtracting the extra amount.

Hare's (2004) example is a bit more complex. In this study, students learned implicit differentiation by reinforcing and expanding their concept of a function. Students had varying misconceptions of the definition of function and thus could not take the implicit derivative of an equation properly. Through guided questions and activities, the students were not only able to use their prior knowledge but also improve on it, while at the same time gain new knowledge.

References

• Hare, A. & Phillippy, D. (2004). Building mathematical maturity in calculus: Teaching implicit differentiation through a review of functions. Mathematics Teacher, 98(1), 6–12.
• Sloyer, C. W. (2004). The extension-reduction strategy: Activating prior knowledge. Mathematics Teacher, 98(1), 48–50.
• Stillman, G. (2000). Impact of prior knowledge of task content on approaches to applications tasks. Journal of Mathematical Behavior, 19, 333–361.

A Student Teacher's Curriculum

Gwen Lloyd, Ph.D., is a former faculty member at my alma mater, Virginia Tech. She was my advisor when I initially transferred into the Mathematics Education program here. Her article explores one student teacher's interaction with materials used in a kindergarten mathematics curriculum. Anne, the student teacher participant, had used two different approaches to her use of curriculum, but in both cases, each use was adaptive (p. 63).

I had used this article as one of my resources for my Curriculum Principle Project. The most relevant sections of the article were not those of Lloyd's methods and results, but of the background research. Lloyd talks about the history of views of teachers' curriculum use. Over time, researchers have had a great contrast of views, ranging from the view that teachers see the textbook as a fixed source of any and all information that is to be delivered to the students in a linear manner; to the polar opposite view that teachers are interpreters of information, changing the curriculum to suit their own classes' needs. Further, two specific, independent studies had analogous findings: they each demonstrated that these contrasting views are extremes of a linear spectrum, with any kind of teacher interaction with curriculum falling on any point on the spectrum.

The motive for Lloyd's study was reasonable. As I've stated in my Curriculum project, I think it's important to learn more about how teachers interact with their curriculum because we can use that information, cross-referenced with data about students' responses, to see what works and what doesn't. Knowing this, we can change and develop, and know how to change and develop, a curriculum that fits students' needs.

After reading the research questions, I found out that Anne was using two different sets of materials (abbreviated EM and MTW (p. 71)) in her kindergarten mathematics instruction. What I wanted to know was whether she used these different sets in the same class, or across different class. If the latter is true, I wonder how big of an effect differential Anne will have on her students' lives at such a young age. If she is using different methods on different classes would there be a butterfly effect? This question applies all the time, when we consider different teachers of the same course. I found out later in the Data Collection section that she, and another teacher, were using both of these materials on the same class, with alternate chunks of time (2 to 4 consecutive days) devoted to each set. It turns out that Anne's class is receiving instruction from both material sets.

Another data collection discussion I found interesting was that among the four kindergarten teachers in this class, each teacher saw one-fourth of students each day, and rotated stations. That way each teacher would have to teach the same lesson each day for 4 days in a row, to a different group of students. Lloyd also pointed out that in this system, the students got to participate with each of the teachers, but none of the teachers were able to observe each other.

I think the Findings section was very in-depth and complete. Lloyd covered Anne's use of each set of materials and the Curriculum Design and Curriculum Construction in each set.

The Discussion talked about how Anne fell on the spectrum from the two studies. She initially lay on the middle of the spectrum but her alterations were leaning her to the right (towards the more deviant extreme). The rest of the Discussion was about what factors could have been an influence in Anne's curriculum use. I think this is a major subject to talk about for any kind of data collection. There are so many variables that we must account for when collecting data, and we need to consider how much and what kind of an effect they have on the data. This also helps contribute to suggestions for future research: to conduct a wider, less in-depth, study and to minimize the variables.

Lloyd suggests that future research examine teachers at different levels of experience and us[e] different types of mathematics curriculum materials and textbooks (p. 91). Basically, she suggests that future researchers broaden the scope to get a better understanding of how Anne's case can be generalized. After all, the findings from Anne's class are particular and specific to her class only. Teachers who are reading this study must be careful when interpreting its results.

References

• Lloyd, G. M. (2008). Curriculum use while learning to teach: One student teacher's appropriation of mathematics curriculum materials. Journal for Research in Mathematics Education, 39(1), 63–94.
• Vennebush, G. P., Marquez, E., & Larsen, J. (2005). Embedding algebraic thinking throughout the mathematics curriculum. Mathematics Teaching in the Middle School, 11(2), 86–93.

Classroom Management

While watching the video, I first noticed the huge differences between classrooms in 1947 and classrooms today. All the desks were facing front, in neat rows and columns. I find in most current classrooms have desks clustered and facing in all directions, or arranged in concentric arcs like in an auditorium. When the students answered questions, they stood up and spoke without permission while modern classroom teachers require their students to raise their hands and wait to be called on. If I were the teacher, I wouldn’t have done the math myself when explaining the conversion problem. I would have focused on bigger concepts, or asked the students how to do the math.

I found it familiar that while the teacher was gone, the students talked and goofed around. This was common in many of my high school classes. One of the things I found unfamiliar was the fact that the teacher was working on the board while the silent class watched and took notes. As a high school student most of my teachers would take a more active role and engage the class.

In the first scenario, the teacher expected the students to be well behaved and silent. He expected them to know how to study and raise their grades without knowing how. In the second scenario, the teacher asked more questions and expected the students to participate. He was still expecting good behavior but was more lenient when punishing bad behavior. With this friendlier attitude, the students respected him more and were less likely to act out.

Reflecting on this video in class, we talked about the six NCTM principles.

• Equity was the most apparent issue. The class was probably 100% white. The teacher was using examples that were sexist. Girls were supposed to be good at cooking and boys were supposed to be good at building.
• Regarding Learning, the students were all expected to learn the way the teacher taught. There was no accountability for differences in learning styles.
• The Teaching was not student-centered and very lecture oriented. In a modern classroom, the teaching is supposed to be more interactive.
• In the second scenario, the teacher was more critical of the Curriculum. When demonstrating a problem, the teacher left no “wait time”, or student interaction. The focus was on the conversion from yards into feet.
• Using the Assessment, the teacher could figure out which topics the students had most trouble with.
• The only available Technology in 1947 was the textbooks, chalkboard, and pencil and paper (used traditionally).

References

• How to maintain classroom discipline: Good and bad methods training educational video [video file]. (2007). Retrieved from http://www.youtube.com/watch?v=gHzTUYAOkPM.

Teachers' Conceptions of Equity

The motivation for the article stems from the assumption that the majority of secondary math teachers have largely unexamined, varying conceptions of what NCTM's Equity Principle means in the classroom. The research question asks what equity means and how we will recognize it when we see it.

Teachers participating in the study met monthly over a year, for about 2.5 hours each meeting. They discussed their initial conceptions about equity, findings from reading research about equity, and their final conceptions about equity after the sessions. This reminded me of my Senior Seminar class, when we would reflect on articles that focused on a particular mathematical content or process (e.g. Trig functions, Representation, etc.), and then would discuss our conclusions and reactions. One of the things we were taught to do was to assume everybody in the class has read the assigned article, so that we wouldn't waste time summarizing.

I am trying to simulate that same practice in this blog (Mathematics Education Research). Even though it would be easy to summarize the research that I find interesting for those who haven't read it, I have to remind myself that anyone interested in reading those articles can obtain the resources to do so. Else, the article's abstract provides a summary. Rather, this blog is more about my reactions and thoughts about the readings—or viewings—so that I can expand on it and provide insight for myself and others.

Anyway, the results of the first part of the study showed that the teachers' conceptions of equity fell into four major categories, and that although these categories were remarkably different from one another, the participants agreed that the responsibility of working toward equity falls on the teacher. During the second half of the study, teachers were asked to pick one student in their class, who was struggling mathematically, to get to know on a more personal level. The teachers that succeeded found that those students raised their level of engagement and achievement in the classroom.

Bartell and Meyer (2008) conclude that the first step for teachers to support and maintain equity is to explore and identify their own conceptions of equity. Further, becoming more personal with an under-proficient student can boost morale and achievement, and not to mention, help the teacher form bonds with his or her students. The authors then pose a few open-ended questions at the end, perhaps as motivation for future research.

References

• Bartell, T. G., & Meyer, M. R. (2008). Addressing the equity principle in the mathematics classroom. Mathematics Teacher, 101(8), 604–608.

Graphing Calculator Use: Activity and Affect

The thesis of this article provides an effective method that captures data on how and why individual students choose to use graphing calculators—henceforth, HGT—outside of whole class situations. I believe McCulloch's purpose is clear and valid. Most recent research on HGT focuses on its effect on student achievement, or else focuses on the teaching and learning of a specific topic in content. This past research drives her interests in this paper:

1. When students work independently rather than in a large group, what are they actually doing with HGT?
2. What aspects of HGT do they attend to?
3. How do emotions, values, beliefs affect their HGT activity decisions?

McCulloch mentions several difficulties and drawbacks about traditional methods of data collection, so offering a new method and determining its effectiveness has well-founded intentions.

To address the problem solving strategies and decision making in detail, a task-based interview is typically implemented. Basically, this means the problem solver—the subject—will think out loud in front of the interviewer. However this design is unsuitable if it is used to collect data about emotions and values. McCulloch's study combines the task-based interview with a video-SR (video stimulated response, a procedure in which videotaped behavior is played back to the subject in hopes that he/she will recall his/her activity). With this combination of designs, the subject will recall both cognitive and affective activity. In addition to capturing the subject's activity, a video capture card was used to capture the screen activity on the HGT. When these three components are put together, the event is recreated.

I think this method should be used for an intensive examination of a subject, but would not be appropriate as an extensive study for a large group or class. The resources of time, money, and equipment are simply not available. The article also mentions other limitations.

McCulloch makes several conclusions from her study:

1. The methodology used is very effective regarding capturing data from individual students, and opens many doors to future research.
2. The methodology captures information that was unattainable in the past.
3. The methodology allows students to reflect and recap their activities with the HGT, providing a learning experience.

I'd like to make one more point about the outcomes of this study. In the article, one of the limitations identified was that the subject's emotions played an important role when reviewing the video-SR interview. therefore… the student will associate [them] with that event in the future (p. 80). I do not see this as a drawback though. In my experience, the concepts with which I associated certain emotions were the ones that I remembered the most in school. Teachers should always promote positive emotions in students and never negative ones, but I cannot ignore the fact that I achieved well when I was tested on the concepts for which I had negative feelings.

References

• McCulloch, A. W. (2008). Insights into graphing calculator use: Methods for capturing activity and affect. International Journal for Technology in Mathematics Education, 16(2), 75–81.

Teacher as Researcher

One of the benefits of teachers taking an active part in the field of research is the community formed among educators. Research allows teachers to reflect on their own pedagogical knowledge and to share and revise it with each other. Lankford (2003) provides a step-by-step sequence for novice researchers interested in becoming more active.

Thinking About Research involves simply motivating oneself to read more relavant literature and to stay informed on current trends and recommendations. Research is to be considered as a tool that provides direction when it comes to trying new approaches and techniques in the classroom.

When Reading and Discussing Research, educators should read a wide range of literature and discuss, using personal experience, the implications it has in the classroom. In this way, teachers are able to bounce ideas off of one another and gain broader perspectives.

Designing and Critiquing Classroom Investigations: At a certain point, teacher teams are comfortable enough to discuss suggestions and critiques for new implementation. It is at this point that the research theory becomes practical.

In addition to Lankford's approach, she provides specific examples of different types of informal investigations conducted in her team's classrooms. These illustrations make clear how important the teacher's role is in research.

After reading the chapter, I had a few questions:

1. How would more conservative teachers feel about new ideas involving research and new teaching methods?
2. In terms of grading, educators focus on the processes involved. However, in the ‘real world,’ only the results matter. How can teachers reduce these conflicting pressures in students? Is it fair for teachers to grade processes and results equally?
3. In what ways does research adversely affect students? When testing new techniques, are the students acting as ‘guinea pigs?’ For example, if two different methods were used on two different classes, and later results showed that one method was significantly more effective, is it fair for the students in the opposite class?

References

• Lankford, N. K. (1993). Teacher as researcher: What does it really mean? In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 279–289). New York, NY: Macmillan Publishing Company.

NCTM Principles Overview

After reading NCTM's six Principles for School Mathematics, I obtained a better understanding of the features required to produce a high-quality educational environment for mathematics students. NCTM emphasizes that the Principles are not mutually exclusive, i.e. they address overlapping themes. I believe the Principles are also collectively exhaustive, i.e. they try to encompass all of the features necessary in the math classroom.

I agree that all students are capable of learning mathematics. However I believe some students are more advanced than others. Like sports, some athletes are good at baseball while others are good at hockey. In the academic setting, there are going to be students ahead of the learning curve and behind the learning curve, and these students will vary from math to English. Teachers should be able to accommodate for this aspect of diversity (among others) while keeping expectations high.

Mathematics is a subject that well exhibits the Curriculum Principle. Math itself is cumulative, so I find it easy to make connections between content areas within the subject. As the NCTM says, the strands are highly interconnected (2000, p. 15). The more easily students can see and realize this, the more connections they will make. The NCTM recommendations for curriculum will help narrow in on the tasks I will need to accomplish while constructing lesson plans.

I never realized the impact teachers make on their students until I heard it. One school year might not seem like a long time, but in the minds of the students it can make all the difference. Looking back on my own experiences in high school and reading about the effects of a single teacher on an entire class enforces this view. Realizing that one can change the lives of children forever may be a scary thought but it can also be a good one if teaching is done effectively. I think an effective teacher has a good balance of content knowledge (knowledge about what the students know and what they need to learn) and pedagogical knowledge (knowledge about how to teach). Teachers also need to continually seek improvement on their own part. As a teacher I plan to continue my exposure to research on mathematics and education so that my teaching practices will be continually improving. This is furthermore in the best interest of the students because as generations change, pedagogical methods (along with content) will change.

I believe the best way to learn is to be in one's Zone of Proximal Development. Students need to be challenged and supported. When a task is challenging enough to overcome boredom, but not too challenging as to promote anxiety, the student is in the ZPD and will learn with understanding. Students need to be able to build on previous knowledge, elaborate on new concepts, and organize concepts in a way that helps them remember them the easiest.

New opinions about assessment open up doors that I had not realized existed in the past. Assessment is foremost used as a tool to detect what students learned and how well they learned it, but it is also used as feedback for the students. Students should easily distinguish their place in the curriculum so they know exactly what their strengths and weaknesses are. Assessment should be used as an intermediary to help students learn, not as the end to a unit where students will never have to use that information again. On the other side, assessment can be used to aid teachers. For example, teachers will know which students are excelling or falling behind in certain areas, and will be able to make decisions for future instruction.

The aid of technology helps level the playing field. This goes back to the Equity Principle. Technology helps students focus on the bigger problems at hand, such as those involving decision-making and problem solving. The increase in technology yields changes in curriculum and changes in views on which concepts are essential in the classroom. Technology shifts the students’ attention from thoughtless algorithms to more complex thinking.

There are a few Principles that stand out when regarding mathematics: the Curriculum and Technology Principles. I favorite these two Principles because I believe they are crucial in the mathematics classroom. As I’ve stated above, math is one of the subjects that is a continuous field with overlapping grey areas rather than a collection of discrete facts or figures. There are more connections that can be made in math than any other subject in secondary school (in my opinion), so building on these connections is essential for students to learn effectively. Technology is the other Principle that sticks out to me. It is true that technology can be used in other courses, but I believe it can be most appreciated in a math course. Not only does technology act as a tool that enhances the learning of math, it is also determines the behavior of students and teachers with regards to content. The more (and better) technology is available, the more students will be able to make decisions, think critically, and focus on meaning.

References

• The National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Proofs and Counterexamples about Continuous Functions

This study focused on reasoning and proof from undergraduate math majors in calculus. Proofs of truths and counterexamples of false facts have been shown to help students understand them. Not only do they help explain why a statement is true or false, but students will also be able to see how. However, reading proofs and constructing them are different processes. In my personal experience, constructing proofs is very difficult especially when I don't know what direction in which I need to head.

Before constructing proofs/counterexamples, students and teachers need to decide which one to produce (to prove a truth or to disprove a falsity?), which means they must decide on the validity of the proposition (is it true or false?), which means they must have a deep understanding of the concepts involved. In fact, many students and teachers have difficulty deciding whether a proposition is true or false because of their inadequate understanding of the mathematical content. Still, many mathematicians often struggle with formal proofs. Thus more effective ways of teaching these are needed, at least in the undergraduate math major domain.

Ko and Knuth (2009, p. 69) pose two research questions for this study:

1. How well do undergraduates construct proofs and generate counterexamples in the domain of continuous functions?
2. What problems appear in the proofs students construct or the counterexamples they generate?

One thing that the authors stated that I found striking was the fact that proof serves as a means of communication among mathematicians. This makes sense, because I noticed when learners can explain something in each others' terms, they gain a better understanding of the idea. Informally, they are proving it to one another.

We usually think of counterexamples, as disputes to generalizing statements. For example, the number $2$ would be a counterexample (in fact, the only one) to the false proposition, All prime numbers are odd. Some counterexamples can actually be ‘examples’, if we think of them as a proof to the generalizing statement's negation (e.g., $2$ is an example that proves the conjecture, There exists an even prime number).

The authors also discussed different types of proof schemes and counterexamples and the properties of each. They created a scale or spectrum of proofs, and one for counterexamples, that ranged by robustness. After receiving calculus students' work on a problem set, they rated it using their own spectra.

Since constructing a proof requires sufficient background knowledge and the decision of whether to prove or disprove, Ko and Knuth (2009) suggest that math teachers should provide a learning environment in which students can easily decide the veracity of statements. There are several methods of doing this. Especially in secondary school, formal proof is not even in the question. Therefore teachers must foster students' informal proof and reasoning as often as possible. Fishbein (1982) states, Ignoring students' intuitive representations of proof does not promote their mathematical understanding (as cited in Ko and Knuth, 2009, p. 75). In addition to drawing attention to students' intuitive thinking, teachers should also explicitly identify students' logical flaws and misconceptions so that they will have a stronger understanding of the processes in constructing (even informal) proofs.

References

• Ko, Y., & Knuth, E. (2009). Undergraduate mathematics majors' writing performance producing proofs and counterexamples about continuous functions. Journal of Mathematical Behavior, 28, 68–77.

How Students View Assessment

Assessment seems to have a completely different effect on students' attitudes based on whether they frequently pass or frequently fail.

Students who demonstrate winning streaks are confident and hopeful, demonstrate continual evidence of success, and are excited to learn. These students are likely to seek more feedback and more challenges. On the other hand, students who demonstrate losing streaks are frequently hopeless and give up easily. They do not feel safe at school and feel that they are always being evaluated. They see feedback as criticism and do not seek challenges or new ideas. Stiggins tries to eliminate this gap.

Rather than using assessment to sort students into winners and losers based on performance, educators should use assessment to help student learn. Stiggins offers an alternative notion, Assessment For Learning (as opposed to assessment of learning). This alternative encourages teachers to turn assessment into a process that involves sharing goals and targets with students, provide frequent and continual feedback in student-friendly language, and provide examples of outstanding student work. This causes students to self-assess and notice trends in their own achievement. Students become more aware of their academic progress. They begin to understand what is expected and make decisions on how to become better. They also generate their own feedback and set their own goals. The hope, Stiggins says, is not to eliminate failure but to eliminate losing streaks. This helps boost student confidence and motivates them to try more.

References

• Stiggins, R. (2007). Assessment through the student's eyes. Educational Leadership, 64(8), 22–26.

Learning and Knowledge in the 21st Century

Chapter 3: Learning and Knowledge in the Twenty-first Century

To be completely honest, I thought the article was just short of a textbook. I remember reading about different perspectives of educational psychology in another course, and much of the article was information that could have been left out. A few of the topics deviated from (what I understood to be) the original thesis of the article, which was about how technology is changing learning and teaching in this day and age.

I enjoyed the introduction of the article because it compared different beliefs and views on learning from very different time periods, while relating back to the theme of technology. It seemed as if educators of the late 19th century were preparing their students for the specific jobs they thought they would have. Whereas now, students seem to be more well-rounded. In the past, school was seen as a means to an end, but now, school is seen as preparation for more learning in the future. One thing that prepares young learners today is that they really do learn how to learn with technology. Views on learning nowadays emphasize cognitive processes like critical thinking, problem solving, and decision making over lower arithmetic and computational skills.

A few big questions did come across my mind while I was reading about this. At what point do we draw the line? Students don't need to be able to take the cube root or write the prime factorization of very large numbers anymore, so why should they need to be able to compute the limit of a rational function or find the general solution of a first order non-linear differential equation? At what point do we say, “That's enough, the calculator can do the rest.”? Why are we, as educators, so selective about what we decide students ‘should’ know?

Aside from discussing the differences due to technology of learning beliefs across time, the article also discussed different perspectives on learning. The Behavioral perspective (Skinner) focuses on external, observable responses. Drill and practice are reinforcements for learning, and educational technology can be highly effective (unless it is excessive or premature, etc.). Behaviorists state that learning is sequential and hierarchical, such as an axiomatic system.

Cognitive psychologists (Piaget) accept that learning is a result of adaptation motivated by disequilibrium. Learners apply existing schema to change what they know about new information, but also alter existing schema to fit new information. This push-and-pull balance of assimilation and accommodation is required when transferring from disequilibrium to equilibrium, thus satisfying the learner's drive. Cognitivists also support scaffolding, which requires teachers to guide and assist learners. Through scaffolding, teachers can determine what type of help to offer and when and how to offer it. Discourse is encouraged so that teachers will be able to recognize students' Zones of Proximal Development, the zone in which the transfer from disequilibrium to equilibrium is most effective, and keep them right in that zone to maximize learning. Before the ZPD, students are unchallenged and bored, while after the ZPD, students are intimidated and discouraged.

Constructivists say that teachers should create complex and realistic learning environments, encourage social interaction and communication, present multiple and diverse perspectives and representations, and facilitate student ownership in learning. Researchers today are emphasizing learning environments that take a mix of all three perspectives. Instruction should be student-centered, multisensory and multimedia-involved, collaborative as well as competitive, active and exploratory, critical, logical, and both theoretical as well as practical.

References

• Niess, M. L., Lee, J. K., & Kajder, S. B. (2008). Guiding learning with technology. Hoboken, NJ: John Wiley & Sons, Inc.

Assessment with CAS

The article on using a Computer Algebra System for teaching mathematics had several major points. The introduction was about how CAS changes teaching and learning of math. In a classroom where students are using CAS, the topics, focus within topics, and goals of lessons will change. Lessons are no longer algorithmic but focus on using operations to understand meaning. The article also mentioned two goals for teaching: to develop theory of mathematical concepts, and to use mathematical concepts in real-world models and applications. Teachers should try to construct exam questions to meet these goals. There were two schemes for analyzing exam questions.

1. The first scheme was based on testing skills and abilities of students, and took a more educational approach. It measured the educational value of problems as questions on an exam. The educational scheme had five categories.
   1. The CAS-Insensitive Questions did not make very much use of the CAS, and computation plays a minor role.
   2. The Questions Changing with Technology made a big difference once the CAS was introduced because time needed to solve the problem was greatly reduced.
   3. The Questions Devalued with CAS involved rare tricks and hard-to-remember equations to solve, so the CAS appeared insignificant.
   4. The Questions Testing Basics became trivial when using CAS because the answer would be produced immediately; however, the student needed knowledge about syntactical structure.
   5. The last category, Rediscovered Questions, are geared to support creativity, fluency, and flexibility, but are rare because of the difficulties in evaluating and grading.
2. The second scheme was based on the usefulness of CAS and took a rather technical approach. It measured the role of technology for answering a question. The technical scheme also had five categories, grouped into how significant CAS was in solving the problem, and how well the student should be familiar with CAS.
   1. In terms of significance, Primary Use needed CAS as a major activity,
   2. while Secondary Use did not facilitate CAS as strongly.
   3. Regarding familiarization, Advanced Use required in-depth knowledge of CAS,
   4. while superficial knowledge of CAS suffices for Routine Use.
   5. CAS is of very little help for questions in the last category, No CAS Use.

After discussing different types of problems in different categories, the article compared the two and made some observations about CAS. It facilitates the two teaching goals discussed above, it reveals educational value of exam questions, forces teacher to be conscious about exam questions, and revives the “forgotten” questions. When choosing exam questions, teachers need to keep the aforementioned goals in mind, but also question how they test the student. Exam questions should test general abilities rather than computational skills. In any testing environment the act of understanding and the act of overcoming an obstacle are equally important in the learning process. In addition, intellectual concentration and emotional tension are present and culminating, which creates a learning situation per se.

References

• Kokol-Voljc, V. (year unknown). Exam questions when using CAS for school mathematics teaching. publication unknown.

Teachers and the Internet

I think it's a bit unfair that teachers are rated based on online activity. One's personal life need not interfere with one's professional life. We are all entitled to do as we please outside of the work place (provided it's legal). If the administration hadn't seen the picture, Sydner shouldn't have been fired. Personally, I have my online profile set to private so that only friends can see it. I have also not made any efforts to befriend my subordinates (be they future students or the swimmers I coach). To the people who argue that employers could hack the privacy settings of social networks, I would counter that if they're using illegal methods to see my profile, they have no room to talk. The moral of the article is that current and future teachers (or any role models) should be careful of what they upload to the Web. But the fact of the matter is, we are unable to control what our friends post.

The results of the Facebook study conducted conflicted with my beliefs about online networking. Reading the material on the Net Generation earlier this semester, I was convinced that social networking and online communication enhanced human interaction. Even when we're physically alone, we're not ‘alone’ due to instant messages, text messages, video/audio chat, direct messages, emails, and status updates. It's hard to believe that these kinds of phenomenon are linked to anxiety, anger, and depression.

References

• Michels, S. (2008, May 6). Teachers' virtual lives conflict with classroom. ABC News. Retrieved from http://abcnews.go.com/TheLaw/story?id=4791295.
• College Facebook users have lower GPA's, more neurotic behavior. (2009, September 29). News Channel 9. Retrieved from http://www.newschannel9.com/news/facebook-985115-college-students.html.

Lesson Structures, Methods, Solutions

When reading the article, I kept in mind that although there is a big difference in Type 1 and Type 2 lessons, there is no “right answer” in choosing a lesson type to model after. Type 1 lessons are narrower and don't elicit as much critical thinking and decision making, but they are more structured and make a classroom more organized. Type 1 lessons may be necessary depending what kind of students I will teach. If I am teaching students that are “behind the curve” and may need extra assistance with math, Type 1 lessons may be required. On the other hand, Type 2 lessons may be more beneficial for gifted students who can handle a more independent learning environment.

As I read the section describing Type 2, a lot of questions popped up in my mind: If the Type 2 lesson allows students to be independent and explore on their own, how do the goals of the lesson (and curriculum) fit in? How do they learn what they're expected to learn? What happens to classroom management if students get off track? What is my role as a teacher if technology allows students to learn on their own? Is my purpose solely that of a ‘marshall’—I'm just there to make sure students are doing what they're supposed to be doing? These questions have significant purpose not only tonight but throughout all the readings. I want to be able to use technology to guide students' learning, not to replace my teaching. My purpose as teacher is to help students investigate and understand relationships among objects, and to complexify this investigation so that the students can simplify (p. 309) it. Rather than making decisions about what and how to investigate, my job is to guide students to make those decisions, and reaffirm that they're making the right (or wrong) ones.

A miscellaneous topic: I noticed that teachers and students often throw around the words ‘lesser,’ ‘greater,’ ‘bigger,’ and ‘smaller’ a lot without realizing their meanings. ‘Bigger’ and ‘smaller’ can be quite confusing in the domain of negative numbers. (Positive numbers are easy to understand in this regard). As teachers, we should be careful when choosing these words to describe relationships and ordering of real numbers. We should use the terms ‘lesser’ and ‘greater’ to make explicit the ordering of numbers. For example, negative five is definitely less than negative three. When we ask, “Is negative five bigger or smaller than negative three?”, students will most likely get confused because they will think of absolute value. Negative three is closer to zero (also known as ‘nothing’) than negative five is, therefore it must be ‘smaller.’

One last point: Tonight's reading enforced my belief that the main goals of any lesson should be investigation and discovery. When I first transferred to the Math Ed program here at Tech, I was excited to teach in a math classroom and lecture to the students, much like many college professors do today. However over time I realized that in a high school environment, students do not want to be lectured to. They want to investigate and discover things on their own. So rather than giving out all the answers, I hope to guide students to create and find their results and own them, so that way they will learn better, remember easier, and maybe even be enthusiastic about math.

References

• McGraw, R., & Grant, M. (2005). Investigating mathematics with technology: Lesson structures that encourage a range of solutions. In W. J. Masalski & P. C. Elliott (Eds.), Technology-supported mathematics learning environments (pp. 303–317). Reston, VA: NCTM.

Teaching Strategies for Technology

When I first watched the instructor talk about technology in the hands of students versus in the hands of teacher, I suddenly imagined a classroom full of students with open laptops working on a geometry problem together. I realized that even though technology is great to demonstrate, the students would get a more beneficial (and intriguing) experience if they could discover things for themselves. I remember the feeling of playing around with dynamic figures and devising my own conjectures. I couldn't wait to see if I was right or not, so I tried to prove them right away. I want to instill this feeling in my students. The instructor also talked a little about equity. I agree that the aid of technology helps level the playing field. It equalizes opportunities for all students.

In the reading, there was a focus on teaching strategies that should be used to implement technology in the classroom. The main focus was that technology should extend math and enhance learning. It should promote higher-order outcomes, such as reflection, reasoning, problem posing, problem solving, and decision making (NCTM, 2005, p. 1). I want my students to be able to be able to develop these processes without worrying about technical difficulties or syntax errors. I want to be able to teach students how to use technology to help them, not do math for them. Technology should also be a tool or aid for students, not their brain. One really interesting argument against the use of technology is that it does the work for the students. One really good example is prime factorization (Fundamental Theorem of Arithmetic). If the CAS can do it for the students, do they really need to know how to do it? Some might say no, but I think the students should at least learn how to do it first, and then use the calculator for more complex problems. That way, it's no magic trick. Once the students learn how to do something, they can use the CAS to do it for them afterward so they can focus their energy on the bigger picture, e.g., when should prime factorization be used? This example illustrates the white box-black box strategy. First teach the students how to do it by hand, and then allow the technology to do it for them. (For those computer science folks who know what “information hiding” is, this is a really interesting subject to talk about.)

References

• Ball, L. & Stacey, K. (2005). Teaching strategies for developing judicious technology use. In W. J. Masalski & P. C. Elliott (Eds.), Technology-supported mathematics learning environments (pp. 3–15). Reston, VA: NCTM.
• The National Council of Teachers of Mathematics. (2005). The use of technology in the learning and teaching of mathematics. In W. J. Masalski & P. C. Elliott (Eds.),Technology-supported mathematics learning environments (pp. 1–2). Reston, VA: NCTM.
• Technology in mathematics education [video file]. (2006). Retrieved from http://www.youtube.com/watch?v=W58ReRyNYp8.

The Net Gen, Continued

Chapter 4: Using Technology as a Learning Tool, Not Just the Cool New Thing
Chapter 5: The Student's Perspective

Chapter 4 emphasized further what was discussed in the first three chapters. It's interesting to read about how the Net Gen learns differently than other generations do. They use computers and other most recent technology in everyday life including work, school, play, and hobbies. But at the same time, the Net Gen is similar in work ethic and ambition to the Greatest Generation: their grandparents. NetGeners like to be creatively challenged.

I don't necessarily agree with the way the article mentions how NetGeners should study. Although a stimulating study environment is essential, I think it's up to the student to decide where and how to study. Personally, I like a quiet room with not much distraction. I find that I can get more done, and get it done faster, with a concentrated attention span.

Human interaction was also mentioned. This section really brought light to how I think about online classes. Social interaction is extremely important, and online classes need to do as much as possible to enhance it, not replace it. I get angry at people who notice trends from statistics of Math Emporium classes and think that it must be working because grades are higher. But it never occurs to them that maybe the reason grades are higher is that cheating is easier, which brings me to my next topic.

Technology is definitely helping cheating. This was made obvious, what with text messages, copy/pasting, internet sourcing, etc. However, technology is also helping teachers (and software) detect cheating. This wasn't mentioned in the article. Cheating detection software has gotten better through time, just like all technology has. Technology makes it easier to cheat, but it also makes it easier to get caught.

References

• Oblinger, D. & Oblinger, J. (2005). Educating the net generation. Retrieved from http://www.educause.edu/educatingthenetgen/.