The Promise of a Dynamic Mathematics Classroom: **Technology at Work**

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by

Kacey Sensenich, K-5 Mathematics Facilitator, NC-PIMS

Eleanor Pusey, 6-12 Mathematics Facilitator, NC-PIMS

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Abstract | Paper | References | Appendix A | Appendix B | PDF Version (without video)

The Promise of a Dynamic Mathematics Classroom: Technology at Work

Abstract:

Technology is an essential tool for the 21st century mathematics classroom. The value of multiple technologies in the K-12 classroom produces multiple payoffs for students’ learning and understanding. The use of technology helps facilitate models of a visual form—an important step for children in moving from the concrete to the abstract. A shift in roles for teachers and students is to work with the mathematics that is presented in a manner that allows for exploration of concepts. Public education should reflect this practice, allowing students to focus more on the analysis and interpretation of results. Throughout the paper, video clips are inserted to showcase student and teacher interaction with a variety of technology tools as a vehicle for exploring mathematics content. Given the documentation from the literature and the compelling evidence seen in the videos, there is no doubt that technology can and does support, enhance, and extend the quality of mathematics instruction in our schools. It is recommended that school systems should maximize their current resources in technology, sustain those resources with ongoing professional development, and consider some new resources in technology for their future..

The Promise of a Dynamic Mathematics Classroom: Technology at Work

Technology is an essential tool for the 21st century mathematics classroom. Technology comes in many forms. It can be as simple as a pencil, or more complex like computer application and hardware. For our purposes, this paper focuses on the more complex—software (applets, applications, etc.) and hardware (computers, calculators, SMART Boards, etc.). Technology is slowly permeating the walls of our country’s public education mathematics’ classrooms. Professional development for teachers related to the implementation of such technology has become more readily available, particularly as state licensure renewal requirements are mandating that teachers’ continuing education includes some units of technology.

Why are teachers not better utilizing these technology tools to improve K-12 mathematics instruction%3f Is it because the worth of technology is not understood or valued%3f Is it because access to technology during classroom instruction is still an insurmountable obstacle%3f Do teachers fear the technology itself%3f Or do they fear the ‘unknown’ (i.e., not being able to address potential student questions, not knowing every aspect of a particular medium, etc.)%3f

The value of multiple technologies in the K-12 classroom produces multiple payoffs for students’ learning and understanding. Technology use in the classroom, as a tool, allows teachers and students to work with the mathematics directly. Students are able to explore areas of mathematics that are challenging to grasp in a two-dimensional world. Technology applets, applications, and hardware give students a chance to broaden their mathematics. Teachers move towards being a facilitator of the students’ learning. This paper will describe the value of using technology in the mathematics classroom, provide compelling evidence of this value as captured in some unrehearsed and unedited video footage from classrooms, and suggest some next steps for school districts and their teachers that could build upon and extend their current efforts.

All video footage takes place in classrooms or professional development sessions of Lead Teachers participating in the North Carolina Partnership for Improving Mathematics and Science (NC-PIMS). NC-PIMS is a comprehensive Mathematics Science Partnership jointly funded by the National Science Foundation and the U.S. Department of Education. The vision of NC-PIMS is to improve the science and mathematics learning of all students, while simultaneously closing the achievement gap between racial and ethnic groups. The vision is supported by three goals: to develop leadership and policies that support instruction in science and mathematics, to create and deliver high quality professional development for teachers, and to design and implement activities which encourage parental and community involvement and student engagement in science and mathematics learning. This paper is a product of the work and success of the project’s Lead Teachers and Facilitators in carrying out the vision and goals of NC-PIMS and the authors would like to acknowledge the partnership’s funding agencies for making this effort possible.

The National Council of Teachers of Mathematics (NCTM) made public their vision for K-12 mathematics education in a document called Principles and Standards for School Mathematics (PSSM). Technology is one of the six defining principles included and is described as “essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning” (NCTM, 2000, p. 11).

How does technology influence the mathematics that is taught%3f George Bratton reports “…the impact of technology on instruction and curriculum have often indicated that the proper implementation of technology does three things: it makes teaching some topics unnecessary, it permits teaching some topics better, and it allows teaching some topics that have never been taught” (Bratton, 1999, p. 666).

View the following clip taken from a second-year algebra class where the teacher had spent the previous day having students work with a calculator-based ranger (CBR). Students walked in front of the CBR trying to match a given graph, as it plotted a graph of their distance (from the ranger) over time. Here the teacher is recapping what the students gleaned from this experience, connecting how they physically modeled rates of change to match certain graphs with varying slopes. **(Play Clip 1) **This concrete experience that students engaged in to model abstract mathematical ideas has no doubt given them a stronger grasp about rates of change and thereby permitted teaching of this topic better. Note also the natural connection made to piece-wise functions, which are not part of her curriculum but a more advanced topic that the students will see in later course work and hopefully recall more readily as a result of this exercise.

How does technology enhance student learning%3f PSSM contends that one way is by making the process of creating a variety of representations (e.g., tabular, graphical, symbolic, etc.) easier. In addition, technology helps facilitate models of a visual form—an important step for children in moving from the concrete to the abstract. For instance, in using virtual manipulatives, teachers are better able to reinforce an abstract concept that students might otherwise struggle to visualize. Also, PSSM states that technology extends the range of problems accessible to students and permits them to execute routine procedures quickly. Finally, it suggests that technology can free up the teacher to observe and attend to student thinking.

Douglas Clements has written many articles, and completed research studies, supporting the use of technology in the mathematics classroom. One article, “Concrete” Manipulatives, Concrete Ideas (Clements, 1999) explores the movement from concrete mathematical concepts to the abstract through the use of technology. The article also states that students’ attitudes are affected with the use of manipulatives. “Attitudes toward mathematics are improved when students have instruction with concrete materials provided by teachers knowledgeable about their use” (¶ 3). He states, “Good manipulatives are those that aid students in building, strengthening, and connecting various representations of mathematical ideas” (¶ 19).

Teachers must focus on teaching strategies that enhance the students’ learning. “To support these curricula goals, teachers are encouraged to see students as active constructors of knowledge, which students build up through exploration, invention, and discourse with other members of their mathematical learning communities, as well as through interactions with various manipulatives and forms of technology” (Nathan and Koedinger, 2000). Through these experiences, teachers are able to help students better understand the mathematical concepts.

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**A shift in roles for teachers and students is to work with the mathematics that is presented in a manner that allows for exploration of the concepts. “The teacher must shift from dispenser of knowledge to facilitator of knowledge” (Quinn, 1998). The following clip demonstrates how students interact with technology using a SMART Board. The students create the graph based on the information they have collected in the classroom. In this instance they have counted the number of pockets they have on their pants. After the graph has been labeled, the students each fill in a box on the graph. During the activity the teacher probes the students for information about why they color certain boxes, what the graph is showing as they fill it in, and about earlier predictions made. She uses questioning to facilitate the discussion. The SMART Board will display the results of the data gathering and the students will be able to continue to interact with the SMART Board and graph while discussing the results with the guidance of the teacher. **(Play Clip 2) **

The use of many different types of technology tools allows students this opportunity for discovery. The SMART Board has given the students an opportunity to be part of the creation of the graph in a dynamic and interactive way. This data can be saved on the computer and used again throughout the week’s lesson and referred to during the year. We now examine some of the literature that supports and describes some of these tools for instruction applicable to specific strands: statistics, probability, geometry, and algebra.

The work of statisticians depends heavily on the use of technology in handling the plethora of complicated formulas and tedious number crunching (Bratton, 1999). Public education should reflect this practice, allowing students to focus more on the analysis and interpretation of statistical results. This is not to say that computing means and medians with paper and pencil are out of place, but certainly in the upper grades, where the formulas become more complex, this level of computation seems less necessary than deciding what to compute (i.e., is the mean or median a more appropriate measure of center to summarize this data set%3f) or interpreting its meaning in the context of the data set. It does not serve our students well to resort to memorization of such complicated formulas when statisticians themselves depend on computers for such. In addition, the ability to display data is a critical skill for students in giving them a “picture” of the data; technology helps students to do this efficiently and gives them the opportunity to focus more on what type of display is appropriate and what can be inferred from the data.

Gary Kader and Mike Perry describe the problem solving of statistics as “detective work”. They say “the interactive and graphical capabilities of the microcomputer offer the opportunity to create a laboratory environment in which students experience statistical problem solving in a dynamic fashion. The appropriate integration of technology enhances the development of statistical concepts and methodologies, statistical thinking, and alternative problem-solving strategies” (Kader and Perry, 1994, p. 130). Therefore, statistical technology tools can provide opportunities for students to engage in higher order thinking such as making comparisons and predictions about data. Finally, engaging students in the use of such tools provides them the strategies they will need for solving problems.

Simulation is an important activity for students to engage in to build a strong understanding of probabilistic concepts. On a larger scale, simulation is how we try to make sense of patterns to predict various phenomena (i.e., weather models, economic supply/demand, etc.). Technology is efficient—it affords students the chance to run many trials of a simulation multiple times, sometimes with a single click of the mouse. This is essential for constructing conceptual meaning of a complex idea—the Law of Large Numbers—that will not make much sense without children having the ability to conduct and keep track of a large number of trials. Technology also makes it possible to simulate more complex situations that would be difficult to set up otherwise.

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View a clip from a professional development session where middle and high school teachers have just finished designing ways they might simulate a context having a 40/60 and 30/70 percent chance of occurrence (see appendix A, problem #1 for a full description of the task). In a whole-group discussion, the workshop Facilitator is soliciting the different ways the teachers did this. **(Play Clip 3)**

Note several appropriate techniques were shared that did not require the use of technology; these included drawing cards out of a prescribed deck or pulling color tiles from a bag with replacement. Nonetheless, the idea of weighting a coin to simulate a probability other than 50/50 would be challenging to accomplish otherwise.

Another important notion in probability is that simulations should produce random results. The next clip shows some confusion that arose in the same professional development session from the calculator’s simulation producing identical results on two different calculators. **(Play Clip 4)** The calculator is programmed to perform an algorithm that produces a random number; yet, every calculator of this same brand and model has been programmed with this same algorithm. If this is the first time, the algorithm has been retrieved (like with two new calculators), then the random numbers produced by either calculator will match each other and therefore, not appear “random” to the students. This is an instance where technology could cause potential confusion for students and teachers should be prepared to foster discussion to help them understand why this is occurring. On the other hand, this could also serve as an instructional aid that can only be provided by technology—that is, the ability to control such randomness for the purpose of whole-group instruction. It may be that having every student generating and working from the same data set will make it easier to talk about and use the data as a large group.

Dynamic geometry tools like Cabri™ and Geometer’s Sketchpad® (GSP) allow the user to investigate relationships in geometry and algebra through construction, transformations, and animation to name a few. Brad Glass contends that these kinds of technology for geometry are valuable because the dynamic nature of the programs force students to reason at higher levels (Glass & Deckert, 2001). For instance, dynamic software gives the user the ability to “drag” points and yet retain the constructed properties of a figure; this gives students a venue for recognizing patterns, leading to conjectures or theorems. So much of geometry necessitates the ability to apply a number of theorems and yet, often students have the most difficulty simply recalling what those theorems are. Environments like GSP® and Cabri™ provide the user a setting where they can discover and visualize those properties as opposed to being told to memorize them devoid of the sense-making or personal ownership that comes with direct interaction with the tool.

The power of these tools is in their dynamic capability through “dragging” to produce many examples efficiently prompting the use of inductive reasoning. Inductive reasoning leads students to conjectures, which provide opportunities for the teacher to question whether students’ conjectures will always be true. This sets up the occasion for deductive reasoning and helping students to see the need for proof. For instance, students could construct a triangle, measure each of its angles, and compute the sum. By dragging any vertex and creating numerous triangles, they might note that the sum seems to remain invariant. At this point, the teacher can begin to ask, “can we create every possible triangle%3f” or “how can be sure that this will work for all triangles%3f” as a precursor for proof.

For many, dragging the triangle and seeing the sum remain 180 degrees is enough to convince them that this conjecture will always be true. Important dialogue can follow about the distinction in mathematics between proving a conjecture and convincing us that something is true. This is a vital discussion that rarely comes up in the geometry classroom and could explain why many people never understood a need for geometry or proof, in particular. Yet students’ interaction with these technologies will motivate these conversations very naturally and unobtrusively.

Graphing calculators are another medium of technology used frequently in algebra classes to study functions. They are particularly useful because of their capability to produce multiple representations of a function with ease and efficiency. This can facilitate a stronger connection between the representations—the function, table, and graph—thereby helping students establish a better understanding of functions.

Penelope Dunham and Thomas Dick reviewed research on graphing calculators and report a number of
payoffs for
using this particular technology (Dunham & Dick, 1994). They reported that the calculator can improve students’ ability to problem-solve by placing less emphasis on algebraic manipulation. Also, they said the calculator provides students with additional tools for problem-solving and checking their solutions. They reported that students had a higher level of engagement in lessons with a graphing calculator focus and teachers tended to do less lecturing. The following clip occurs at the end of a class where some seventh graders have been working with motion detectors connected to graphing calculators to model distance/time graphs. **(Play Clip 5**) This kind of excitement is refreshing to see in a mathematics class where more commonly there exists a low level of student engagement and enthusiasm.

One pitfall with calculators is that students can view it as an “expert,” incapable of making errors. Of course, this is not entirely true as the user can input numbers incorrectly or misinterpret the results on the screen. However, a child that operates the calculator correctly and interprets their findings appropriately will have increased confidence and can begin to see himself/herself as the “expert.” In the next section, we provide four clips of students ranging from elementary age up through high school but all engaged differently in some form of calculator use.

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**The first clip shows the work of three students with the calculator. These students are exploring the basic multiplication facts. The calculator is showing them a problem to solve. The students then use paper or colored chips to model and solve the problem. They then type the answer in the calculator to see if they are correct. If they are not, they go back to the problem and reexamine their solution.
When the boy in the clip is presented with the problem 9 x 0, he believes the answer is 9.
When the calculator indicates it is wrong, he is baffled. The group then discusses why it could be wrong. During this scenario, the students try to provide rationale for why the answer is zero. The use of the calculator has prompted this discussion, about zero, that continues when the students are in line for lunch.
**(Play Clip 6) **This group of students will no doubt continue to try and make sense of the complexity of zero and its properties.

The second clip returns to the seventh grade classroom and highlights the work of one group of girls working to match the given graphs by walking to and from a motion detector. **(Play Clip 7)**
It is interesting to see how diligently they work to match the graphs as closely as possible. They use the labels on the bookshelves as points of reference to estimate their distance from the motion detector, and coach each other about how long to model each piece of the graph. Finally, it is encouraging to see them recognize and verbalize the limits of the technology as evidenced by the one girl who realized that there was obviously a distance at which it was too far for the motion detector to pick up their motion.

The third and fourth clip follow the direction and discussion in a second year algebra class, where students are collecting data to measure the height of a stack of cups against the number of cups added, plotting the data, and computing lines of best fit (see Appendix B for a complete description of the task). Here the teacher attempts to focus students by asking them to find the equation of best fit in two ways. **(Play Clip 8)** This directs the students to think at a higher level, drawing comparisons between the equations and noting the similarities and differences. It also motivates a rich discussion that follows about whether an equation of line of best fit computed by hand is more accurate than an equation of linear regression computed by the calculator. One pair of students has disagreed about this issue; here they open the discussion up to the rest of the class to hear their viewpoints on the subject. **(Play Clip 9)**These are critical discussions for teachers to initiate with students, particularly as calculators become tools that students use on a regular basis.

Recent changes in North Carolina’s Standard Course of Study are raising new issues for middle grades teachers. As algebra is gaining emphasis in earlier grades, rate of change is one big idea that can be facilitated through students’ work with motion detectors. This is a valuable tool because it sends out sonar waves to detect the movement of an object or person, measures and records the distance of that object from the motion detector, and then plots a graph in real time. This provides a completely different experience that meets the needs of the kinesthetic and visual learner. View another clip from the seventh grade math class where a different group is working with the motion detector to model various rates of change. **(Play Clip 10)** Notice how the first student had an “aha moment” after looking at his graph compared to the original graph—“Oh that’s right, I need to be three feet away not six feet away.” This clip occurred towards the end of the lesson that day and illustrates the power of immediate feedback (from the technology) in correcting the students’ thinking. This group’s efforts provide a powerful argument that all three of them are making the critical connections between the graph’s features and their motion.

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**The emphasis on probability has also seen changes in the middle grades curriculum where the majority of concepts are showing up initially in the sixth grade with reinforcement in the seventh and eighth grades. This resulted in changes to the state department’s division of testing and accountability, as North Carolina’s End of Grade Test for eighth graders has become completely calculator active. **(Play Clip 11)** This presents a new direction for school systems that must take steps to prepare their middle grades teachers by providing high-quality staff development. Training should include not only the practical use of operating the graphing calculator but more importantly the more advanced content coming down from the high school curriculum, which is likely to be unfamiliar, or forgotten, from their previous teacher preparation course work.

Given the documentation from the literature and the compelling evidence seen in the videos, there is no doubt that technology can and does support, enhance, and extend the quality of mathematics instruction in our schools. It seems natural to reflect now on what school systems might consider as next steps to build upon and extend their current efforts to use technology in classroom instruction. This is best summed up in three essential categories: to maximize existing resources in technology, sustain those resources with professional development, and consider the addition of new resources in technology.

In maximizing current resources, schools and school systems should inventory the various technologies they have available. A pitfall that occurs in education, especially in the secondary schools, is that the different disciplines become more departmentalized with less connections occurring across the curricula. The connections are there naturally and yet each subject tends to function in isolation of the others in the school. For instance, mathematical modeling can be described as the phenomenon of trying to describe and predict scientific behavior. Data collection devices such as like motion detectors and probes that hook directly to a graphing calculator are becoming more standard equipment in science laboratories. These resources are in a lot of science classrooms already, but it is doubtful that both content areas are benefiting from this resource simply due to the lack of professional dialogue and collaboration between the departments. If departments of mathematics and science would simply pool their resources, they could afford these technology media and improve the instruction for both of these high-need core disciplines without taxing the budgets of both. This is one strategy that schools could investigate to support the deficits occurring in science and mathematics student achievement.

Elective courses like drafting, carpentry, and art have embedded connections to mathematics. These departments often have their own pot of money to support their technology needs and are able to furnish complete computer labs for their use. Unfortunately, often times these labs are reserved solely for their use while the remaining departments compete for space and instructional time in a limited number of non-vocational labs. Consider how a tool like Geometer’s Sketchpad® could facilitate understanding of key geometry concepts needed in drafting. Art classes could utilize the transformations and animation features of Sketchpad® in their study of tessellations and perspective drawings.

Another vehicle for maximizing existing resources is to inquire of the local universities’ mathematics and education departments. They are increasingly looking for more opportunities to partner with their community school systems. For instance, the Science and Mathematics Education Center out of the University of North Carolina at Wilmington offers a technology loan program where teachers can check out equipment like calculators, probes, and cables to use in their classrooms.

Sustaining current resources is also a critical piece that cannot be overlooked. Technology-based professional development must continue not just when new tools are purchased and installed; trainings should be offered on a recurring basis to combat teacher turnover and the loss of skills over time.

School systems could identify a cadre of teachers who are already having success incorporating tools like Sketchpad® and graphing calculators in their classroom instruction. This group of Lead Teachers could participate in train-the-trainer sessions that prepare them to provide future trainings for the district’s teachers. They should also provide the follow-up support to assist their peers in making the transitions necessary to begin implementing the technology tool during instruction. Finally, the Lead Teachers could be contracted in the summers to write model lessons with the technology that address different content areas or grade levels. It is essential that these leaders be compensated for their time, preparation, and service to the district, as it will maximize the chances of sustaining their efforts over time.

Lastly, school systems should consider the addition of new technologies to supplement their existing technologies. As districts, schools, and teachers make decisions regarding the purchase of the technology, they should consider all of the issues that have been raised. Technology influences the mathematics that is taught by making some content unnecessary to teach, some content easier to teach, and some content possible to teach that would not be otherwise. Technology enhances student learning because it provides a medium for visualizing and interacting with the mathematics, extends the range of problems accessible to students, and frees up the teacher to serve as a facilitator to observe and attend to student thinking. Technology is like a manipulative in that it helps the learner move from the concrete to the abstract in understanding mathematics, yet it has the added bonus of being dynamic unlike other manipulatives (i.e., algebra tiles, base ten blocks, etc.). Technology allows the learner to focus on higher order thinking such as comparing, analyzing, predicting, and proving while the technology executes the more routine procedures of mathematics. Finally, technology improves students’ attitudes about doing mathematics and increases their level of engagement and interest.

By weighing the many options, school systems are better prepared to make decisions that will benefit students and teachers in the mathematics classroom. Schools and districts are faced with the challenge of keeping their hardware and software current despite the ever-changing face of technology. In order to change the trend of limited computers and computer networking, school board members, administrators, teachers, parents, and students need to understand the benefits of a networked school.

Having access to computers and the Internet is a great tool to motivate students and help teachers with instruction. “Network use is highly motivating for both students and teachers: kids interact with computer networks with energy and enthusiasm often missing in more conventional classroom structures, and teachers are stimulated by the ability to share ideas, concerns and solutions with colleagues across the country as easily as if they were in the next room” (Eisenberg & Ely, 1993, ¶ 7).
Networked classrooms allow students access to the Internet and intranet during class when time permits. Students are able to use the many technology tools for reinforcement, extra practice, remediation, extensions, and research. The following clip shows a group of students working on the computer with a program to help them practice the basic multiplication facts. The students have an opportunity to work with multiplication that is responsive to the level where they are performing. **(Play Clip 12)** “By allowing learning and teaching to take place at any computer terminal at any time of day, education becomes more responsive to the needs of lifelong learners in the information age" (Eisenberg & Ely, 1993, ¶ 8). The possibilities are open and endless.

Another medium that districts should consider purchasing for the classroom is a document camera. These can begin to replace the standard overhead projectors as a primary means of presentation display. These can be connected to an LCD projector, are easy to set up, and allow the teacher to display student work readily. View a seventh grade classroom discussion that took a different turn when one student was able to quickly display her answer to a prompt from the previous night’s homework. The homework problem asked students to sketch a graph of the following situation: the height above ground of the air valve on a tire of a bicycle ridden on flat ground. **(Play Clip 13)** A student misconception was made public, the teacher was able to address it, and all students benefited. This example reinforces several ideas mentioned already—that is, the technology enhanced student learning and freed up the teacher to attend to errors in student thinking.

Technology is an essential tool for the 21st century mathematics classroom. This paper described the value of using technology in the mathematics classroom, provided compelling evidence of this value as captured in some unrehearsed and unedited video footage from classrooms, and suggested some next steps for school districts and their teachers that could build upon and extend their current efforts.

The NCTM standards “call for a focus on the process, rather than the product of mathematics so that students become better, persistent problem-solvers in their everyday lives” (Vinson, 2001). The literature supports the use of technology as a tool for teaching and learning mathematics. NC-PIMS Lead Teachers have demonstrated the positive impact of technology for mathematics classrooms in spite of the many common technology barriers that exist for teachers universally. NC-PIMS Lead Teachers face challenges beyond their control when implementing technology into their mathematics classrooms. Those challenges include, but are not limited to: lack of available computers for all students, access to labs, and reserving the use of LCD projectors, SMART Boards, calculators, and other equipment. These are challenges that continue to be worth fighting, despite the frustration they cause—the benefits for teaching and learning will pay off in the long run. In spite of these obstacles, NC-PIMS Lead Teachers continue to succeed in the delivery of high-quality mathematics lessons that are enhanced by their use of technology.

References:

Bratton, G. N. (1999). The role of technology in introductory statistics classes.

*Mathematics Teacher, 92* (8), 666-669.

Clements, D. H. (1999). Hands-on math: Learning with computers and math

manipulatives. *Arithmetic Teacher, 40 *(9), 528-530.

Dunham, P. & Dick, T. (1994). Research on graphing calculators.* Mathematics *

*Teacher, 87* (6), 440-444.

Eisenberg, M. B., & Ely D. (1993). Plugging into the ‘net. *Emergency Librarian, 21 *(2),

8-9.

Glass, B., & Deckert, W. ( 2001) Making better use of computer tools in geometry.

*Mathematics Teacher, 94 *(3), 224-228.

Kader, G. & Perry, M. (1994). Learning Statistics with Technology. *Mathematics *

*Teaching in the Middle School, 1* (2), 130-136.

Nathan, M. J., & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of

students’ algebra development. *Cognition & Instruction, 18* (2), 209-238.

National Council of Teachers of Mathematics (2000). *Principles and Standards for *

*School Mathematics.* Reston, VA: author.

Quinn, R. J. (1998). The influence of mathematics methods courses on preservice

teachers’ pedagogical beliefs concerning manipulatives. *Clearing House, 71 *(4),

236-239.

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and after a methods class emphasizing manipulatives. *Early Childhood Education Journal, 28* (2), 89-94.

Appendix A

Design your own Simulation

(from NCTM Academy for Professional Development: Data Analysis & Probability)

1. Full of Hot AirJoe Bob’s hot air balloon company,

Rise Above it All,books customers in advance with the following warning:

We only fly in fair weather. In fact, only 60% of the time is the weather acceptable. In addition, we can only fly if the National Guard is not flying which happens (they are not flying) 40% of the time. If those two restrictions are met, we try to take off in town and land in the country. That only occurs when the wind is blowing in an easterly direction, which is about 70% of the time.

If you have a reservation for a balloon ride with Joe Bob, what is the probability that you will fly and that you will land in the country%3f

- Conduct a simulation to find the experimental probability of this event. Run your simulation at least 20 times.

- Calculate the theoretical probability of this event.

2. Fast Serve in TennisMost of us have played tennis (or attempted to play) and realize the difficulty in making a good, strong service. The tennis rule makers realize this also and allow a second service attempt when the first one is out:

Suppose that Ted Tennis has a strong first serve. When it is good, it wins him 75% of his points. However, it is only good 50% of the time. His second serve is good 3/4 of the time, but he wins only 50% of the subsequent points.We can use our ideas of simulation and expected value to calculate the percentage of points Ted wins when he is serving, remembering that when he fails on both serves he loses the point.

a) Conduct a simulation to determine the experimental probability of this event. Run the simulation at least 25 times.

b) Calculate the percentage of points that Ted expects to win while serving using the theoretical probability.

Appendix B

© 2007 The North Carolina Partnership for Improving Mathematics and Science

Visit the NC-PIMS Website at: http://www.ncpims.org/