Teacher Discussion of Project 4

A three-act play is a high-demand task broken up into three distinct stages. The first act introduces the media to students, thereby generating questions that the class as a whole wants to answer. Next, in the second act, students start to formulate their solutions by asking questions and receiving more information. Finally, in the third act, the teacher reveals the solution to the original problem in act one and allows students to compare their predicted answer to the actual one. By having students work through these three stages, teachers have the opportunity to have students employ a variety of SMPs as they complete each act. Below is an example is different SMPs that can be utilized for each act.

Act 1	Act 2	Act 3
SMP 1 SMP 5	SMP 4 SMP 6 SMP 2	SMP 3

From the above table, it is apparent that each stage of a three-act play has students use different skills in order to progress through the play. From creating models (perhaps with technology!) to critiquing another student’s argument for an answer, three-act plays force students to employ several different analytic skills in order to arrive at a solution (National Governor’s Association, 2010) . Having students use these different analytical skills for real world problems develops their relational understanding of the material, whether it be for geometry, algebra, or probability and statistics.

As a result, three-act plays can be a meaningful and exploratory task that enables students to apply their reasoning skills to real world situations. Another aspect of three-act plays is that it can be adapted for any grade level or course content. Any standards from the Alabama Course of Study can be used as a framework from which the three-act play can build from. Add in the usage of technology and you truly have a task fit for any mathematics student.

Tasks like these are extremely important for student growth in mathematics. According to Principles to Actions: Ensuring Mathematical Success for All, high-level tasks like the three-act play, “encourage reasoning and access to the mathematics through multiple entry points, including the use of different representations and tools, and they foster the solving of problems through varied solution strategies” (National Council of Teachers of Mathematics, 2014, p. 17). Using the open ended nature of act one, students can create their own questions for solving. This also leads to the varied solution strategies and multiple entry points that the quote mentions.

Implementation

If I were to implement this three-act play into a mathematics classroom, I would do it so that every act took up one class period. Students need ample time to analyze, formulate questions, and collaborate with one another in order to form a relational understanding of the content of the three-act play. Limiting the entire three-act play to one or two days means students have less time to collaborate and form their varied strategies, defeating the point of implementing a three-act play to begin with.

The first class period would just have students analyze the sink video. I would alternate having students formulate questions and strategies in their group with full class discussions about their small group discourse. By the end of the class, the students will have a question to answer in addition to ideas about what additional information they would like to know.

The second class period would involve students formulating their solution strategies to the central problem of the play. In addition, students will take their ideas from the day before and begin asking me for information that was not given in Act 1. Similar to day 1, students will spend a majority of their time in their small groups. Full group discussions will be placed incrementally so that students can air their group’s ideas to the entire class. The goal of this full group discourse is not in having every student adopt one strategy. Instead, this gives students the chance to have their solution strategy heard by other students outside of their group. This allows other students to analyze and possibly critique that student’s strategy, an example of SMP 3. By the end of this class period, I would expect students to be confident in their group’s solution strategy as they solve the central question of act 1.

The third and final class period would be centered around students sharing their solutions for the question from day 1. I would also show the real world answer from the original sink video. Afterwards, I would spend the rest of the class period having students explain their solutions, and how why their solution might have been different from the actual answer. During this time, the floor would also be open to other students who have questions about a given student’s answer. The goal for the end of the class period would be having a majority of the class understand different students’ solutions and why their own solution might have been different from the actual answer. This way, students have an understanding of the underlying standards behind this specific three-act play.

Here were several standards that fit the essential problem behind this three-act play.

• Relate volume to the operations of multiplication and addition, and solve real-world and mathematical problems involving volume. [5-MD5]
• Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems. [6-G2]
• Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems. [8-G9]

References

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

National Governor’s Association. (2010). Common Core State Standards for Mathematics. Washington, D.C. Retrieved from http://www.corestandards.org/Math/  

Alabama State Department of Education (2016). 2016 Revised Alabama Course of Study: Mathematics.  

Student Version Act Three

From the picture we can see that it took 5 minutes and 43.61 seconds to completely fill the sink.

Three Act Play, Third Act

Here is the student version of the third act

The third act of the three act play both verifies the solution for the original problem as well as extends the problem for different solutions. In verifying our solution, we first had to consider all of the information given to us.

• The sink is 16″ by 14″
• The sink is 7″ deep
• The diameter of the stopper is 3.5″
• The depth of the stopper is 1.5″
• It takes 7.16 seconds to fill a 20 oz bottle

To start in verifying our solution, let us look at the volume of our sink. Since we are given the dimensions of the sink, finding the volume is fairly simple. By multiplying the length and width of the sink by the depth, we find that the sink is . Due to the fact that we are dealing with volume, we will be using cubic inches as our unit of measure. Next, we need to find the volume of the stopper. Because we are given the diameter of the stopper, we can find the area of the circle it forms. Recall that the formula for finding the area of a circle is . Using this formula we see that the area of the circle formed by the stopper is 9.62″. Then, in order to find the volume of the stopper, we multiply the area of the circle by the depth of the stopper, 1.5″. As a result, the area of the stopper is . By adding the volume of the stopper to the area of the sink, we can then find the total area of the sink, which ends up being .

Now that we have the total volume of the sink, let’s look at the rate at which the sink is being filled with water. Using the rate at which the sink filled the 20 oz bottle, we can determine how many cubic inches of water is being poured every second. To do this, we need to find the conversion rate between fluid ounces and cubic inches of water. By doing a quick search using Wolfram Alpha, we see that 1 fluid ounce is of water. Next, we multiply the 20 oz by the conversion factor of 1.805 to convert the ounces into cubic inches, which ends up being . Now that we have converted the flow rate of water from fluid ounces to cubic inches, we can divide the new figure by 7.16 seconds (the time it took to fill the 20 oz bottle), in order to see how many cubic inches of water flows out of the sink in 1 second. After doing some division, it turns out that for every 1 second, of water flows out of the sink.

Finally, now that we have the total volume of the sink and the flow rate of the sink in the proper unit of measure, we can find exactly how long it is expected to take in order to fill the entire sink. To find this, we divide the total volume of the sink ( ) by the rate of water coming out of the faucet every second ( ). This results in us expecting the sink to be filled in 313.912 seconds. By dividing this figure by 60, we see that it will take 5.232 minutes. We can also change the decimal from the 5.232 minutes to seconds by multiplying .232 by 60, which results in the final expected time to fill the sink being 5 minutes and 13.92 seconds.

Comparing this result to the time found in the picture, we see that there is a 29.69 second difference. While this is slightly off of our actual time, first consider the different factors that may have contributed to our predicted time being different from our observed result.

• The rate of water flowing out of the faucet is not always constant
• The measurements we took did not considered the curved corners of the sink and the sloping bottom. Our measurements assumed that the sides were perfectly straight.
• The stopper being plugged is not a perfect countermeasure for preventing water from draining out as we fill the sink.

Given the differences between our simulation and the reality we observed in the video, it should be no surprise that our result should be different from the one we observed. However, from our simulation we did find that a sink of that approximate size should take around 5 minutes to fill.

An extension or sequel for this type of problem could involve finding the area of the curved curved corners of the sink in order to have even more precise measurements for the volume. Another option would be finding how long it take to fill a sink with different dimensions or different rate of water coming from the faucet.

My Final Thoughts on Technology

Over the course of the semester we’ve explored how technology works in conjunction with mathematics education. We’ve looked at how technology changes the manner in which students learn as well as what technologies can be used for various high school mathematics courses. It turns out that the capacity to use technology in a mathematics classroom extends far beyond graphing calculators and smart boards. Using technology in the classroom is a precise process of finding appropriate technological tools, verifying their fidelity to the intended content area, and seamlessly integrating it into the classroom. While this process of finding and implementing technology can be difficult, being able to follow through means presenting mathematical content in potentially new ways to students that deepen their understanding of the material.

            The most important thing to consider when talking about implementing technology in mathematics classrooms is how it opens new pathways for student learning. Technology is often viewed as a computational aid, or task servant. By using technology to do computations, construct models, and data collection, teachers are now able to pose previously impossible types of problems to students (Dick & Hollebrands, 2011). These new types of problems allow for the rise of new types of questioning that seeks to promote mathematical reasoning and sense making. Having students engage in mathematical reasoning and sense making is the ultimate goal that we as mathematics teachers should be continually striving for. This quote from Focus in High School Mathematics: Technology to support reasoning and sense making accurately reflects this goal, “For students to learn mathematics with understanding, they must have opportunities to engage on a regular basis with tasks that focus on reasoning and problem solving and make possible multiple entry points and varied solution strategies” (National Council of Teachers of Mathematics, 2014, p. 23). This sentiment is echoed across various research backed literature about mathematics education as well as in the SMPs, MTPs and essential concepts.

            Perhaps the biggest boon for having technology integrated into the classroom is in the ability to have students create mathematical models. Programs such as Desmos, Geogebra, CODAP, Excel, and TI-Nspire allow students to model real world situations, a growing focus for all secondary mathematics courses. Having these interactive visual modeling programs enables teachers to create high-demand tasks that let students explore different content areas in mathematics through the creation of models. Having this initial exploration for different content areas is vital to students as they develop a relational understanding of the material. In addition, having students create mathematical models of different situations builds their procedural fluency. Building a procedural fluency for a topic can lead to students have better retention of the material as well as enable them to apply their understanding to new situations in a flexible way (National Council of Teachers of Mathematics, 2014, p.42). Without having this initial exploration, students are rushed to build up their fluency in mathematics. This leads to major consequences for students in that it drops a student’s confidence in their abilities and is considered a cause of mathematics anxiety (National Council of Teachers of Mathematics, 2014, p.43). As a result, having technology is a must for all secondary mathematics courses.

            In conclusion, integrating technology into the mathematics classroom adds a whole new dimension to teaching mathematics. For teachers, it unlocks previously impossible high-demand tasks for students, allowing for more opportunities to have students develop their mathematical reasoning and problem solving skills. As a result, students will have a much stronger relational understanding of the content than ever before. This leads to students growing more confident in their skills as mathematicians as their mathematical identities grow. Therefore, mathematics teachers should strive to seek out meaningful technological tools to augment the way students approach mathematical content.

References

• Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to Support Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics.
• National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

Finding the Best Technological Fit for Statistics and Probability

When students first start thinking about statistics and probability, a few things usually come to mind. Flipping a coin, rolling a die, or dealing a hand for Texas Hold’em. Students encounter these examples from the middle school level all the way into an undergraduate course in statistics and probability. Exploring all three of these real world scenarios can offer a chance for a deep understanding of both probability and statistics. The difference between learning about a coin flip in middle school compared to high school and beyond is often exactly how deep students delve into the realm of statistics and probability. For example, in middle school students would expect the probability of getting heads and tails to be equal. However, when they’re asked to flip a coin twenty times they’ll most likely observe that there were not equal frequencies of heads and tails. In a college scenario, this same idea might be taken and explored using more samples and different statistics. In most cases, to really understand and see some of these different statistical measures at work, there needs to be a larger sample size. In the middle school example, the sample size was a small, manageable amount that would most likely be gathered by hand. Due to time limitations and efficiency, there would be a limited amount of samples that could be taken from students conducting the experiment by hand. This limitation for the amount of samples hinders students from observing important patterns that can grant insight into probability and statistics. This is where technology can play a huge role.

In previous posts, we’ve explored how using technology allows students to explore and form a relational understanding of mathematical concepts. Using technology helps eliminate the tedious nature of seeing some mathematical concepts in order for students to focus on identifying patterns and seeing the structure of whats in front of them. Similarly, technology can be used to explore probability and statistics. However, while there may be go to programs for things like graphing (Desmos) and geometry (Geogebra), finding the right program for statistics is a little trickier.

There are a lot of different options for technology use with statistics and probability. What makes it harder to choose one definite choice is that each option has distinct pros and cons as far as functionality goes. For the purpose of this post, we’ll be looking through three very good options.

The first option that should be considered is the TI-84 or similar handheld graphing calculators. The TI-84, while an older form of technology compared to other options, is still a solid option for exploring probability and statistics, especially in school environments where iPads and other devices are not available for individual student use.

Figure 1

As far as capabilities goes, the TI-84 largely matches other options in terms of the statistical functions available. In addition, the TI-84 offers a variety of graphs for representing your data. In Figure 1 you can see an option for graphs such as a boxplot, histogram, dot plot, and line graph. However, where the TI-84 falls short of the other options is in its ability to generate data. While it is still a major step up from manually generating data through flipping a coin, it lacks the capability to create and manipulate data that can be with spreadsheets or other technology.

The next option we’ll be looking at is a familiar one! That’s right, spreadsheets can also be used for exploring probability and statistics. This option can of course can utilized with a variety of programs like Microsoft Excel, Apple’s Numbers, or Google Sheets. The benefit of having these various programs for each platform is that regardless of what technology your school has, there will be a spreadsheet program that can be utilized.

Figure 2

Compared to the TI-84, spreadsheets make it incredibly easy to generate data using formulas. In addition, you can create different tabs in each spreadsheet that can reference each other’s data, which is very useful for organizing your work space. Where spreadsheets fall slightly behind is in the kinds of graphs that are pre-loaded into the program. For example, there is no readily available option for creating a box and whisker plot in Numbers. However, spreadsheets still retains all the various statistical functions that can be found in the TI-84. While it is very easy to generate data with formulas, it still takes effort in dragging down the bounds of your sheet size and dragging down all the formulas for more data.

The final technological option that we’ll be discussing is one that you may not have been exposed to before. CODAP, which stands for Common Online Data Analysis Platform, is a free educational resource that can be used for exploring statistics and probability. Rather than being a separate program that needs to be installed, CODAP works in any web browser. The only downside as far as accessibility goes is in working with CODAP on devices such as smart phones or iPads. In my experience, editing attributes and moving the various tables around take considerably more work on the iPad compared to on a standard desktop or laptop.

Figure 3

In terms of capabilities, CODAP is perhaps the best at generating data in a fun and visual way with their built in sample plugin. The plugin provides three ways for collecting data: a spinner, a mixer, and a collector. Each of these options includes a cool little animation that visually shows how each point of data was collected. CODAP also makes it very easy to collect very large samples of data compared to the TI-84 and spreadsheet programs. By adjusting the sampler parameters, you can repeatedly generate hundreds of points of data. This data can then be represented in the various graphs that CODAP provides. By dragging over the attribute you want to visually show, you can manipulate your graph to produce a variety of results. Another interesting feature of CODAP is in being able to create formulas for the various attributes in your tables, a feature very similar to one found in spreadsheets. The one negative aspect of having this dynamic work space is that it can often become cluttered once you add multiple graphs or attributes to your chart.

For teachers, using technology to teach probability and statistics is an absolute must. This need is reflected in the essential concepts under statistical inference and visualizing and summarizing data. Both of these focuses under statistics and probability repeatedly mention technology as a viable and encouraged means of demonstrating concepts in statistics as well as probability (National Council of Teachers of Mathematics, 2019, p.60). Focus in High School Mathematics: Technology to support reasoning and sense making also strongly supports the use of technological tools in order to create probability models. “Using technology tools can help students tremendously in building models and exploring the behavior of these models.” (Dick & Hollebrands, 2011, p.69).

In conclusion, using technology to help students create a deeper understanding of probability and statistics is highly encouraged. Using technology for statistics and probability can allow students to create models that would be very tedious or inaccurate by hand. With this models, students can then draw statistical inferences that would have been difficult to see otherwise. In addition, the processing power of technology allows for thousands of data entries for students to manipulate and extrapolate conclusions from, a practice which is nigh impossible by hand. In this post we mentioned the pros and cons of three separate technologies in terms of their use for statistics and probability. Compared to other content areas, statistics and probability is unique in that there is no cut and dried program for teachers to utilize when approaching this material. However, the three options discussed earlier all have their own benefits and disadvantages.

References:

• Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to Support Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics.
• National Council of Teachers of Mathematics. (2018). Catalyzing Change in High School Mathematics: Initiating Critical Conversations. Reston, VA: Author.

Experience Geometric Nirvana with Geogebra!

We’ve established in two previous posts how mathematical action technologies aid in teaching mathematics to students. Today we’re going to be looking at another one of these technologies in the form of Geogebra! Geogebra is a wonderful program that can be used with any mathematical discipline. While the focus of Geogebra is on it’s dynamic geometry software (DGS), which, “allows students to move beyond the specifics of a single drawing to generalizations across figures” (Hollebrands & Dove, 2011, p. 33). Geogebra also includes other programs that mimic what we’ve explored in our other blog posts. For example, Geogebra has a tab that routes you to it’s graphing program, which looks and operates like Desmos. In addition, Geogebra includes programs for CAS capabilities, spreadsheets, 3D graphics, and probability. For the purpose of this post we’ll be primarily focusing on the bread and butter of Geogebra, the DGS functionality.

Figure 1

I often think that one of the constraints with geometry is in efficiently showing precise representations for shapes. For example, let’s say that you are learning about the properties of trapezoids. The teacher wants to show a property is always true for a trapezoid. To show this they have to create several example pictures, thereby showing the property holds for different trapezoids. Drawing out these representations and making them mathematically accurate takes valuable class time. Having mathematically accurate drawings in your geometry class is important when learning geometry. Having drawings that do not properly show angle and length measures makes it harder to follow along and more difficult to uphold the properties of the shape. Therefore, precision should be a priority when making geometric figures, which is where Geogebra comes into play. Geogebra allows for one single geometric construction that can be used to show different examples, which can lead to generalization.

Some other positives about Geogebra include but are not limited to:

• It’s free!
• Easy to use.
• Flexible for both simple and complex tasks.
• Places an emphasis on precision, which aligns with the standards of mathematical practice shown detailed in the common core (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, 6).
• Able to include multiple figures on the same screen with options for grid backgrounds, axes, or a blank workspace.
• Can be loaded up on any device, whether it be a desktop, phone, ipad, or SmartBoard.
• Has different perspectives for your different needs, including geometry, statistics, CAS capabilities, graphing, spreadsheets, and 3D graphics.

Using the DGS functionality, teachers and students can now create efficient, precise example figures. To demonstrate how amazing Geogebra is lets go through the steps in constructing a rectangle using the iPad app.

1. Draw any sized segment using the segment tool.
2. Use the perpendicular line tool to create two legs from the endpoints of your original segment.
3. On any point of one of your perpendicular legs, add another perpendicular line.
4. Using the point tool and create a point at the intersection of the two perpendicular lines. This point should automatically turn black instead of blue.
5. Using the polygon tool, click on all four vertices of your rectangle, including the first point that you start with. Thus creating a construction of a rectangle.
6. Clean up your construction by hiding the perpendicular lines. This can be done by clicking the line and unchecking the “Show Object” line.

Figure 2

Now that we’ve constructed our rectangle we can use it as a precise manipulative in order to explore the properties of a rectangle. To help find these properties we can also show the precise measurements of both the angles and sides of our rectangle using the measurement tool. By creating this construction we eliminate the need to spend copious amounts of time drawing imprecise figures.

Figure 3

An important thing to note about Geogebra is that it is a manipulative for students. As a result, students are able to explore geonetric figures with the Geogebra app by constructing their own figures through their own methods. The way that I constructed my rectangle is just one method, students might come up with their own unique way using what they know about rectangles. As a result of this exploration, students are able to have a wholly organic understanding of geometry through their own findings in the Geogrebra app.

For geometry teachers, this app will become your new best friend. The possibilities are simply endless for creating high cognitive demand tasks. And again, having tasks that allow students to explore geometry through the construction tools is so important for their conceptual understanding as geometry students. Having this initial exploration is vital to students developing a relational understanding of the material in addition to building their procedural fluency. Building a procedural fluency of a topic can lead to students have better retention of the material as well as enable them to apply their understanding to new situations in a flexible way (National Council of Teachers of Mathematics, 2014). Without having this initial exploration, students are rushed to build up their fluency in mathematics. This leads to major consequences for students in that it drops a student’s confidence in their abilities and is considered a cause of mathematics anxiety (National Council of Teachers of Mathematics, 2014).

Using Geogebra also allows teachers to demonstrate the difference between a geometric construction and a drawing. Drawing a geometric figure is defined as using “primitive objects” such as the segment tool to create approximations of shapes. When you then go to move any part of this drawing, not all of the properties of your intended shape will be preserved (Hollebrands & Dove, 2011). However, with constructions of shapes, the properties of the intended shape are preserved regardless of how the shape is manipulated (Hollebrands & Dove, 2011). This testing of a geometric figure is sometimes referred to as the “drag test”. As a teacher using this app, demonstrating these differences with students will help them create precise constructions, which is a focus of SMP 6 (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010).

In conclusion, Geogebra is a fantastic DGS which opens the doors for teachers to create any multitude of high demand tasks for students. Having this software be integrated into geometry classrooms means giving students the perfect manipulative for geometry. By having students explore geometry through construction of figures, students will be able to organically learn geometry in a way that is fun and interactive. It’s wide availability on a variety of platforms makes it the supreme DGS software. As a result, Geogebra is a must have in any geometry classroom!

• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics.Washington, DC: Authors.
• Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to Support Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics.
• National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

The Wonders of Desmos!

Despite existing for over 30 years, not much has changed in the landscape of graphing utilities for students. Maneuver your way to the graphing tab of your TI-84, enter your equation into the table, and then proceed to look at tiny, misleading, pixelated screens in order to observe the behavior of a function. Sure, new graphing calculators have been released with all manner of probability and statistics capabilities, but the essential graphing function has largely stayed the same since 1985. That is until the creation of Desmos in 2011. Desmos fundamentally changed the way students look at graphs. With it’s user-friendly interface and endless graphing capabilities, Desmos soon became the gold standard for free graphing applications. Even the Texas Instrument’s own graphing app, the TI-Nspire, closely resembles the interface of Desmos. So, in today’s post we’ll be discussing the wonders of Desmos, the graphing program that 9 out of 10 students recommend for graphing functions!

I feel that a problem that most mathematics teachers run into is the resources that are available to use in the classroom. Depending on the budget of the school system, technology like graphing calculators are an expense that some school cannot afford to purchase. For example, a TI-84 graphing calculator runs at about $112 per calculator. If each mathematics classroom has an average of 20 students, then the school can end up expending $2240 or more on graphing calculators. Even in cases where students do have graphing calculators available, teachers are often not proficient enough in their usage, leading them to collect dust instead of being used by students.

On the other hand, Desmos has several qualities that appeal to both educators and students

• It’s free!
• Easy to use.
• Places an emphasis on precision, which aligns with the standards of mathematical practice shown detailed in the common core (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, 6).
• Complex enough to model all manner of things, including inequalities, integrals, statistics and all kinds of functions.
• Able to include different types of visual models, including tables and charts. All loaded and available in the same screen
• Can be loaded up on any device, whether it be a desktop, phone, ipad, or SmartBoard.
• Has different sections dedicated to geometry as well as custom made or preloaded activities that cover a wide variety of topics.

While I do endorse the use of Desmos both in and out of the classroom, that’s not to say graphing calculators should entirely be done away with. Studies have shown that students that have calculators integrated into their classrooms show improvement in both computational skills as well as advancing their ability to problem solve (Ronau et al., 2011, 2). For example, in a school with no access to graphing calculators or similar technology, using Desmos could serve as an effective, affordable substitute. And in a classroom that has graphing utilities, Desmos could be used as an introductory visual to help students dive into a topic, with graphing calculators being used in a supplementary, do it yourself, kind of capacity. So rather than choosing between one or the other, keeping both as viable options in the classroom would be best, both for the teacher as well as the students. More options is always better than having no options!

To best showcase the wonders of Desmos, why don’t we go through an example problem to see just how useful Desmos can be! For this example, I’ll be using the ipad app for Desmos available for free on the app store.

The problem asks us to consider the situation: “Florida tosses a ball off the top of the Haley Center, the tallest building on Auburn University’s campus. From physics, we know that the function h(t) = -16t^2 + the initial velocity *t + the initial height gives us the height of a falling object (in feet) at a given time in seconds.”

To get a visual representation of what this function is doing, let’s graph it in Desmos!

Something that should be immediately apparent are the two sliders underneath the function that we graphed into Desmos. These sliders are options that Desmos creates whenever a variable is entered into our function. Since we have two variables, initial velocity and initial height, Desmos created two sliders that we can edit so that we can view how changing certain variables affects our graph. Clicking on the play button on the far left of the slider enables students to see how the graph changes with the different values in real time. The GIF below highlights what that looks like.

For teachers, having this interactive visual model enables them to create high-demand tasks that let students explore specific functions and their unique behaviors. Having this initial exploration is vital to students developing a relational understanding of the material in addition to building their procedural fluency. Building a procedural fluency of a topic can lead to students have better retention of the material as well as enable them to apply their understanding to new situations in a flexible way (National Council of Teachers of Mathematics, 2014, 42). Without having this initial exploration, students are rushed to build up their fluency in mathematics. This leads to major consequences for students in that it drops a student’s confidence in their abilities and is considered a cause of mathematics anxiety (National Council of Teachers of Mathematics, 2014, 43).

In conclusion, we’ve demonstrated the usefulness of Desmos as a free, easy to use visual tool for teachers. Using this widely available mathematical action technology enables all students to explore graphing using an interactive visual model (Dick & Hollebrands, 2011, xiii). Having this model enables teachers to build student’s confidence in mathematics in addition to increasing their retention of the content. Having this technology opens doors for both students and teachers to explore mathematics in previously cumbersome ways, making it a vital tool for any mathematics classroom.

Below are links to previously mentioned Desmos extensions, including the geometry tool as well as the Desmos activity page.

Click here for the geometry page!

And here for the activities page!

• National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
• Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to Support Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics.
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics.Washington, DC: Authors.
• Ronau, R. N., Rakes, C. R., Bush, S. B., Driskell, S., Niess, M. L., & Pugalee, D. (2011, September 30). Using Calculators for Teaching and Learning Mathematics. Retrieved September 29, 2019, from https://www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Calculator-Use/.

The Magic of Spreadsheets!

Technology is a tool that can aid in solving a wide variety of problems, from the most basic addition problems to complex word problems. What largely guides our ability to solve a problem in mathematics is the tool we choose to use. Using the wrong tool can mean needlessly increasing the difficulty of a problem, creating misconceptions for students and impairing their ability to think analytically. However, choosing the right tool for the mathematics classroom can mean opening up a whole new depth of problems for students to solve through the usage of technology.

I feel that choosing the wrong technology often hamstrings a teacher’s ability to offer problems that promote problem solving and relational understanding to students. As we’ll see later in this post, there’s a big difference between utilizing a calculator to solve a word problem and instead having students use spreadsheets.

While the most common mathematical tool of choice is the calculator, I increasingly believe that teachers and students alike should seek other, more effective, tools. The answer that I’ve found in the last few weeks to that dilemma is the spreadsheet.

Spreadsheets are a mathematical action technology that allows students to solve problems through their own analytical reasoning and problem solving skills. It’s a magical tool whose only limitation is how creative you’re willing to be in finding the answer. Unlike a calculator, there’s a depth to solving problems with spreadsheets. You can use it as a basic calculator or you can create a multitude of variables, constants, and sequences in order to model a situation. Recently I heard spreadsheets referred to as a, “what if machine.” To me this is the most accurate description you can attribute to spreadsheets. It gets right to the heart of what you would use a spreadsheet to do, answer questions analytically. The keyword of that wonderful description is “what.” This is because unlike a calculator that answers “what is the solution”, spreadsheets answer the who, what, when, where, and whys of mathematical problems, a sentiment found in the Standards for Mathematical Practice, specifically in standard one which states, “Make sense of problems and persevere in solving them” (“Common Core State Standards for Mathematics”, 2010, pg. 6) . And precisely because a technological tool can answer those questions is the reason why we should use it. Being able to truly understand mathematics means being able to relationally understand what’s going on, as echoed by this quote in Focus in High School Mathematics, “Teachers…must shift their perspectives about teaching from that of a process of delivering information to that of a process of facilitating students’ sense making about mathematics” (Dick & Hollebrands, 2011, p. vii). To be able to connect algebra and functions abstractly in order to problem solve and develop solutions. So, needless to say, I believe this 40 year old logic software is the answer to our initial dilemma of finding the right mathematical tool to use in the classroom.

The best way to showcase the capability of this technology is to show it being implemented in the classroom setting to solve problems.

This first problem has students trying to find the number of hits And total at bats in order to find a specific batting average through the use of spreadsheets.

This example spreadsheet comes from Principles to Action pg. 86.
(Leinwand, 2014, p.86)

To preface this, let’s identify how students were introduced to using spreadsheets to solve this problem and others like it.

1) The activity is first introduced to the students in the form of a whole class discussion about the real-world context surrounding the problem (Beigie, 2017, p. 28) .

2) Students are asked to work in pairs or in other group arrangements in order to troubleshoot problems in addition to allowing them to collaborate through social means (Beigie, 2017, p. 28) .

3) Students generally start thinking about their formulas through the use of pen and paper but eventually start tinkering with the spreadsheets initially as they grow more comfortable using the tool (Beigie, 2017, p. 28).

Looking at the problem itself we can also identify key attributes that makes it a high-level cognitive demand task.

• The problem is centered around a real world context.
• The problem asks students to take their initial results to form predictions and conjectures that can be tested and refined in order to arrive at the answer.
• While not represented in the problem, the values taken from the columns of the spreadsheets can be modeled using graphs, another key function of using spreadsheets. A student’s ability to recognize the need to model a situation, such as a column of spreadsheet values, is a value found in standard four in the SMPs, which states that “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace” (“Common Core State Standards for Mathematics”, 2010, pg. 6).

As students delve into this problem in their groups, they utilize several important functions of spreadsheets to gain a better grasp of the mechanics of the question. For example, in their initial table in figure (a), students learn to have multiple combinations of values for hits and total at bats in order to compute the different batting averages. Creating and implementing these columns of values for their variables is the core function of spreadsheets, justifying the “what if machine” description we talked about earlier. Rather than individually computing the batting average for each number of hits and total at bats, students get to plug these values into their batting average formula in a manner that allows them to show what happens when the variables being plugged into the formula are changed. This highly visual aspect of using spreadsheets allows students to identify patterns that give a deeper understanding of their formula than they would receive if they were simply asked to solve for the solution.

This figure above demonstrates another creative way to use spreadsheets in solving problems. In this case, students were asked to solve for x in the equation 2^x = 2x+5. To do this, students utilized skills previously learned from the prior problem. Namely being able to show different values of x in their columns for each side of the equation. Using this information, you can solve the equation for x in a variety of different ways. The most apparent one, shown above, is by taking the x-value data from both sides of the equation and constructing a graph, visually showing where both lines intersect and thus giving the solution to the equation. Another way, while less visual, still arrives at the same answer. This method involves seeing where the values of x in both columns match or are approximately the same. While I only showcased two methods for solving this equation, there’s many different ways to arrive at the same answer, thus showcasing the creative nature of spreadsheets for solving problems. This idea of multiple entry points and various strategies is found in Principles to Action in the quote, “For students to learn mathematics with understanding, they must have opportunities to engage on a regular basis with tasks that focus on reasoning and problem solving and make possible multiple entry points and varied solution strategies” (Leinwand, 2014, p. 23).

In conclusion, utilizing spreadsheets in mathematics classrooms opens the door for students to use their problem solving skills to analytically think about problems with real world contextS. The benefits of using spreadsheets include promoting relational understanding of a topic for students by abstractly identifying patterns for the formulas they are modeling. This leads to students having a deeper understanding of mathematics that far outshines any instrumental understanding of mathematics. In short, having students use spreadsheets in conjunction with mathematics is teaching them to be better mathematicians and problem solvers, the penultimate goal of every math teacher.

• Darin Beigie. (2017). Solving Optimization Problems with Spreadsheets. The Mathematics Teacher,111(1), 26-33. doi:10.5951/mathteacher.111.1.0026
• Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics. Reston, VA: National Council of Teachers of Mathematics.
• Leinwand, S. (2014). Principles to actions: ensuring mathematical success for all. Reston: National Council of Teachers of Mathematics.
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics.Washington, DC: Authors.

Modernizing Mathematics for Students in the Modern Age

At first glance, the title “Modernizing Mathematics for Students in the Modern Age” can seem to imply a very daunting task for teachers. But while modernizing math does imply a great deal of change, it should not appear that the goal is to reinvent the wheel for students. Rather, think of technology as the grease that allows the wheel to be more efficient as it rolls along.

Technology that is used in the classroom can be sorted into two categories, conveyance technologies and mathematical action technologies. Conveyance technologies are those those used to convey information (Hollebrands & Dick 11). Examples of this would be the internet and presentation software like Power point. These technologies do not explicitly help students solve problems faster but they serve as an efficient means for teachers to convey information to their students with the addition of sound and animation.

The other type of classroom technology is one we are all intimately familiar with. Mathematical action technologies are technologies that can actually perform mathematical tasks, such as calculators, smart phones, and problem solving software (Hollebrands and Dick 12). When teachers question the role of technology in the classroom, this is often the type of technology they are referencing due to it’s ability to help students solve math problems without necessarily knowing the material.

Technology is not inherently a positive or negative addition to education. Its use can either invigorate or hinder a student’s comprehension depending on its use. Recently I had the chance to operate the Wolfram Alpha app, a mathematical action technology. Wolfram Alpha is an app that serves as a Google like search engine for math related information such as answers to equations or information on famous mathematicians. While I was exploring the range of actions it could perform, I thought about the pros and cons of introducing this app to students in a mathematics classroom. What I came up with was a list of pros and cons for its use

Example of Mathematical Action Technology
Courtesy of the Wikipedia page on Calculators

Example of Conveyance Technology
Courtesy of the Wikipedia page on Microsoft Powerpoint

Pros

• It can show various types of information about a given equation such as it’s graphical representation and all it’s possible solutions. This can serve to give students a more comprehensive view of an equation than they would normally receive.
• It allows students to check their answers as they work by seeing a step by step guide on how to solve a given equation, allowing them to teach themselves to a certain degree.
• Since it allows students to solve most simple problems, it puts the onus on the teacher to craft more complex and innovative problems for students to solve.

Cons

• Depending on the school’s access to technology, you may not be able to use this application with students.
• With so much information being shown for each equation entered, you run the risk of confusing or overwhelming students.
• Can possibly condition students to look up solutions to problems without trying to solve it themselves first.

So what can be gleaned from this list of pros and cons is that depending on its implementation and usage in the classroom, usage of this app can really help or hinder student learning. And while technology is something that should be used in the classroom to aide in student comprehension, it does not necessarily mean that every assignment should utilize it. Rather than dealing in absolutes, use your experience as an educator to determine its usage in your classroom.

References

(2019, August 21). Microsoft PowerPoint. Retrieved from https://en.wikipedia.org/wiki/Microsoft_PowerPoint

(2019, August 16). Calculator. Retrieved from https://en.wikipedia.org/wiki/Calculator

Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Welcome to my Page!

I just wanted to take this opportunity to first of all welcome you to my website! In the future I plan to use this space to post and write about subjects related to mathematics or education.

While my knowledge on both subjects is somewhat limited I hope that we can explore and become more proficient in both as the semester progresses.

If u have not had the chance to explore the rest of the website, I would encourage you to look through the ones available. While there is not much here now, that will certainly change in the future.