“What is the definition of a trapezoid?” Previously in my CTSE 5040 class, the discussion livened when this question was presented. Many of us thought this was a trick question and many of us gave the question much thought. Hesitantly, we replied with bases are parallel and each lower base angle is supplementary to the upper base angles. There was a long pause because the question seemed too easy and we felt there had to be a catch. Our instructor grinned when he asked, “does a trapezoid have exactly one or at least one pair of parallel sides?” Some of us responded quickly with exactly one, but, again, the long pause caused some recoil of the statement. Everyone in the room began to question what they previously thought the properties of a trapezoid were. Then someone stated, “if it is at least one, then that means every parallelogram is a trapezoid”, this promoted a whole new level of controversy.

The most common definition of a trapezoid in U.S. textbooks is exactly one pair of parallel sides, but others use the inclusive definition of at least one pair. Our instructor provided us with an NCTM discussion link on this topic, so we could read various arguments and decide which definition we believed to be most valid. The discussion brought up many interesting points and opinions, but one stood out to me. The writer stated that they discussed this definition with NCTM around 2007-2010, and at that time the United States used exactly one pair to define a trapezoid. However, Canada and other parts of the world use at least one pair of parallel sides to define a trapezoid. I like the analogy used in this discussion post, the writer said that this dissimilarity can be compared to how the United States does not regularly use the metric system, but the rest of the world does. Other posts were from teachers asking what definition standardize tests use because they wanted to prepare their students. It seems like many simply want to be on the same page, but others are set in their ways and closed off to opposite opinions.

After reading the discussion posts, I still could not draw a conclusion on which definition I believed. Each side had valid arguments, so I turned to someone who’s opinion I respected, thinking it may help me clarify and draw an accurate conclusion. I asked my Euclidean Geometry teacher. After class, I cornered him and before I could finish the introduction of the question he replied confidently with at least one. I was shocked at his confidence and I am sure he saw the shock on my face. He reiterated, “A trapezoid has at least one pair of parallel sides”. Just to be sure, I said that with the inclusive definition, all parallelograms would be considered trapezoids. He nodded. With that nod, I had my answer.

When back in class, I told my instructor who presented the controversial question, what my Euclidean Geometry teacher proclaimed. My instructor, who was not surprised, said that many mathematicians prefer the inclusive definition because they enjoy connections. Having things relate back to something else is satisfying and convenient. I then asked my instructor what he believed to be the definition of a trapezoid to be, he grinned. This question of inclusive or exclusive would be interesting to ask students in a geometry class because it allows them to formulate their own opinions which results in high cognitive demand and feelings of independence. Then, from their decisions, a teacher may adapt their lesson to fit the definition chosen by students. I believe this is an effective way of involving students with the curriculum and assessing student thinking.

I also enjoy how this discussion topic can be related to real-life events. I had not thought about this before my instructor brought the idea up in class. In life, there will always be different opinions on different matters. However, having a different opinion does not make you wrong or right, it simply means you have a different viewpoint or way of thinking. This is an important lesson to educate students on. I appreciate how, on this discussion topic, mathematics can be compared with the real world.

References

National Council of Teachers of Mathematics. (2018, October 3). myNCTM. Retrieved from myNCTM: https://www.nctm.org/login/SSOReturnUrl=https%3a%2f%2fmy.nctm.org%2fcommunitis%2fcommunityhome%2fdigestviewer%2fviewthread%3fMessageKey%3d215ddaecc6ba4ede965e3a73ba5bda8b%26CommunityKey%3d0a10e70-ceb6-4514-ac1c413a77367143%26tab%3ddigestviewer#bm

Let’s be honest. Graphs can be confusing because they can vary in numerous ways. For example, the equation of a line can have different slopes, y-intercepts, x-intercepts, domains, and ranges.

Graphs of parabolas can be transformed in similar ways as well, and they are more complex than lines.

Trigonometric functions are also tricky to graph, especially when the amplitude is changed and there are vertical and horizontal shifts.

In a classroom, it is time consuming and challenging to have students graph various functions and compare them with other functions. However, the images I used above as examples are from a website called Desmos. I was able to show various functions, all color coordinated, and ways they can be changed and transformed on a graph.

Is it not satisfying to see various functions graphed so neatly, so you can compare and contrast each one?

Desmos is an advanced graphing calculator available as a web application and a mobile application. The Desmos application found on your mobile device is a more condensed version of the web version. However, it is beneficial when a desktop or laptop is not readily available.

Desmos does not stop at lines, parabolas, and trigonometric functions. There are many more capabilities found on this powerful mathematical tool.

There is an option to add a table in Desmos, so you can find points with a given equation.

Also, if you click on the functions button in the lower right-hand corner of the graph, a menu appears displaying many more options to choose from.

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Desmos, both web and mobile version, can be extremely helpful when used in the classroom. Students can add sliders when exploring various functions to see how different values change the graph, and they can check their previous work. If students create an account, they can save their work and return to it later.

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“Graphing applications can allow students to examine multiple representations of functions and data by generating graphs, tables, and symbolic expressions that are dynamically linked” (NCTM, 2014).

Desmos is not only beneficial for the students, but the teacher as well. Desmos offers a teacher version that allows teachers to assign tasks for the students to complete. From the teacher account, teachers can create their own assignments, or they can use one of the templates provided. When students are completing the assignment, their progress can be monitored from the teacher’s account. This is extremely helpful for teachers who want to assign activities on Desmos, but also want to make sure students stay on task and understand the material.

You will need to explore Desmos to find what all it has to offer, so here is the Learn Desmos website to help you become more familiar with this tool.

There are many Mathematical Action Technologies available that can be implemented in the mathematics classroom. “Students should have opportunities to use Mathematical Action Technologies in all content domains to explore mathematical relationships and deepen their understanding on the Essential Concepts, to interpret mathematical representations, and to employ complex manipulations necessary to solve problems” (NCTM, 2018). I believe that Desmos can be extremely effective in the mathematics classroom and I look forward to taking advantage of what it has to offer for my students and myself.

References:

Desmos. (2018). https://www.desmos.com/

Desmos. (2018). Learn Desmos. Retrieved from Learn Desmos: https://learn.desmos.com/

Desmos. (2018). https://teacher.desmos.com/

“Can we use a spreadsheet?”

When I first heard this in my mathematics education class, I know my facial expression showed my extreme confusion. I had never heard of using a spreadsheet in mathematics until then. I know that sounds crazy, but it’s true.

Thankfully, it was not long before I became familiar with spreadsheets through the Numbers app on the iPad, which we used for numerous mathematical computations. I now realize what I have been missing. I cannot imagine my life without spreadsheets, and I find new ways to use them each day.

The great thing about spreadsheets is their many practical uses. Mathematics and business majors are not the only people who can use a spreadsheet. They can be used for lists, budgeting, organization, decision making, and planning.

Spreadsheets are also practical in the classroom.

There are a variety of ways you can implement them. Students can make lists of books they want to read, jobs they want to apply for, or colleges they want to apply to. Students can use spreadsheets when learning how to budget and keep track of their money. When the new semester is shortly approaching, students can plan their new semester schedules.

The program I have become most familiar with is the Numbers app. It is downloaded onto my iPad, so I can easily access it when it is needed.

Simply, download the app. This is what the icon looks like:

To create a new spreadsheet, tap on the green “+” in the right-hand corner. (Circled)

There are various templates you can choose from. For example, there is basic, personal finance, personal, business, and education. (Highlighted)

At first, the various options can be overwhelming. However, after you have explored and experimented with each one, the app will become more familiar.

Here is a link to Apple’s Numbers help page: Numbers Help

 Spreadsheets were once foreign to me. However, now that I have been introduced to this time saving instrument, I will implement it into not only my everyday life, but my classroom as well. I am confident anyone can find a way to use spreadsheets to simplify and organize aspects of their everyday life.

References: 

GECAWICH, M. (2017, September 6). The Top 5 Real World Applications for Spreadsheets. Retrieved from iAcademy: https://blog.iacademy.com/computer-apps/spreadsheets/

Apple. (2018). Numbers Help. Retrieved from Numbers Help for iPad: https://help.apple.com/numbers/ipad/4.1/#/tan72790d40

To Wolfram or Not to Wolfram

Madison D. Varnadore

8/27/2018

     Technology, if the students’ usage is regulated and monitored, can provide many opportunities for students to learn and understand the material presented to them. However, if the usage is not regulated and monitored, many students tend to use technology as a crutch and do not learn the necessary problem-solving skills. Technology can be used to further the students’ education by showing different ways of performing tasks. For example, some students learn better when they are lectured to, but others may better understand the material once they have gone through and worked problems at home. The only drawback to working problems at home is the lack of resources available to answer questions that may have come up. This is where a website like Wolfram Alpha comes in. Wolfram Alpha is a computational knowledge engine or answer engine. Possibly the simplest way to describe it would be to compare it to Google, but this is more mathematically condensed.

Now, how can we implement Wolfram Alpha when teaching and learning mathematics? When working with functions, it is helpful for students to see what is going on graphically. Implementing Wolfram Alpha when describing a function allows students to see an accurate representation rather than trying to imagine or draw the graph on their own. A more controversial action of Wolfram Alpha would possibly be the various techniques shown when solving certain mathematical problems. However, some teachers tend to dislike when students work problems differently than what was taught in class.

How do teachers feel about implementing websites like Wolfram Alpha in their classrooms? I spoke with a high school and a middle school teacher to see what they thought about Wolfram Alpha, and I asked what ways they have implemented this answer engine into their own classroom.

I first talked to a practicing 8^th grade honors mathematics teacher who had never been introduced to Wolfram Alpha. He took a few minutes to look through the site and was pleased with its capabilities. He believes that it has potential to be beneficial to the classroom, admitting that first he would need to familiarize himself with the website. I followed up by asking if he has used any similar mathematical tools in his classroom. He discussed ways that he has implemented desmos.com when discussing various functions, but none other than that. However, Wolfram Alpha also has graphing capabilities like desmos, and can be implemented in the classroom as well.

Here is a comparison of the graphing capabilities of desmos and Wolfram Alpha:

Continuing the conversation, he began discussing the opportunities available with technology. However, he believes that the use of technology needs to be monitored. He strongly believes that technology is beneficial, if students do not become reliant on it. It’s interesting that we previously discussed this reliance on technology in class. Each of us agreed that engines like Wolfram Alpha can be beneficial but can also be used as a crutch for students. He ended our discussion by saying something that resonated with me. He said “I sometimes think that good, old-fashioned thinking skills get lost in the overload of technology that our students have at their disposal. The school where I teach is very limited in what technology is available to students, and I still teach classes where there is more focus on students trying to learn fundamental skills and less on applications with technology. I think there must be a balance between teaching students how to effectively use the technology resources that are available while also teaching them to learn to think for themselves” (J. Wilkerson, personal communication, August 23, 2018).

We have often discussed this balance in class, but how can teachers find a balance if the opportunity isn’t there? Many of the readings mentioned a need for change, yet still today there are schools with limited access to technology, and therefore are reliant on lecture style teachings. In Principles to Actions, there is a list titled ‘Guiding Principles for School Mathematics’. In this list it has a section that discusses technology usage. In this section, according to the National Council of Teachers of Mathematics (2014), it states, “An excellent mathematics program integrates the use of mathematical tools and technology as essential resources to help students learn and make sense of mathematical ideas, reason mathematically, and communicate their mathematical thinking” (p. 5). If there is going to be a change from lecture style teachings to more student involved teachings, there might need to be a focus on making more technology available to rural school districts. Conversely, many schools are too reliant on technology and students lose the ability to problem solve. Finding balance is the ultimate end game, but it’s not as easy as it seems.

I wanted to receive opinions from both a middle school mathematics teacher and a high school mathematics teacher. The second teacher I talked to is a 9^th -12^th grade mathematics teacher. She was also unfamiliar with Wolfram Alpha, so she looked through the website to see what all it offered. She saw Wolfram Alpha as a great tool to use for higher level mathematics homework. She acknowledged the possibility of dishonest students using it to simply copy down the correct answer, but for others she believed it could be added instruction. With the step by step solutions, students can see where they might have made a mistake in their computation. I was interested in what she thought about implementing technology in a mathematics classroom, so I asked. She responded by saying “I am starting to use more apps like the ones with graphing calculators, and the protractor app for geometry is cool. However, I believe technology should be supplemental” (J, Harbison, personal communication, August 23, 2018). She too doesn’t want her students to become reliant on technology, but rather use technology to enhance the instruction present.

  Overall, the mathematics teachers I talked to were impressed by Wolfram Alpha. They expressed interest in using it in their classrooms and see it as a possible valuable tool for students. The only concern is the likelihood of students misusing the engine. However, with some monitoring, the tool can help students better understand the material presented in class. It can also show students different ways in which mathematics is used in everyday life. Since I do not have the knowledge an experienced teacher possesses, I enjoyed hearing about different ways these teachers implicated technology in their classrooms. I agree that technology can be extremely useful for students, and I am looking forward to finding new ways to utilize it in my classroom.

References