Summary:
As I listened to General Conference, a worldwide broadcast put on by the leaders of the Church of Jesus Christ of Latter-day Saints every six months, I noticed a few connections to technology in both the content of the speakers’ talks and also how the conference was broadcasted using technology. One speaker, Elder Quentin L. Cook, a member of the Quorum of the Twelve Apostles, said, “One adjustment that will benefit almost any family is to make the internet, social media, and television a servant instead of a distraction or, even worse, a master.” In addition to this powerful thought Elder Cook provided, I thought about how unified everyone needed to be in order to translate the conference into hundreds of languages and broadcast the conference in hundreds of countries across the globe. Unity is key in settings both in and out of the classroom when dealing with technology.
As a class at Brigham Young University who all participated in watching the conference, we discussed some of the ways technology was used. Dynamic images during talks were pedagogically valuable, as in Elder Neil A. Andersen’s talk when a dynamic image of a scattered group of dots were turned three-dimensionally and created a pristinely clear image of an eye. This dynamic image provided a powerful analogy to using the principles found in the document entitled “The Family: A Proclamation to the World” as the true perspective that comes from living the principles of the restored gospel of Jesus Christ. In addition, we talked about four components of technology: Technology as a Master, Technology as a Servant, Technology as a Partner, and lastly, Technology as an Extension of Self. We described the differences of each of these and stressed the importance of understanding the math itself rather than using the technology to just get by, or to get the answer and not really care to learn the mathematics conceptually.
Critique:
We discussed how the fact that the words to all the hymns the congregations across the world are asked to join in on are not displayed as subtitles at the bottom of the screen, as they once were. This technological change is not conducive to a wide and varied audience speaking a plethora of languages and dialects. For English as a Second Language (ESL) learners, it makes following along and singing the English words rather difficult to participate. This is something I would change if I were involved in the technological crew in broadcasting General Conference.
One thing I would like to continue to study out and research further is the idea of Technology as being my Partner. While we discussed technology being my partner in helping me understand the mathematical concepts as I go, and using its capabilities combined with my own, I could not quite pinpoint the difference between using it as my Partner and using it as my servant. However, despite my confusion about this point, I will always seek to use technology as my servant and dabble in having it become the extension of self, depending on whether or not I am learning about the technological power of a certain tool, or if I am using it as a tool in my future classroom.
Connections:
I know that when technology is used as a servant, it is simply there to aid in portraying the message I am trying to teach. It makes it easier for the teacher to explain concepts and students to learn because there is a visual to which the teacher can refer. Students are better able to visualize what the teacher is saying, and they can ask more direct questions having to do with the topic. As a future math educator, I will use technology as my servant. I will be its master instead of it mastering me and my entire lesson. Furthermore, based on the class discussion, I will hopefully reach a point with all the technological mediums I will use, where they will all be simply an extension of myself. It will be second nature to use them all, and students will be edified by each lesson involving technology.
This is the last of the discussions I will have with this particular group of students all learning about technology in the classroom together. As a result, I wanted to conclude with something I always want to remember as a future educator regarding technology. That is—I will become thoroughly familiar with the technology I choose to use as a teacher so that I do not spend class-time troubleshooting it. I want the technology to be helpful and not hinder the precious learning environment.
Tuesday, April 16, 2019
Spreadsheet Data Exploration
For a long time, I have been at a loss as to the kinds of things Microsoft Excel can do. Spreadsheets full of data are not only organized in specific rows and columns based on categories of some sort (i.e. year, country, etc.), but the data can be duplicated, copied, and analyzed using multiple techniques that are programmed into Microsoft Excel. For this exploration, I took data from the web regarding demographics about Texas and used the abilities of the spreadsheet to create and explore relationships between different sets of data. I was able to see if there were any pieces of data that seemed inaccurate, continue patterns down columns, and analyze how accurate they are compared to actual measured data, and analyze possible relationships between different categories.
Here is the spreadsheet I created to analyze this specific data.
All of the data gathered (and the data I created from the given data) are all from the years 1984 through 2001. In the year column, I expanded the column down all the way to 2024 to hopefully create estimations of certain data in the future. I researched and found the actual data for the year 2017 and compared it to the estimations, and while they are similar, they are relatively fairly different from each other. I gathered marriages, divorces, births, deaths, infant deaths, and maternal deaths. In addition, I created columns to calculate the marriage rate, divorce rate, birth rate, death rate, deaths from deliveries, and deaths not from deliveries. I created these rates based on the population to make these data sets more accurate, as the population is dynamic throughout the years. In each of these columns of data I created, I took the number of [marriages, births, etc.] and divided it by the product of the corresponding population in column B and 1,000,000, as the values in column B are in millions. I then multiplied this entire value by 1,000 to get the number of individuals per 1,000 in the population that would get married, or be born, etc. I then dragged the cell down and created an entire column with this formulaic pattern.
To format my spreadsheet, I double-clicked the tops of each column to make the width of the column dependent on fitting all the data to make it all visible. I then selected B2 to freeze the top row and the far left column in order to make them stationary as you scroll through the spreadsheet. This helps interpret the data with the years and the category always visible.
Overall, I learned a lot of different strategies to create a helpful Excel spreadsheet in order to analyze data, create my own based on the data given, and make the spreadsheet look presentable.
Here is the spreadsheet I created to analyze this specific data.
All of the data gathered (and the data I created from the given data) are all from the years 1984 through 2001. In the year column, I expanded the column down all the way to 2024 to hopefully create estimations of certain data in the future. I researched and found the actual data for the year 2017 and compared it to the estimations, and while they are similar, they are relatively fairly different from each other. I gathered marriages, divorces, births, deaths, infant deaths, and maternal deaths. In addition, I created columns to calculate the marriage rate, divorce rate, birth rate, death rate, deaths from deliveries, and deaths not from deliveries. I created these rates based on the population to make these data sets more accurate, as the population is dynamic throughout the years. In each of these columns of data I created, I took the number of [marriages, births, etc.] and divided it by the product of the corresponding population in column B and 1,000,000, as the values in column B are in millions. I then multiplied this entire value by 1,000 to get the number of individuals per 1,000 in the population that would get married, or be born, etc. I then dragged the cell down and created an entire column with this formulaic pattern.
To format my spreadsheet, I double-clicked the tops of each column to make the width of the column dependent on fitting all the data to make it all visible. I then selected B2 to freeze the top row and the far left column in order to make them stationary as you scroll through the spreadsheet. This helps interpret the data with the years and the category always visible.
Overall, I learned a lot of different strategies to create a helpful Excel spreadsheet in order to analyze data, create my own based on the data given, and make the spreadsheet look presentable.
Thursday, March 28, 2019
Understanding Geometric Transformations
Summary:
In preparation for a discussion in class on Karen F. Hollebrands’ article “High School Students’ Intuitive Understandings of Geometric Transformations,” I read it and took notes that provided a lot of insights into the research that was conducted and how it relates to my future mathematics classroom. High school students who had taken algebra 1 in eighth grade were chosen in order to analyze how much knowledge of geometric transformations they were able to remember. The geometric transformations in question included reflections, rotations, and translations. Many conclusions were analyzed and discussed, such as the fact that students seemed to think about reflections as flipping an image over a certain line instead of that line being the perpendicular bisector of all the lines connecting each point of the image on each side of the line.
The professor and students studying to be educators in the future in my Teaching With Technology class at Brigham Young University (BYU) extensively discussed the ideas presented in Hollebrands’ article. We discussed the implications of reflecting an entire plane of points rather than just the preimage. This idea is not typically seen when analyzing how students perceive reflections, so this was an interesting conjecture—something I have never thought about before! We talked about things we learned from the article, like reflecting points across a line of reflection simply means the points are equidistant from the line. The same thing happens when reflecting lines or entire images—each part of the line or image is exactly the same distance from the line of symmetry as its preimage.
Critique:
It was mentioned in the article that the transformations that the students struggled the most with were translations. This was due to the complete inability to appropriately use the given vector describing the translation required. Because of this, I wholeheartedly disagree with the conclusion the author made from the results of this study—that students understood translations the least simply because they could not use vectors exactly how they were intended to be used for the transformations. I believe this was not an accurate conclusion that could be made from the information gained from the study.
In addition, it was stated that students are typically more successful when the transformation requires a reflection over horizontal or vertical lines. This reminds me that rotations are usually easier to visualize (from my own experience) when it requires 90, 180, 270, or 360 degree-rotations instead of any other rotations, such as 45-degrees or 135-degrees. This was an intuitive conclusion made from the study’s results, and I agree with it.
Connections:
It was stated that students who heavily relied on the visual aspect of the preimage instead of the specific properties that define certain transformations were more likely to have transformed the preimage incorrectly. Knowing this fact, I plan on teaching the fundamental properties of each transformation first in the unit so students understand the underlying mathematics of each one instead of an artsy new configuration of a picture. Most of my experience with transformations growing up in school involved flipping or turning an image on its side. I never learned the actual properties of transformations—such as each point being equidistant from the line of reflection as its preimage. Knowing this fact would not only help me visualize the new image better (even if the line of reflection was not vertical or horizontal), but it would help me to check my answer mathematically to ensure accuracy. Overall, I learned a great amount about transformations as I analyzed the study done with these brave eighth graders.
In preparation for a discussion in class on Karen F. Hollebrands’ article “High School Students’ Intuitive Understandings of Geometric Transformations,” I read it and took notes that provided a lot of insights into the research that was conducted and how it relates to my future mathematics classroom. High school students who had taken algebra 1 in eighth grade were chosen in order to analyze how much knowledge of geometric transformations they were able to remember. The geometric transformations in question included reflections, rotations, and translations. Many conclusions were analyzed and discussed, such as the fact that students seemed to think about reflections as flipping an image over a certain line instead of that line being the perpendicular bisector of all the lines connecting each point of the image on each side of the line.
The professor and students studying to be educators in the future in my Teaching With Technology class at Brigham Young University (BYU) extensively discussed the ideas presented in Hollebrands’ article. We discussed the implications of reflecting an entire plane of points rather than just the preimage. This idea is not typically seen when analyzing how students perceive reflections, so this was an interesting conjecture—something I have never thought about before! We talked about things we learned from the article, like reflecting points across a line of reflection simply means the points are equidistant from the line. The same thing happens when reflecting lines or entire images—each part of the line or image is exactly the same distance from the line of symmetry as its preimage.
Critique:
It was mentioned in the article that the transformations that the students struggled the most with were translations. This was due to the complete inability to appropriately use the given vector describing the translation required. Because of this, I wholeheartedly disagree with the conclusion the author made from the results of this study—that students understood translations the least simply because they could not use vectors exactly how they were intended to be used for the transformations. I believe this was not an accurate conclusion that could be made from the information gained from the study.
In addition, it was stated that students are typically more successful when the transformation requires a reflection over horizontal or vertical lines. This reminds me that rotations are usually easier to visualize (from my own experience) when it requires 90, 180, 270, or 360 degree-rotations instead of any other rotations, such as 45-degrees or 135-degrees. This was an intuitive conclusion made from the study’s results, and I agree with it.
Connections:
It was stated that students who heavily relied on the visual aspect of the preimage instead of the specific properties that define certain transformations were more likely to have transformed the preimage incorrectly. Knowing this fact, I plan on teaching the fundamental properties of each transformation first in the unit so students understand the underlying mathematics of each one instead of an artsy new configuration of a picture. Most of my experience with transformations growing up in school involved flipping or turning an image on its side. I never learned the actual properties of transformations—such as each point being equidistant from the line of reflection as its preimage. Knowing this fact would not only help me visualize the new image better (even if the line of reflection was not vertical or horizontal), but it would help me to check my answer mathematically to ensure accuracy. Overall, I learned a great amount about transformations as I analyzed the study done with these brave eighth graders.
Parabola Paradise
There are many things you can create using any line in space and any point not on the line. One of which is a parabola! In fact, with every line and every point not on that line, there exists a unique parabola. The parabola is created by connecting the accumulation of the infinite number of points that lie exactly equidistant to the line (the shortest distance to this line) and the given point. Parabolas of varying narrownesses and widenesses have been created and analyzed as a result of this construction. It was fascinating to me to construct a parabola in this way in Geometer's Sketchpad alongside some of my classmates who are all studying how to best use technology in the classroom as future educators.
Once again, you will have to download Geometer's Sketchpad in order to view the attached exploration.
Once again, you will have to download Geometer's Sketchpad in order to view the attached exploration.
Wednesday, March 27, 2019
Creating Quadrilaterals From Quadrilaterals
Summary:
In preparation for a discussion in class on Wayne Nirode’s article, “Creating Quadrilaterals from Quadrilaterals,” (the link takes you to a place where you can choose to purchase the full article) I read it and took notes that provided a lot of insights into how to best create tasks that engage my students and enlarge their minds, no matter the level of intellect. This quadrilaterals task began with students creating new convex quadrilaterals from the given convex quadrilaterals by connecting the four midpoints of each side in a successive order, for example. Each student was asked to create his or her own 10 rules for creating new quadrilaterals, and by the time the class gathered together all of their rules, there were up to 150 unique ideas! With all of these rules, groups of students were assigned a few to prove on their own, analyze each of them, and create posters presenting the rule and its proof. Each proof analyzed which kind of quadrilateral could be made using the given rule based on the starting quadrilateral. As I read about this task and the successes that came from it, I learned that I could create tasks with both a low floor and a high ceiling (LFHC). LFHC is used to describe educational tasks that are easy to begin and think about at first, but that the same task could encourage deeper thinking within students who finish quickly, or quickly learn the fundamental mathematical concepts for the task. Creating Quadrilaterals from Quadrilaterals is the perfect example of a LFHC task.
The professor and students studying to be educators in the future in my Teaching With Technology class at Brigham Young University (BYU) extensively discussed ideas that stemmed from this task. While we talked a little bit about the benefits of the quadrilaterals task, we branched off for the majority of the discussion and shared ideas about the best ways we could implement geometric proofs in our future classrooms. The task given provided multiple opportunities for students to grapple with different quadrilaterals and trying to prove them. If they got stuck, the task gave an opportunity for students to move onto a different rule to try to prove together. If proofs are wrong, students can discuss all the ways instead of just the right ways to complete the proofs. This would enlarge their minds as they consider why certain reasoning is wrong. This will surely solidify their understanding of why certain proofs are correct as well. In addition, my classmates and I all reminded each other about some critical geometric definitions, such as orthocenters, centroids, and incenters, and how they all relate to exploring relationships with triangles. Someone noted that if there is this much variety (and more) in exploration of characteristics of triangles in geometry, how many more characteristics could be explored in quadrilateral characteristics!
Critique:
In our class discussion, it was mentioned that students will distract themselves with anything, no matter what task you give them, and that the only thing you can do to help engage the students is to give them an especially engaging task. While I do believe this is helpful (and vital) in keeping students’ attention, I also believe my responsibility to help them stay on task only goes so far. I want to instill within each of students a personal responsibility and accountabliltiy for their learning. However, I also believe that it is my job as their teacher to create the best environment possible to facilitate meaningful learning experiences. I can do this with what I put on my walls, my established rules in the classroom, and a trusting teacher-student relationship with each pupil.
While I love the part of the task when students are able to get together in a group and prove different rules for creating new quadrilaterals because they can bounce ideas off of one another, I think this task would be far too long, and students might start to not like it after about 2 class periods. Making posters fosters creativity and engages the right side of the brain in a mathematics classroom—which does not happen very often! However, I believe making posters would take up too much valuable time in the classroom.
Connections:
The article provided an excellent example of the kinds of tasks I want to use in my future classroom. This task is similar to a practice we have discussed as a class in previous discussions that I would love to implement—that is, I would like to use the textbook as a resource in creating meaningful tasks for students as I search the example problems in the back or at the end of the section, and begin my lesson with a question towards the end. These questions are written for an audience who has already learned the material and is simply thinking about the concepts deeper for further enlightenment. If I choose to begin my lessons with these types of questions, students can grapple with it and come to their own conclusions. They can seek to learn the processes and procedures on their own with a facilitated discussion. This task seems to be something Nirode took from the back of a textbook to begin with, and ran with it, hoping students would learn powerful facts and concepts about quadrilaterals through their own exploration of it rather than a list of formulas and factual characteristics of quadrilaterals.
Something profound that my professor suggested was to never force students to prove conjectures using the traditional two-column format, where the first column involves the line of logical statements from beginning to end and the second column includes the reasons why the logical statement can be said. In his extensive experience teaching geometry to high school students, he found that they would often write down the “given” and the conclusion but not understand what to write in between. The point of the proof is to encourage creating an argument as to why a certain conjecture is correct (or incorrect, as discussed previously), and not in arguing through a specified two-column format that often gets in the way of the desired purpose of geometric proofs. This discussion changed how I plan on teaching geometric proofs in the future. It is refreshing to remember the true purpose of a proof and how my teaching strategies and tasks can illustrate that.
In preparation for a discussion in class on Wayne Nirode’s article, “Creating Quadrilaterals from Quadrilaterals,” (the link takes you to a place where you can choose to purchase the full article) I read it and took notes that provided a lot of insights into how to best create tasks that engage my students and enlarge their minds, no matter the level of intellect. This quadrilaterals task began with students creating new convex quadrilaterals from the given convex quadrilaterals by connecting the four midpoints of each side in a successive order, for example. Each student was asked to create his or her own 10 rules for creating new quadrilaterals, and by the time the class gathered together all of their rules, there were up to 150 unique ideas! With all of these rules, groups of students were assigned a few to prove on their own, analyze each of them, and create posters presenting the rule and its proof. Each proof analyzed which kind of quadrilateral could be made using the given rule based on the starting quadrilateral. As I read about this task and the successes that came from it, I learned that I could create tasks with both a low floor and a high ceiling (LFHC). LFHC is used to describe educational tasks that are easy to begin and think about at first, but that the same task could encourage deeper thinking within students who finish quickly, or quickly learn the fundamental mathematical concepts for the task. Creating Quadrilaterals from Quadrilaterals is the perfect example of a LFHC task.
The professor and students studying to be educators in the future in my Teaching With Technology class at Brigham Young University (BYU) extensively discussed ideas that stemmed from this task. While we talked a little bit about the benefits of the quadrilaterals task, we branched off for the majority of the discussion and shared ideas about the best ways we could implement geometric proofs in our future classrooms. The task given provided multiple opportunities for students to grapple with different quadrilaterals and trying to prove them. If they got stuck, the task gave an opportunity for students to move onto a different rule to try to prove together. If proofs are wrong, students can discuss all the ways instead of just the right ways to complete the proofs. This would enlarge their minds as they consider why certain reasoning is wrong. This will surely solidify their understanding of why certain proofs are correct as well. In addition, my classmates and I all reminded each other about some critical geometric definitions, such as orthocenters, centroids, and incenters, and how they all relate to exploring relationships with triangles. Someone noted that if there is this much variety (and more) in exploration of characteristics of triangles in geometry, how many more characteristics could be explored in quadrilateral characteristics!
Critique:
In our class discussion, it was mentioned that students will distract themselves with anything, no matter what task you give them, and that the only thing you can do to help engage the students is to give them an especially engaging task. While I do believe this is helpful (and vital) in keeping students’ attention, I also believe my responsibility to help them stay on task only goes so far. I want to instill within each of students a personal responsibility and accountabliltiy for their learning. However, I also believe that it is my job as their teacher to create the best environment possible to facilitate meaningful learning experiences. I can do this with what I put on my walls, my established rules in the classroom, and a trusting teacher-student relationship with each pupil.
While I love the part of the task when students are able to get together in a group and prove different rules for creating new quadrilaterals because they can bounce ideas off of one another, I think this task would be far too long, and students might start to not like it after about 2 class periods. Making posters fosters creativity and engages the right side of the brain in a mathematics classroom—which does not happen very often! However, I believe making posters would take up too much valuable time in the classroom.
Connections:
The article provided an excellent example of the kinds of tasks I want to use in my future classroom. This task is similar to a practice we have discussed as a class in previous discussions that I would love to implement—that is, I would like to use the textbook as a resource in creating meaningful tasks for students as I search the example problems in the back or at the end of the section, and begin my lesson with a question towards the end. These questions are written for an audience who has already learned the material and is simply thinking about the concepts deeper for further enlightenment. If I choose to begin my lessons with these types of questions, students can grapple with it and come to their own conclusions. They can seek to learn the processes and procedures on their own with a facilitated discussion. This task seems to be something Nirode took from the back of a textbook to begin with, and ran with it, hoping students would learn powerful facts and concepts about quadrilaterals through their own exploration of it rather than a list of formulas and factual characteristics of quadrilaterals.
Something profound that my professor suggested was to never force students to prove conjectures using the traditional two-column format, where the first column involves the line of logical statements from beginning to end and the second column includes the reasons why the logical statement can be said. In his extensive experience teaching geometry to high school students, he found that they would often write down the “given” and the conclusion but not understand what to write in between. The point of the proof is to encourage creating an argument as to why a certain conjecture is correct (or incorrect, as discussed previously), and not in arguing through a specified two-column format that often gets in the way of the desired purpose of geometric proofs. This discussion changed how I plan on teaching geometric proofs in the future. It is refreshing to remember the true purpose of a proof and how my teaching strategies and tasks can illustrate that.
Monday, March 11, 2019
Geometer’s Sketchpad Toolkit
There’s a handy program called Geometer’s Sketchpad that is especially important to future math educators in constructing geometric shapes quickly and easily using digital versions of a compass and a straightedge. For example, one can quickly create a regular hexagon given the length of one of its sides, or a right isosceles triangle given one of the legs. While staying in Euclidean geometry, I was able to construct 10 tools that are saved onto one Sketchpad document.
Unfortunately, you will need to download Geometer’s Sketchpad to access this “toolbox” of mine. It is fairly easy and does not take up a lot of storage space. You will need a desktop or laptop computer, as a downloadable version of Geometer’s Sketchpad for tablets and mobile devices either does not exist or is not yet available to the public.
Happy geometering!
Unfortunately, you will need to download Geometer’s Sketchpad to access this “toolbox” of mine. It is fairly easy and does not take up a lot of storage space. You will need a desktop or laptop computer, as a downloadable version of Geometer’s Sketchpad for tablets and mobile devices either does not exist or is not yet available to the public.
Happy geometering!
Wednesday, February 13, 2019
Desmos Discourse
Summary:
In preparation for a class discussion, I read an article entitled “Increased Participation and Conversation Using Networked Devices” by Christopher Danielson and Dan Meyer. It outlined some of the content provided by the online desmos graphing utility and how teachers can utilize it to reach the students to help them learn in the best ways.
While other online graphing utilities are adequate at best (like this one or the calculator app found here), desmos.com, or more specifically teacher.desmos.com, provides tools teachers can use to access previously created activities, or create their own. Two of these activities Danielson and Meyer analyze and promote. First, they outline Central Park, which is a simple virtual manipulative in which students can estimate and eventually perfect the exact amount of space needed between each of the parking space lines so that all four cars can fit. It walks students through guess-and-check strategies and eventually creating their own algebraic expressions to describe the situation. The second activity they describe is called Polygraph, where partnerships of students guess each other’s graphs by eliminating options based on a series of yes-or-no questions. The authors explored why this particular activity helps teachers gauge the level of vocabulary and mathematical understanding of particular concepts having to do with graphs.
In our class discussion, we extensively reviewed the strengths and weaknesses of the Central Park and the Polygraph tasks. We discussed appropriate amounts teachers should be able to allow students to guess and check until they find the correct answers. We concluded that the Central Park task could be greatly improved in a way that actually teaches the students how to write mathematical expressions in more productive ways than just trying to guide the student to figure it out completely on their own. The jump between guessing and checking their answers and developing an accurate mathematical expression is too big that this task alone cannot accomplish. Much of our class discussion revolved around teaching strategies using these two tasks specifically as well as more general strategies to be used when, students get stuck and cannot proceed on their own, hindering their learning. The Polygraph task is much better because it is set up in such a way that teachers can monitor student thinking based on their strategies for asking questions about the graphed polygons in order to eliminate them down to one chosen graph. Teachers can see all questions and responses to see not only different vocabulary words being used, but also different pathways of thinking about these polygons that students might use (i.e. how many sides, orientation, size, regularity, etc.). We discussed how we as future educators can help students learn through their independent (and partner) exploration by giving them just the right hint. The “right hint” will enhance learning, rather than be unhelpful or give away the solutions too quickly without sufficient student contemplation.
Critique:
Originally, after reading the article on my own, I thought the simple Central Park manipulative was a good way to help students learn how to create their own expressions after realizing guessing and checking the answers was no longer a good strategy. However, as the class discussed this task and its limitations (including the students too easily sliding into just guessing-and-checking instead of dividing, creating expressions, incorporating variables, etc.), my opinion shifted negatively towards this task, and my eyes were opened to this fundamental flaw. It seems to me that this task, developed in the early stages of Desmos.com, was sufficient for the time being, but since its technology has drastically improved in the last couple of years, it will no longer suffice as a helpful tool for instruction.
The class collectively agreed that the Polygraph task provided much better results. This is because this task informed the teacher about what kinds of vocabulary each student was able to use when describing characteristics of simple polygons graphed on common Cartesian planes. I agree with the fact that students are much more apt to engage in a mathematical problem if they see a need to do so. It is much harder to have students engage in a problem that they do not already have. This task puts them in a situation in which they need something (i.e. formal mathematical vocabulary), so they are more likely to learn the formal language required.
On a more general note, one suggestion someone made in our class discussion when students get stuck, is to scale back to an easier “back-up” assignment. While I did not agree with this strategy for my own teaching techniques, I do agree with the next suggestion: Instead, ask a question that makes the current difficult assignment a little more doable. This does not allow students to get in the habit of “playing dumb” or giving up too easily or often or soon because they know they will have to engage in the task anyway—it is already established over time in the classroom’s expectations for the learning environment. However, we teachers must be sure the cognitive demand of the task is not too high before administering it to the class if we do not have an easier “back-up” task to administer if students begin to struggle.
Connections:
A feature included in this website that thoroughly impressed me as a future educator was the ability to monitor all the students’ screens at the same time. In each of these two tasks described above, I will be able to observe all the questions students are asking, all the responses, all the expressions students come up with to try to describe the space each car needs in the parking lot, etc. By having all of this information so easily accessible, teachers can identify the specific students who might be struggling with the concepts, or who might just be guessing-and-checking. In the class discussion, we mentioned that there is a stark contrast between the students who are guessing-and-checking because they do not want to engage in the task (a motivation problem), and the students who are doing so because they do not know how to engage in the task (a mathematical problem that could be a result of the task shortcoming, a misunderstanding of a fundamental concept, or a misunderstanding of instructions). In addition, this also allows teachers to observe whether or not it is a class-wide problem or just an issue with certain individuals. Teachers can then decide how to address the issues at hand to facilitate the best learning environment within the classroom.
Students are also able to monitor their own progress as the program checks their answer and suggests new ways to accomplish the task. In the Polygraph task, students bounce ideas off of their partner to eliminate different graphs through the use of yes-or-no questions. The series of these questions and their affirmative or negative answers teach both students involved what the other might be thinking and how he or she might go about solving the task. This diminishes the busy work of the teacher going around individually to check answers and explain to each student where they went wrong and why. In my future classroom, I will give my students in partnerships the opportunity to talk together about the conclusions they reached on their own and the conclusions they reached as they worked together to complete the task.
Another point discussed was the benefit from helping students by giving a little hint—perhaps giving an idea of the direction a student might solve a problem. Often there are students who need just a tiny push in the right direction, and then they can hit the ground running, figuring out the answer completely on their own! How satisfying that is for them! This is a strategy I plan to use in my future classroom as I get to know my students and identify the specific students who need subtle hints every once in a while instead of the whole procedure, or the lack of any help whatsoever.
In preparation for a class discussion, I read an article entitled “Increased Participation and Conversation Using Networked Devices” by Christopher Danielson and Dan Meyer. It outlined some of the content provided by the online desmos graphing utility and how teachers can utilize it to reach the students to help them learn in the best ways.
While other online graphing utilities are adequate at best (like this one or the calculator app found here), desmos.com, or more specifically teacher.desmos.com, provides tools teachers can use to access previously created activities, or create their own. Two of these activities Danielson and Meyer analyze and promote. First, they outline Central Park, which is a simple virtual manipulative in which students can estimate and eventually perfect the exact amount of space needed between each of the parking space lines so that all four cars can fit. It walks students through guess-and-check strategies and eventually creating their own algebraic expressions to describe the situation. The second activity they describe is called Polygraph, where partnerships of students guess each other’s graphs by eliminating options based on a series of yes-or-no questions. The authors explored why this particular activity helps teachers gauge the level of vocabulary and mathematical understanding of particular concepts having to do with graphs.
In our class discussion, we extensively reviewed the strengths and weaknesses of the Central Park and the Polygraph tasks. We discussed appropriate amounts teachers should be able to allow students to guess and check until they find the correct answers. We concluded that the Central Park task could be greatly improved in a way that actually teaches the students how to write mathematical expressions in more productive ways than just trying to guide the student to figure it out completely on their own. The jump between guessing and checking their answers and developing an accurate mathematical expression is too big that this task alone cannot accomplish. Much of our class discussion revolved around teaching strategies using these two tasks specifically as well as more general strategies to be used when, students get stuck and cannot proceed on their own, hindering their learning. The Polygraph task is much better because it is set up in such a way that teachers can monitor student thinking based on their strategies for asking questions about the graphed polygons in order to eliminate them down to one chosen graph. Teachers can see all questions and responses to see not only different vocabulary words being used, but also different pathways of thinking about these polygons that students might use (i.e. how many sides, orientation, size, regularity, etc.). We discussed how we as future educators can help students learn through their independent (and partner) exploration by giving them just the right hint. The “right hint” will enhance learning, rather than be unhelpful or give away the solutions too quickly without sufficient student contemplation.
Critique:
Originally, after reading the article on my own, I thought the simple Central Park manipulative was a good way to help students learn how to create their own expressions after realizing guessing and checking the answers was no longer a good strategy. However, as the class discussed this task and its limitations (including the students too easily sliding into just guessing-and-checking instead of dividing, creating expressions, incorporating variables, etc.), my opinion shifted negatively towards this task, and my eyes were opened to this fundamental flaw. It seems to me that this task, developed in the early stages of Desmos.com, was sufficient for the time being, but since its technology has drastically improved in the last couple of years, it will no longer suffice as a helpful tool for instruction.
The class collectively agreed that the Polygraph task provided much better results. This is because this task informed the teacher about what kinds of vocabulary each student was able to use when describing characteristics of simple polygons graphed on common Cartesian planes. I agree with the fact that students are much more apt to engage in a mathematical problem if they see a need to do so. It is much harder to have students engage in a problem that they do not already have. This task puts them in a situation in which they need something (i.e. formal mathematical vocabulary), so they are more likely to learn the formal language required.
On a more general note, one suggestion someone made in our class discussion when students get stuck, is to scale back to an easier “back-up” assignment. While I did not agree with this strategy for my own teaching techniques, I do agree with the next suggestion: Instead, ask a question that makes the current difficult assignment a little more doable. This does not allow students to get in the habit of “playing dumb” or giving up too easily or often or soon because they know they will have to engage in the task anyway—it is already established over time in the classroom’s expectations for the learning environment. However, we teachers must be sure the cognitive demand of the task is not too high before administering it to the class if we do not have an easier “back-up” task to administer if students begin to struggle.
Connections:
A feature included in this website that thoroughly impressed me as a future educator was the ability to monitor all the students’ screens at the same time. In each of these two tasks described above, I will be able to observe all the questions students are asking, all the responses, all the expressions students come up with to try to describe the space each car needs in the parking lot, etc. By having all of this information so easily accessible, teachers can identify the specific students who might be struggling with the concepts, or who might just be guessing-and-checking. In the class discussion, we mentioned that there is a stark contrast between the students who are guessing-and-checking because they do not want to engage in the task (a motivation problem), and the students who are doing so because they do not know how to engage in the task (a mathematical problem that could be a result of the task shortcoming, a misunderstanding of a fundamental concept, or a misunderstanding of instructions). In addition, this also allows teachers to observe whether or not it is a class-wide problem or just an issue with certain individuals. Teachers can then decide how to address the issues at hand to facilitate the best learning environment within the classroom.
Students are also able to monitor their own progress as the program checks their answer and suggests new ways to accomplish the task. In the Polygraph task, students bounce ideas off of their partner to eliminate different graphs through the use of yes-or-no questions. The series of these questions and their affirmative or negative answers teach both students involved what the other might be thinking and how he or she might go about solving the task. This diminishes the busy work of the teacher going around individually to check answers and explain to each student where they went wrong and why. In my future classroom, I will give my students in partnerships the opportunity to talk together about the conclusions they reached on their own and the conclusions they reached as they worked together to complete the task.
Another point discussed was the benefit from helping students by giving a little hint—perhaps giving an idea of the direction a student might solve a problem. Often there are students who need just a tiny push in the right direction, and then they can hit the ground running, figuring out the answer completely on their own! How satisfying that is for them! This is a strategy I plan to use in my future classroom as I get to know my students and identify the specific students who need subtle hints every once in a while instead of the whole procedure, or the lack of any help whatsoever.
Thursday, January 31, 2019
Puzzling Jugs
I was introduced to a virtual manipulative recently called the “Jugs Puzzle Game.” It outlines a riddle that can be manipulated virtually in order to solve it. Basically, there are 2 jugs of a given size that has a certain goal amount of water to put into one jug. You have an unlimited amount of water, and you can fill up jugs, pour them out, and transfer water between the jugs. It seems simple enough, right? Except you can’t fill up a jug part of the way with the water faucet because at the faucet, you can only fill the jug to its full capacity. This makes it difficult to achieve the desired amount of water all in one jug because it requires a certain order of filling, draining, and transferring water between jugs to achieve the desired output.
I struggled for a long time to complete the first level the first time I tried completing the task. I felt helpless because I thought it was impossible, but I knew that it was a solvable problem, and that others had done it before. This boosted my confidence in completing the task, so I kept clicking around until I eventually got the hang of the process. With each jug, there are only two options with what I could do next: I could drain the water, or transfer it over, perhaps leaving a partial amount of water in the jug. As I tried each level again and again, I would gradually improve my skills in deciding which move would be a progressive move to help me get closer to my desired output rather than something that wouldn’t get me anywhere. After a while, I was addicted.
I was thinking about the mathematics behind the task and trying to figure out what exactly I was doing in order to achieve my desired amount of water in one jug. I thought about other combinations of jug capacities and thought the solution to each problem would have something to do with the size of the given jugs—I couldn’t have just any two jugs with arbitrary capacities to receive a certain desired amount of water in one of them. I realized the capacities of each of the two jugs must be relatively prime to each other, making each of the two jugs lack the same amounts of water that might factor into their capacities, as this would make it impossible to fill a jug with a desired amount of water.
When a “solvable” problem is presented, I first outline where I am, what my goal or destination is, and what means I have available to me in order to get there. In this case, I had 2 jugs of different sizes, an unlimited amount of water, a faucet, and a drain. I also knew that I had to fill the jugs all the way to the top instead of partially. I knew I needed a given amount of water in a jug (either one that could hold that given amount or more). The means I had to get there included 3 options for each of the jugs. I could fill them completely, drain them completely, or transfer water between the jugs to leave partial amounts of water in one of them or both. In my understanding of the simple terms of completing the task, I was able to push past any uncertainties and fill, drain, or transfer water over and over again. In doing this, I found new ways of filling the jugs to have the exact amount of gallons of water I needed. In teaching this concept, I would likely not use this virtual manipulative, as it was difficult and frustrating to figure out. After a long time of thinking about it, I finally thought about the jug capacities being relatively prime, and I felt that this fact was not particularly enlightening. Overall, it was a frustrating way to attempt to teach a concept that could have been taught in an easier manner without the struggle of this virtual manipulative.
When I do not know whether or not the given problem even has a solution, I can tell by the number of possibilities and probable amounts of water each jug could have. If I could add and subtract the given jugs’ full capacities, zero them out, or fill them completly, I could tell if a given amount of water needed in one jug was possible to achieve before trying it.
This virtual manipulative helps students to better understand critical addition and subtraction skills and improves mental math abilities. Numerical fluency increases as students will understand what “moves” they should make in the virtual manipulative to obtain their desired answer. As students’ skills improve and they move onto higher levels, they will be better able to keep track of more anticipated addition and subtraction answers ahead of time. These abilities make me think of accomplished chess players, who need to keep track of their pieces on the board, their opponents pieces on the board, what moves each piece can make, and anticipate their opponent’s moves in order to attack the king.
Overall, as a future teacher, I would not use this virtual manipulative to teach such an important concept such as prime factorization. I felt the concept was not fully realized in my experience using the virtual manipulative, and it could have been improved using perhaps another mathematical problem or riddle.
There are a few examples of lesson plans focused on teaching prime factorization that I find much more useful pedagogically. Click here or here to find them.
I struggled for a long time to complete the first level the first time I tried completing the task. I felt helpless because I thought it was impossible, but I knew that it was a solvable problem, and that others had done it before. This boosted my confidence in completing the task, so I kept clicking around until I eventually got the hang of the process. With each jug, there are only two options with what I could do next: I could drain the water, or transfer it over, perhaps leaving a partial amount of water in the jug. As I tried each level again and again, I would gradually improve my skills in deciding which move would be a progressive move to help me get closer to my desired output rather than something that wouldn’t get me anywhere. After a while, I was addicted.
I was thinking about the mathematics behind the task and trying to figure out what exactly I was doing in order to achieve my desired amount of water in one jug. I thought about other combinations of jug capacities and thought the solution to each problem would have something to do with the size of the given jugs—I couldn’t have just any two jugs with arbitrary capacities to receive a certain desired amount of water in one of them. I realized the capacities of each of the two jugs must be relatively prime to each other, making each of the two jugs lack the same amounts of water that might factor into their capacities, as this would make it impossible to fill a jug with a desired amount of water.
When a “solvable” problem is presented, I first outline where I am, what my goal or destination is, and what means I have available to me in order to get there. In this case, I had 2 jugs of different sizes, an unlimited amount of water, a faucet, and a drain. I also knew that I had to fill the jugs all the way to the top instead of partially. I knew I needed a given amount of water in a jug (either one that could hold that given amount or more). The means I had to get there included 3 options for each of the jugs. I could fill them completely, drain them completely, or transfer water between the jugs to leave partial amounts of water in one of them or both. In my understanding of the simple terms of completing the task, I was able to push past any uncertainties and fill, drain, or transfer water over and over again. In doing this, I found new ways of filling the jugs to have the exact amount of gallons of water I needed. In teaching this concept, I would likely not use this virtual manipulative, as it was difficult and frustrating to figure out. After a long time of thinking about it, I finally thought about the jug capacities being relatively prime, and I felt that this fact was not particularly enlightening. Overall, it was a frustrating way to attempt to teach a concept that could have been taught in an easier manner without the struggle of this virtual manipulative.
When I do not know whether or not the given problem even has a solution, I can tell by the number of possibilities and probable amounts of water each jug could have. If I could add and subtract the given jugs’ full capacities, zero them out, or fill them completly, I could tell if a given amount of water needed in one jug was possible to achieve before trying it.
This virtual manipulative helps students to better understand critical addition and subtraction skills and improves mental math abilities. Numerical fluency increases as students will understand what “moves” they should make in the virtual manipulative to obtain their desired answer. As students’ skills improve and they move onto higher levels, they will be better able to keep track of more anticipated addition and subtraction answers ahead of time. These abilities make me think of accomplished chess players, who need to keep track of their pieces on the board, their opponents pieces on the board, what moves each piece can make, and anticipate their opponent’s moves in order to attack the king.
Overall, as a future teacher, I would not use this virtual manipulative to teach such an important concept such as prime factorization. I felt the concept was not fully realized in my experience using the virtual manipulative, and it could have been improved using perhaps another mathematical problem or riddle.
There are a few examples of lesson plans focused on teaching prime factorization that I find much more useful pedagogically. Click here or here to find them.
Tuesday, January 22, 2019
Teaching Youth with Technology
Brian K. Ashton, in his article entitled “Teaching with Tech: Engaging Youth in a Digital World,” speaks to a general audience comprised mainly of members of the Church of Jesus Christ of Latter-day Saints who often find themselves teaching religious subjects to youth in Sunday School classes. While his audience does not exactly include those aspiring to be secondary mathematics teachers, many principles he discusses throughout his article are relevant and applicable to any classroom seeking to learn using technology. Ashton gives advice on how to best utilize technology in the classroom. First, we must learn about the technology. “As... teachers [become] more skilled in using technology, they also became more excited about using digital devices to study... and the issue of digital devices being a distraction in the classroom largely [goes] away.” By learning more about the technology as teachers, many positive effects are likely to ensue, including a renewed excitement to learn as well as less technological distractions during class. In addition, Ashton says, “I rarely find students using their cell phones inappropriately in classes where teachers [help] the students [become] involved in the lesson.” Typically students start to zone out or try to multi-task (i.e. pulling out their smart phones to scroll through social media) when they are not thoroughly engaged enough in the subject matter. Overall, as Ashton asserts, there must be a balance in the use of technology in the classroom. If technology starts to be the focus of the lesson instead of the important mathematical concepts to be learned, the students are missing out on precious learning opportunities.
After reading and studying Ashton’s article, a group of my classmates gathered together and discussed some of his most striking assertions. I was reminded that my students can teach me about the different ways to use technology if I do not already know. I should embrace the technological advances as they come and always seek ways of incorporating them in my classroom. In our discussion, we heavily emphasized the need to help students engage with the material in productive ways, such as looking up a definition of a word they might not know, message a friend to ask for notes or due dates, or explore alternative manipulatives to describe the same mathematical concept in a different way. One last point we discussed was the importance of incorporating technology in our own homes to ensure our future children are better prepared in their classrooms at school to use technology in productive ways. This will help facilitate and support their learning both at school and in the home.
Critique:
One student pointed out that establishing trust with students is one of the most powerful ways we can help students use technology in productive ways. She suggested we explain principles conceptually first, then teach them a faster and easier way to do it on a calculator. In this way, she said, the students can complete the mathematical task both ways. I disagree with this idea based on personal experience as a student. If a teacher spent thirty minutes explaining something conceptually and then follow up the explanation with a simple technique on the calculator, I would relax and stop trying to learn how to complete the problem conceptually and rely on the calculator the entire time. I would then give myself permission to forget the “why” behind the concept from my mind, knowing I can instead “arrive at the correct answer” in a faster and easier way, which hinders my learning in the long run. However, along the same lines, this student also suggested something I do agree with—that teachers should build trust with students by being real. For example, a student might complain, “Why do we have to learn this the long way, when I can just punch these numbers into a calculator?” If we respond with, “Well, you’re not always going to carry around a calculator in the real world!” then we lose all trust with the students because today, most of us carry around our smartphones with a built-in calculator on it. The students might then consider us out-of-date, or irrelevant, and stop trusting us with accurate information on which they can rely. On the other hand, if we explain that it is important to know how to compute by hand because it further develops mental math capabilities, estimation accuracy, and more, then the students are more likely to trust that you are teaching them to assist in their lifelong learning rather than punishing them with busy work.
Connections:
In considering even more ways I can apply what I have learned from this article and class discussion, something stuck out to me that I want to always remember when I am applying for jobs as a mathematics teacher is to ask whomever is interviewing me, “What kinds of technology is available to me in math classrooms?” Based on their response, I can gage not only how much technology I would have available to me to use in my class, but I would know how much the school cares about providing their teachers with appropriate resources to teach the students. I know as I make a sincere effort to incorporate technology in the best ways in my classroom, learning processes will develop instead of being hindered.
Thursday, January 17, 2019
Online Professionalism
Royce Kimmons, in his article about online professionalism for educators throughout the United States, describes the specific difficulties teachers (and other school faculty) face in their personal and professional lives online. He expounds his thesis through his list of teachers who have faced severe repercussions from their inappropriate use of technology, his list of ways educators can improve their virtual footprint, his five scenarios and analyses of educators making severe mistakes, and his commentary throughout the article. Most notably, Kimmons asserts that relationships between teachers, students, and parents must be kept professional at all times. Even the slightest question about how a teacher acts in what they say or do might make the student talk negatively about the teacher to others in an exaggerated way. Simple and seemingly harmless activities such as friending a student on facebook may have serious negative consequences, and teachers often lose their jobs and/or teaching licenses as a result.
In one of my math education classes at BYU, we had an insightful class discussion on the use of technology in both the personal and professional lives of educators. Most of the students in the class are preparing to be educators in the future, but all of us have different backgrounds and experiences with the use of social media. One student voiced a concern about students approaching her in inappropriate ways rather than the other way around. This concern sparked a fascinating conversation about different ways to handle it. As a class, we concluded the best way is to pretend to be oblivious, ignore it altogether if possible, or just brush it off in the moment. However, later it is best to contact the principal directly to see if he has any alternate accommodations for those particular students. We all agreed it would not be the best idea to react in the moment to the student(s), get angry or emotional in any way, or even contact their parents regarding the situation, but that contacting the principal would be in everyone’s best interest. In addition to discussing how to deal with specific cases when students might start making moves on the teacher, as a class we discussed how to behave online in general. This included making sure privacy settings were set according to how much we would want future employers, students, and/or their parents to see if they were to search our name. Many of my classmates decided to test out what kind of information they would find on themselves if they were to log off all of social media outlets and simply search their name on google. They shared the astounding results of finding links to silly videos they recorded or embarrassing pictures or statuses they wrote years ago. The discussion opened my eyes to the types of things I can be on the lookout for in my own personal virtual space.
Critique:
As a result of reading the article and participating in the discussion in my math education class, I was convinced that as a future teacher myself, I will be keenly aware of the posts, comments, and texts I send to everyone. Cyberspace is a vast, virtual space that carries a lot of my personal information. As a result, I will thoroughly analyze my privacy settings on all social media outlets to ensure the information available to the public is professional enough to where any future employers, students, and their parents would be able to find it and I would not be embarrassed or feel the need to explain myself. This will help me keep my job.
Connections:
When I am an educator, I will establish the kind of classroom environment that encourages respect for everyone in hopes that students will not approach me in a disrespectful manner. In high school, my math teacher and I established a very good relationship with the perfect balance between professional and personal conversation. As I approached her and asked her advice about my future education plans, she gave me great advice that I still hold dear to my heart. I have never been able to find her on facebook, and we only correspond if I reach out first. I hope to someday follow her example of the perfect balance she helped us both establish in our student-teacher relationship.
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