Fall 2017 - Professor Kate Poirier

Author: Kate Poirier (Page 1 of 4)

Final exam topics

  1. Desmos Activity Builder – You will create a short activity for students. I’ll include some instructions below.
  2. Geogebra – This component will be similar to the Geogebra component on your midterm.
  3. Graphing calculators – I’ll ask you one of the 6 questions you considered for Project #3. Don’t forget your calculator!
  4. Research article – You will write a short essay based on your classmates’ presentations for Project #4.

Desmos Activity Builder:

  1. Head to teacher.desmos.com
  2. Sign in using your usual Desmos account.
  3. From the left-hand side of the screen, select Custom.
  4. From the upper-right-hand side of the screen, select New Activity.
  5. Give your activity a name.
  6. Now you can start building your activity! Start adding different elements to your first screen.
  7. To add another screen, select Add New Screen by clicking on the green plus sign from the left-hand side of the screen.
  8. To see what the students would see, select Student Preview from the upper-right.
  9. Your work is saved automatically but to share your work you must create a class code. One way to do this is to go back to the main screen, select Custom again, and then select your activity. From that screen, select Create Class Code.


Last year’s research article discussion

Like this year, last year’s research article projects ultimately led to the questions on the corresponding component of the final exam.  As part of your presentations on December 14th, we’ll have a discussion about common themes in your articles that can find their way into final exam questions. For reference, I’ll include my own post-discussion thoughts from last year here.

I really enjoyed everyone’s presentations today! I wish we had more time to discuss each of the research articles in detail; they’re all fascinating!

I have not yet nailed down exactly what questions you’ll see on the research article component of your final exam yet, but I wanted to capture some themes and issues that we discussed in class today and that appear in your OpenLab posts. Feel free to add your own ideas/questions/comments in the comments on this post.

These are the themes that I will have in mind when putting together this component of the final exam. My goal is to develop questions about any of these themes, phrased so that each of you will be able to answer using supporting evidence from these articles. Ideally, you could use evidence from the article that you read in detail yourself, but evidence from another group’s article might be more appropriate. Be sure to read each group’s OpenLab post to remind yourself of their summary of the article, their question and answer about the important detail, and their discussion questions.

  • There are barriers to implementing technology in the classroom. These barriers are both personal (for example, teachers and students may be apprehensive about using unfamiliar tools) and logistical (for example, not all classrooms have computers for every student; not all students have internet access at home).
  • There are components of the role of technology in math learning and teaching. I’ll spell out the four that appeared in Jodel and Josiel’s article here, though we discussed how these appeared in other articles (and in our own experiences) too:
    • exploring
    • conjecturing
    • verifying
    • generalizing
  • There are levels of technology use from the educator perspective that vary from non-use (before learning how to use the tool) and renewal (after having become comfortable using the tool and reexamining how to use it).
  • Students struggle to connect the calculations they do in their math classes with real-world applications. We saw in Tyniqua and Armando’s presentation as well as in Mei and Majid’s that students struggle with proportion/ratios/fractions. Technology can be used in intervention activities that allow students to make this connection through exploration and discovery by beginning with something students are familiar with.
  • Technology can be used as a pedagogical tool in different ways. Jodel and Josiel’s article showed us that Korean textbook questions for junior high school students used technology to foster conceptual understanding while questions for senior high school students used technology mainly in a technical/mechanical/procedural way. From Evelin and Sonam’s article, we learned that more successful students gravitated to using a graphing tool when it was appropriate; we noted that these student already have a conceptual understanding of the problem and are using the tool to do the mechanical work for them. The less successful students didn’t necessarily see how the tool could be used to answer the question. On the other hand, from Tyniqua and Armando’s article, we learned that Google Maps can be used to exploit students familiarity with technology in their lives to help them grasp an abstract mathematical concept.
  • For technology used as a mechanical/procedural tool, it’s important that students have to think in order to solve a problem. There must be a balance between thinking/developing technical or procedural skills and relying on technology.
  • Students’ perception can be exploited using technology to help them recognize patterns and learn more quickly. From Luis and Gary’s presentation, we saw that this use of technology can be more effective than traditional pencil-and-paper work. This helps develop fluency in mathematical language, ideas, and content.
  • Technology can be used to assess students’ strengths and weaknesses  and to deliver instruction that is tailored to individual students’ needs.

Last year’s research article reports

Last year’s version of the research article assignment wasn’t exactly the same as this year’s, but it was similar. I’ll copy the OpenLab posts from last year so you can get a sense of how others interpreted this assignment.

Source: EBSCO Database from Citytech Library
Published: October 1st, 2014
Title: Key factors for Successful Integration of Technology into the Classroom: Textbooks and Teachers
Author(s): Hee-Chan Lew, Seo-Young Jeong

The article is about the use of technology in a classroom; however, the author accentuates the most in Korean secondary school mathematics. From the article the author states that Korean Mathematics teachers and Korean Mathematics textbooks are the two primary reasons why it is difficult to implement technology in Korean Mathematics education. Furthermore, she talks about the four main components of the role of technology in math learning and teaching as well as noting that the role of technology in mathematics education requires careful distinctions between two different types of mathematical activities.

In fact the author defines the levels of technology-use in mathematics classroom within eight types of level of use in sequence. In addition, the author shows that technology mainly plays a technical role in activities of Korean senior secondary mathematics textbooks. There are 193 exercises with technology in Korean senior secondary Mathematics textbooks. However, there are 124 activities with technology in junior secondary textbooks.

Does Mathematics education benefit without technology benefit the teachers?

Title: Map, Scale, proportion, and Google Earth

Author(s): Martin C. Roberge and Linda L. Cooper

Published: April 2010


This article is based on the concept of using sources such as Google maps and Google Earth in order to teach students proportions in terms of large and small-scale scenarios. The tools are used in a pedagogical sense, by enabling students to use what they already know about places, ratios, and measurements and apply it to real life situations where they make connections with geography and proportional reasoning calling for a higher level of thinking. So, not only are the students learning how to read a map, use its key to measure the distance between places or objects, but they are able to convert the maps measurements to real ground measurements through proportioning to get approximates of distances with a very small margin of error. This activity also forces students to look past the ideas of proportionality typically taught in the classroom by making them use their reasoning skills to come up with the basic format taught in the class and other ways that also lead to correct answers.


Question: How do can we incorporate a tool such as Google Earth into our lesson plan without it being overwhelming to our students?

In order to incorporate a tool such as Google Earth into our lesson plan we can guide our students to recognize the different types of images on the computer from different perspectives and then zoom in and out and encouraging the students to think about what words can they use in order to describe what is happening as you manipulate the picture of the object in relation to size and distance. Once the students realize the action that you are doing you can relate this to “zooming in and out,” on a phone or computer (everyday activities), and then introduce maps of different scale factors starting out with things that are familiar—such as, their neighborhood, the area surrounding their school and once they grasp the concept of proportions based on a large scale, then we can broaden the area of the map based on an entire city, then region and so forth using a smaller scale this way the students will not be overwhelmed with converting the measurements on the map with real life measurements. This really forces students to critically think and analyze each situation and also forces them to create and answer many questions on any type of picture.


Discussion questions;

  • What is the relevance of using a source like Google Earth in a mathematics classroom?
  • How can we adapt the activity for a younger audience (6th graders)?
  • How can we adapt the activity for an older audience (high school students)?
  • What are some ways to access students learning using Google earth as a pedagogical tool?
  • What other concepts can we have students learn by using Google earth?
  • What other ties does this topic have to other fields in STEM?

Title: Slower Algebra Students Meet Faster Tools: Solving Algebra Word Problems With Graphing Software

Author(s): Michal Yerushalmy

Journal Name: Journal for Research in Mathematics Education

Published: 2006

In this journal article, they conducted a study on middle school students to see how each pair used the graphing tool to solve word problems.  In this study, they conducted three groups, three pairs of successful students (U25), three pairs of average students and three pairs of low successful students (L25). Each group was given the same word problems to complete. Both groups had access to the computer and graphing tool.  The graphing tool is a software with capabilities similar to a basic graphic calculator that presents two-dimensional graphs and numerical values for any single variable expression. As each pair completed the word problem, there were interviews being done as they worked. The interviewer was not allowed to give the answers only to assess their thinking process. The main goal of the interviews was to observe the processes by which the L25 constructed mathematical meaning while solving traditional word problems with the function graphing software. From this study, they noticed how each pair had a different approach when it came to using the graphing tool. Some students wanted to use the tool to get an answer, others use the tool to check if their answer was correct and others students just wanted a visual representation of the function. Overall, the tool was part of the students’ reasoning and argumentation and was used to reflect on conjectures.

Question: How can we guide students to use graphing tools properly to solve algebra problems?

Graphing tools were constantly used in classrooms, and students were encouraged to use them. The techniques involved in using the graphing tool were practiced routinely in the eighth and ninth grades, including values, reading linked representations of function, and reading the values at intersection of two function graphs. Graphing tools are very helpful when students are dealing with functions. But if a student does not use them properly or over rely on these graphing tools, it may cause them just use the graphing tools to get the answers of the problems instead of trying to reason and understand the concept behind the problem. So, in order to help student to use the graphing tools properly, we need to set rules that they can only use them when they complete difficult operations and to confirm conjectures. Also make the assessment “uncheatable,” students need to be able to understand the concept, validate it and apply it. That is goal of solving a problem.

Discussion questions:

  1. Should middle school students be allowed to use graphing tools in class?
  2. What kind of graphing tool are helpful for middle school students?
  3. What are the benefits for students to use graphing tool in the class?
  4. How do you know if your students are over relying on graphing tools?
  5. How does a graphing tool help a teacher teach her/his students understand algebra?
  6. Should you allow student use graphing tool during the test?

Title: Conditions for Effective Use of Interactive On-line Learning Objectives: The case of a fraction computer-based learning sequence

Authors: Catherince D. Bruce & John Ross

Journal: The Electronic Journal of Mathematics and Technology

Year of publication: 2006

  • This paper focuses on the challenges of students’ understanding about fractions from students’ perspectives and teachers’ perspective. Not using factions daily is one of the factors that makes it difficult to embed the significance of learning fractions as part of students’ life. The success of supporting students’ understanding lies on the design of instructions. The traditional teaching methods lack the emphasis at students’ conceptual understanding with little connections to students’ existing knowledge. However, technology-assisted learning is introduced as a successful model in enhancing students’ understanding with challenging math concepts.

The paper takes a main point on a computer-based learning package named CLIPS-Critical Learning Instructional Paths Supports. The package consists of its own characteristics and learning tasks for students. Even though students make meaningful progress in understanding of fractions under CLIPS, there are limitations and exceptions that students would not benefit from the program. Through case studies, the paper concludes the importance of building the direct relationships between online learning tasks and in-class learning tasks. The necessity of having in-class activities that are within students’ zone of proximal development. The full participation or involvement in the CLIPS will make a difference, and the pair work between students will support each other in completing the CLIPS tasks. Last but not the least, since the CLIPS program is computer-based learning, students can keep their own pace and go back for checking their work. The educators believe that students go with the sequence order to understand the content better than those who were absent and chose the tasks randomly.

  • Why do you think learning fractions is challenging for middle grades students in your own opinion?

First of all, there are different ways to represent fractions: division sign, colon, and fraction bar. Fractions are divided into proper fraction, improper fraction, and mixed fractions. They will have questions involving mixed fractions, but what they need to do first it to convert them to improper fractions to make computation easier. If a teacher cannot make his or her students understand the meaning of proportionality, it is going to be extremely hard for students to complete a task associated with fractions or understanding the significance behind ratio. From the reading, I learned that a computer-based learning might be a possible way to assist students to have a better understanding of something that was not clear to them through vivid images and audio. At the same time, there are challenges to implement technology in a classroom. The learning objectives from the sites should be correlated to the lesson itself. Schools need financial support to provide students’ access to computers. There are also technical issues along with computers that might happen in the classroom, which will make it unsuccessful for students to keep a consistent attention during the tasks. In conclusion, I agree that students need some technology in their learning if students can use it wisely with their goals of learning in mind.

When students are learning fractions, they will be able to understand what a ratio is. How to complete a ratio table is considered one of the basic and important tasks for students when they learn fractions from my observation experience. Thus, there are a lot of definitions that students need to know in order to understand fractions.

Without access to the reading, I learned that students struggle to factions because they are familiar with whole numbers. They are good at simple operations with these numbers, but students will have difficulty with whole numbers with different signs. They are likely to make conceptual errors when they subtract negative whole numbers. It is going to be a higher level when students learn fractions.

  • What are the strategies that you think can help students build a good habit of using internet?

What are possible ways that we can negotiate with students’ parents’ involvement with students’ online assignments at home? (like sit there with the students for half-hour)

Title: Perceptual Learning Modules in Mathematics: Enhancing Students’ Pattern Recognition, Structure Extraction, and Fluency


The article that we choose is about perceptual learning software where the technology can help produce rapid and enduring advances in learning. In the article it talks about the positives about using perceptual learning software in the classroom with the different experiments conducted with students. In the first experiment 68 students (9th and 10th grade) participated in the study in which the students were broken up into 2 groups, a PLM and a control group. The PLM group used the computers while the control group didn’t use the computers. When the PLM group used the computers they were able to see in recognizing algebraic functions and to transfer of Perceptual Learning improvements in information extraction to algebra problem solving to help the students be able to see and know what the question were saying. The result of the experiment showed that the students that used the perceptual learning modules did better than the control group that were using paper and pencil. Then in the other experiments conducted using students from different grades and different topics. The results were the same as the first experiment that was conducted. Then the authors did another experiment comparing the students that used paper and pencil to the students using the perceptual learning module software. In the third experiment conducted the students had a ball on top of the ruler and a billiard cue poised to strike it. The students had to find the distance traveled and the endpoint. The perceptual learning modules software was a way to help the students to find the cues to help students be able to understand the problems easier. In conclusion using the perceptual learning module software can be a positive thing for students but what about college students.


Would the perceptual learning module software work well for college students taking high level mathematics courses?

When it is appropriate to use technology as a tool-what determines the need?

Project #4: Research Article – due Thursday, December 14

For project #4, you will report on an academic journal about technology in math education.


  1. Choose an article from one of the journals listed below. The article should be around 10-20 pages long and should have been published between 2007 and 2017.
  2. Your article must be approved by me. Comment on this post with your choice; include the title and author(s) of the article, the journal name, and year of publication. Each pair must choose a different article, so make sure to check others’ posts before you claim yours. Post your claim by midnight on Tuesday, December 5.
  3. Submit an OpenLab post with the following:
    1. The title and author(s) of the article, the journal name, and year of publication.
    2. A 1- or 2-paragraph summary of the article.
    3. Details about one important point made by the article. Write this as a question with a short essay response. (The reason for writing it as a question and response is that these questions will serve as inspiration for one of your final exam questions.) Make your question and essay response as clear as possible as it will serve as a study guide for your peers.
    4. One discussion question about the important point from the item 3 above (or more discussion questions, if you like).
    5. Add the category “Project #4: Research Article” to your post.
  4. Prepare a 15-20 minute presentation based on your OpenLab post and prepare to lead a short discussion with the class about the important point you chose to report on above.

Due date: Thursday, December 14


  • Journal for Research in Mathematics Education

  • Educational Studies in Mathematics

  • Mathematics Teacher

  • Mathematics Teaching in Middle School

  • For the Learning of Mathematics

  • Research in Mathematics Education

  • Mathematics Education Research Journal

  • The Australian Mathematics Teacher

  • College Mathematics Journal

  • Journal of Mathematics Education at Teachers College

Journal access

These journals may be accessed through the CityTech library. You can view them online from anywhere by following the directions here.

Graphing Calculators Project #3 due Monday, November 28

One of the most familiar technological tools in the classroom is the last we’ll discuss in this class: the graphing calculator. You are probably already aware of at least the basic functions of whatever calculator you have used in your own classes, but you might not have thought about the calculator as a pedagogical tool.

I’m of two minds about graphing calculators. On one hand, they are surprisingly powerful machines and, when used the right way, they can help a student understand a concept or an example without being distracted by rote computation. On the other, they’re clunky and old fashioned; we have much more powerful and user-friendly tools available now (for example, the software we’ve discussed in this class).

In addition to the benefits of using the calculator as a pedagogical tool (see the graphing calculators homework), you should become familiar with the pitfalls as well. There is a nice chapter on Lies My Calculator and Computer Told Me from Stewart’s Calculus book. The examples listed in it aren’t the most relevant for us (many of them deal with rounding errors) but the chapter contains a nice quote:

Computers and calculators are not replacements for mathematical thought. They are just replacements for some kinds of mathematical labor, either numerical or symbolic. There are, and always will be, mathematical problems that can’t be solved by a calculator or computer, regardless of its size and speed. A calculator or computer does stretch the human capacity for handling numbers and symbols, but there is still considerable scope and necessity for “thinking before doing.”

For project #3, choose one of the following exercises. (Comment on this post to indicate which exercise you have selected—do not choose an exercise that has already been selected by someone else.) Prepare a short lesson introducing the topic and explaining the issue the calculator encounters in this case. Your presentation must include the use of the virtual TI calculator (available on our classrooms’ desktop). You may also include hand-drawn graphs and/or Desmos graphs if they are relevant for your topic. In addition to your presentation, include a post on your ePortfolio outlining your presentation. Include screenshots or photographs of the graphing calculator.

  1. Imagine you are trying to help your students understand \lim_{n \to \infty} (1+ \frac{1}{n})^n. Try substituting larger and larger numbers for n in your calculator. What do you expect to see? What do you notice?
  2. Graph the function f(x)=\sqrt{4-\ln(x)} on your calculator. What behavior do you expect near the y-axis? Do you see it on the calculator’s graph? Compare the graph your calculator gives you with the graph Desmos gives you.
  3. Graph the functions f(x) = \sin(10x), g(x)=\sin(100x), h(x)=\sin(1000x) on your calculator. Do you see what you expect to see? Do you notice anything weird? What happens if you graph the same functions on Desmos?
  4. Graph the function f(x) = \sin(\ln(x)) on your calculator in the window [0,1] for x and [-1,1] for y. How many roots does it look like there are in [0,1]? Change the window to [0,0.1] for x and then to [0,0.01] for x with the same y-values. What has happened to the roots? Try graphing the same function in Desmos.
  5. In the standard window on your calculator, graph the piecewise defined function f(x)= 3x - 2 if x < 1.5 and x^2 if x \geq 1.5. Ask the calculator to tell you the derivative at x=1.5. Is this what you were expecting? Try graphing the function on Desmos.
  6. Use the equation solver on your calculator to solve \frac{\sin(x)}{x} = \frac{1}{x}. How many solutions do you expect?

Due date: Tuesday, November 28

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