Most popular posts from one year of blogging

The Math Switch began one year ago, in July, 2013. In that time, I have enjoyed sharing ideas on inquiry-based learning and on educational technology. In the last few months, I have been able to post regularly, at 3 Tuesdays per month. Over that time, the most popular posts have been:

1. 9 Ways to Engage Reluctant Students, aka Tackling the Startup Problem 
2. Harnessing Your Personality 
3. Dealing with misconceptions, Part 1: Seven ways to handle misconceptions in the moment 
4. Engage! 
5. A Critical Examination of my Transition to Higher Mathematics course, inspired by Grant Wiggins 

Thank you to all who have stopped by to read the posts. A special thanks goes to those that have re-shared, or commented on the blog. 

An Introduction to 6 Apps for Quizzes and Polls

In this post, I discuss 6 apps and websites for quizzes and classroom polls. This is not a deep look, but I will tackle some critical basic features: the types of questions available, the kinds of resources that can be embedded in the questions, and what students or participants need in order to respond. All of these 6 apps are free, at least up to a certain usage level.

Readers may also wish to consult the comparison chart at http://www.polleverywhere.com/vs (the chart dates from November, 2012), and to look at some of the information provided by Richard Byrne at http://www.freetech4teachers.com/2013/03/four-good-alternatives-to-clicker.html and elsewhere on his site.

1. Edmodo: https://www.edmodo.com is a course management system, with quizzes and polls as embedded tools. Edmodo has multiple choice, true/false, short answer, fill in the blank, and matching quizzes, as well as multiple choice polls. Quizzes have a number of nice features, including the ability to embed links, video, images, and LaTeX (by enclosing the mathematics with [math]…[/math]). Polling is simpler, with just the multiple choice mode and no embedding. Students should have accounts and be set up in a class in order to use either polls or quizzes.
2. Socrative: http://socrative.com has multiple choice, true/false, and short answer formats for both polls and quizzes. Quizzes can have embedded images, but not video or links or LaTeX (unless you create an image with LaTeX in it). Quizzes can be run as a game called Space Race, where getting answers right moves a rocket across the screen in a race with other participants. In a quiz, students can get immediate feedback on whether their answer was correct if the quiz is set up with the correct choices marked. Alternatively, if the correct choices are not marked, students do not immediately know if they responded correctly. Polls (“Single Question Activities”) can be run instantly, with no need to pre-load questions. In a poll, the idea is to set everything up without Socrative, and just use Socrative to collect votes of A/B/C/D/E, where the instructor can designate what each response means. Student accounts are not needed. Just recently, accounts have been switched over to Socrative 2.0. Socrative 2.0 adds a feature, Exit Ticket, which is pre-formatted with three questions: a multiple choice question about how well the student feels he/she has learned the day’s lesson, and two short answer responses, one a request to describe what was learned, and the second to answer the teacher’s question (which allows the teacher to pose a specific question, i.e., outside the app, in addition to the general one). 
3. Google Forms are part of the suite of Google Drive tools. Forms support multiple choice, multiple correct, short answer, and fill in the blank. Forms can have embedded images, video, or links, but not LaTeX (unless you create an image with LaTeX in it, as I described earlier). Students do not get immediate feedback about the correctness of their answer choices. Auto-grading of the responses can be accomplished by installing the Flubaroo script in Sheets. Students do not need accounts. However, to get the maximum benefit from Flubaroo, it is a good idea to collect student emails in the Form.
4. Quiz Bean is web-based, and not an app. It has multiple choice, true/false, and multiple correct formats. Quiz Bean supports embedded images, but not video or links or LaTeX (unless you create an image with LaTeX in it). Students get immediate scoring feedback as they progress through the quiz. Students need accounts and accounts should be set up into a class by the instructor.
5. Quizlet is built more as a study tool. After setting up an account, users build virtual index cards and then practice quizzing themselves, matching the items in one of a few ways. The index cards can include images or text. 
6. gFlash+ is similar to Quizlet in that it is designed for building virtual index cards. The “g” indicates that the index cards can be created from Google Sheets. There is no need for a gFlash+ account, but this app works best if connected to a Google Drive account.

Besides the ones listed above, there are many, many more. An incomplete list of them includes:

1. Exit Ticket: http://exitticket.org
2. Kahoot: https://getkahoot.com
3. Mentimeter: https://www.mentimeter.com
4. ParticiPoll: http://www.participoll.com
5. Poll Everywhere: http://www.polleverywhere.com. 
6. TAPit: http://theanswerpad.com
7. Flisti: http://flisti.com
8. Infuse Learning: http://www.infuselearning.com
9. Quiz Socket: http://www.quizsocket.com
10. Geddit: http://letsgeddit.com
11. Top Hat: https://tophat.com

I hope this spurs some ideas. There are so many ways to collect feedback from students!

Doing math on iOS

In this post, I describe my experience using various apps to do mathematical computations. This is focused on the kind of mathematics that arises in K-14 classes, and not research-level work.

Here is the list of iOS apps I have tried for doing math of various sorts on my iPad:

• TI-Nspire CAS is the most valuable app for the iPad. Although it is pricey at $29.99, it is designed for extended exploration in a way that most other apps are not. This has been my go-to app in my work doing mathematical modeling (e.g., linear regression) with middle school teachers. Some of my favorite features include the ability to graph multiple functions or multiple regressions on the same graph and the ability to export files to Dropbox or elsewhere. The export feature allows me to input data to a spreadsheet and share it, thereby saving everyone else from entering data (and making typos).
• Wolfram Alpha is versatile, as long as one is interested in looking at one object at a time. By this I mean that one can easily graph any function or set of functions, plot a data set and perform regression, or do standard calculations, but it is not possible to store the results within the app. Instead, it is necessary to take screenshots or copy-paste information to another location (Evernote, for example). The app also makes it difficult to edit information because it is not possible to scroll through a long command line that has been entered. On the other hand, if given an equation, it can show the steps involved in solving the equation. The app can also serve as a search tool to answer questions or provide information. The app requires an active internet connection at all times.
• MyScript Calculator is a lot of fun for basic calculations. It transforms hand-written mathematics into typed math script and performs the calculations indicated. It should be noted that getting formatting correct is sometimes difficult, say if there is a rational expression with exponents in the denominator, but it works well for quick scratch calculations.
• Geogebra is a spectacular app for the desktop or laptop, but the iOS app has a long way to catch up. What is missing are the settings. For instance, I have never found a way to use a non-square scaling, such as I might need for an exponential function, where the outputs grow much faster than the inputs. Neither does there seem to be a way to adjust the labels (e.g., to show the label on a function), or to display a table of values. Unlike the Nspire or Wolfram, Geogebra does not render 3-dimensional graphs. Still, the app is free, and is good for a lot of Euclidean geometry and 2-dimensional graphing, and it offers sliders for dynamic exploration as well.

The following are apps that I have used, but not extensively:

• Geometry Pad uses the freemium model. I have used only the free version, which includes the ability to draw basic geometric objects. The premium version adds a lot of features, including the ability to do calculations, graph functions, and a lot more.
• Sketch2Graph takes a hand-drawn graph, converts it to a plot of a linear or quadratic function or conic section, and outputs the equation describing the plot. The function graph can then be manipulated by hand. This enables some nice exploration of these graphs and the relation between the graph and the equation.
• Algebra Tiles is designed for illustrating or manipulating algebra tiles in an app. The interface has three modes, basic, equations, and factors. This app works as a tool, and is not built to give practice problems nor does it show how to use the tiles. It does serve as a functional replacement for using actual tiles.

Readers, what have I missed?

Taking notes on iPad: Evernote, NotesPlus, and Notability

In this post, I briefly describe situations in which I find I need to take notes on my iPad, and the apps that I find most useful in these situations: Evernote, NotesPlus, and Notability.

There are two recurring situations in which I find I need to take notes with my iPad. The first is in meetings. In these situations, usually I am able to type notes as the meeting is happening. In this case, I use Evernote. Evernote is perfect for typing notes because I can open the app and start a note almost immediately. Occasionally, there may be a one-sheet handout as well. (Generally, if the handout is longer than one page, the presenter will share it via email.) I have scanned a number of handouts and the scans have been clear. In addition, items scanned into Evernote become searchable. Occasionally, someone hands out a business card, in which case I scan that in too. For certain recurring meetings, I have a particular notebook where I keep all my meeting notes, or I have a tag for the committee that I use to make sure I can find the note later.

The second situation in which I take notes is during class. In this case, I find I prefer to handwrite my notes, rather than typing, because I may need to write mathematics. Sometimes I am writing something that I want to share with the class, and other times I am making notes about what is happening during presentations or group work that I want to remember for later discussions or follow-up. This includes the possibility that I photograph student work and then annotate it. I use Notes Plus or Notability for class notes. I can recommend both apps. For NotesPlus and Notability:

• Both can be backed up to Dropbox. 
• Both offer an eraser as well as an undo button. 
• Both offer a close-up box for writing. 
• Both apps have always retained everything I’ve created. I have never experienced disappearing notebooks or pages.
• Both apps offer a variety of pen thicknesses and colors as well as a highlighter.
• Both apps offer the ability to add audio recordings to notes.
• Line segments are handled differently. In NotesPlus, drawing line segments is integrated into the note. For instance, if I want to draw a rectangle, I begin drawing it, and it appears where I put it. Usually, NotesPlus auto-detects the line segments and gives me control points to adjust the placement of the segment. In contrast, in Notability, when drawing segments, the app takes me out of the space where I am working, and then I have to insert the line drawing back onto the page. When I insert the drawing, there is white space around it, so it feels like inserting a picture into a document. 
• Typed notes are handled differently within each app. NotesPlus offers text boxes that can be inserted anywhere, whereas in Notability the options are to insert stickies or to move the cursor around on the page. 
• NotesPlus has a built-in web browser, in case one is looking to clip information from websites to insert into notes, a feature not present in Notability. 

Together, this set of tools has really helped me get the most productivity from my iPad.

9 Books to Read and Reread

In this post, I offer some suggested readings that I find help inform my approach to teaching. The books are listed in no particular order. 

For any teacher, I recommend:

• What Works in Schools? Robert J. Marzano, Debra J. Pickering, and Jane E. Pollock. (Note that there is now a second edition available with a substantially different organization. Either edition is valuable.) This book (first edition) discusses nine strategies shown by research to be effective in improving student learning outcomes. Sometimes the strategies are “obvious,” but it can still be helpful to be reminded that they are important teaching tools. For instance, summarizing and note-taking are effective. However, for me, many of my students have never been taught how to take notes, or have never discussed strategies for taking notes. So I make an effort to tell students when someone states an idea that I think everyone should write down, and I set aside some time for students to discuss what they should write down during class. Other strategies take a more concerted and planned effort to implement. For instance, generating and testing hypotheses is another strategy. While this is a natural part of doing mathematics, this reminds me to include tasks in which students do more investigative work. More than a list of nine ideas, the book has specific recommendations that are helpful. For instance, what are some important features to make cooperative learning successful? These are the kinds of specifics that are discussed in the book.
• Why Don’t Students Like School? Daniel Willingham. Willingham is a cognitive psychologist who poses some key questions and answers them from the perspective of his discipline. There are a few things that I like about this book, and that make me go back to it. One of the things I like is that each chapter closes with implications for the classroom. For example, one chapter discusses our human tendency to prefer and make sense of things as stories. In a course like precalculus, this might be used to frame “telling the story of a function,” where a function has properties like limits as x goes to infinity, asymptotes, periodic behavior (or not), symmetry, and so on. In calculus or analysis, the story idea might be put in terms of the central “conflict,” will a sequence converge or not, or another, is a function continuous or not. Rereading (or skimming) this book and thinking about the implications often inspires me to find ways to improve my day-to-day plans.
• What’s the Point of School? Guy Claxton. Claxton describes what he believes are the core goals of an education. These are big-picture concepts like developing people who are curious and are lifelong learners. While this is not a book that I return to for help in thinking through the details of teaching, I find that it helps to remind me of what is really important in my role as an educator.
• Switch. Chip and Dan Heath. This book inspired the name of my blog. The Heaths describe how to make a switch—a change—either in yourself or others. The single most important idea is that a lot of what we do is driven by emotion, and so we need to think in those terms when looking to effect change. The authors go through several ways of activating the emotions that will enable a switch to happen. I have returned to the book many times, for example, to remind me of how to approach students who are struggling, to help them find the emotion that will drive them to turn around their performance in my classes.
• Understanding By Design. Grant Wiggins and Jay McTighe. This is a book that puts forth a framework for thinking about curriculum design by starting with the end results, then thinking about how those results will be measured, and only then moving into designing the learning activities that will produce the desired end results. I return to this book from time to time to remind myself of how to frame my goals, and how to find ways to measure progress towards those goals.
• Mindset. Carol Dweck. Dweck has done significant research into the power of having a growth mindset, a mindset in which one believes that through hard work, one can get smarter or better. In the book, she describes some of this research and how it can make a difference across different domains of school and life. The book helps to remind me of why a growth mindset matters, and serves up examples that I use in explaining the power of the growth mindset to students.

For math teachers at any level, I recommend:

What’s Math Got To Do With It? Jo Boaler. Boaler has studied high school students experiencing problem-based curricula and compared them with those in traditional curricula in two different countries, the US and the UK. This book describes some of what was learned in those settings, and distills for a general audience—including parents of schoolchildren—some of the key ideas of what mathematics learning is, or should be, about. From the perspective of a math teacher, this book is less likely to offer ideas for day-to-day decisions, but like Claxton’s book, helps to remind me of the goals of teaching mathematics.

For college teachers, I recommend:

What the Best College Teachers Do. Ken Bain. Bain’s book centers how the select group of highly-respected teachers he studied approach teaching, from preparing for class, to setting expectations for students, to conducting class, and so forth. Each chapter holds a wealth of good advice, like seeking the commitment of the students: asking them to consider whether they are willing to do what it takes to succeed in the class, and therefore have them commit to the effort required. I find I sometimes return to the questions he poses in the chapters as a way of gaining a fresh perspective on my courses.

Finally, for college math teachers, I recommend:

The Moore Method: A Pathway To Learner-Centered Instruction. Charles A. Coppin, W. Ted Mahavier, E. Lee May, and G. Edgar Parker. The four authors of this text each describe how to implement the Moore method, as they see it. The book offers the reader a chance to consider various aspects of teaching in a learner-centered environment, and benefits from the approach of the authors, which is essentially to offer their individual responses to the key questions in setting up and operating a Moore Method course. This variations-on-a-theme approach has the effect of providing the reader with a canvas and a palette, rather than promoting a specific paint-by-number prescription. The authors take on a wide variety of issues associated with implementing the Moore method, including such topics as, What if no one has anything to present? How do I grade? and many others. I have returned to the book many times to seek out new ideas of how to handle syllabus construction, or to remind myself of ways to approach managing an IBL classroom. (In full disclosure, I should mention that I am personally acquainted with the authors, and have worked closely with Ed Parker.)

What are some of your favorite or most inspiring reads from the educational realm?

Typesetting Mathematics in Google Drive

In a previous post, I mentioned that in the native Google formats on Google Drive, it is difficult to typeset mathematics. Here I will describe how to work around this issue. 

Options for creating images with mathematical symbols are:

1. In general, the first ways that come to mind to typeset mathematics are via LaTeX, MathJAX, by using Equation Editor for Word, or by using Math Type. However, in order to adapt any of these to a Google Doc or Form, an image (JPEG or PNG) is needed. Since I prefer LaTeX, I use a LaTeX to image compiler to create an image that can be inserted into a Form or Doc. I have been using SciWeavers. The site is not perfect, as it can be difficult to get justification and alignment right, but it does the job. You can copy-paste the image or copy the URL and let the Google Form retrieve the image that way. For a Form, it makes the most sense to typeset your entire question and copy it.
2. Typeset mathematics in your favorite way. Then take a screenshot and crop it so that it has only the math that you want. Finally, copy-paste the image into the Google Doc, Form, or whatever you are working on.
3. Create the entire document outside of Google Drive, but load it into Drive for the purposes of sharing or whatever other reason you want the item in your cloud account.
4. g(Math) is a Docs add-on that renders math formulas and graphs in Docs. I have not tried it, but this at least solves the problem for Docs (but not Forms). 

These are either incomplete or inelegant solutions. I am hoping that by posting, either someone will suggest a better alternative, or that Google will address the issue.

By the way, for those of you using Forms with your math classes, I am working on a list of some alternatives to Forms that are out there—and there are a LOT of options. This will be the subject of a future post.

After I composed this post, I wanted to make sure everything was up-to-date, and I found this thread https://productforums.google.com/forum/#!topic/docs/mMQl4IkKG2c that includes the same suggestions for Forms. 

Joys of Teaching

In this post, I explore the question: What are the joys of teaching?

Teaching is something I have been doing for a long time, going all the way back to when I was a student and a tutor. Early in college, I decided that I wanted to teach at the college level. (Immediately before that I had planned on a career in engineering.) If I had to put my finger on what made me want to teach then, I think it was that, as a tutor, I had seen the joy of connecting people to mathematics, a discipline that is associated with a lot of negative experiences for many students. But, as with so many things in life, as we grow older and gain more experience, what we appreciate changes. Nonetheless, seeing students light up when they grasp an idea remains a particular joy for me.

I enjoy teaching because I enjoy the challenge. Each semester, I find myself faced with challenges: I am challenged to react to students’ misconceptions that I have not seen before, or I am challenged to get students to invest their best effort, or I am challenged to find new ways to keep students engaged in class. Every year, I reflect back on what has happened in my classes, I review the evidence of student learning, and I think over the learning experiences that students had. I always see a need to do better. I always rethink my course problem sets and grading scheme, and I look for ways to remake them in ways that will encourage students to learn more from the course.

I enjoy teaching because I enjoy connecting with students. More than just teaching content, teaching is a coaching and mentoring relationship. I have been at my school long enough that I have had a number of students at multiple points in their careers, sometimes across lower and upper division courses, or in their undergraduate major and in master’s courses or teacher professional development institutes. Sometimes, students complete a course with me and continue to come back for advice or assistance. I enjoy seeing the students grow and gain new perspectives on what they have learned, or seeing them begin to transition from thinking about the classroom from the student’s view to thinking of themselves as teachers.

I enjoy seeing the impact of a positive learning experience for students in a way that I cannot see through the other things that I do. Being in front of, in the middle of, and generally in the presence of students gives me opportunities to impact students in ways that are not visible when I am in the role of researcher, or serving the university.

This is a time of year for reflection, and for planning. I encourage my fellow instructors to take a moment to enjoy the fact that teaching is an awesome responsibility, and a great privilege. Enjoy teaching in spite of, and perhaps because of, the challenges! 

Feel free to share your favorite joys of teaching in the comments.