Express Activity 4: Graphing and Documenting Math on the Monarch

Expanded Core Curriculum Mastering the Monarch
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Now that you have graphed an equation, inserted it into a document, and previewed it in print mode, reflect on the experience and share your insights.

Discussion Prompt:

What equation(s) did you graph? Describe your experience exploring zooming, panning, and navigating graphs.

How did the exported graph appear in the Word Processor?

What challenges did you encounter while transferring and viewing the graph?

How do you see the Monarch’s graphing tools benefiting students or clients in CIE settings?

If you’ve used other accessible graphing tools, how does the Monarch compare?

Optional Extension:

If you have experience with embossed tactile graphs, how does digital graphing on the Monarch compare? What are the advantages and challenges of each?

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Experience

In general, I am very impressed by the graphing calculator on the Monarch. I wish it had existed when I studied analytic geometry and calculus at secondary school.

On the Monarch, I created graphs of various equations familiar from secondary school mathematics, such as $x^2+y^2=1$, $y=e^x$, $y = \sin x$, and $y = \frac{1}{x}$. (The equations are here given in LaTeX notation.)

I didn’t need to use panning or zooming operations, since the graphs fit the display well enough to be readable without manipulation. More complex graphs might need panning and zooming to be performed during exploration, of course.

I also succeeded in exporting an equation from KeyMath to the word processor, which worked as documented.

Implications for Students and Professionals

So far as I know, the graphing calculator should be sufficient to support secondary school mathematics, except possibly multivariable calculus, which was available at the secondary school level when I was a student (the Australian school curriculum could take students further in mathematics than seems typical in the U.S.). If I am reading the User Guide correctly, it suggests that KeyMath does not support three-dimensional graphs, for instance.

I haven’t studied mathematics at the university level. However, my general understanding is that larger and more fully-featured software packages are used, at least in applied courses, including symbolic algebra systems and statistics tools. For example, [SageMath[(SageMath - Tour) is an open-source example of a symbolic algebra system. The R statistics package is an open-source option for statistics that has been extended to support accessibility features for students and professionals who are blind. Both of these tools include graph plotting capabilities.

Thus, when students need more complex mathematics software than the graphing calculator available on the Monarch, the question becomes how well these tools can be supported by the Monarch’s tactile viewer and any other relevant features. In addition, it isn’t clear what the limitations are of the KeyWord word processor in supporting mathematical notation - in particular, whether it can handle notations used in undergraduate and graduate-level mathematics. If not, the student would need alternative means of entering and editing mathematical expressions that support the full range of undergraduate and graduate-level material.

I would expect a professional with an undergraduate degree or higher in mathematics or a science discipline to require specialized software in the workplace, although there may arise situations in which a graphing calculator such as that found on the Monarch would be useful. However, this is a speculative claim and I think it’s an open question for those participants in Monarch RISE with mathematics and science degrees what software they would expect to use with the Monarch in workplace environments.

Comparisons

I think the graphing calculator as implemented for the Monarch is a major advance over what has previously been available. Sonification has been used, of course, as have embossed graphics, but what the Monarch offers are instantly available and very readable graphs of mathematical equations. This helps to give students who are blind a similar opportunity to their sighted peers in the mathematics classroom, for whom graphing calculators have long been readily available.

Although embossed graphs of equations offer higher resolution and hence more precise images, it doesn’t seem to me that the graphs displayed by the Monarch would be of substantially less educational value, especially once students become accustomed to working with them. The graph of a circle displayed on the Monarch, for example, is clearly an approximation, but it’s still easily recognizable. It is probably best for students to be familiar with both embossed graphs and dynamically displayed graphs (as on the Monarch). However, I think the Monarch’s advantages in generating graphs instantly and in providing full access to mathematics and science textbooks are considerable, and these benefits also need to be carried into workplace settings.

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I graphed two linear equations—y = 2x + 3 and 3x - 4y = 1—and explored how they intersect. Navigating the graph was intuitive. I used the zoom buttons to get a closer look at specific parts of the graph, and the D-pad let me pan around easily. Using the point-and-click feature to find the intersection was particularly impressive—it was responsive and gave immediate audio feedback. After copying the graph to the clipboard, I pasted it into a Word Processor document. When you insert a math equation into a document, the Monarch places it between special markers: EDMS (start of math) and EDME (end of math). These markers clearly show where the math section begins and ends. Then, using Enter with V to preview the document allowed me to see exactly how the equation and graph would appear both on the tactile display and visually—giving me confidence that the formatting was correct before sharing or printing.

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Thanks, Jason — your post was a thorough and thoughtful overview of the Monarch’s graphing potential, and it immediately made me think of rhythm as one of the clearest intersections between mathematics and music.

In my work teaching music to blind students, rhythm is often one of the most abstract concepts to convey tactually, especially when it comes to proportional relationships between note values, time signatures, or tempo modulation. I see strong potential in the Monarch’s graphing features to represent these relationships in a tactile format.

Like you described with plotting equations such as sin(x), we could graph rhythms over time: for instance, showing how four sixteenth notes span the same duration as one quarter note, or how a triplet divides the beat differently than straight eighths. Even just plotting beats along an x-axis could help illustrate tempo changes or rhythmic syncopation. What I appreciate from your post is the clarity with which the Monarch renders these graphs — that tactile immediacy is crucial for reinforcing rhythmic ratios physically rather than abstractly.

I also share your interest in how the Monarch could be extended to support more advanced representations. While current tools are promising, the ability to graph polyrhythms or nonstandard subdivisions could further bridge musical and mathematical learning for students in both academic and CIE settings.

Thanks again for such a rich and forward-looking post.

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I graphed the equations in the Guided Notes and Activities file. I found that navigating the graph was quick and easy to do. I also found that when I pointed and clicked on a spot it gave me the coordinate in braille. I was super excited to see how the graph exported into the Word Processor. Once, I won my battle with getting it to save to a USB drive, I found the process incredibly easy. I see this tool benefitting students with allowing them to share their graphs with their teachers more independently and accessing the same graphs their peers are in real time. These can easily be added prior to the lesson if previewing is needed and then toggled off and on (which was a cool feature).

I’ve used Desmos on the computer in the past in braille mode, however, I found that it was not very easy for the students to use because it was only describing it.

Optional Extension:

I have made embossed graphs, as well as, used graphing boards and other materials to graph for students. I think the Monarch’s digital graphing will be that it allows the student to see the shape and the coordinates easily without the risk of the braille getting ruined and in real time with their peers. They don’t need to have multiple graphs on paper. However, I can see the challenge being with shapes (for Geometry).

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I found this section to be the most difficult part of the tech training by far. I followed the module and things appeared to work correctly but definitely over my head. I do not have practical braille math skills as I learned braille as an adult. That being said, I can certainly see what an invaluable teaching and learning tool this will be to the benefit of blind students in the stem areas. THe panning and zooming features provide great perspectives both at the detailed level as well as the “big picture’“ view. I was not able to use the visual preview feature.

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Hi Matthew!

I can relate to what you said about the panning and zooming — I also found those features really useful for moving between the fine details and the overall shape of the graph. That shift in perspective makes it much easier to connect the “big picture” with the smaller elements. I didn’t use the visual preview feature either, but I agree it could be a great way for instructors or colleagues to quickly confirm a layout before producing a hard copy. Even without advanced braille math skills, it’s easy to see how this tool could make complex spatial information much more accessible.

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Hi Matthew,

Did you make another attempt to use the visual preview feature? I appreciate you persevered through this activity and see the value Monarch has for improving access to STEM. Alicia Wolfe

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I think the graphing calculator feature is very neat and I’m excited about the potential of what is possible. I graph the equation Y equals X+3. Viewing the graph and panning around it was very easy. I like that I can easily adjust the size of the graph using the zoom keys.

Exporting the graph to a KeyWord file was not as intuitive as I was expecting. I was expecting to use the copy and paste functions and had to refer back to the user manual in order to figure out how to do it properly. I like that when the graph has been pasted into the word document, the equation is accompanied by the letter I to indicate that there is an image there. When I tried to view the graph in print preview mode, it just said the word “graph.”

Overall, I think this graphing capability has a lot of potential. I think the advantages to using this rather than graphing on paper are that you can adjust the size of the graph more easily. You can also re-label axes and you can create graphs for your instructor that are printed. I do think this would also help make courses such as calculus accessible if tactile graphics are not available, because you can type the equation into the braille display and have the graph created without having to try to figure it out yourself using paper.

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Graphing First Equation

Entering Equation Text

Ihad only ever eperienced using embossed tactile graphing paper when I had to graph equations back in high school. My math skills are not the strongest. I did quickly find the setting to turn on the grid, though. This brought back a lot of what I remembered.

I was surprised to find no tick marks after I graphed my first equation (I tried y = 3x-2 for the first one).) It was difficult to find the braille curson when editing the equation.

Graph Feedback

The graphs should an option for turning on tick marks and/or braille indicators for various features like intersection points. Without any kind of specifics about where you are on the axes, you only get a sense of a plot’s generic shape. Tick marks along the top and left that are persistent might be useful.

Feedback From Sighted Colleagues

There are a few other people in my office who know braille and have produced tactile graphics. They would be in a position to help a student if anything were to happen to me, since they work with me in the office. I talked about Brian in my introductory post. Destin and Chris are the other two people. I showed them about graphic and they have asked me to pass along some feedback.

When you hook the HDMI cord to a monitor, some of the things sighted people see are different than what is in braille.

  • When you turn on the Monarch and the monarch butterfly logo appears in braille, the screen visually says Android.
  • Desmos calculator says desmos on the screen but not on the monarch. this happens when you open Math - KeyMath.
  • Desmos Calculator shows visually with square pixels which they feel gives an illusion of inaccuracy. Visually reading braille where the dots have been turned into squares is extremely difficult. Circular pixels would be preferred.

this is my first time trying to use lists and headings in a discussion. Please excuse any errors, because I’m still learning.

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I must admit it’s been a while since I did complicated math or even looked at graphs. But I was very impressed with the graphing calculator. It did take me a while to figure out how to enter math equations, in the equations list. I do know nemeth and remember it but somehow it was not very intuitive with entering equations. Once I was able to get an equation entered, I was able to get the graph to show. I do wish there were less keystrokes to get graphs to show and the context sensitive help was only so helpful. I had to keep flipping back and forth to find the keystrokes I needed. I graphed several equations, including complicated ones with sin and cosin. I really liked the ability to scroll through the graph to find how things changed from different points on the axis. I like how you can point and click somewhere on the graph to get the coordinates. I started with simple graphs and worked my way up to more complicated graphs. Viewing two equations on the same set of axes was helpful in comparing equations. I think this would have been helpful when I was learning complicated math in high school. I did have braille math books but no access to a talking or tactile graphing calculator. In my specific job, graphs are not used very much so I am not sure how I’d integrate this in to a CIE setting, but perhaps others can comment on that part of it. I did try to copy adn paste in to a keyword document and that worked fine, but I don’t see myself using this particular feature very much.

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This is my favorite app so far. I had a wonderful time graphing the linear equations from the slope intercept formulas. I think the students will love being able to touch their graphs but also make changes digitally. It will be much easier than ripping off Wiki Sticks from tactile graph paper. I had trouble using the point and click feature of the cursor for a single coordinate point, but it got better with practice. I think some students will encounter this issue as they first begin to use the Monarch but practice will be the key to improvement.

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I think it would be useful to have at least a mode in which the coordinates of intercepts and maxima/minima are labeled in braille. When I was a secondary school student, including braille labels adjacent to these points was essential to creating good graphs of mathematical functions.

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This portion of the training was a little bit over my head. I was able to folloe the lesson, but I will never use the graphing functions of the Monarch.

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Back in June, I actually attended a training on the Monarch that my state was holding. This was the first time I got to play around with both the keycalc application, as well as moving my results into the word processor. I had a lot of fun with it I think the equation I messed around with at the time was something like GE 2X+Y = 7 or something like that. Once it brought the graphic up onto my display, I spent time moving my finger to various coordinates on the screen and pointing and clicking on them. It felt good honestly to be on par with my sighted piers in being able to do something like this.

I was not really impressed with how the results were moved into the word processor though. With my equation at least, the equation itself was just fine, but I feel like the graph was better inside of keycalc than it was in the word processor. I do not know the best way to work on that currently, but I would love to play around with it more to see what I can do to make myself more comfortable with it.

Math is probably one of the most winning features on this device, both for my school-age students, as well as my working students. While we can do a lot with tactile graphics with this device, being able to play around with graphing, and advanced mathematical equations is going to be huge for my students, especially the ones who are in more advanced math classes.

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I was able to successfully graph y=2x+3 and copy the graph into a Word document. I really like how the Monarch can integrate text, math expressions, and graphics into a single document. I admit I haven’t formally studied math for about 20 years (I’ve studied statistics but don’t quite consider that math in the same way as algebra and calculus), but I recall having to use my own shorthand in high school to write out math equations on my Braille Lite (which made no sense to my teachers, only to me because they would display as Nemeth on my braille display). So obviously this is a huge leap forward. In the future, I think it will be really cool if we could use the Monarch to generate a scatterplot of data. I don’t think it’s possible to do that in the KeyMath app since a scatterplot requires an entire data set and not just an equation. I’m looking forward to using the Monarch in terminal mode (once we have JAWS support and bluetooth) and seeing if it will display graphs I have open on my computer, such as those generated by R.

In terms of using KeyMath in CIE settings, I agree with others who said it seems like a robust graphing calculator for secondary school (high school) and maybe some university level math. In a workplace, it will depend on what software the others in that workplace use and what kinds of data need to be graphed. I think the Monarch has a lot of potential, but there will likely be many cases when it will work better as a braille attachment to a computer rather than as a stand-alone device.

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What a great idea! Graphing rhythm over time to show that four sixteenth notes span the same time as one quarter note! As a former music teacher, that resonates with me.

I may also need to explain fractions in Braille and how to represent them tactually to a non-native English speaker. Having a simple graph or ruler to feel would be so helpful.

As an aside, I did graph an equation I remembered from long ago, and the resulting graph made sense to me. I’d say the ruler or representation of musical time values is most exciting, though. (I don’t have to dust off that knowledge base like I do my algebra!)

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So I graphed some equations with their slope. Honestly, I thought it was really neat, but I also felt like it could be kind of hard to interpret just by touch. I think having a lesson on how to actually read graphs is super important. I did like how easy it was to put in all the information. When I exported the graph, it turned out perfect, which really showed me how valuable the Monarch can be. Especially for math in school or even at work—it’s something a sighted teacher or employer could look at too. The thing is, graphs can be tough. If you don’t already understand what the graph is showing, then trying to feel it is even harder. And since you can only feel a small part at a time—or sometimes the lines are just too close together—you can lose some of the information. That’s why it really has to be zoomed in or adjusted so the important parts actually make sense. Graphs create a visual for interpreting information, and the Monarch opens the door for that information to be understood and experienced, too.

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I wish I had had access to the Monarch last year when I had a totally blind student taking Algebra 2. I love how the Monarch can let you explore graphs of functions in real time. It is extremely helpful that you can use the click and point feature to find out where functions intersect. I will be using this with my students in the future.

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I went through the exercise in the video and also the study guide and was successful, and the graph was very clear. But I haven’t thought about algebra since I struggled through it in seventh grade which was back in 1968!!! So I’m not making a lot of progress there. I find using Keymath as a calculator to be very helpful as it saves all calculations in a list, but as far as graphing goes, I am like an untutored elementary school student.

What we really need is some accessible math textbooks in ebrl format. If I get access to these, I will study algebra and practice more with KeyMath. I’m just too ignorant to comment on this exercise intelligently.

If anyone knows of access to a book teaching these skills that is fully accessible, and costs nothing, please reply. Note that NLS does have “Hot X, Algebra Exposed” in Braille to download but the figures are not described and the math is in UEB and not Nemeth.