GeoGebra – Empowering Learners With Tools of Math Construction

Presentation by Michael Borcherds, CTO at GeoGebra


Do you want to take your math instruction to the next level? Use GeoGebra to create endless interactive works and assessments that have the ability to give your students a true intuitive understanding of the topics in your curriculum. This is no simple online graphing calculator, but a powerful mathematical tool capable of expressing concepts as simple as the quadratic formula to more high-level ones such as Möbius transformations and non-Euclidean geometry with the same level of clarity.

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Don’t have the time to watch the whole thing? Here are some highlights

Authored by Kyle Hovey, Software Developer


GeoGebra’s suite of tools can work with any math topic. Whether you are giving instruction on algebra, calculus, differential equations, statistics, or topology, GeoGebra will offer a formidable suite of tools to aide you in expressing concepts.


Constructions in GeoGebra are constraint-based, meaning that you can add certain qualities to your diagrams that must hold, and any remaining degrees of freedom are available for interactivity. For instance, you could create a parabola, a point on that parabola, and a line that goes through that point that is tangent to the parabola. The line is now fixed to the curve, but notice that nothing is specifying where the point must be on the curve. This means that you may move the point around and observe the consequences of the movement you impose.


With GeoGebra Notes you can use a smart board, touch screen, or tablet to give a lecture on an overpowered whiteboard capable of adding graphics that you can annotate freely. If a student asks a question about something on the whiteboard, rather than re-drawing you could simply interact with the graph or include another one on the fly.


With GeoGebra Classroom, you can leverage the power of custom interactive constructions in your assessments. Instead of assigning simple fill-in-the-blank style math assessments, you can have students interact with the problem and have GeoGebra determine the correctness of the answer. GeoGebra Classroom also lets you ask your whole class a question and see their responses in real-time as they interact with the problems.


Give GeoGebra a try in your classroom. It will quickly become your go-to resource for conveying mathematical topics in a clean and understandable fashion.

Making Mathematics Worthwhile

Authored by Kyle Hovey, Software Developer

Math stigma

Despite the apparent overwhelming stigma against mathematics education, according to a survey of 1,000 students done by Texas Instruments, 46% of students report that they liked learning about mathematics in school. Perhaps the stigma stems from something deeper than preference: practicality. Sixty-eight percent of students responded that they would appreciate mathematics more if they knew where it would be applicable to their own lives. It is odd, then, that no one would question the usefulness of critical thinking in their own lives. Therein lies the current problem of math education: math is viewed as machinery that will provide some immediate and topical utility to your own life.

Much like critical thinking, however, mathematics skills are more abstract in their benefit and the value is impossible to measure directly. The ties between critical thinking and mathematics go deeper than this fact. Mathematics is built upon an ancient edifice of logic, constructed by some of humanity’s most brilliant logicians. In a way, the reasoning employed by mathematics is dual to that used in science. In science, we rely on the principle of uniformity of nature to uphold that empirical evidence will provide enough certainty about the structure of the world around us and how it will continue to behave in the future. Mathematics, however, relies on deductive reasoning from some very simple truths (axioms) that we accept on the basis of faith alone (there exists a number 0, representing nothing etc…). Along with some rules of inference, we grow these axiomatic seeds into ornate theorems powerful enough to develop understanding of even things we could never hope to interact with.

Sadly, this structure is not revealed to most students until graduate-level courses. This can make sense if most students will only need to use the results of mathematics for STEM careers, but perhaps participating in it is what would drive a better understanding of its place in our lives and endow learners with an enhanced ability to synthesize heuristics, and not only recall/compose algorithms.

Procedure or creativity?

In this way, in order to truly appreciate mathematics, you need to partake in it, not simply absorb its results. Ironically, creativity is not often associated with mathematics. More often, curriculum is based around memorization of procedure, and pattern matching to learn when to apply them. It is important to be adept in these procedures, after all, just like it is important to be fluent in a language before you can speak it. Teaching literacy (despite being the main focus of modern math education) is not the problem, giving students agency in their own creative endeavors is. The challenge, then, is to balance this added context with application. For instance, if a teacher were to explain a concept such as fractal dimension, they could approach it by stating the definition:

The proportion that a manifold’s hyper-volume scales when all metrics are adjusted by a scalar amount.

It would be absolutely forgivable to see a class full of bright students glaze over when hearing this, and it could practically be expected that not many would understand. Still, the teacher could proceed with the algorithm of how to calculate this concept given a structure, and the class could pass an exam with flying colors without ever knowing what it was that they were even being tested on. What if, instead, you began with an example:

Fractal dimension is akin to a measure of roughness. If you were to measure the coastline of Britain with a meter-stick, you would measure a shorter distance than if you were to use a foot-long ruler (due to being able to measure more fine details). Notice that instead of measuring with a smaller ruler, you could also just scale Britain up and use the meter-stick to get the same change. Because of this, the coastline of Britain can be said to be more than one-dimensional, but less than two-dimensional. That is to say that if Britain were to be made n times larger, then the coastline would be n^d times larger where d is the fractal dimension of the coastline, and d is more than one, but less than two.

Cultivating an appreciation for math

What if, after explaining this, you let the students converse with one-another and gave them some tools to find other instances of fractal dimension? Perhaps the first assignment could be to simply research this topic, and have students come up with their own thoughts on the matter. Perhaps the teacher could offer an interactive software tool to build fractals and measure their fractal dimension. The next assignment could ask the students to create a fractal with a given fractal dimension. By the time this second assignment was done, maybe the students wouldn’t even need to memorize a formula. They themselves could create the formula themselves, reconstructing it from their total and personal understanding of the concept. Imagine not having to study for a math test, but instead having to come prepared to think creatively about problems and develop solutions based solely on understanding.

I can offer personal testament to this technique. There was a time I struggled with mathematics. I was profoundly fascinated, but always seemed to fall short of the requisite ability to receive the grade I was aiming for. Then, one day, a close friend of mine walked me through one of their favorite theorems. It was one that was not formally taught in school until graduate classes, yet the way it was explained made it easier to understand than finding the focus of a parabola, or computing the determinant of a matrix. I realized that, armed with this theorem, I could easily derive the 30+ expressions I had diligently written down on flashcards in my backpack and had spent hours memorizing. More than that, I understood at a fundamental level how each of those expressions were derived. Still, I couldn’t tell you the exact value of learning that theorem, or exactly where it would be used in my life from that point on. In fact, the man who developed that theorem (Leonard Euler), though being one of the most brilliant minds to ever grace this earth, could not have expected where this knowledge would be used. We now rely on that theorem to build our bridges so that they won’t collapse in a windstorm, so that we can store thousands of songs on an SD card, so that we can edit photos and view them on our devices, so that we can have wireless communication and broadcast signals into space, and so that we may understand the quantum world that we cannot see.

I’m omitting much, but it is complete and simple enough to say this: the value of mathematics education does not lie in the heuristics we associate with it, but rather in the development of organized abstract thought that powers the technological advancement of our society. If students can come home with one belief about mathematics, I hope it can be that one.


Presentation by Jared Smith, Associate Director at WebAIM


We met with Jared Smith from WebAIM to learn about their online accessibility offerings.  WebAIM trained us on making our websites and apps more accessible to those with disabilities (hearing, visual, and other physical disabilities).  Enlightening!  We wanted to share some highlights and FREE resources, like a free website evaluation tool that identifies potential accessibility issues.

If you’re short on time, feel free to jump to the topics that most interest you:

Authored by Rich Kingsford, Software Development Manager and Adjunct Instructor

What’s accessibility?

Online accessibility strives to make websites and apps easier to use for those with disabilities or need special accomodations.

My background with accessibility is mostly with the federal government (503 compliance).  My fellow contractors and I usually viewed this as jumping through a thousand pedantic hoops.  The most annoying one, by far, was we had to make our websites functional even if someone turned off javascript (a very popular scripting language you probably enjoy about a hundred times a minute without realizing it).  Some of the rules were completely reasonable and it felt good to make our site easier to use for those with hearing and visual disabilities.  But the other 90% of our efforts were about jumping through annoying hoops (compliance and such). 

WebAIM and the Web Content Accessibility Guidelines (WCAG) from seem mostly about helping people who really need it.  I like it.  Far more motivating than just boring old compliance.  And WebAIM’s tools make it much easier to identify user stories (software developers often write user stories, or short stories from the user’s perspective, to understand and manage requirements, or the features, enhancements, and bug fixes customers want most). 

WebAIM crawled a million websites and found 98.1% of them had detectable WCAG failures (average of 60 violations on just the homepage).  “One of our goals is to put ourselves out of work.  We have a long way to go.”

Do you have any questions about accessibility?

A quick note on standards

There are several accessibility standards (e.g. 503, WCAG) and sometimes these can be pretty daunting.  The cool thing about standards is they have more power when more people abide by them.  In this instance, a developer can begin learning the standards as she applies them.  She can reach out to the community for help or with a proposal.  She can transfer her mastery from project to project or even to another company.  

I’d like to share a limitation with standards; sometimes they get stale (old, outdated, and confining).  Let’s remember to periodically challenge the status quo and think outside the box.  A great way to strengthen a standard is to think up a good idea, test its hypothesis, and then build a persuasive proposal (or maybe a new product or service that will make big bucks for you ). 

What’s your take on accessibility standards?  Are there other standards you like?

Top violations

WebAIM shared several common violations.  Let’s walk through a few.

Low contrast text

Please remember some people are more sensitive to low contrast than others. Check your own content through WebAim’s Contract Checker.

Missing alternative (alt=””) text

Alternative text (or alt text) should be placed on any image (or image tag) on your website or app.  That way, a blind person can better visualize what the image is, even if she can’t see it with her eyes.  Just like captions on a TV show help someone who is hearing impaired, alt text can help someone who is blind (partially or otherwise) enjoy the content on your website or app.  You might ask, “How can a blind person consume (or read) a website or app?”  Usually they use a screen reader tool that reads the content out loud.  Imagine you’re partially blind and are having a hard time reading this article.  Now imagine your personal assistant sitting next to you reading whatever is under your mouse.  If you clicked on an image, the screen reader would read the alt text to you.  Pretty neat, huh? Explore additional alternative text basics here.

Missing form control labels

We use forms all the time.  But oftentimes websites and apps are missing form labels, making it hard for screen readers or navigation tools to “see” what’s going on.  Maybe you control your device audibly instead of through a mouse or touchpad.  You might use a navigation tool that lets you go into a text field by saying its name. Learn how to create accessible forms here.

Missing document language

Language tags help search engines and content aggregators (e.g. Google or understand the language of your content.  If you don’t have them, the various systems who are ‘wanting’ to consume your content might not be able to guess. Learn more about document language here.

What are some common violations you’ve seen?

WebAIM showed us’s tools.  You can send your website through this tool and it will highlight accessibility violations (kind of like spell check).  Don’t forget about the Chrome and Firefox extensions!

Learn more about’s tools.

A few questions

“Do you have any device recommendations for those with disabilities?”

  • iOS devices such as iPhones and iPads are usually pretty good.  They were first to really get accessibility right.  Android devices have also come quite a ways.
  • Several of our customers who have trouble seeing use external bluetooth keyboards with their phones.  Pretty cool.

“How can we, as a software development company, give assurance we’re compliant?  Or that we will be compliant?

  • There’s no “one and done” type solution.  Just like your usability and security concerns, you should regularly identify and address your accessibility health.

Do you have questions for WebAIM


WebAIM trains and helps you strengthen the accessibility features for your website or apps.  Thanks for the great presentation, Jared Smith!