Desmos Graphing Caluclator

Desmos Graphing Caluclator

Desmos is an education startup, currently with one major offering – an online graphing calculator. Graph any equation, cartesian or polar coordinates, exponential and trigonometric functions, and multiple equations at the same time. Using MathQuill (a what-you-see-is-what-you-get LaTeX editor for HTML5) and/or a built-in key pad users can easily enter and seamlessly edit any equation.

Users can also create a free account to save graphs and use a projector mode to display on in-class screens.

Desmos Graphing Calculator - On Screen Controls

A screen shot of the calculator in-browser, here you can see the on screen controls.

Aesthetics: 7
Entertainment: 2
Usability: 10

I am currently trying to rebuild my own mth skills after years of neglect, starting from precalculus on through multivariable calculus. My main tool for this has been Khan Academy, which has a great set of exercises arranged in a knowledge map, but a TERRIBLE calculator interface. I haven’t had a chance to play with the two simultaneously yet, but I strongly suspect that Desmos’s calculator is going to be a hugely useful tool for visualizing and manipulating complex functions.

Desmos Graphing Calculator Screen Shot 2

In this screen shot you can see the display options, including a projector mode that makes everything a little bolder for better display on class room projectors.

Additional awesomeness include the explorations, where users can explore graph-able functions ranging from different forms of linear equations, through derivatives; and the graph gallery, where users can peruse other users recently saved graphs.

Desmos Online Graphing Calculator - Trig. Functions Exploration

Here is one exploration, which graphs any/all of the trig functions. On the right some points of intersection have been selected showing their coordinates, on the left is the menu for selecting different explorations.



Khan Academy

I almost feel like reviewing Khan Academy is unnecessary, it has been so prominent over the past few years. However, it is one of the most impressive online resources for learning, and I like it, so this site would be incomplete without it. Khan Academy is an enormous, diverse, and free collection of online lectures across an equally enormous number of subjects – in their own words “from math and finance to history and art.” Plus math exercises, an interactive computer science section, and the potential for much more. Khan Academy is a rapidly growing . Because they are constantly adding new lessons, and expanding features, this article will probably be out dated by the end of the year, if not the week.

Aesthetics: 8
Entertainment: 6
Usability: 9
Bonus: Huge number of subjects!

Note: Almost exactly the day after I posted this Khan Academy updated their interface. I have updated the first image, but the rest of the screen shots below still show the old interface.

If there is a subject you want to learn about, it is probably here. The majority of the material is video lectures – usually 4-10 minutes. In the videos, the teacher, often Salman Khan himself, speaks over a visual presentation to explain the subject. While I haven’t seen many, those I have seen are quite good. There has been some back lash (see the links at the bottom of the page), and it is key to note that Salman is not a professional teacher, nor a certified expert in every subject he covers. However, he is smart and he speaks in common language that I think is accessible and accurate. Khan academy shouldn’t replace school, or college, but these thousands of lessons are a great place to start with any subject. But wait – there’s MORE!

Khan Academy Home Screen Shot

The home page – look at all that stuff! In the top left you’ll see a drop down menu, with a link to the knowledge map.

Mathematics exercises, with a map! These can be accessed by searching for the material, or via the knowledge map (see above to find your way to the map; look below for a picture of the map).

Khan Academy exercises are great for sticky subjects students are stuck on, need to review, or just practice. Problems come as straight numbers, and sometimes word problems, with randomly generated values. In my experience one of the most mechanical and time consuming tasks for a teacher or tutor is generating problems that make sense and solve with a clean/neat answer. Not only do these exercises vary the numbers, they include “hints” that gradually walk though the problem, and then reveal the answer.

Khan Knowledge Map Screen Shot

Look at all that knowledge!  One complaint I have is that the knowledge map doesn’t zoom out very far – to get this screen shot I zoomed out by pressing Ctrl and the minus key, use Ctrl + to zoom back in.

The problems come in “stacks,” illustrated by the stack of cards (see below, top right).  Student can score a 1-3 leaves on each card. 3 leaves for a correct answer on the first try, 2 for a second attempt or with hints, 1 if the answer has been revealed by showing all the hints. At the end of a stack the number of leaves is shown, plus the number correct in a row, plus the number answered “quickly”.

A Khan Exercise Screen Shot

The Scratch pad allows students to do work on the screen, and then they enter their answer on the right. Here I have shown all the hints to the problem, just to illustrate, but showing all the hints results in a maximum score of 1 leaf.
Khan Results
The fire cards shows the number answered quickly, while lightning indicates the longest streak of correct answers, and leaves shows the number of points earned.

For more information on Khan Academy check out the links below.

If you like Khan Academy, you might also like:

Some of coolest videos on Khan Academy – under construction – suggest one in the comments!

Criticisms of Khan Academy

Among teacher and education innovators there is a huge amount of criticism of Khan academy. The reason for this is the mainstream media’s unchecked exuberance for the site, which overlooks major flaws and growing pains inherent to the website. Some examples below:

World Education Games

Want to rev up the competition in your class? You can pit your students versus the world at the World Education Games. The games themselves will take place on

5 March – World Literacy Day
6 March – World Maths Day
7 March – World Science Day

Note: These dates are set relative to the host city Sydney – I’m not sure what that means for their start times in the US.

I tried out the math and literacy categories, but the science required a plug-in (for more info on science check page 9 of the teachers guide). Because it’s the age group I work with, I tried out the middle school bracket. In both math and science the ‘game’ is a one on one race with another participant from around the globe. The user is presented with the question at the top of the screen, a graph below showing their opponents progress, and a timer with the time left in the round.

World Education Games Maths Level 4 Screen Shot

Me losing to a nice Canadian girl at level 4 Maths, where they include negative numbers. The [x] next to my profile means I have gotten one question wrong – after 3 questions wrong you “strike out”.

In the Math games Level 1 the questions started with adding and subtracting single digits from two digit numbers [77-6]. Level 3 escalated to multiplication, division, and complete the sequence [43, 52, 61…], while level 5 included functions as advanced as cube roots, including find the cube root of 6859 [it’s 19].

Maths Level 4 Results Screen Shot

After each round the questions asked, answers given, and correct answers are shown.

In the Literacy game students race to spell a word after hearing it alone, and then in a sentence. Essentially it is a one on one spelling be. Note: I am a terrible speller. At level 1 i got timetable, plain, rarely, shelves, soil. At level 3 carafe, adept, artisan, enzyme, intense, abyss archery eligible, and at level 5 sortsango, caterwaul, centrifuge, mastectomy, relegation, paradigm, penicillin. To be honest the only reason I got that many in level 5 was because I got a bunch of science words, and I still only got to 7 words.

Literacy Level 3 Screen Shot

A screen shot of Literacy Level 3. A voice reads the word, then the sentence using the word. Speakers are necessary for this game.

Normally this blog will focus on creative, well programed, or otherwise exceptional free educating software. The software on WEC is nice enough, with some interesting structures for the questions, but it is sponsored and driven on a host of for-sale educational software. That said, I find the use of competition in educational games worth a post.  Below you will find links to and about the World Education Games, including a teacher’s guide to getting involved.

World Education Games Home Page

WEG Teacher’s Guide

Wikipedia on the WEG

Donna Román’s post on her blog  “Map without Borders” where I first read about the WEG

The Guiness World Record Book entry for the 2010 Maths Day – “The largest online Maths competition” a record they intend to shatter this year


My first recommendation is, for memorizing basic addition, subtraction, multiplication, and division facts.

Entertainment: 6/10
Ease: 9/10
Aesthetics: 8/10

Intended users: Teachers, Students, Parents – separate interface for each. Designed for both at home and computer lab use.

A screen shot of my class - all names removed and data scrambled.

A screen shot of my class – all names removed and data scrambled.

This website does one thing, and it does it exceptionally well: fact fluency.  Fact fluency is the ability to quickly recall an answer from memory without computation. Specifically, Xtramath works on users ability to recall addition, subtraction, multiplication, and division facts from 0+0=0 to 144/12=12.  The reason this website is so great, is that fact fluency is fundamental to learning math, and also excruciating to teach – but Xtramath is a fast, effective, and easy solution.

The ability to quickly recall these answer is critical, not only because it speeds more complicated operations, but because it also instills confidence. One of the greatest things a math teacher ever did to/for me was make me take timed tests converting fractions to decimals. That was fifth grade, and, of course, I hated her for it. I complained about it for years, all the while reaping a huge advantage over less lucky peers who struggled with both fractions and decimals because they failed to see the connection. In contrast, I thought I was naturally gifted with fractions and decimals. At least I was .5 right.

The advantage to fact fluency versus re-computing arithmetic every time is hard to overstate.  Fluent students spend 0-2 seconds solving 7*8, while non-fluent middle school students can spend 40-90% of their time for each problem performing basic arithmetic functions – or worse, running to a calculator to punch it in. That lag time slows learning and prolongs misunderstanding. Half the time they’ve forgotten the original problem before they retrieve the fact.

In neuro-geek-speak: the lag time weakens reinforcement of correct intuitions, while errors build an unreasonable aversion to practicing advanced concepts. This can be especially painful during class practice, when teachers cannot spare the time to locate the error behind a verbal wrong answer. Suddenly, despite understanding the concept at hand, a student’s arithmetic error can become a full-blown public failure. This is where Xtramath comes in.

The traditional method of teaching fact fluency is a combination of chanting, flashcards, games, and timed test – all of which can be painful. Don’t get me wrong, they work, but they are boring, or stressful, or both at the same time. Xtramath has three major advantages over the old ways.

  • It is shame-less
  • It is less boring, because it is tailored to the student, and
  • It feels like learning, because you are learning

To elaborate, unlike answering a question out loud in class, chanting with a group, or competing in teams, there is no one standing over your shoulder when you do Xtramath. Unlike a timed test, there is no grade and no one grading. That means no pain in getting it wrong – just a quick prompt of the right answer, type it in, and the mistake is behind you.  Stress is bad for memory, so why should learning be stressful?

Screen shot of a mistake.

A mistake in Xtramath prompts the user to put in the correct answer immediately, and then move on to the next question.

At first I wasn’t that impressed with Xtramath because it was “just” another flashcard web site, and on some level it is.1 However, Xtramath is distinguished by its laser-like focus, and a smarter algorithm. Flash cards are perfect for fact fluency, because recognizing visual cues and regurgitating the answer is the objective. More significantly, Xtramath repeats problems that are students are slow to answer and those they get wrong. Not only is this a more effective use of students time – it’s less boring, and boring is also bad for learning.2

Which brings me to my third point, Xtramath is brief. Every time students log in, they are prompted to do 2-3 sets of “cards”. This means that students don’t get tired, yet they get the reward of being finished if they stick through 2-3 sets of 25-50 questions.3 I don’t know if they tested and optimized it, or if the teachers who made it just knew from experience – maybe it detects boredom via slow-down and errors. In any case, Xtramath manages to push students into a substantial amount of practice, with out being overwhelming. Learning theorists have long known that splitting up practice is more effective than the same amount of time spent in a single session. This is known as the spacing effect, and it is one of the oldest, most robust, and most substantial principles of behavioral science.  If need be, students can quit at any time by closing the window, and they want they can log back in for another session as soon as they want.

(a list of flash card websites is on my to-blog list)

Craig, S., Graesser, A., Sullins, S., & Gholson, B. (2004). Affect and learning: an exploratory look into the role of affect in learning with autotutor. Journal of Educational Media29(3), 241-250. doi: 10.1080/1358165042000283101

a rough estimate – I have tried the system my self, and I know it is a sizable number per set, but I don’t have any exact measurements.

**** Pavlik Jr., P. I., & Anderson, J. R. (2008). Using a model to compute the optimal schedule of practice. Journal of Experimental Psychology: Applied14(2), 101–117. (Text).

*****Dempster, F. N. (1996). Distributing and managing the conditions of
encoding and practice. In R. A. Bjork & E. L. Bjork (Eds.), Memory (pp.
317–344). New York: Academic Press.