Students missing test questions due to computer interface issues

I’ve had a series where I’ve been looking at Common Core exams delivered by computer looking for issues. Mathematical issues did crop up, but the more subtle and universal ones were about the interface.

Part 1: Observations on the PARCC sample Algebra I exam
Part 2: Observations on the PARCC sample Algebra II exam
Part 3: Observations on the PARCC sample Geometry exam
Part 4: Observations on the SAGE sample exam

While the above observations were from my experience with design and education, I haven’t had a chance to experience actual students trying the problems.

Now that I have, I want to focus on one problem in particular which is on the AIR samples for Arizona, Utah, and Florida. First, here is the blank version of the question:


Here is the intended correct answer:


Student Issue #1:


In this case, it appears a student didn’t follow the “Drag D to the grid to label this vertex” instruction.

However, at least one student did see the instruction but was baffled how to carry it out (the “D” can be easy to miss the way it is on the top of a large white-space). Even given a student who missed that particular instruction, is the lack of dragging a letter really the reason you want students to miss the points?

Also, students who are used to labeling points do so directly next to the point; dragging to label is an entirely different reflex. Even a student used to Geogebra would get this problem wrong, as points in Geogebra are labeled automatically. I do not know of any automated graphical interface other than this test which require the user to add a label separately.

Student Issue #2:


Again, it appears possible the full directions were not read, but a fair number of students were unaware line connection was even possible, because they missed the existence of the “connect line” tool.

In problems where the primary activity was to create a line this was not an issue, but since the primary mathematical step here involves figuring out the correct place to add a point, students became blind to the line interface.

In truth I would prefer it if the lines were added automatically; clearly their presence is not what is really being tested here.

Student Issue #3:


This one’s in the department of “I wouldn’t have predicted it” problems, but it looks like the student just tried their best at making a parallelogram and felt like it was fine to add another point as long as it was close to “C”. The freedom of being allowed to add extra points suggests this. If the quadrilateral was formed automatically with the addition of point “D” (as I already suggested) this problem would be avoided. Another possibility would be to have the D “attached” the point as it gets dragged to the location, and to disallow having more than one point being present.


Induction simplified

When first teaching about the interior angles of a polygon I had an elaborate lesson that involved students drawing quadrilaterals, pentagons, hexagons, etc and measuring and collecting data and finally making a theory. They’d then verify that theory by drawing triangles inside the polygons and realizing Interior Angles of a Triangle had returned.

I didn’t feel like students were convinced or satisfied, partly because the measurements were off enough due to error there was a “whoosh it is true” at the end but mostly because the activity took so long the idea was lost. That is, even though they had scientifically investigated and rigorously proved something, they took it on faith because the path that led to the formula was a jumble.

I didn’t have as much time this year, so I threw this up as bellwork instead:


Nearly 80% of the students figured out the blanks with no instructions from me. They were even improvising the formulas. Their intuitions were set, they were off the races, and it took 5 minutes.

Three design puzzles from The Psychology of Arithmetic

Edward L. Thorndike’s book The Psychology of Arithmetic (1922) is the earliest I’ve seen containing criticism of the visuals in textbook design. I wanted to share three of the examples.


Fig. 47.—Would a beginner know that after THIRTEEN he was to switch around and begin at the other end? Could you read the SIX of TWENTY-SIX if you did not already know what it ought to be? What meaning would all the brackets have for a little child in grade 2? Does this picture illustrate or obfuscate?


Fig. 51.—What are these drawings intended to show? Why do they show the facts only obscurely and dubiously?


Fig. 52.—What are these drawings intended to show? What simple change would make them show the facts much more clearly?

Animated mathematical notation and the genre of the mathematical video

While it is still common (and frankly, necessary) to rail at the limitations of learning mathematics via watching videos, my personal umbrage has more to do with presentation than with educational philosophy.

The mathematical video genre is still in its infancy. I am reminded of early films that were, essentially, canned plays.

Duke of Guise

(From L’Assassinat du Duc de Guise in 1908.)

Oftentimes in videos teaching mathematics with notation they simply duplicate what could be done on a blackboard, without fully utilizing the medium.

However, there are techniques particular to the video format which can strengthen presentation of even mundane notation. For instance, in my Q*Bert Teaches the Binomial Theorem video I made crude use of a split-screen parallel action to reinforce working an abstract level of mathematics simultaneously with a concrete level.


For now, I want to focus on applying animation to the notation itself for clarity.

First, the Project MATHEMATICS! video The Theorem of Pythagoras from 1988.

The video is chock-full of interesting animated moments, but I want to take apart a small section at 5:43. In particular the video shows some algebra peformed on \frac{x}{a} = \frac{a}{c}.

Step 1: Multiply the left side by a. The variable “falls from the sky” and is enlonged to convey the gravity of motion.


Step 2: Once the variable a has fallen, the equation “tilts” to show how it is imbalanced. A second a falls onto the right side of the equation.


Step 3: The equation comes back into balance, and the two a variables on the left side of the equal side divide.


Step 4: The a variables on the right hand side start to multiply, conveyed by a “merge” effect …


Step 5: … forming a^2.


Here’s a much more recent example from TED-Ed:

When adding matrices, the positions are not only emphasized by color but by bouncing balls.


When mentioning the term “2×2 matrix” meaning “2 rows by 2 columns” the vocabulary use is emphasized by motion across the rows and columns.


The second matrix is “translated up a bit” by doing a full animation of the matrix sliding to the position.


When the video discusses “the first row” and the “the first column” not only are the relevant numbers highlighted, but they shrink and enlarge as a strong visual signal.


When discussing the problem of why matrix multiplication sometimes doesn’t work, the “shrink-and-enlarge” signal moves along the row-matched-with-column progression in such a way it becomes visually clear why the narrator becomes stuck at “3 x ….”


These are work-heavy to make, yes, but what if there was some application customized to create animation with mathematics notation? At the very least, there’s a whole vocabulary of cinematic technique that has gone unexplored in the presentation of mathematics.

Some prototype pages of Visual Algebra

While I theorized a book called Visual Algebra it was non-obvious what such a beast would look like; perhaps this prototype from my work in progress gives an idea. The original is a single page (intended for web or iPad or Kindle Fire or whatever gizmo happens to be around when I finish this thing) which I have split up to be blog-margin friendly:

Instead of applying the Rule of Four

…or the less common five representations (“pictoral” is like my example above with the boxes)…

…I am trying to apply a sort of Rule of Six, adding my tree structure matching the mathematical symbolism. I am attempting to show representations simultaneously as much as possible. This is a very image-intensive approach and would not be remotely practical in dead-tree book form, but given the book is meant to be electronic I can use as many images as I want.

However, I’ve got terrible angst in that it’s very difficult to write in a relaxed and friendly matter while still being exact enough for mathematics. Here is a page that pains me (click for full size):

Would I ever really say “the numerical value on one side of the equal sign must be balanced with change on the other side”? I can’t just say “you have to keep them balanced”; that would be unclear and vague enough to cause potential misconceptions. Such a statement also shuffles under the rug manipulating an expression without changing it.

Also, when I turn the tree diagram from a puzzle into a pedagogical tool, it starts to get clunky and confusing. I’m unclear the best way of patching it up or if I should just pitch it.

(For those worried about the fate of the long-awaited video Giant Space Whales vs. Logarithms, it won’t be too much longer, I promise.)

Teaching manipulation of expressions and equations before introducing variables

In addition to the video I’ve been toying with a sample first chapter of this project idea.

It is quite usual in Algebra books to introduce variables early — they seem to be what makes algebra Algebra, that is. However, a good amount of early stages of manipulating expressions and equations is not dependent on having variables. I am wondering about possibly spending longer than usual on concrete number manipulation with problems like–

Isolate the 253:
253 + 23 = 276

Isolate the 3:
2(3 + 4) + 5 = 18 + 1

Show why the two sides of the equation are equivalent:
6 * 4 = 3 * 4 + 3 * 4

Theoretically, one could provide a strong enough background in manipulating expressions and equations to allow variables to be introduced while hitting the ground running, with students quickly able to isolate variables in multi-step equations.

This has the advantages of:

1. It’s clear that (using the first problem as an example) 253 = 253 is the goal. The point becomes the process, and the usual student tactic of writing the answer without explanation (I did it in my head!) would look silly.

2. Students who have “symbol overload” can use their concrete sense to help focus solely on operations, a la Bret Victor (sort of).

3. The meaning of the equality sign is reinforced; it’d appear much odder to a student to perform an operation on only one side of the equation when starting with 20 + 1 = 21. Also, the restrictions to operations on an expression would make more sense when starting with, say, 5 rather than 5 + 2x.

4. Answers would have a built-in sanity check.

Unfortunately, it means applications would be forestalled. While some of the manipulations represent tricks for mental arithmetic, I don’t know if that would be enough motivation to carry a struggling student through.

Has anyone tried something like this, even on a small scale?

Visual Algebra?

In recent discussion about Bret Victor’s Kill Math project Ben Blum-Smith brought up the books Visual Complex Analysis and Visual Group Theory which as he puts it “all arguments are geometric and illustrated by diagrams”. (I’m not familiar with the latter, but Visual Complex Analysis is fantastic and I highly recommend it.)

I feel like these sorts of books will eventually create a revolution in upper-level mathematics — I’m eagerly awaiting someone to write Visual Linear Algebra — but could we re-conceive lower level mathematics in the same way?

By Visual Algebra I’m not meaning graphs, I’m meaning the more mundane symbolic “solve for x” manipulation.

Solve for x: 2x + 3 = 5.

In the same vein as my puzzle equivalent to solving a quadratic, solid lines mean multiply, dotted lines mean add.

Solving for the highlighted circle is equivalent to solving for x in 2x + 3 = 5.

(I swear I have seen something closely resembling this elsewhere for equations, and I think it even has a buzzword attached — anyone know?)

I originally thought of these sorts of puzzles as a gentle introduction to the topic, but would it be possible to integrate this kind of visual-symbolic thinking in every part of an algebra course?

ADD: Here’s an image where the puzzle is closer in look to the equation:

This sort of thing is risky because rather than applying inverses and so forth students may make it a general method to draw circles and arrows everywhere.

Proof of the Fundamental Theorem of Arithmetic (with fancy design)

This is a test case for my mission to reform the design of mathematical notation. The original proof was a very clear one, from Niven’s Numbers: Rational and Irrational; I wanted to see if I could make it even clearer.

This includes: syntax highlighting, specific lemma naming as long “procedure names” rather than arcane numbers, longer variable names when appropriate, and indents on suppositions leading to contradictions. It likely can be improved, so feedback is appreciated.

The design of mathematical notation

The above page is from an 1847 edition of Euclid made by Oliver Byrne.

I bring it up to show that the way we present mathematics is a cultural construct. There are alternate methods. Perhaps (as is sometimes the case in the above volume) they are not always clear, but when something in mathematical notation design is confusing to new students it should be looked at with a skeptical eye and we should be willing to break new territory: new fonts, new interactivity, new colors.

In The Design of Everyday Things Donald Norman argues that many things attributed to “human error” are really design error.


It’s labelled as PUSH, it’s the fault of the user if they pull instead, right?


Just as if a student does the calculation on the right, it must be entirely their fault, right?

Brent Yorgey calls this “quite possibly the most horribly chosen mathematical notation in the history of the world” and writes:

I would like to find whoever made up this notation for inverse trig functions and make them pay $10 to every student who has ever been confused by it. The only problem is they probably don’t have A GAZILLION DOLLARS.

When discovering a design that is inherently confusing, the proper response is to change the design.

I. An actual textbook example

If m < n, the line y = 0 is a horizontal asymptote.

If m = n, the line y = \frac{a_m}{a_n} is a horizontal asymptote.

If m > n, the line has no horizontal asymptote. The graph’s end behavior is the same as the graph of y = \frac{a_m}{b_n}x^{m-n}

The letters m and n aren’t the two most visually and verbally distinguishable choices, especially in small fonts. The first time I taught this it was directly from the textbook version, and students were wildly baffled; I consider it the fault of the design, not the students.

One improvement is to change the letters:

Now, at least, eyes won’t strain and blur just to tell the difference between the letters. More experimentally, words can stand for variables (just as in computer programming):

It’s much easier for students to think of TOP < BOTTOM, TOP = BOTTOM, and TOP > BOTTOM instead of m < n, m = n, and m > n.

There is even some precedent in logic for words (in an all caps font) standing in for the logic symbols.

II. On color

Subtle changes in color can affect clarity of a text.

While textbooks have introduced color almost universally, the changes in color have not touched the mathematical notation itself. This is understandable when pencil and paper still predominate, but if the layout is done with computers there’s no reason even the most extreme color changes can’t be made . . .

. . . just as editors for programmers use syntax highlighting.

III. Interactive design

Text design no longer needs to be static. Consider the example above, rendered any way the user so chooses:

The settings could be consistent, as in the syntax highlighting example, or it could be a design where the equation starts “plain” but the user can click to modify the colors of specific parts: all constants, all exponents, all operations.

IV. What is possible with design?

Here’s a new rendition of Alice in Wonderland for the iPad:

When will mathematical texts reach the same level?

(To be continued in Textbooks of the Future.)

Quadrilateral Property Combo Chart

One thorn in the side of geometry teachers is difficulty untangling the relationship between squares, rectangles, rhombuses, and parallelograms. I’ve seen it done with flow charts and I’ve seen in done with Venn diagrams, but neither fully satisfied what I wanted. So I combined the two:


Here’s a version with trapezoids and kites: