Abstract and concrete simultaneously

In most education literature I have seen going from concrete to abstract concepts as a ladder.


The metaphor has always started with concrete objects before “ascending the ladder” to abstract ideas.

I was researching something for a different (as yet unrevealed) project when I came across the following.



That is, someone used the same metaphor but reversed the ladder.

This is from a paper on the Davydov curriculum, used in parts of Russia for the first three grades of school. It has the exotic position of teaching measuring before counting. Students compare objects with quantity but not number — strips of paper, balloons, weights:

Children then learn to use an uppercase letter to represent a quantitative property of an object, and to represent equality and inequality relationships with the signs =, ≠, >, and B, or A<B. There is no reference to numbers during this work: “A” represents the unmeasured length of the board.


A later exercise takes a board A and a board B which combine in length to make a board C, then have the students make statements like “A + B = C” and “C – B = A”.


Number is eventually developed as a matter of comparing quantities. A small strip D might need to be used six times to make the length of a large strip E, giving the equation 6D = E and the idea that number results from repetition of a unit. This later presents a natural segue into the number line and fractional quantities.

The entire source is worth a read, because by the end of the third year students are doing complicated algebra problems. (The results are startling enough it has been called a “scam”.)

I found curious the assertion that somehow students were starting with abstract objects working their way to concrete ones. (The typical ladder metaphor is so ingrained in my head I originally typed “building their way down” in the previous sentence.) The students are, after all, handling boards; they may be simply comparing them and not attaching numbers. They give them letters like A and B, sure, but in a way that’s no less abstract than naming other things in the world.

After enough study I realized the curriculum was doing something clever without the creators being aware of it: they were presenting situations that (for the mind) were concrete and abstract at the same time.

For a mathematician’s perspective, this is impossible to do, but the world of mental models works differently. By handling a multitude of boards without numbers and sorting them as larger and smaller, an exact parallel is set up with the comparison of variables that are unknown numbers. Indeterminate lengths work functionally identical to indeterminate number.

This sort of thing doesn’t seem universally possible; it’s in this unique instance the abstract piggybacks off the concrete so nicely. Still it may be possible to hack it in: for my Q*Bert Teaches the Binomial Theorem video I used a split-screen trick of presenting concrete and abstract simultaneously.


Although the sequence in the video gave the concrete example first, one could easily imagine the concrete being conjoined with an abstract example cold, without prior notice.

(For a more thorough treatment of the Davydov curriculum itself, try this article by Keith Devlin.)


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.

Telling left from right

I had a discussion last week when reviewing slope that went like this:

Student: Wait, how can you tell if the slope is positive or negative just by looking?

Me: Well, if you imagine traveling on the line from left to right, if you’re moving up the slope is positive and moving down the slope is negative.

Student: …What?

Me: (points) So, starting over here … (slides hand) … and traveling this way … this slope is moving up. Starting over here … (slides hand) … this slope is moving down.

Student: But I don’t understand where you start.

Me: You start on the left.

Student: I’m still confused.

Me: (delayed enlightenment) Wait … can you tell your right from your left?

Student: No.

This isn't picture that was up at the time, but it's in the same genre.

This isn’t the picture that was up at the time, but it’s in the same genre.

Left-right confusion (LRC) affects a reasonably large chunk of the population (the lowest estimate I’ve heard is 15%) but is one of those things teachers might be blissfully unaware is a real thing. (Note that LRC is at something of a continuum and affects women more than men.)

My own mother (who was a math teacher) has this problem, and has to use her ring finger whenever she needs to tell her right from her left. She reports that thinking about the graph as “reading a book” lets her get the slope direction correct.

Answers to the priming questions

To follow up on this post about the psychological affect of priming on students taking math tests

1.) What would happen if these two questions on a recent test of mine were reversed (that is, #4 was given as #5 and #5 was given as #4)?

Sam Alexander essentially got this one right in that students would be more likely to apply a difference of cubes to problem #4. The students would likely pull out a 6x to get 4x^3-1 and claim the cube root of 4 is 2. As the problem is originally written, the more common issue is to attempt difference of squares instead. It’s possible to do the factoring with difference of cubes if one does not insist the factors are rational (use 4^{\frac{1}{3}}) but the flat-out mistake is more likely.

2.) The following is a multiple choice question with the image removed. If a student were to ignore the math content of the answers (that is, guess), which would be the most common answer?

The conic section depicted can be categorized as:

a.) Parabola
b.) Ellipse
c.) Cone
d.) Plane
e.) Hyperbola

This answer is nearly creepy, but the sort of thing that happens with priming all the time. Note the strong iteration of a single letter.

The Conic seCtion depiCted Can be Categorized as:

a.) Parabola
b.) Ellipse
c.) Cone
d.) Plane
e.) Hyperbola

Hence the most likely choice by randomness would be C.

EDIT: As adroitly pointed out in the comments the word “cone” is also suggested by the setup word “conic”.

Once you’re aware of priming you will start to notice its effect more often. I stay on the lookout for situations where a student gets a problem wrong not because they didn’t know the mathematics but because they were influenced by the priming in surrounding problems.

Priming on math tests

To continue my thinking about getting math problems wrong for cognitive science reasons, I want to discuss priming.

Derren Brown is a mentalist (that is, a “psychic” who admits up front he is doing nothing more mystical than applied psychology) who is particularly good at priming. In this video he asks a teacher to read a story and think of an image; the story influences a teacher into thinking of a teddy bear through subtle cues.

For example, the title of the story is “Near” where the “N” is drawn to somewhat resemble a “B”, and placed next to an animal which resembles a bear.

Also, the story incldues the phrase “red beak to wear” where the “k” resembles a “r” if the top is removed, and the close proximity of the rhyming word “wear” also suggests a bear.

A group of children painted as they listened to the story, and came up with an image resembling a teddy bear simply through the priming in the story.

(Derren Brown has done this many times; try his Subliminal Advertising to see how far priming can go.)

Keeping the technique in mind, answer these two questions:

1.) What would happen if these two questions on a recent test of mine were reversed (that is, #4 was given as #5 and #5 was given as #4)?

2.) The following is a multiple choice question with the image removed. If a student were to ignore the math content of the answers (that is, guess), which would be the most common answer?

The conic section depicted can be categorized as:

a.) Parabola
b.) Ellipse
c.) Cone
d.) Plane
e.) Hyperbola

Analyzing the bellboy puzzle

Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person.

Now each person paid $10 and got back $1. So they paid $9 each, totaling $27. The bellboy has $2, totaling $29.

Where is the remaining dollar?

There’s a nice post on the answer now at Partially Derivative, but I want to discuss my meta-question:

Can you generalize the errors made in the puzzle? Can you give a textbook, not-designed-as-a-puzzle example where this happens?

That is, is there some general principle at work here, perhaps something to do with psychology, revealed by the puzzle?

The operations in the first paragraph use (more or less) a number line.

While I drew this as in the negative direction, I expect mentally most people would be thinking in positive numbers (as they are given in the problem) so I colored them accountant-style rather than put negative signs in front.

The operations in the second paragraph hew closer to an allocation model:

Dissecting it this way, it’s fairly clear the “manager” line is missing, and the bellboy’s take is in the wrong direction.

I would consider the psychology “trick” to be the jump between two mental models, made worse by the majority of people considering the first part of the problem on a positive number line. In an allocation model the idea of “negative allocation” doesn’t occur to most people.

While a $1 disparity would be difficult to find “in the wild” — the numbers and the omission of the $5 are carefully engineered for the puzzle — I would expect to find real textbook problems that ask students to transition from one mental model to another with potential confusion of signs. Here’s an example of what I mean:

Rick’s bank account is overdrawn by $219. What will the new balance be if he deposits $196?

Elementary Algebra: Equations and Graphs by Yoshiwara, Yoshiwara, and Drooyan

This isn’t a perfect match, though. Does someone have a better example?

Are learning styles dead?

If you’re in education you’ve likely heard things about different “intelligences”: Bodily-kinesthetic, Visual-spatial, Verbal-linguistic, and so forth. You may have also heard about the necessity to change one’s teaching to accommodate the different learning styles. For example, a class proving the Pythagorean theorem might justify it once as a flip-book animation (to accomodate the Visual-spatial and Kinesthetic intelligences) and once as written prose (to accomodate Logical-mathematical and Verbal-linguistic intelligences).

I was therefore startled by this comment in a Tom Henderson interview:

I don’t know if you saw the article I posted here at Technoccult a few weeks back, but it looks like the whole “learning style” thing is complete bunk.

So, are learning styles dead? Is this yet another failed education idea to toss onto the pile?

Well, sort of.

I traced the links back to their source, and came across this (publically available) journal article entitled Learning Styles: Concepts and Evidence, which includes the following:

The most common—but not the only—hypothesis about the instructional relevance of learning styles is the meshing hypothesis, according to which instruction is best provided in a format that matches the preferences of the learner (e.g., for a “visual learner,” emphasizing visual presentation of information).

The article isn’t addressing the idea of learning styles as a whole, but rather the idea that one can “teach to” the different learning styles with different methods.

This was a meta-study, which surveyed the literature on learning styles searching for scientifically verified studies. What the researchers found is very few studies of the studies done count as scientific:

To provide evidence for the learning-styles hypothesis—whether it incorporates the meshing hypothesis or not—a study must satisfy several criteria. First, on the basis of some measure or measures of learning style, learners must be divided into two or more groups (e.g., putative visual learners and auditory learners). Second, subjects within each learning-style group must be randomly assigned to one of at least two different learning methods (e.g., visual versus auditory presentation of some material). Third, all subjects must be given the same test of achievement (if the tests are different, no support can be provided for the learning-styles hypothesis). Fourth, the results need to show that the learning method that optimizes test performance of one learning-style group is different than the learning method that optimizes the test performance of a second learning-style group.

In essence, here would be a successful experiment:

A random group of subjects is tested for learning style. They are divided into two groups, A for the auditory learners, V for the visual learners.

The groups are further divided (randomly and with double-blind conditions, so the experimenters who have direct access to the subjects don’t know which group is which) into subgroups A1, A2, V1, V2.

Groups A1 and V1 are taught a lesson auditorially.

Groups A2 and V2 are taught the same lesson visually.

All groups are then tested.

Auditory learners taught auditorially (A1) do better than the auditory learners taught visually (A2).

The visual learners taught visually (V2) do better than the visual learners taught auditorially (V1).

The Learning Styles paper includes a graphical explanation:

Of the surveyed papers that met this criterion:

. . . we have been unable to find any evidence that clearly meets this standard. Moreover, several studies that used the appropriate type of research design found results that contradict the most widely held version of the learning-styles hypothesis, namely, what we have referred to as the meshing hypothesis (Constantinidou & Baker, 2002; Massa & Mayer, 2006).

However, the study emphasizes that this is just the current state of the research, but that:

Future research may develop learning-style measures and targeted interventions that can be shown to work in combination, with the measures sorting individuals into groups for which genuine group-by-treatment interactions can be demonstrated. At present, however, such validation is lacking, and therefore, we feel that the widespread use of learning-style measures in educational settings is unwise and a wasteful use of limited resources.

In other words, the meta-study did not debunk learning styles so much as debunk the current research and industry built around them, but that does not preclude the possibility (which hasn’t been well-tested enough) that some sort of learning style individuality may help.

Pashler, H., McDaniel, M., Rohrer, D., & Bjork, R. (2009). Learning Styles: Concepts and Evidence Psychological Science in the Public Interest, 9 (3), 105-119 DOI: 10.1111/j.1539-6053.2009.01038.x

How 10% of people see the number line

Estimates have historically varied from 4.5% to 12%, but the best current estimate is around 10%.

People with synesthesia experience at least one sensation simultaneously with another. For example, they might hear music and see colors with particular notes, or when they see words they might associate particular tastes. Many who have synesthesia aren’t aware it is a special condition, and can go all the way to adulthood before realizing other people don’t think that way.

One of the most common types of synesthesia is “number forms”, also known as “spatial sequence synesthesia”. These are people who when hear some sort of sequence (numbers, months, years) they associate and physically see it as some exact location in space.

[Image source: Visualised Numerals, an 1881 essay by Galton.]

Keep in mind this is not some sort of artificial mental mnemonic: this is a genuine and involuntary mental effect.

The above is an approximation of the number forms of a particular person; note how the numbers positioned to give some difficulty in telling time. “Curving” initial numbers are quite common. Also note how the numbers mentally start sorting themselves by 10s; this is also quite common.

[Statistics source: Wednesday Is Indigo Blue: Discovering the Brain of Synesthesia by Cytowic and Eagleman.]

Misuse of Bloom’s Taxonomy

You may be familiar with this method of categorizing the depth of different types of thought:

bloom taxonomy

(It’s the revised version; the original categories were labeled Knowledge, Comprehension, Application, Analysis, Synthesis, and Evaluation. Note the top two rows in the revised version have been swapped.)

In principle this is a perfectly sensible arrangement. In practice there is an inane focus on the verbs being used to phrase the question, with little or no attention paid to the thought process in between.

For example, this list of verbs represents “Understanding”
classify, describe, discuss, explain, identify, locate, recognize, report, select, translate, paraphrase

while this represents “Creating”
assemble, construct, create, design, develop, formulate, write.

Which of these prompts will result in deeper thought?

A. Explain why the graph of the tangent has asymptotes.

B. Construct a diagram of the unit circle.

Now, prompt A does have a simple end product (“division by zero”) but the thought process in between can be vast, whereas the latter prompt is clearly nothing higher than the “Remembering” level of the taxonomy.

If one sat and pulled apart the different methods used the taxonomy seems more sensible (clearly if extrapolated from scratch prompt A enters the realm of creation) but that isn’t how the taxonomy is generally used or taught, in my experience. The list of verbs is a checklist, a way for teachers to escape having to genuinely think about a problem. Ignoring the content is a poor way to construct higher-order questions.

The Dark Side of Positive Reinforcement

Still flustered and busy, but this news tidbit was too good to pass up.

From a press release regarding a new study of Joanne V. Wood and John W. Lee:

The researchers asked participants with low self-esteem and high self-esteem to repeat the self-help book phrase “I am a lovable person.” The psychologists then measured the participants’ moods and their momentary feelings about themselves. As it turned out, the individuals with low self-esteem felt worse after repeating the positive self-statement compared to another low self-esteem group who did not repeat the self-statement. The individuals with high self-esteem felt better after repeating the positive self-statement–but only slightly.

The psychologists suggested that, like overly positive praise, unreasonably positive self-statements, such as “I accept myself completely,” can provoke contradictory thoughts in individuals with low self-esteem. Such negative thoughts can overwhelm the positive thoughts. And, if people are instructed to focus exclusively on positive thoughts, they may find negative thoughts to be especially discouraging.

Typical teacher classroom poster, from Teacherstorehouse.com:


It’s not an equivalent situation, but it’s worth thinking about.