Put not your trust in princes.Psalm 146, KJV
Triangles and pyramids do to teachers what catnip does to cats.
Put just about any idea in the form of a three-sided polygon and watch teachers adopt it en masse as an article of faith. And, boy, have we as a profession unquestionably and uncritically adopted some stinkers.
What follows is a countdown, from least-worst to worst (in my estimation), of what I would collectively call . . . [PLAYS SINISTER ORGAN NOTES, ACTIVATES VOICE-ECHO] . . . PERFIDIOUS PYRAMIDS!
Number 4: Maslow’s Hierarchy of Needs
Why bring this one up? Firstly, Maslow never put his hierarchy in the form of a pyramid. This implies that all of a student’s ‘Deficiency Needs’ must be met before the ‘Being (growth) Needs’ can be addressed; Maslow was more nuanced in his original writings.
The analogy of psychological needs to vitamins was drawn by Maslow. Like vitamins, each of the needs is individually required, just as having much of one vitamin does not negate the need for other vitamins. All needs should independently contribute to subjective well being.Tay and Diener 2011
Secondly, the methodology by which Maslow arrived at the characteristics of a ‘self-actualized’ person was by looking at the writings and biographies of a number of people (including Albert Einstein and Mother Theresa) whom he considered to be ‘self-actualized’: this is a qualitative and subjective approach that would seem highly open to personal bias and hard to characterise as ‘scientific’ (McLeod 2018).
Number 3: Bloom’s Taxonomy
As with Maslow, there is little to argue with the intent behind Bloom’s Taxonomy, which was an attempt to classify educational objectives without — repeat, without — arranging them into a formal hierarchy.
Bloom’s Taxonomy is often missapplied in education because the ‘higher’ levels are deemed more desirable than the ‘lower’ levels.
As Sugrue (2002) notes:
It was developed before we understood the cognitive processes involved in learning and performance. The categories or ‘levels’ of Bloom’s taxonomy … are not supported by any research on learning.
the popular misinterpretation of the taxonomy has led to a multi-generational loss of learning opportunities. It is a triumph of philosophy over science, of populism over rigour, of politics over responsibility.James Murphy, The False Dichotomy
Number 2: Formula Triangles
The main issue with formula triangles is that they are a replacement for algebra rather than a system or scaffold for supporting students in learning how to manipulate equations.
Koenig (2015) makes some trenchant criticisms of forumula triangles, as does Southall (2016) who argues that they are a form of ‘procedural’ teaching rather than the demonstrably more effective ‘conceptual’ teaching. Conceptual teaching encourages students to understand why a particular technique is used rather than applying it as a ‘magic’ formula. Borij, Radmehr and Font (2019) also have an interesting and nuanced discussion on these types of teaching (in the context of learning calculus).
Workable alternatives to formula triangles are the FIFA and EVERY systems.
But the winner for the educationally worst pyramid or triangle is . . . [DRUM ROLL]
NUMBER 1: The Learning Pyramid
To put it bluntly, there is no research to support the percentage retention rates claimed on any version of this pyramid.
Modern versions of the learning pyramid seem to be based on Edgar Dale’s ‘Cone of Experience’ first published in 1946 in his influential book Audio-Visual Teaching Techniques.
Dale’s main argument was to encourage
the use of audio-visual materials in teaching – materials that do not depend primarily upon reading to convey their meaning. It is based upon the principle that all teaching can be greatly improved by the use of such materials because they can help make the learning experience memorable…this central idea has, of course, certain limits. We do not mean that sensory materials must be introduced into every teaching situation. Nor do we suggest that teachers scrap all procedures that do not involve a variety of audio-visual methodsDale 1954 quoted by Lalley and Miller 2007
The peculiarly neat percentage increments in retention rates on the learning pyramid are first found in Treichler (1967). As Letrud and Hernes (2018) note:
Treichler asserted that these numbers came from studies, but he did not say where they could be found. […] A set of learning modalities similar to those distributed by Treichler were at some point fused with a misreading of Edgar Dale’s Cone of experience as a hierarchy of learning modalities, and these early categories were supplemented and partly replaced with categories of presentation modalities like “audiovisual”, “demonstrations”, and “discussion groups”.
The final word is perhaps best left to Lalley and Miller:
The research reviewed here demonstrates that use of each of the methods identified by the pyramid resulted in retention, with none being consistently superior to the others and all being effective in certain contexts. A paramount concern, given conventional wisdom and the research cited, is the effectiveness and importance of reading and direct instruction, which in many ways are undermined by their positions on the pyramid. Reading is not only an effective teaching/learning method, it is also the main foundation for becoming a “life-long learner”
Borji, V., Radmehr, F., & Font, V. (2019). The impact of procedural and conceptual teaching on students’ mathematical performance over time. International Journal of Mathematical Education in Science and Technology, 1-23.
Dale, E. (1954). Audio-visual methods in teaching (2 ed.). New York: The Dryden Press.
Koenig, J. (2015). Why Are Formula Triangles Bad? Education In Chemistry, Royal Society of Chemistry.
Lalley, J., & Miller, R. (2007). The learning pyramid: Does it point teachers in the right direction. Education, 128(1), 16.
Letrud, K., & Hernes, S. (2018). Excavating the origins of the learning pyramid myths. Cogent Education, 5(1), 1518638.
McLeod, S. (2018). Maslow’s hierarchy of needs. Simply psychology, 1, 1-8.
Southall, E. (2016). The formula triangle and other problems with procedural teaching in mathematics. School Science Review, 97(360), 49-53.
Sugrue, B. Problems with Bloom’s Taxonomy. Presented at the International Society for Performance Improvement Conference 2002
Tay, L., & Diener, E. (2011). Needs and subjective well-being around the world. Journal of personality and social psychology, 101(2), 354.
Treichler, D. G. (1967). Are you missing the boat in training aids? Film and Audio-Visual Communication, 1(1), 14–16, 28–30,48.
I think the recent problems of the Scottish Education system was its expansion in the use of different explanatory graphic shapes to sustain the momentum of CfE. There was the concentric pizza, & I forget the rest.
The real problem was that if the process of CPD is going to be “train the trainer” you need a good choice of cartoon strips to sustain interest. Sadly Calvin and Hobbes was too close to the bone
“Further reflection, in part related to comments on the essay on the FQXi website, has led to the further development of the above ideas and also in some instances their reformulation. A comment to the effect thatthe arguments would be better founded not on the little known semiotics but on physics led to the realisationthat Peirce’s thirdness, or equivalently triadic relationships, while it forms the basis of semiosis, has wider relevance and is indeed found in situations such as that of Jupiter’s satellites, it being the case in that situation
that a linear relationship exists between the three orbital phases of Io, Europa and Ganymede …”
Brian Josephson (Nobel Prize winner) via “Going beyond Physics Paradigm” on YouTube Dec2020
Semiotics is a can of worms I have so far failed to dive into, as I find it akin to being in a hall or mirrors — you are never quite sure what is real and unreal. That said, it’s interesting that a Nobel laureate put forward the idea. For my own part (way below the Nobel laureate level) is that there’s a lmit to what can be achieved by manipulation of symbols, although I still think there’s a lot of mileage in A. J. Ayer’s contention that all the truths of mathematics are essentially tautologies.