While it might seem a tad heretical to my colleagues in education, I’ve never been terribly fond of Bloom’s Taxonomy. As I watched students show their thinking in class, the demarcations sometimes felt a bit artificial and too fine-grained to me; I would have a difficult time telling the difference between “knowledge”, “comprehension”, and “application”, for example.
Recently, I’ve been reading Charlotte Danielson’s Enhancing Professional Practice: A Framework for Teaching, and I’m now thinking of a different taxonomy, inspired in part by the first chapter in the book. Danielson does a nice job of describing constructivism, and in that explication, she hints towards a distinction between factual knowledge and conceptual knowledge, and suggests that each requires a different kind of teaching.
I can get on board with this. Getting students to memorize facts is often just a function of time, and frequency of exposure. Conceptual knowledge includes the relationship between those facts, and getting students to understand those relationships takes more deliberate practice in the classroom.
Further, I think we can elaborate on this idea of factual and conceptual. As a teacher of physics and math, I would say that there is also procedural knowledge: how to apply a recipe or algorithm. Math teachers love procedures.
Additionally, influenced by the great text How People Learn from the National Academy of Sciences, (“Key Finding 2: To develop competence in an area of inquiry, students must: (a) have a deep foundation of factual knowledge, (b) understand facts and ideas in the context of a conceptual framework, and (c) organize knowledge in ways that facilitate retrieval and application”) I think we can zoom out from simple concepts to a larger Conceptual Framework – the sort of knowledge that takes the relationship between smaller concepts, and organizes them into a larger set of relationships that ties the concepts together, that could help define the entire discipline or significant parts of it.
I’m not sure that this can apply to all disciplines, actually, but it really helps me think about teaching something like physics. Here is how I might break this down within my discipline, and writing, according to some of my ELA teaching friends:
|Factual||Gravitational acceleration is 9.8 m/s/s
The mass of a typical human head is around 3 kg
A force is a push or a pull
|A noun is a person, place or thing.
Use question marks to indicate a question.
|Conceptual||Newton’s 2nd law: an object accelerates more when there’s more force on it, and accelerates less when there is more mass. As an equation: a = F/m. It is also written as F=ma||Active voice is more powerful and more readable than passive voice.
An introduction to a piece of writing works best if it immediately connects to the reader.
|Procedural||If Force= 6 Newtons, and mass= 2 kg, then you can determine the acceleration: a = F/m = (6 N)/(2 kg) = 3 m/s/s||Sentences should be written, such that they include a noun, a verb, and punctuation.|
|Conceptual Framework||Systems can be described either by the forces acting on them (Newtonion mechanics), or by their energy (Lagrangian mechanics), or both.||Persuasive writing usually involves an introduction, evidence, analysis, and a conclusion that are all connected|
I think there is a fluid relationship between all of these kinds of knowledge, and particularly “facts” and “concepts”. As we become more sophisticated in our knowledge of a thing, we start to move “concepts” of that thing into our internal reservoir of “facts”. For example, while I recognize that Newton’s second law has several parts to it and internal relationships (making it a “concept”), I am so fluent in using and thinking about it that I often think of it as a single entity, as a building block for understanding more complex systems. The same is true for our multiplication tables. We can understand 5 x 3 as a concept of adding three groups of five (or five groups of 3), but eventually 5 x 3 = 15 becomes reflexive, and it becomes a fact that we can recall very quickly. So in a way similar to how we move new ideas from working memory into longer-term memory, so can concepts become factual chunks for us.
And so it goes: we then turn groups of concepts into conceptual frameworks of increasing size and complexity.
We need to be wary. People can learn and recall as “facts” things that just aren’t true. And when they string together untruths, the resulting concept’s problems are multiplied. An observation of the “flat earth” can yield all sorts of incorrect concepts about the nature of our planet and the way it moves, for example. Part of teaching involves diagnosing misconceptions; we should gauge our students around not only their understanding of the relationships between the facts, but also the assumptions (what they consider facts) they make when discuss the relationships.
Taking a page from Danielson’s book, different kinds of knowledge suggest different kinds of acquisition. Here are some ideas:
|Knowledge type||Notes, and means of acquisition|
|Factual||Often doesn’t require the teacher – can be found from several sources. Can be achieved through multiple exposures, like remembering someone’s name. Often done with association: name with face; number with operation (3 x 5 = 15); lyrics with music, etc.
|Conceptual||Brings together facts and finds a relationship. Sometimes the pattern IS the concept. Often requires some pattern-finding and analysis. When people think of a concept, they very frequently share with others to check their own understanding (like what Vygotsky said)|
|Procedural||Like factual, but includes a specific sequence. Requires practice, and feedback. Often referred to as a “skill”.
Easily attained if there is only one recipe/algorithm, but for any kind of elaboration or permutation, or “if-then” decisions, it often requires a conceptual understanding of what created the basic algorithm.
This is often what is tested in math exams.
|Conceptual Framework||Usually found by patterns and relationships among concepts, so it requires exposure to many concepts. Again – the relationship often IS the framework.
Teachers who don’t have a strong command of the subject they are teaching often don’t address this level of knowledge, because they don’t know it, and don’t know how to address it.
Again, different kinds of knowledge/acquisition suggest different approaches to instruction, and Danielson suggests that constructivism is an important approach to building concepts. I agree. Below are some examples and non-examples of constructivism. They are in a table, because I love tables.
|Concept||Example of Constructivist teaching||Non-Example of Constructivist teaching|
|Meter in poetry||Teacher gives students two poems with the same meter and tells them “these have the same meter. Please read them aloud to each other”. Teacher then gives them two different poems that are the same meter as each other, but different from the first two, and has the students do the same thing.
Finally, the teacher holds a class conversation about what they believe meter to be, based upon similarities and differences in the poems they read. If necessary, provides corrections and clarity, and finally a definition.
|Teacher might describe Iambic and Trochee meter to students with diagrams on the board, with students reading aloud.
The teacher then provides some examples of each to reinforce the idea of the kinds of meter previously introduced.
|Slope = velocity on position vs time graph||Teacher gives students three different position graphs of moving cars, moving at a constant speed. The graphs have numbers on both axes. Teacher asks the students to rate the cars according to which is going fastest, without telling them how to do so.
Teacher also asks the students to find the slope of the lines of the three graphs, and asks them to compare the slopes to the speeds.
Different students share out their findings. Teacher provides corrective feedback as necessary, and formalizes connection between slope and velocity.
|Teacher draws a position vs time graph on the overhead, explaining the process the entire time, calculates the slope of the line, and then shows that the slope is the same as the velocity. Students take notes. Teacher answers questions.|
I’ll let the reader develop his/her own working definition of constructivism. (See what I did there?) What I’ve come to discover is that while some students can individually perform the tasks asked of them in a constructive lesson, several other students may find it very frustrating; they aren’t sure where to start. Much more effective, I’ve found, are constructive lessons done with collaborative groups, where students bring several perspectives to bear in the pattern-finding or knowledge-building, and try ideas out on each other. In essence, the students construct the concept together. (This is, by the way, how much of knowledge in the world outside of school is created, particularly in science. Data is observed, analyzed, and conclusions are made, then checked with colleagues.)
Eventually, the concept must be owned by each student, and that requires individual effort, and perhaps some practice. So I’ve come to discover that the following seems to work:
- Collaboration is best for concept development
- Individual effort is best for concept mastery
I use this pair of ideas particularly when it comes to assessment, but also as I am designing my lessons, and considering – how far along the spectrum are the students in their concept formation? Should my lesson be designed for more individual focus, or more group interaction?
The following table is not at all complete, but meant to provide examples of approaches well-suited to acquisition of different kinds of knowledge.
|Knowledge type||Possible Instructional Approaches or Techniques|
|Factual||Any kind of repetitive process that provides multiple exposures to an idea.
Synchronous or asynchronous instruction.
Lecture followed by Reading
Reading on its own
Quizzing for mastery
Lab activities where students verify what is already known (e.g. “verify that the acceleration due to gravity is 9.8 m/s/s”)
|Conceptual||Constructivist, Student-centered teaching techniques that allow students to try new ideas on each other. On a day to day basis, that might include
– Socratic Seminar
– Card Sorts
– Concept Attainment
– Chalk talk
– Interactive Lecture
– Concept maps
On a unit basis, it might include problem based learning and project based learning (PBL).
Pattern-finding using data or sets of numbers, sets of facts, sets of concepts.
Lab activities where students use data to determine a relationship.
|Procedural||“I do/we do/you do”
Approaches that require student practice with a lot of just-in-time feedback.
Strategic use of non-examples (what did Johnny do wrong is his problem-solution)
Some lab activities, like titration, where the technique is target idea.
|Conceptual Framework||Pattern-finding using data or sets of numbers, sets of facts, sets of concepts.
Project Based Learning
Reflective writing after several units or projects that require connections between the units.
Ineffective classroom instruction is often the result of a mismatch of a teaching approach with the kind of knowledge that is desired as an outcome. Among the primary perpetrators of this: “I do/we do/you do”. This is quite effective when you need to build skill in a procedure, like “completing the square” in algebra. But if you want students to understand that acceleration is inversely proportional to mass, then a different technique (like a lab where students can discover that relationship) is much better suited than “I do / we do / you do”.
As a teacher, then, planning for your instruction should include a consideration of the type of knowledge you want to impart. You might have in your mind that you want to teach the parts of the cell in a biology course. There are two parts to that, at least: (1) the factual names and functions of the various parts, e.g. the cell membrane acts as a barrier and filter, and mitochondria who provide energy for the cell, and then (2) the conceptual nature of how those parts interact, given the various roles of cells. This might suggest that you could so some parts of an activity that include the factual acquisition (like with flashcards and quizzing for mastery), but then some interactive group work that might have a group create a concept map that connects all of the functions of a cell together.
I hope this is useful to someone. By one recent measure, there 305 trillion, yes trillion, choices to be made about instruction, so planning the classroom environment and learning experience can be extraordinarily complex, leaving a teacher with choice paralysis. By considering the type of knowledge, and considering a limited number of means of imparting or creating that knowledge, perhaps this framework could help a teacher become just a little more efficient and effective.
(post script: A follow-on to “How People Learn”, prosaically titled, “How People Learn II” has come out, and is free for a PDF download. I plan on reading that here soon to see if I need to adjust this framework)