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Professor of Education at the University of Texas at Arlington and Director of the Southwest Center for Mind, Brain, and Education
Dr. Michael Connell
Ed Tech Designer & Visiting Researcher at the University of Texas’ Southwest Center for Mind, Brain, and Education
Introduction
In Part 1 of this article I posed a challenge you may still be considering. If you remember the Iceberg Challenge, your goal was to decide what would happen to the water level after all the ice melted. For many years, what nearly all my students found particularly irritating about this challenge (and me) is that I stopped providing the answer. You might be feeling that irritation too.If you read Part 1 and tried the challenge, are you feeling this irritation?
My goal – then or now – was not to be irritating. My goal is to use our collective experience of the iceberg challenge to clarify what we mean when we use the word understanding, so that we’re all talking about the same phenomenon in the same way with the same expectations.
In Part II we introduce an understanding scorecard to help expose the Illusion of Understanding and in turn define what understanding means in the area of math, and finally consider what choices may be available to Khan and all educators, especially those who work online, to better support authentic understanding.
What do we mean by understanding?
At one extreme, an understanding might mean that we know something (anything) about a subject, so that we can participate in a cocktail conversation. For example, imagine a person said to you, “I think it makes a difference to coast lines if all the polar ice floating on the oceans melts, but I’m not sure how…” Would you say that person understands Archimedes Principle?Alternatively, would you say a person understands Archimedes Principle if they can provide a definition or use a mathematical formula to solve for a missing variable?
At the other extreme, would you say a person understands Archimedes Principle who can:
 Recognize deeper connections between situations that seem unrelated on the surface  such as what happens to ice melting in a glass and what happens to a balloon full of oxygen released on Mars (whose atmosphere is predominantly carbon dioxide),
 Solve a variety of novel challenges like the Iceberg Challenge,
 Explain their reasoning and articulate why they believe their answers are correct across different contexts, and
 Recognize how a new concept or formula relates to what they have learned previously, so they can start using it quickly?
These three points of view (let’s call them “low,” “medium,” and “high” understanding, respectively) map out positions along a continuum that begin to portray understanding in a richer and more complex way. We may all discover that in the past we have been holding different assumptions when using words like “understanding” (or “learning,” for that matter).
Do these three points of view frame a continuum that feels useful to you?
Using this continuum as a shared point of reference, we can ask a couple of distinct but related questions:
 What outcomes are possible? What is the highest level of understanding that students can theoretically achieve in a given subject area on a large scale in a particular formal education system, given the available resources in that system?
 What outcomes are expected? What level of student understanding should we hold the formal education system accountable for in practice?
Even though the two questions are distinct, they are related. The first question (what is theoretically possible) puts a hard limit on reasonable answers to the second question (what the community demands of its educational system). Two common mistakes that people make when reasoning about education are:
 They assume a low level of understanding is the best that can be achieved at scale in an education system, and – without checking that assumption – they decide to set a low bar for student understanding based on it.
 Conversely, they ignore the ceiling on what is theoretically possible and make impossible demands of educational institutions.
How do we determine what level of understanding is possible?
Formal education systems are so complex that it is difficult to analyze them to determine what kind of results are possible from them. How should we measure student understanding given the complexity and unique features of different formal education systems? One way is to create a "scorecard" based on what the learning sciences claim will lead to high levels of understanding. Recall, in particular, the five principles from learning science about the conditions required to develop authentic understanding: Authentic understanding depends on hierarchically organized knowledge.
 Authentic understanding is grounded in direct experience.
 Authentic understanding is stabilized by practice (generally at every level within the hierarchy).
 Authentic understanding requires formative feedback.
 Authentic understanding is contextsensitive.
Watching the video takes about 8 minutes. Afterwards, see if rating the video as Low, Medium, or High on each of the five principles helps you summarize your reflections on the overall level of understanding we might expect from students using the video as an instructional tool. Of course, the more videos you watch, the easier it will be to generate a summary evaluation of the arithmetic curriculum.
Evaluation
criterion

Examples
of arithmetic activities supporting “high” understanding

Your_Rating of Khan_Academy
Arithmetic

Learning is grounded in experience

Handson learning experiences using [familiar
objects like] chips, dice, or paper clips to associate physical objects to
ordering, counting and symbols used to represent numbers.


Knowledge is hierarchically developed from the
student’s point of view.

Concepts learned in a hierarchical way: Understandings
begin as actions (as above), which precede and eventually support
understandings that are representations of actions (writing, speaking or
drawing), which in time support understandings that coordinate numerous
representations to form abstractions (like justice or calculus). If you want to know more about hierarchies
of understanding see this article (pages 34).


Provides scaffolded practice (preferably at
every level within the hierarchy)

The curriculum covers fewer concepts, so students
can spend significant time practicing with physical objects (chips, dice,
etc.) then with drawing pictures, then with symbols. The teacher helps them as necessary
(provides scaffolding) during this practice at every level of the hierarchy.


Provides formative feedback

As students practice with physical objects
(chips, dice, etc.) then with drawing pictures, then with symbols, their
level of understanding is made visible to themselves and the teacher, which creates opportunities for providing very
specific corrective feedback when a student gets stuck or misunderstands
(this is formative feedback).


Develops connections between abstract
principles and realworld contexts

The abstract principles are numbers,
operations, and the other symbolic formalisms of math. Students spend a lot of time developing
connections between these abstract principles and real world scenarios that they
are used to model.

How useful does the scorecard seem to you?
As you complete the scorecard, it might also help to consider some of the following questions (from a first grader’s point of view):
 Do you need to know what an avocado is to make sense of the instruction?
 How important is it that the avocado looks like an avocado on video?
 How comfortable does the child already need to be with the idea that the number “2” has a special relationship with the two avocados that Khan draws?
 What do the symbols “+” or “=” mean as used in Khan’s lesson?