Understanding is

• the ability to connect a representation to many other representations. If you understand something in only one or two ways, you scarcely understand it at all.¹⁾

• a psychological process related to an abstract or physical object, such as a person, situation, or message whereby one is able to think about it and use concepts to deal adequately with that object. ²⁾

• a relation between the knower and an object of understanding.³⁾

Other suggested aspects of understanding are⁴⁾:

• Understanding has many operational deffinitions based on subject area. For example: understanding language, understanding mathematics, understanding one's behaviour. One specific definition is therefore not the only possible method for measuring.
• Teaching, learning and assessment methods depend on the definition of understanding that wants to be achieved. A broader repertoire of teaching styles and strategies = broader understanding.
• Understanding of a concept is a continuum, not a dichotomous state.
• Understanding is to complex to be assessed by only single style of test, or satisfactorily represented by one single numerical score.

1. Understanding can be discussed only for concepts that already are a part of one's acquired knowledge (schemata). In other words, knowledge is prerequisite for understanding.
   • For example, one can not understand how a plane can fly if one does not know what is a plane i.e. does not have a concept of plane in his schemata.
   • On the other hand, even if one has a concept of plane in his schemata i.e. knows what a plane is, that does not necessarily mean that he understands how a plane can fily.
2. Understanding is a state defined by the relation between a concept of understanding and other concepts in schemata as well as the structure of the concept of understanding.
   • For example, one can prove to know the principle of osmosis by stating the definition: “Osmosis is the net movement of solvent molecules through a partially permeable membrane into a region of higher solute concentration, in order to equalize the solute concentrations on the two sides”⁵⁾. Yet to understand it, one has to not just know the definition but also (at least) the meaning of all the words (concepts) it includes, for example the concepts of a solvent and concentration. Once the concept of osmosis is acquired in one's schemata, understanding of that concept reflects how it is integrated into the concept map: how developed and interrelated are (among others) subconcepts of conceptration and solvent and how interrelated is the very concept of osmosis into ones schemata, for example related to concepts of plants or roots since osmosis is responsible for the ability of plant roots to draw water from the soil.
3. Understanding is a continuous state, proportional to the number and strength of connections between the concept of understanding and other elements of knowledge as well as within the concept of understanding. This can also be described as the meta-knowledge about the object of understanding.
   • There are more than just two states (yes/no) of how a concept can be integrated into one's concept map or schemata, therefore there are more than just two states of understanding. The concept of osmosis can include only the basic subconcepts like partially permeable membrane, solvent and concentration, or it can include additional detailed subconcepts like osmotic pressure, biological membranes, lipids, solubility, and charge. The concept of osmosis can also be related to any number of other concepts like cell turgor, thermodynamic free energy, diffusion, active transport, etc.
4. Understanding derives from ones schemata and cognitive reasoning and reflects in the ability to:
   • define a set of simple rules that apply to the concept of understanding,
   • predict its behaviour,
   • explain, interpret or conclude about its relation to other concepts or knowledge.

White and Gunnstone suggest the following forms of understanding⁶⁾:

• Conceptual understnading - set of memory elements one associates with the label; improved by better memory, better connections or more clarity in elements' formulation
• Understanding whole disciplines (?)
• Understanding single elements of knowledge - grammar, procedures, rules
• Understanding extensive communication - a poem, speech, painting, ballet, block of text
• Understanding situations - seeing paralells between a situation and previous experiences; having a script for it
• Understanding people - seeing paralells between a person's actions and their previous actions, and being able to predict actions

Wiggins and McTighe suggest, as an alternative (or supplement) to Bloom's Taxonomy of Learning outcomes, the “6 Facets of Understanding”, claiming students can demonstrate their understanding if they can:

• Explain
• Interpret
• Apply
• Have perspective
• Empathize
• Have self-knowledge

Diagnosed failures of understanding can be caused by⁷⁾:

• Too few connections
• Wrong connections (misunderstanding)
• Not having an index for retrieval (i.e. not knowing a word)
• Wrong symbols used in index (a communication failure)

We suggest that, since understanding is a function of acquired elements of knowledge and relations between them, all failures of understanding are caused by:

• Lacking connections or lacking elements of knowledge
• Wrong connections or wrong elements of knowledge

Since understanding generally has various definitions

Assessment of understanding is usually reffered to as assessment of conceptual understanding. This is achieved using the carefully developed multiple-choice tests (concept inventories) that examine conceptual understanding on a narrow set of topics.⁸⁾ Questions used in the assessment are supplied with potential answers addressing common student misunderstandings. Some authors combine multiple-chioce and open-ended questions.⁹⁾

• “The questions were intended to assess conceptual understanding of the learning goals rather than simple factual recall.”¹⁰⁾

Concept maps and ontologies are another mean often utilized to assess understanding since they explicitize relations between domain concepts. Still, it most oftenly remains unclear in such methods what is the difference between the knowledge and understanding, i.e. why (and how) do concept maps assess understanding¹¹⁾.