**A Map of Operationally Connected Categories**

**as an instrument for classifying physics problems**

**(and a basis for developing a universal standard for measuring learning outcomes of students taking physics courses;**

**a novel tool for measuring learning outcomes in physics).**

A shorter version of this
paper was presented at 2016
AAPT-NES meeting (for slides, please visit http://www.teachology.xyz/prnes.htm or www.GoMars.xyz/prnes.pdf

or a short video).

or a short video).

**Abstract**

Currently there is no tool for measuring
learning outcomes of students, which would be broadly accepted by teachers,
schools and district officials, by parents, policymakers, and researchers.
Educational standards cannot provide a basis for such a tool, since for an
educator “a standard” means a

*verbal description*of skills and knowledge which students should be able to demonstrate. The actual realization of the use of that verbal description has not been standardized.
In physics a standard (also called an etalon, or a prototype) is
an actual object, or a feature of an object, accompanied by a specific
procedure which allows comparing similar features carried by other objects with
the standard one. The existence of such standards had allowed physics to reach the status of a developed science and
become a foundation for significant engineering and technological advances.

Educational research, though, in its current
stage, cannot be assessed as a developed, but rather as a developing science.
On of the main reasons for that is the absence of the scientific-type standard
for measuring subject knowledge.

There is however an approach to standardization
of measurement of physics knowledge similar to standardization of measurements
in physics.

This approach is based on a specific technique
used for classification physics problems. At the core of such classification is
the use of graphs (in the sense of the graph theory, or nets, networks), such
that:

**1. every physical quantity represented by a vertex/node of a graph must have a**

*numerical representation*, i.e. has to be measurable (capable of being measured, i.e. there has to be a procedure leading g to a numerical value of the quantity represented by a vertex/node); and**2. every link between any to vertices must have an**

*operational representation*: i.e. for any quantity represented by a vertex, if its value is getting changed, and the values of__all but one other quantities__represented by other vertices connected to the changing one are being kept constant, the quantity represented by the remaining vertex linked to the changing one must change its value.**The main article**

“What can’t be measured - can’t be managed”. Outside
of science, this sentiment is also widely accepted by the members of a business
community. In education, however, the discussion of “accountability”, and
“measurability” brings many controversial views and opinions on absolutely all
levels of education, research, and educational policy making. What should not
and what should be measured, when, by whom, what conclusions and decisions should
be derived from the results? But the most controversial question is “how”?
Currently there is no tool for measuring learning outcomes of students, which
would be broadly accepted by teachers, schools and district officials, by
parents, policymakers. An open to public, consistent with the taught and
learned content, and uniform by the procedure and results tool for measuring
learning outcomes does not exist. The situation is like all 50 states had
different currency, and there were no exchange rates, hence there was no way to
use money from one state in another one, the only way to compare the value of
different goods would be barter.

As the consequence, any data on the quality of
teaching are usually readily adopted by one “camp” and ignored or even vigorously
disputed by another one (for example, read “Boston’s charter schools show striking gains”, The Boston Globe,
March 19, 2015 issue).

As a physics teacher I am concerned with having
a reliable tool for measuring learning outcomes of students taking physics
courses. In educational sciences such a tool is usually called “a standard”.
However, the meaning of this term in educational sciences is very different
from its original meaning originating from physics.

For an educator “a standard” means a verbal
description of skills and knowledge which students should be able to
demonstrate, i.e. it is a verbal description of the achievement expectations
for students graduating from a certain level of education. There is no
guaranty, however, that when different assessment developers use the same
standard for preparing testing materials (or other assessment tools) the
materials would be equivalent to each other.

In physics term “standard”
has a very different meaning; a standard is an actual object, or a feature of
an object, accompanied by a specific procedure which allows comparing similar
features carried by other objects with the standard one (that is why “a
standard” is also often called “a prototype”, or an etalon). For example, a
standard of mass is an actual cylinder. A verbal description such as: “A
standard of mass looks like a cylinder with diameter and height of about 39 mm,
and is made of an alloy of 90 % platinum and 10 % iridium” would not work as a
standard, because it is impossible to compare the mass of another object with a
sentence.

Over the decades of intensive research, many
instruments for measureing learning outcomes of students taking physics courses
have been proposed, developed, and used as possible standards, including but
not limited to: the Force Concept Inventory (Hestenes et al.), the Force and
Motion Conceptual Evaluation (Thornton, Sokoloff), the Brief Electricity and
Magnetism Assessment (Ding et al.), SAT physics subject test, AP physics exam,
MCAT (physics questions). However, all current tools for measuring learning
outcomes suffer from at least one, or several deficiencies listed below:

1. the list of concepts depends on the
preference of a developer and usually not open to a user and to a public (when
using an inventory one can derive an assumption about the concepts analyzing
the questions in the inventory, but that is only an assumption, and it also
depends on who uses the inventory).

2. The set of questions depends on the
preference of a developer.

3. The set of questions is limited and cannot be
used to extract gradable information on student’s skills and knowledge.

4. The fundamental principles used for
development of a measuring tool are not clear and not open to public, teachers,
instructors, professionals.

5. The set of questions becomes available for
public examination only after being used, hence it only reflects the view of
the developers on what students should know and be able to do, there is no
general consensus among the users, instructors, and developers on what should
be measured and how to interpret the results.

6. The set of questions change after each
examination, which makes virtually impossible to compare the results of
students taking different tests (public should just rely on the assertion of
the developers that the exams are equivalent, having no evidence of that).

Based on a physics approach to what “a standard”
is, I propose that a tool and a procedure used to measure learning outcomes of
students taking physics course (or, for that matter, any STEM course) must
satisfy the following conditions:

(a) Every aspect of the
development and the use of the tool has to be open to public and be able to be
examined by

*anyone*.
(b) The use of the tool must
lead to gradable information on student’s skills and knowledge.

(c) The use of the tool must
lead to gradable information on student’s skills and knowledge, AND must not
depend on any specific features of teaching or learning processes.

(d) The use of the tool must
lead to gradable information on student’s skills and knowledge, and must not
depend on any specific features of teaching or learning processes, AND must
allow to compare on a uniform basis the learning outcomes of any and all
students using the tool.

(e) Any institution adopting
the tool should automatically become an active member of the community
utilizing the tool and can propose possible alternations to the tool to
accommodate changes in the understanding of what students should know and be
able to do.

The latter condition mirrors the history of
adaptation of standards in physics, when the community of scientist agreed on
the common set of standards and measuring procedures (if follows the
consensus-building process).

We can make a reasonable assumption, that after
the initial adoption of the tool by a number of participated collaborators (a
school, a college) and demonstration of its accuracy and uniformity, this
approach will attract attention of officials of different levels, (school
principals, district, city, state officials, staff of philanthropic
foundations). The demonstration of such a tool and its applicability would have
a profound positive effect on the whole system of education, leading to a
better comparability, measurability, accountability without having negative
effects which had led to a resistance of the use of standardized tests. Such a
tool (or rather a set of tools developed for all STEM subjects and for all
levels of learning) would be adopted voluntarily and gradually by schools and
districts forming a community of active co-developers. Members of the community
who adopted the tool would be issuing regular updates, keeping it in agreement
with the current understating of what should students know and be able to do.

It might seem impossible to develop a tool,
which would satisfy all the conditions above. However, in fact, there is a
singular example of the use of a tool similar to that: at the time without a
solid theoretical foundation but rather as a practical instrument, a similar
tool for measuring the level of physics knowledge of prospective students (high
school graduates applying to an institute) had been used at Perm Polytechnic
Institute (Russia, ~ 1994 - 2003).

There is no common view on how to probe
student's understanding of physics. Term “understanding” is being understood
differently, has various interpretations and cannot be used as a basis for
development of a reliable measuring instrument.

However, it is much easier to find a consensus
between physics instructors on

**what type of problems an A student should be able to solve**. This idea has been used as a basis for a tool used to measure the level of physics knowledge of prospective students (the approach can be called “a driving exam approach” because it focuses on probing specific skills form a set of well-defined skills require to graduate).
In early ninetieths Perm Polytechnic Institute
experienced a large amount of high school graduates applied to becoming a PPI
student (the competition to become a PPI undergraduate was fierce).

*Every*applicant had to take a physics entrance exam, and administration was in the need of a relative quick, objective and equivalent to all students procedure to measure the level of physics knowledge of the applicants. To answer this need, a set of about 3000 problems had been developed and used to compose an exam constituted of 20 problems with different difficulty level to be offered to high school graduates applied to become PPI undergraduates. Every exam was graded according to the same scale leading to a usable grade distribution. The philosophy behind the approach was simple: it is just impossible to memorize the solutions of all 3000 problems in the book (a book was openly sold via PPI book store), and even if there was a person who could memorize all the solutions, PPI would definitely wanted to have that person as a student.
The methodology for development of the tool was
based on a “driving exam” approach, which requires - instead of a verbal
description of skills and knowledge a student has to display after taking a
course (a.k.a. “educational standards”) – a collection of exercises and actions
for which a student has to demonstrate the ability to perform, assuming the full
level of learning (a.k.a. “physics standards”).

Currently this methodology is based on the
following four principles.

1. In physics every component of student’s
physics knowledge or a skill can be probed by offering to a student to solve a
specific theoretical or practical problem (to probe rote knowledge a student
can also answer a specific question like “what is ...”).

2. For a given level of learning physics there
is a set of problems, which can be used to probe student’s knowledge and
skills.

3. For a given level of learning physics a set
of problems, which can be used to probe student’s knowledge and skills, has a

*finite*number of items.
Hence, it is plausible to assume that to test
the level of acquired physics knowledge and skills a collection of problems can
be developed which - when solved in full - would demonstrate the top level of
achievement in learning physics (an A level of skills and knowledge). A subset
of a reasonable size composed from problems selected from the set can be used
as a test/exam, which can be offered to a student to collect gradable
information on student’s skills and knowledge.

The path to the theoretical foundation of the
describe approach had been laid out by the author in 1996 – 1998 (a shorter version
of this work has been published in 2015 at http://stacks.iop.org/0031-9120/50/694 or visit http://www.cognisity.how/2018/04/LernAids.html).

This paper represents a short summary and a
further advancement of the approach and its application to developing a framework
for standardizing knowledge measurement in physics.

4. The fourth principle states that all physics
problems can be classified by (a) the set of physics quantities which have to
be used to solve a problem; (b) the set of mathematical expressions (which by
their nature are either definitions or laws of physics, or derived from them)
which have to be used to solve a problem; and (c) the structure of connections
between the quantities, which defines the sequence of the steps which have to
be enacted in order to solve a problem (an example of a general “algorithm” for
solving any physics problem is at http://www.cognisity.how/2018/02/Algorithm.html).

The latter principle leads to the introduction
of the new terminology needed to describe and classify different problems (in
physics many similar problems use very different wording which makes them very
similar to math word problems; learning how to solve physics problems automatically boosts the ability to learn word problems).

All problems which can be solved by applying (a)
the

*exactly same sets of quantities*and (b)*expressions*, and (c) using the same sequence of steps are__congruent__to each other. Problems which use (a) the same set of quantities and (b) expressions (b) but differ by sequence of steps are__analogous__problems. Two problems for which set of physics quantities differ by one quantity are__similar__.
Four problems below are offered to illustrate
the terminology. Among three problems below problem A is analogous to problem
B, and congruent to problem C. Problem D is similar to problems A, B, and C.

__Problem A.__

“For a takeoff, an airplane needs to reach speed
of 100 m/s. The engines provide acceleration of 8.33 m/s

^{2}. Find the time it takes for the plain to reach the speed.”__Problem B.__

“For a takeoff, an airplane needs to reach speed
of 100 m/s. It takes 12 s for the plain to reach this speed. Find acceleration
of the plain during its running on the ground.”

__Problem C.__

“A car starts from rest and reaches the speed of
18 m/s, moving with the constant acceleration of 6 m/s

^{2}. Find the time it takes for the car to reach the speed.”__Problem D.__

“For a takeoff, an airplane needs to reach speed
of 100 m/s. The engines provide acceleration of 8.33 m/s

^{2}. Find the distance it travels before reaching the speed.”
It is important to stress, that all congruent
problems can be stated using

*a general language not depending on the actual situation described in a problem*. Problem E below is congruent to problems A and C and stated in the most general language not connected to any specific situation.__Problem E.__

“An object starts moving from rest keeping
constant acceleration. How much time does it need to reach a given speed?”

Based on the fourth principle we can also state
that for every problem a unique visual representation can be assign to it,
which reflects the general structure of the connections within the problem.
This visual representation is a graph (an entity of a graph theory): each node
(vertex) of the graph represents a physical quantity without the explicit use
of which a problem cannot be solved; each edge of the graph represents the
presence of a specific equation which includes both quantities connected by the
edge (link). Each graph represents a specific example of a knowledge mapping,
but has to obey to two strict conditions (which makes each graph unique and
this approach novel):

1. every quantity represented by a vertex/node
of a graph must have a numerical representation, i.e. has to be measurable
(capable of being measured, i.e. there has to be a procedure leading g to a
numerical value of the quantity represented by a vertex/node).

2. every link (edge) between any to vertices
must have an operational representation: i.e. for any quantity represented by a
vertex, if its value is getting changed, and the values of all but one other
quantities represented by other vertices connected to the changing one are
being kept constant, the quantity represented by the remaining vertex linked to
the changing one must change its value.

Graphs, which satisfy the above two conditions,
represent a novel visual technique for representing a specific scientific
content, called “a map of operationally connected categories” or MOCC.
Knowledge maps usually represent the thought process of an expert in a field
solving a particular problem (analyzing a particular situation). MOCCs
represent

*objective*connections between physical quantities, which imposed by the laws of nature.
The second condition eliminates possible
indirect connections/links (otherwise the structure of a graph would not be
fixed by the structure of the problem, but would depend on the preferences of a
person drawing the MOCC). However, even in this case there is a room for a
discussion: should the links represent only definitions and fundamental laws or
also expressions derived from them? This question should be answered during the
trial use of the developed tool.

Expanding the language introduced above, we
define that all problems which MOOCs include exactly same set of quantities are
called

__like__problems (they compose a set of like problems). A problem which is stated in a general language and which is like to all problems in a set of like problems is the__root__problem of the set; any specific problem in a set can be seen as a variation of the root problem.**A set of root problems can be used to describe desired level of learning outcomes of students.**
Members of the community using the tool for
measuring learning outcomes of students taking physics course would agree on
the set of root problems used as the basis for the tool. Then each member could
use its own set of specific problems
congruent to the set of root problems. This approach would ensure the complete
equivalency of the results when measuring learning outcomes.

One of the consequences of the graphical representation
of a problem is the ability to assign to it a

*n objective numerical indicator of its*__difficulty__:*D = N*

_{V}*+ N*, which is equal to the sum of the number of vertices included in MOCC,

_{E}*N*, and the number of unique equations represented by the links of in MOCC,

_{V}*N*. This indicator can be used for ranking problems by their difficulty when composing a specific test to be offered to a student. This indicator of difficulty does not depend on a perception of a person composing a test and provides uniformity in composing tests (this is the simplest objective indicator of difficulty of a physics problem which is based on the topology of MOCC corresponded to a problem).

_{E}
For example: every specific problem congruent to
problem E corresponds to MOCC in Fig. 1 with difficulty

*D*= 5 (four quantities and one equation which relates them).**Appendix I: Further applications of MOCCs**

Physics knows how important visualization can be for its development. Teachers constantly tell students to start their problem-solving process with drawing a picture or a diagram. Invention of diagrammed approach to visualize mathematical structure of calculations in quantum electrodynamics via Feynman's diagrams had led to the Feynman's Nobel Prize.

Nowadays MOCCs represent the most accurate instrument for classification of physics problems and developing a tool for objective measuring learning outcomes of students taking physics courses. The same principle also can be used for developing an analogues tool for measuring learning outcomes of student taking any STEM course.

Nowadays MOCCs represent the most accurate instrument for classification of physics problems and developing a tool for objective measuring learning outcomes of students taking physics courses. The same principle also can be used for developing an analogues tool for measuring learning outcomes of student taking any STEM course.

It is worth to note of the use of MOCCs as a
pedagogical instrument. The visualization of logical connections between
quantities helps students to grasp the logic of creating of the solution of a given
problem. Some students develop a habit of using MOCC when study physics. MOCC
also can be used as an end of a topic assignment, when students should use it
to visualize all the important connections they have learned. Below is an
example of such a MOCC developed by a high school student after topic “Direct
Electric Current”. On the pedagogical use of MOCCs and other learning aids, please,
visit http://www.cognisity.how/2018/04/LernAids.html.

All like problems have a MOCC with the same vertices but different links used to solve a problem. Each MOCC describes a set of like problems. Each like problem (in general – any problem) can be stated in a general (i.e. abstract) language which does not depend on the specific physical situation describe in the problem (click here for the example). A problem which is stated in a general language and which is like to all problems in a set of like problems is the

All like problems have a MOCC with the same vertices but different links used to solve a problem. Each MOCC describes a set of like problems. Each like problem (in general – any problem) can be stated in a general (i.e. abstract) language which does not depend on the specific physical situation describe in the problem (click here for the example). A problem which is stated in a general language and which is like to all problems in a set of like problems is the

__root__problem of the set; any specific problem in a set can be seen as a variation of the root problem.**A set of root problems can be used to describe desired level of learning outcomes of students.**

I believe the time has
come to create a coalition of individuals and institutions who would see the
development of objective and universal tools for measuring learning outcomes in
physics as an achievable goal.

The first step would be agreeing on the set of
root problems and classification them based on the difficulty. We need a group
of people who would collected all the problems from all available textbooks and
routinely converted each problem into a root problem, draw a MOCC for each
problem, calculated its difficulty.

All individual MOCCs combined together would
represent the map of objective connections in physics as a field of a natural
science.

This map could have been used to teach AI to solve physics problems in a way it uses
geographical maps to “solve driving problems” (in addition with other learning
tools this AI could replace eventually an average physics teacher). But for us
MOCCs represent the most accurate instrument for classification of physics problems
and the nucleus of the tool for measuring learning outcomes of students
objectively and uniformly (i.e. comparably).

Thank you for visiting,

Dr. Valentin Voroshilov

Education Advancement Professionals

GoMars.xyz

Dr. Valentin Voroshilov

Education Advancement Professionals

GoMars.xyz

To learn more about my professional experience:

The voices of my students

"The Backpack Full of Cahs": pointing at a problem, not offering a solution

Essentials of Teaching Science

The voices of my students

"The Backpack Full of Cahs": pointing at a problem, not offering a solution

Essentials of Teaching Science

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**References**

Thornton R.K., Sokoloff D.R. (1998) Assessing
student learning of Newton's laws: The Force and Motion Conceptual Evaluation
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Hallouin I. A., & Hestenes D. Common sense
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Hestenes D., Wells M., Swackhamer G. 1992 Force
concept inventory. The Physics Teacher 30: 141-166.

Hestenes D., 1998. Am. J. Phys. 66:465

Ding L., Chabay R., Sherwood B., & Beichner
R. (2006). Evaluating an electricity and magnetism assessment tool: Brief
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“Learning aides for students taking physics”,
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Voroshilov V., “Universal Algorithm for Solving
School Problems in Physics” // in the book "Problems in Applied
Mathematics and Mechanics". - Perm, Russia, 1998. - p. 57.

Voroshilov V., “Application of Operationally-Interconnect
Categories for Diagnosing the Level of Students' Understanding of Physics” //
in the book “Artificial Intelligence in Education”, part 1. - Kazan, Russia,
1996. - p. 56.

Voroshilov V., “Quantitative Measures of
the Learning Difficulty of Physics Problems” // in the book “Problems of
Education, Scientific and Technical Development and Economy of Ural
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