Community
Studies: Publications
The Problem Solving Process
Seven
Core Thinking
Skills for Mathematics
Expectations:
The
Student will:
- Describe seven key structured thinking skills and explain how these skills are used to apply and enhance the problem-solving process.
- Identify important prerequisites to becoming a good thinker.
- Synthesize (ie “output”) Portfolio entries connecting Thinking Skills, the Learning Process and the Mathematics Problem-Solving Process.
- Apply the seven key structured thinking skills to a problem in geometry.
- Summarize learning through organized reflection.
Assignment:
This
is individual work. The questions are in Section 2 below.
Mark:
___ /
15 Due Date: Today
or Homework Tonight
What is the
single biggest difference between life and your personal computer? There is no “undo”
command in life.
This
Sub-Module is Important Because:
- To
keep or advance in a job you must be able to think clearly -- to propose
better ways of doing something; to save your employer money; to be able to
use new tools quickly… and so on…
Other
Resources / Instruction Aids to Use -- Files Will be In Pickup Folder:
- Note:
The following files are not necessary for the assignment below, but are
required for the Math ISU project:
- Keywords_Structures_Geometry.doc
- math_Problem_solving_Process.doc
- Thinking_Geometry.doc
1 Thinking Skills In the Design, Problem-Solving and Learning Processes
1.1 Learning About Thinking
To be complete, the learning process
must involve a great deal of “quality thinking”. After “doing”,
“seeing” and “hearing”, the human mind must do a certain amount of
processing in order for a module of learning, knowledge or skill development to
be solidified. This is generally called “thinking”. Note that reflection
“after the fact” is not sufficient (i.e. not good enough). The approach to
the development of thinking skills must be radically improved. This is the
purpose of the Seven Key Thinking Skills shown in Figure
1
below. Structured thinking will be rigourously “taught” by this teacher.
Seven basic forms of thinking (in order from simplest to most demanding) are: recall, translate, interpret, apply, analyze, synthesize, and evaluate. Recall (memory) is what many people mistakenly believe learning (and school) is “all about”. This is a very unfortunate perception. While most would classify thinking as a skill, to become a good thinker first requires the learner to develop sound attitudes such as persistence, flexibility and openness.
It is
very important to recognize that the goal of making the Design and
Problem-Solving Processes of paramount (crucial) importance in school today does
not infringe on student imaginativeness and creativity. A good design process
actually fosters (supports) creativity and the use of imagination by encouraging
brainstorming, consideration of many varied inputs and generation of multiple
alternate solutions.
Creativity drives improvements in our quality of life. Creative thinking is best used in multi-disciplinary efforts in very close connection with thorough critical thinking that reaches for a solid understanding of a sustainable social, economic or political system. Indeed, creative thinking can be used to improve the multifaceted (having many aspects) socio-economic-political environment which we all share.
Creative thinking is, very generally, a collection of skills, attitudes and concepts centred on the Synthesis form of thinking. Critical thinking is, very generally, a collection of skills, attitudes and concepts centred on the Analysis form of thinking. Skilled analysis, synthesis and evaluation are at the heart of any successful design project. “Reverse engineering” a piece of technology is a powerful tool for improving analytical skills.
The use of these seven thinking skills is very much a learning process or cycle. It may be seen that the design process and the learning process follow a similar and perfectly natural pattern -- input, processing, output and then, finally, evaluation of the output and another cycle of improvement. The cerebral cortex of the brain is dedicated to three functions known as sensory, integrative and motor. We could label Figure 1 below as follows:
- Input = Sensory (Observe/experience the world around us using our senses)
- Process = Integrative (Put / connect our observations into a bigger picture)
- Output = Motor (Act or do something with our enhanced understanding; simplify if necessary)
1.2
The Seven Thinking Skills -- A Closed
Loop
Process
Figure 1: Information Processing Using Seven Structured Thinking Skills
All students must actively engage in the following forms of thinking where relevant to their assignments (in order from most complex to simplest). Note that the terms in brackets are merely approximations or “simplifications” of that thinking skill.
- Higher Forms of Thinking:
- Evaluate (judge against criteria).
- Synthesize (to output or put together; one aspect of creativity)
- Analyze (to break down or take apart; one aspect of critical thinking)
- Apply (use knowledge in a new situation)
- Lower Forms of Thinking:
- Interpret
(summarize, sort, establish classifications, reveal relationships, make
connections with other courses etc.)
- Translate (“do not copy”) (put into your own words) (transform from English words into a math equation)
- Recall (Memory)
It is very important to note that, in using a higher form of thinking, you are also generally using some aspects of each lower form. Critical thinking is not just analysis as it involves the thinker having certain well-developed attitudes as well. Critical thinking speaks more to the quality of the thinking that is performed.
Note that “copying” from a text or elsewhere is NOT evidence of learning or understanding. When “describing” something, you must “translate” into your own words, also using higher forms of thinking as much as possible depending on the question. Accurate translation coupled (along) with necessary interpretation and analysis is typically acceptable evidence of some degree of understanding or learning. You must practise, demonstrate and document all seven forms of thinking as you engage in the problem-solving process.
Note that thorough problem-solving
involves cycling through all of the seven thinking skills several times.
1.3 Thinking Skills: Definitions and Applications
In the following table, column 2 lists some general characteristics of each form of thinking. The purpose of column 3 is for you to describe how you used this form of thinking on a particular assignment or project. The sample data in italics in column 3 relates to the grade 9 Integrated Technologies popsicle stick bridge project. Delete these samples and then describe your own applications of each thinking skill. This exercise is a relatively complete summary of or "reflection on" any assignment or project.
|
1 Keyword |
2 An
Initial Definition |
3 Student
Translation / Application / Additional Context --- Describe
Your Best Example of Using This Thinking Skill in (eg) Your Bridge
Project |
|
Thinking |
A set of
mental skills that put humans above all other animals. We can use these
mental skills to make the world a better place for all creatures. The
following seven structured thinking skills comprise a simple yet powerful
model of Thinking that is particularly useful in learning. |
|
|
Recall |
-Form of
thinking -- Lowest -- the least effective form in terms of learning -Recite,
remember or pull pieces of knowledge or sequences from memory. Outputs may
be a keyword definition or match, procedure, statement of a theory or law
etc. -These
“recalled facts” may either be correct or incorrect. -Several
cycles through all seven thinking skills may be required before a
“fact” is “correctly comprehended” or understood in a particular
context or situation. -Recall
simply pulls information from memory without necessarily having any degree
of comprehension or significance -Use
cues, mnemonics, concept maps etc. to trigger recall -Paying
attention to input you are given is very important -The
human brain continuously takes in feedback from the other forms of
thinking in order to more completely develop some “understanding”
surrounding a package of information |
Delete this Teacher Sample Answer: Before I started working on my bridge design I first pulled out of
my memory some information from my science class in elementary school. The
topic was structures and we learned about forces such as tension and
compression. I will use this information as I continue to think about my
bridge design. I also recall from my Structure class in elementary school that
the shorter the distance between supports or piers, the more force the
structure can withstand. |
|
Translate |
-Form of
thinking -- Second lowest -Re-word
the idea or statement -Student
translates input information into a meaningful sentence or paragraph using
his or her own words -Promotes
learning by preventing the student from
copying input information -Paraphrase
a statement, law, procedure etc. -Summarize
a paragraph in your own words. -Convert
a verbal statement into a mathematical formula or a simple diagram. -Focus
only on the limited scope of the information piece that is in front of you
-- don’t question the information itself -Above
all, engage with or “dive into” the input information that you are
given -- don’t just read it -Outputs
may be another paragraph, combination of paragraphs, mathematical formula,
concept map or mind map or other visualization etc. -Note:
DO NOT COPY from someone or somewhere. -DO
NOT translate one word at a time using synonyms. |
Delete this Teacher Sample Answer: The Bridge Design Scenario letter was not clear about the length
of the span that the bridge had to cross. The letter said “maximum
of 30.5 cm long”. So I translated
that sentence in the scenario letter into “the bridge must cross a 30.5
cm span from pier to pier” and showed it to my teacher for his
interpretation. My teacher
agrees that my translation is a much more clear requirement. |
|
Interpret |
-Form of
thinking -- Third lowest -Explore
the input you have been given -Student
assesses the collection of input information and determines relationships
and classifications within the information set(s) -summarize
a paragraph or a page or an entire article -Look for
the “essence” of the information set -Look for
patterns in the data -“If I
only remember one thing from this article, this is it…” -Question
the information by using the 80/20 principle -- “80% of the value of this article will be found in only 20% of the
article’s information” -- find and then translate and re-organize
that 20% -sort,
classify, categorize or group data sets -make
simple preliminary or initial connections with other courses or subject
areas -- eg “how does the concept of structure relate to database
programming?” -- this is the beginning of extra fact-finding that may
prove useful -transform
some confusing but important input information into “a more useful to me right now” format -- eg a table of due dates
for unit assignments -Outputs
may be a concept map, flow chart, table of structured information, written
paragraph, summary of several paragraphs, prediction etc. -Note:
Recall, Translate and Interpret are generally mental processes but, of
course, making a concept map is “doing” |
Delete this Teacher Sample Answer: I read about types of bridges on the internet. A beam bridge has a
thick heavy member spanning the river. The top portion is in compression
-- getting shorter when it bends under a load -- and the bottom portion is
in tension -- getting longer. According to other web pages, a truss bridge
is similar, but it has only slender top and bottom members that are
connected by slender diagonal struts. I interpret a truss bridge as being
a special case of a beam bridge, with the top member in compression and
the bottom member in tension. I should probably verify my suspicion
through a test of some kind. |
|
Apply |
-Form of
thinking -- Middle -Application
of information to solve problems in new or unknown situations -Use
information, concepts, skills, procedures, scientific laws, principles and
other learnings in a new concrete situation. -“Transfer”
your knowledge and skills to other areas of study -This
thinking skill focuses on “making connections” -- more detailed than
the simple connections made using the Interpret thinking skill -Explain
how a familiar concept relates to a new topic -Value-added
fact-finding begins in earnest as an important component of “making
connections” -Outputs
may be a performance, demonstration, physical product, procedure, essay,
mathematical solution, concept map etc. -Note:
“Doing” may generally be thought of as beginning at the Apply thinking
level |
Delete this Teacher Sample Answer: I remembered from last week that my teacher stressed that the glue
joints are very important. I used my apply thinking skill as follows. I
made sure we used the best glue for our Popsicle stick truss joints. I did
some tests on several kinds of glue. Carpenters glue was the strongest if
some pressure is applied while it dried. I needed to use my evaluate
thinking skill here too. |
|
Analyze |
-Form of
thinking -- Third highest -An
important aspect of “critical thinking” -More
complete and detailed than the Interpretation form of thinking -Student
“takes apart” collections of information using logic, intuition,
emotion, deductive and inductive reasoning, lateral thinking and other
formal and informal techniques (eg, “gut feeling”) -Reverse
Engineering is a very valuable form of analysis for Technological Studies
-- “how does this contraption work?” -Find
gaps in the information set -- what other information do you need in order
to fill those gaps? -Where
there is uncertainty, the good designer obtains more data -- this may
require further physical experiments or detailed research -Sometimes
what is not stated is just as important as what is stated -Respond
to your own “what if” questions -Be the
“devil’s advocate” making initial checks for weaknesses and faults -Expose
logical interfaces between sub-systems -Explore
cause and effect patterns or relationships -Compare
and Contrast 2 concepts is a simple form of analysis |
Delete this Teacher Sample Answer: Going back to my interpretation that a truss bridge is maybe just
a smart way to make a beam bridge, I decided to explore this idea in a
rather formal, organized manner. Here is my analysis. I
found the above visual model of a beam in bending in the
Keywords-Structures document. The arrows pointing toward each other
represent compression forces and the arrows pointing away from each other
represent tension forces. Suppose I model compression as positive force
and tension as negative force. Then I would say that there must be zero
force at the horizontal central axis of the beam. So I now ask:
if the middle of a beam bridge has zero force on it, then why would
I waste good material there? This mass of material in the middle of the
beam cross section is just adding dead load to our bridge and not helping
us at all! |
|
Synthesize |
-Form of
thinking -- Second highest -An
important aspect of “creative thinking” within the context of a
greater system or an integration of sub-systems -More
complete and detailed than the Application form of thinking -Student
uses the creative process to carefully render a product or other output -Outputs
may include a physical or written product, process, performance,
demonstration or any other form of communication etc. -A set of
Requirements and other planning procedures for a possible product solution
must also be synthesized in the early stages of product development -Goal-setting
is a strong consideration -- think about “how to get there” -A list
of possible alternative solutions may also typically be synthesized during
a brainstorming session -Propose
alternative viewpoints and hypotheses -Elaborate
on a product design idea -- integrate further details that make the
product more valuable -Elaborate
on the interfaces that emerged during analysis -Create
metaphors and analogies to underscore the importance of a concept or to
assist you in transferring skills and knowledge to a new topic or subject -Define
the level of “control” that must be integrated into your product,
procedure or process -Combine
seemingly independent concepts into a workable value-added idea |
Delete this Teacher Sample Answer: Here is where I “put it all together”. The teacher said that truss joints are weak spots in any bridge --
the interfaces between Popsicle sticks in my truss bridge for example. I
made a very strong pair of trusses using carpenters glue -- see
“Apply” above. So I reason
that the interface between the entire truss and my floor deck could also
be a weak spot. It sure would be if I just use glue to stick the floor
deck to the underside of my pair of strong trusses! I want to avoid just
having the strength of glue as the only thing keeping my floor deck from
ripping away from the strong trusses. In my design, the trusses provide
the strength function while the floor deck provides the transport function
of my bridge -- getting people from point A to B. I simply laid my floor
deck Popsicle sticks on top of the strong bottom members of my trusses.
Just a little dab of glue will stop them from sliding around. The glue
does not have to provide any strength at all. To summarize, the synthesize thinking skill is vital to taking
apparently isolated individual ideas and concepts and weaving them
together in a unique or special way to yield a better solution. |
|
Evaluate |
-Form of
thinking -- Highest -Student
makes judgements against various established standards and other criteria -Synthesize
your own scoring system to assist the evaluation process -Decision-making
-- what do we do next? -Engineering
Judgement is particularly important when safety is an issue. -weigh
the advantages and disadvantages of each alternative solution idea under a
variety of clear criteria (eg risk, build-ability; test-ability; cost etc) -Note:
Several cycles through all seven thinking skills may be required before a
“fact” is “correctly comprehended” or understood in a particular
context or situation. Problem-solving also requires several cycles through
the 7 thinking skills. |
Delete this Teacher Sample Answer: For version 2 of my bridge I will experiment with making some
I-beams as the horizontal bottom members of my truss. This is because I
think that the bottom members are not in pure tension. They actually bend
under the traffic loading as well. |
Table 1: Thinking Skills: Definitions and Applications
This
is individual work. Teacher’s
sample answers are in blue font.
- In
about 30 words, Interpret,
or summarize section 1.1 “Learning About Thinking”. What is the
“essence” of section 1.1? (3 marks)
The
essence of Section 1.1 is to clearly point out that “learning to think” is
crucial in school. Memory work is a very small part of learning. It is good
analytical and creative thinking that will make our world a better place. The
design process follows the same natural pattern as the learning process. The
cerebral cortex of the brain works in much the same way as the design process --
input, process and output.
- What
do you need to do before you can become a “good thinker”? Which thinking
skill did you just use to answer this question? Using a different form of
thinking, state some other examples of prerequisites to becoming a good
thinker. (2 marks)
In order to become a good thinker
you must first become good at being open-minded, flexible, persistent,
inquisitive and curious. For example, it was only 2 years ago that I first even
“thought about thinking”. I am now a better learner myself because I
constantly question my own thinking.
- In
20 or so words, use your Interpret thinking skill to explain what this
Special Insight means to you:
What is the single biggest difference between life
and your personal computer? There is no “undo” command in life.
(2
marks)
You must think clearly before you
act or speak. Your actions can hurt others and this hurt cannot be reversed. The
computer’s undo command is teaching us to be careless.
or
Some
things that you do can hurt others -- you cannot undo these actions. You have to
constantly think about impacts and consequences of your actions -- before you
act!
- At what Thinking Skill does “Doing” actually begin? Explain in about 20 words using examples from what you know so far. (3 marks)
“Doing” most typically begins when you start to Apply some knowledge to a new situation in order to solve a problem or secure an opportunity. For example, I know some things about a few special students. I know some things about a software program. I will apply what I know in order to start to develop a learning opportunity for these students using that software. This will ultimately involve my higher forms of thinking as well.
- In about 50 words -- your own -- compare and contrast the two thinking skills, Analyze and Synthesize. Explain with examples connected to your experiences in real-world problem solving assignments. (5 marks)
When you Analyze something, you
break it apart in a logical sort of way in order to better understand what each
part or sub-system “does”. Synthesize is much the opposite -- we put things
together to make a new whole. To properly analyze a system or other examples is
an important prerequisite to doing a good job of synthesizing something new. It
is important to be critical during design, so you must carefully use your
Analyze thinking skill. It is also important to be creative during design and
this requires use of the Synthesize thinking skill.
When I translate a physical object
into a 3 view orthographic drawing, I am also breaking up the object into 3
particular carefully selected views. Someone else could synthesize an isometric
drawing (a single 3D representation) by “putting together” the information
from those 3 orthographic views.
Safety
Reminder:
If you
will be working with some tools -- “Think
Safety”.
Inputs
/ Knowledge / Understanding That I Still Need or Connections that I Want to Make
For This Unit:
(give each a #)
Peer
Assessor’s Name and Notes:
|
|
The Scarboro Heights Record V15 #1