Running head: MATH AND SCRATCH PAPER
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Math and Scratch Paper:
A Method to Enhance Problem-solving Ability in Math
Sibo Wang
Northwestern University
MATH AND SCRATCH PAPER
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Abstract
This paper explores whether the proper use of scratch paper can enhance the problem-solving
abilities of people in mathematics. According to cognitive science perspective, the
problem-solving abilities of people are capped by the limited capacities of working memory.
As a result, writing down crucial details in thinking process on scratch paper should diminish
the limitation caused by insufficiency of working memory. This paper constructs a new
method called Scratching to Assist Thinking Process (SATP). This method uses scratch paper
to apply theory about mathematical methodology and perspectives in cognitive science to
problem-solving tasks, and examines the effectiveness of SATP by experiments.
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Math and Scratch Paper: A Method to Enhance Problem-solving Ability in Math
My high school in Beijing emphasized on mathematical skills. As a result of such
emphasis, many students in our high school were good at math, and some of them even got
IMO (International Mathematics Olympiad) gold medals. Though I studied in such a
competitive school and wanted to be good at math, I did not want to study too hard and to be
a nerd. Thus, I expected to become a smarter but less nerdy person who has strong
mathematical problem-solving ability.
Because I do not know how to become smarter, I decided to seek alternative ways to
enhance my mathematical problem-solving ability. Thus, I studied computer science.
According to theories of computability in computer science, all modern computers, which are
Turing complete, are capable to perform the same tasks, but at different speeds. For example,
an old 286 machine is able to perform the same tasks as the most advanced super-computer,
but at a much slower speed. Consequently, 286 seemingly will never finish the task and have
weaker problem-solving ability. But improving either its CPU or RAM will increase its
problem-solving ability. Changing the CPU is the same as changing the brain of a person, so
such unrealistic idea cannot be applied on human. But people may adopt the idea of
improving RAM to increase their problem-solving abilities. Though the innate capacities of
working memory are unalterable, people can extend their working memories by scratching
their thoughts on paper. Theoretically, people, with infinite scratch paper used appropriately,
have infinite capacities of working memory. Therefore, using scratch paper efficiently can
improve mathematical problem-solving abilities of people. But how can people use scratch
paper efficiently so that scratch paper can actually improve our problem-solving ability?
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Scratch paper has been used by mathematicians since long ago, and it is still widely used
by people to help with math. However, scratch paper is scarcely used efficiently. People
mostly use scratch paper to do calculation they cannot do in their heads. Such traditional
method of using scratch paper is inefficient, because this method does not enhance the
thinking process of problem-solving. People scarcely write down what theorems are related
to the problem or what formulas they have tried but did not work. I hypothesize that as an
assist for thinking about possible solutions, a new method of using scratch paper, writing
down thoughts of crucial steps of finding solutions on scratch paper, will improve people’s
mathematical problem-solving abilities. Such new method, called Scratching to Assist
Thinking Process (SATP), cannot only be theoretically constructed, but also be tested through
experiments.
General Procedure of Mathematical Problem-solving
To outline SATP, a general model of mathematical problem-solving is necessary. In How
to Solve it, G. Polya (1945) describes a classical four-step-procedure for solving
mathematical problems:
First, we have to understand the problem; we have to see clearly what is required.
Second, we have to see how various items are connected, how the unknown is linked to the
data, in order to obtain the idea of the solution, to make a plan. Third, we carry out the plan.
Fourth, we look back at the completed solution, we review and discuss it. (P. 5)
Such four-step-procedure is a good model for representing the mathematical
problem-solving process. However, because the second step of the traditional
four-step-procedure is hard to perform, the conventional procedure needs to be improved to
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outline SATP. Usually, when people feel that they have no idea about how to solve problems,
they stuck on the step of devising the plan. Therefore, I decide to divide the second step of
traditional model into smaller and more applicable steps.
First, people search for related concepts, theorems, and formulas to find the appropriate
ones to solve the problems, and list the search results out in their working memories. The
process of problem-solving contains following elements: a starting state, a goal state, rules of
transition, and heuristics.1 The starting state is all data and conditions people know when
people start to solve the problems, while the goal state is the conclusion people want to derive
from starting state. Those two elements describe the problems that need to be solved. Rules
of transition are the rules move the system of problem from one state to another. For example,
theorems and formulas make people know more about the problem in mathematics and move
closer to the goal state, so theorems and formulas are rules of transition of math problems.
And the heuristics are principles to determine what operations need to be done. Without
heuristics, people cannot rationally know how to find a solution to problem. In this step,
people find out all possible rules of transitions, so they can test about different heuristics later
by combining different rules of transitions.
Next, people try different heuristics they came up with by combining different rules of
transition in the previous step. If there is an existed heuristics which solve the problem
directly, people will use that heuristics as their plan to solve the problem. Professionals in a
certain field usually have smarter heuristics, so they have great advantage in this step.2
1
See Pretz, J. E., Naples, A. J., Sternberg, R. J. (2003). Recognizing, defining, and representing problems. In Davidson, J.
E.& Sternberg, R. J. (Eds.), The psychology of problem-solving (pp. 1 - 30). Cambridge, UK: Cambridge University Press.
2 See Raab, M. & Gigerenzer, G. (2005). Intelligence as smart heuristics. In Pretz, J. E. & Sternberg, R. J. (Eds.), Cognition &
intelligence: Indentifying the mechanisms of the mind (pp. 188-207). Cambridge, UK: Cambridge University Press.
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Professional who is more efficient at solving problems usually practices more in certain fields.
As a result of practices, the person has better knowledge about what rules of transition shall
be used in various situations. In other words, by practicing, people acquire better heuristics,
so they can solve the problems in the field better. For example, if two theorems are usually
used together in order to calculate a variable, people will realize this fact, and use those two
theorems together after practicing. Therefore, finding appropriate heuristics faster improves
mathematical problem-solving abilities of people.
If people cannot find a heuristics which can solve the problem directly because of the
difficulties of problems, they will choose the seemingly best heuristics to simplify the
problem and to get more information. Repeating the simplification process, people simplify
the problems to easy ones, and solve them. This model also explains why difficult problems
for normal people may seem easy to professionals. Professionals usually skip this step in
problem-solving process, because they have good enough heuristics. As a result, professional
solve problems faster than others.
In a brief summary, the general mathematics problem-solving model we are going to use
in SATP is understanding the problem, brainstorming related information, choosing the best
heuristics, repeating the previous steps to simplify the problem, carrying out all heuristics to
find the solution, and checking the answer.
Specific Steps in SATP
After the general model for mathematical problem-solving constructed, I will specify
how to use scratch paper in each step to enhance the problem-solving abilities of people.
Scratch paper is an extension of working memory. And according to Hambrick and Engle
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(2003), “in order to solve problems successfully, people need ability to maintain goals, action
plans, and other task-relevant information in a highly activated and accessible state, and when
necessary, to inhibit activation of irrelevant or distracting information” (p. 179). Therefore,
the functions of scratch paper in SATP are maintaining and managing task-relevant
information. By using those functions of scratch paper, people can enhance their
mathematical problem-solving ability based on the general model of mathematical
problem-solving.
Understanding problem
The first step of the general model is to understand the problem. This step seems easy,
and most people think they understand the problem smoothly. However, when the problems
are too long and have too much information, people sometimes forget crucial information
about problems after they understood the problems. This phenomenon happens because of the
limited capacities of working memory. People store the information they understood from
problems in their working memories. Working memory, in this situation, is maintaining the
relevant information. When a problem contains too much information, working memory will
fail to maintain all the information because of its limited capacity. People may still run out of
working memory even if working memories have enough capacity to hold all the information
of the problem. People need to maintain the significance of the information they got, the
possible heuristics, related rules of transitions, and comparison among those heuristics in
their mind. Consequently, releasing more capacity to maintain more relevant information will
help with the problem-solving process. Therefore, if people use scratch paper to maintain the
relevant information, they would have less chance to run out of working memory, and putting
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understanding of problems on scratch paper releases more capacity of our working memory,
so such capacity can be used in future.
People may move some of the information from our working memory to scratch paper by
following procedure: write down or highlight the conditions, question and data at first, and
then write down the analysis about the characteristics of those elements. For proofs, please
write down or highlight the statement you need to proof. Also, analyzing the key
characteristics of conditions, questions, or data is important and cannot be omitted. Here is an
example of analyzing key characteristics:
Problem: solve the equation: x3+3x2+3x+1=27
Analysis: on the left side of the equation, it is a cubic equation, with coefficient 1,3,3,1.
The coefficient of the equation is a characteristic regarding the binominal theorem, and
(x+1)3=x3+3x2+3x+1.
Brainstorming related concepts
People will brainstorm about suitable heuristics and related rules of transitions after they
understood the problems. For easy problems, people will get the suitable heuristics and
related rules of transitions in glance. However, for more difficult problems, people need more
actions after they finish the search for applicable heuristics. Therefore, when people do not
have enough working memory to hold the search results, they would not be able to continue.
If they wrote the search results on scratch paper, they could test each heuristics without
forgetting the results. As a comparison to SATP, traditional method of using scratch paper
enforces this step in people’s mind, so people may run out of working memory in this step.
In some problems, the linkage between the known information and possible unknown but
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useful information is not explicit. Sometimes such linkage needs to be found by construction.
For example, drawing a line on a diagram to link the condition and related concepts is a
construction. Here is an example of how to brainstorm and construct in a proof problem:
Problem: (Menelaus Theorem)
Aj
F
E
B
C
D
As the diagram shows, in △ABC, line DF intersects AC at point E. Proof that:
AF BD CE
∙
∙
=1
BF CD AE
Proof: we are trying to use our own method to find the solution. By the first step we have
talked about, the most significant characteristic of the statement we need to proof is that it
contains lots of ratio between segments.
When brainstorming about rations between segments, we first came up with the idea
about similar triangles. There are two basic form of using similarity of triangles:
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We select the second model or heuristics to approach the problem, so we are trying to
construct the second model in our diagram by drawing a parallel line CG.
Aj
F
G
B
E
C
D
With this construction we can get information that:
EC GF
=
AE AF
And:
BD BF
=
CD GF
We can also find out information with the construction of heuristics 1 of similarity.
Try different heuristics and simplify the problem recursively
After the brainstorming process, people need to test different heuristics. If problems are
easy, people try different heuristics directly to see which ones are working. However, when
problems are difficult, people will not be able to try out all heuristics easily, because they
need many heuristics linked together to solve the problems. In this process, working memory
maintains all of possible heuristics and the test results of them. If any of relevant information
was forgotten, people would fail in solving the problem, especially when the heuristics that
would generate the correct solution was forgotten. Therefore, people need a systematic way
to test heuristics and use them to simplify problems on scratch paper.
How to simplify a problem with heuristics on a scratch paper? A typical way to do this is
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to rewrite the original problem to a simplified version using the information you have got.
Following in the example of proofing Menelaus theorem, here is an example of organizing
the information to a simplified problem:
Conditions:
EC GF
=
AE AF
And:
BD BF
=
CD GF
Statement need to be proved:
AF BD CE
∙
∙
=1
BF CD AE
By writing down what we have, we simplify the original complicated geometric problem
to a simple algebra problem. We can prove the statement by plugging in the two known
equations.
If the simplified version of problem is still difficult, people then need to simplify the
problem on scratch paper again. By repeating the brainstorming process and the
simplification process, the problem will become easier and easier. Also such recursion is
indeed creating better heuristics for the problems. Smarter heuristics simplify the problem by
combining worse heuristics together, and recursion process simplifies the problem by
decomposing it to simpler problems. They are both making problems easier to people. After
people finish this process, the problem should be simplified to an easy problem, and plan to
solve a difficult problem is completed at this moment.
Carry out the plan and checking the answer
After the previous steps, people shall have enough information to finish solutions to
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problems and to check their answers. So in this step people just enforce their plans, and get
the correct results.
Testing SATP
Procedure of testing SATP
Now SATP has been constructed, but its effect has not been tested yet, so an experiment
needs to be conducted. The goal of this experiment is to measure how much problem-solving
ability can be improved by SATP. In the test, I will compare SATP to the traditional way of
using scratch paper to do limited calculation. Here is the test procedure:
1) Distribute instructions, problem sets, formula sheets, and handouts of SATP to
test-takers. Problems in the problem sets are arranged in sequence of difficulties.
2) Test-takers start working on the problem sets with the traditional method. Record the
difficulty of problem they first stuck on. This difficulty is the problem-solving ability
of the individual using scratch paper traditionally.
3) Test-takers continue to work on the problems with SATP. Record the difficulty of
problem they stuck on using SATP. This difficulty is the problem-solving ability of
the individual using SATP.
4) Test end after the two difficulties are recorded. The improvements of mathematical
problem-solving abilities by SATP are the differences between the problem-solving
abilities of individuals using SATP and traditional method.
5) No limitation about time will be placed. Test-takers have unlimited time to work on
problems. In addition, breaks are allowed during test.
In the experiment, to simplify the comparison, I presume SATP will improve the
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problem-solving ability of test-takers. So in the test procedure, test-takers will continue to
work on the same problem set instead of starting over on a new problem set after they switch
to SATP. Such test procedure based on the presumption brings two advantages. First this
procedure decreases the time of test. As a result, test-takers’ problem-solving abilities will be
less affected by their impatience. Second, both SATP and the traditional method are working
on the same problem set, so the measurement of problem-solving abilities will be more
accurate.
Some may argue that such presumption might be wrong. Normally, if our presumption is
right that the SATP helped with the problem-solving abilities of people, the test-takers would
be able to solve the problem they stuck on, and do more problems in the test. The difference
reflects the increase in problem-solving ability with the SATP. In the abnormal case, if our
presumption was wrong, the test-taker would not benefit from using the SATP. As a result,
they would not work out more problems after switching to SATP. Instead, they would stick on
the same problem. If this case happens, I will record the SATP does not work well on this
individual. Such accuracy of results is enough for our purpose.
For problem sets, I selected problems from 2D geometry and basic high school algebra to
ensure that the test-takers are capable to solve all problems. I limited problems to 2D
geometry and basic algebra problems because this method may fail to reflect the true
problem-solving ability of a test-taker if the person does not know the related rules of
transitions required to solve the problem. Also, a theorem and formula sheet with all required
rules of transitions will be provided, so test-takers will not stuck on the problems because
they do not know some formulas.
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In addition, I did not set time limit for this test. Some may have less mathematical
problem-solving ability under time pressure, and others may do better under the same
pressure. Therefore, time limit would affect the test results, so setting the time limit is not
appropriate for this test.
Measuring difficulties of problems
The major obstacle in this experiment is measurement of difficulties of problems.
Without a proper quantitative measurement, no one can find the differences between
problem-solving abilities mathematically. To construct such quantitative measurement, I
started with the fact that the difficulties of problems are related to heuristics. Usually
problems, which can be solved by a single heuristic, are easy, because people are able to
devise the plan easily. A problem that needs three different heuristics to solve would be a lot
harder, because people hardly come up with a plan to solve such a problem by staring at it.
According to such fact, I define that the absolute difficulty of a problem is the heuristics that
needed to solve the problem. For example, the absolute difficulty of the problem, calculating
the hypotenuse given the length of two other sides of a right triangle, is one because a
heuristic of using Pythagorean Theorem is enough to solve the problem.
But difficulty of a problem differs by people, because heuristics of people are different.
Therefore problems have the same absolute difficulty are different in difficulties for people.
People, who know more heuristics either by knowing more math theorems or by practicing
more on small tricks, often perceive problems easier than those who know fewer theorems. In
other words, accumulating heuristics makes the relative difficulty decrease. For example, if
one who does not know the formula to solve quadratic equation is asked to solve a quadratic
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equation, the person would perceive this question as a hard question because he or she will
need approximately three heuristics to solve the problem. However, once the person knows
the formula, he or she will perceive this problem as an easy problem. So the absolute
difficulty of a problem does not accurately represent the difficulty perceived by people.
Instead, the absolute difficulty only gives an approximation.
And such approximation is accurate enough for our purpose. Because most people who
have similar education background in mathematics would have similar sets of heuristics,
using the absolute difficulty I defined to measure the difficulty of problem would be
appropriate to our experiment if I pick the samples carefully.
Sampling
To validate my definition of absolute difficulty, the sample of experiment will only
include Chinese students in Northwestern University who studied in China in their middle
school. Because the materials are unified in Chinese middle school, Chinese students have
similar heuristics. Consequently, I minimize the unexpected effect of education background
by narrowing the sample to the specific range of Chinese students in Northwestern.
Such a sampling strategy also has its limitation. Logically I am only able to draw
conclusion from this experiment about Chinese students. But because I only have limited
time and no fund support, I am unable to have larger sample size. But because such
restriction of sampling makes our conclusion more accurately in the changes caused by SATP
than random sampling, I would rather trade generality of the experiment to accuracy.
Data
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Table: Test data
No. 1st stuck
2nd stuck
Improvement
1
4
5
1
2
3
5
2
3
4
5
1
Analysis
In the test, the average improvement after using SATP is 1.3. The result means after
switching to the SATP test-takers on average solve 1.3 problems than originally they could.
This reflects that SATP improved the problem-solving ability of the test-takers. However, the
average improvement of 1.3 is not a great improvement. Such fact demonstrates that SATP
cannot improve the problem-solving abilities of people in a large scale. In short, SATP has an
effect of improving problem-solving ability, but such improvement is not huge enough to
change the problem-solving abilities of people totally. Therefore SATP cannot be relied as an
ultimate weapon to attack the most difficult problems around the world, but can be a good
tool to solve the hard ones in problem sets.
Limitations
The study has some limitations, although I successfully construct SATP and prove that
SATP is effective by experiment.
Redundancy of SATP
People have to write a lot to use SATP. Indeed, people are trading off their speed to
increase their problem-solving ability. So using SATP in a test with time pressure is not a
good idea. Also, the trade off perspective explains why SATP cannot solve every problem.
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Because whenever one wants to improve his or her problem-solving ability, one has to trade
off his or her speed. One, who is working on the problem with an extremely slow speed,
would lose his or her patience to work on the problem. This happens during the experiment,
and becomes the reason that test-taker gave up on the problem. If I have more time and
resource to continue on this research, I will try to delete some redundant step of SATP, so it
becomes a more efficient tool.
Sample size
Because I only tested SATP on Chinese students, the conclusion of the research is not
perfectly right. As I mentioned before, the limitation of time and energy prevents me from
doing a test with large enough sample size. If I have more time, energy, and funds, I will
cooperate with Searle Center of Teaching or SESP to conduct tests with enlarged sample size
and to see the effect of SATP.
Individual differences
Because individuals are different, people are expected to have different performance after
they use SATP. Some may have a habit using scratch paper similar to SATP before they know
it. For those people, the effect of SATP would be limited. Also, for those who know enormous
amount of heuristics, the effect of SATP would be insignificant too. A person, who can solve
any problems with less than three heuristics, does not need to use SATP to enlarge his or her
working memory, because the person will not need to put much information in his or her
working memory.
Individual differences also affect the experiment results if I enlarge the sample size.
Though with the education background fixed this limitation would not affect the result of
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tests too much, differences in education background, intelligence, and patience are inevitable
in large sample size experiment. Those uncontrolled variables will affect the result of further
studies tremendously.
Definition of absolute difficulty
Though I have defined the absolute difficulty of problems, I have not defined a standard
set of heuristics to measure the difficulties of problems. Therefore a problem may have two
absolute difficulty based on two sets of heuristics. For example, some people who know
Menelaus Theorem can solve certain problems easily, but others who do not know Menelaus
Theorem may think the same problem difficult. If we count Menelaus Theorem as possible
heuristics, the problem will be easy. Otherwise, the problem will be hard. Thus, heuristics
should be weighted according to their internal difficulty, so the measurement of
problem-solving abilities will be more precise. But because too much works are needed to
evaluate different heuristics accurately, I cannot do it with limited time, energy, and funds. I
will try to weight theorems if I have chance to continue this study.
Suggestion and emotion
Because in the instruction, test-takers are told that SATP will be used to attack harder
problem. Therefore, the suggestion that people will do better using SATP may affect the
result of the test. However, such effect is inevitable in the current frame of experiment,
because the test-takers shall always know the procedure of experiment. Also, the emotion can
affect the experiment. If test-takers are happy, they may do better on the test. Because those
two factors are hard to control, I decide to ignore such effects in experiment to simplify the
study, or the test would be too complicated to design.
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Conclusion
According to our test results, the construction of SATP is partly successful, because
SATP improves problem-solving abilities of people, although it still has lots of limitation.
Other from limitation, SATP can also be specified to work better. Based on the model of
SATP, I can build other models specified to solve problems in economics, physics, or other
fields. By specification, some relatively useless step of SATP will be discarded, and SATP
can run in a higher efficiency. Also, using part of SATP to do some smaller task, such as
construction and proof of lemma, will be much less time consuming than solving the whole
problem. As a result, using SATP as a part of problem-solving is not bad. In future, with
proper specification and more research, SATP can be a more useful tool for people.
19
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References
Hambrick, D. & Engle, R. (2003). The role of working memory in problem-solving. In
Davidson, J. & Sternberg, R. (Eds.), The psychology of problem-solving (pp. 176 – 206).
Cambridge, UK: Cambridge University Press.
Herrmann, D. J., Yoder, C.Y., Gruneberg, M., & Payne, D. G. (2006), Applied cognitive
psychology: A textbook. Mahwah, NJ: Lawrence Erlbaum Associates.
Pretz, J. E., Naples, A. J., Sternberg, R. J. (2003). Recognizing, defining, and representing
problems. In Davidson, J. E.& Sternberg, R. J. (Eds.), The psychology of problem
solving (pp. 1 - 30). Cambridge, UK: Cambridge University Press.
Polya, G. (1945). How to solve it: a new aspect of mathematical method. Princeton, NJ:
Princeton University Press.
Raab, M. & Gigerenzer, G. (2005). Intelligence as smart heuristics. In Pretz, J. E. &
Sternberg, R. J. (Eds.), Cognition & intelligence: Indentifying the mechanisms of the
mind (pp. 188-207). Cambridge, UK: Cambridge University Press.
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Appendix I: Problem Set of Test
An Experiment on a New Method of Solving Math
Problems
Instruction:
This experiment is dedicated to measure the amount of problem-solving abilities of
people can be improved by a new method of using scratch paper. In this experiment, you will
be asked to solve 6 problems. Problems in this test are arranged in sequence according to
difficulty. You should start with the normal way you deal with math problems until you stuck
on a problem. PLEASE DO NOT MAKE ANY GUESSES ON ANY PROBLEMS. Also in
this test, you are not asked to write a specific solution to the problem. If you are sure you get
the idea to solve the problem, you can continue to work on the next problem. Please report
after you feel you stuck on a problem to me, get a handout explaining a new method of
thinking about math problems that theoretically can improve your ability of problem-solving,
and continue by using the new method mentioned on the handout. If you stuck again, please
report the number of problem you stuck on. Pauses during the test are allowed. You can take
breaks anytime you want. Also, all theorems and formulas needed to solve problems will be
provided.
Good luck on dealing with problems!
Problems:
1) Solve the equation:
x 2 − 16x + 63 = 0
b+c=8
2) For, a, b, c ≥ 0, find all solution of {
bc = a2 − 12a + 52
1
3) In the diagram below, D is the midpoint of AB. ∠AED = 2 ∠C + 90°. BC=3, AC=7. Find AE.
B
D
A
E
C
4) In the diagram below, DF=FE, ∠DGC=∠EHC=90°, ∠GCD=∠HCE. Prove that GF=HF
MATH AND SCRATCH PAPER
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C
H
G
5)
E
F
D
In the diagram below, AB=CD, E, F are midpoints of AD, BC. MN⊥EF. Prove that ∠AMN=∠DNM
A
E
D
N
M
B
C
F
6) In the diagram below, ABDE, ACGF, BCHI are all squares. L, J, K are the centers of three squares.
Prove that AK, BJ, CL pass through the same point
D
F
A
L
J
E
G
C
B
K
I
H
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Important Theorems and Formulas
Algebra:
1) (quadratic equation) For ax2+bx+c=0 (a is not 0), we have:
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
2) x 2 + y 2 ≥ 2xy
Geometry:
I)
For △ABC and △DEF, if∠D=∠A, ∠E=∠B, we have
AB BC AC
=
≡
DE EF DF
II)
For △ABC, if M is the midpoint of AB, N is the midpoint of AC, then MN∥BC, and
1
MN = 2 BC
III)
Two triangles are congruent if:
a) All corresponding sides are equal
b) Two corresponding angles and another pair of corresponding sides are equal
c) Two corresponding sides and the corresponding angle between the sides are equal
IV)
The three heights of a triangle pass the same point called orthocenter.
V)
For right triangle the length of hypotenuse is as twice as the distance between
midpoint of hypotenuse and the right triangle.
VI)
The height of an isosceles triangle on the side which is different to other sides,
bisects the angle.
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Appendix II: Handout of Instructions of SATP
Hand out of a new method using scratch paper
1) Understanding the problem:
Write down or highlight the conditions, question and data at first, and then write
down the analysis about the characteristics of those elements. For proofs, please write
down or highlight the statement you need to proof. Also, analyzing the key
characteristics of conditions, questions, or data is important and cannot be omitted.
Here is an example of analyzing key characteristics:
Problem: solve the equation: x3+3x2+3x+1=27
Analysis: on the left side of the equation, it is a cubic equation, with coefficients
1,3,3,1. The coefficient of the equation is a characteristic regarding the binominal
theorem, and (x+1)3=x3+3x2+3x+1.
2) Devising the plan:
In this phase you are going to develop a plan about how to solve your problem.
a) Brainstorming:
Brainstorm the concepts, formulas, and theorems related to the characteristics of
conditions and questions on scratch paper. In some problems, the linkage
between the known information and possible unknown but useful information is
not explicit. Sometimes such linkage needs to be found by construction. For
example, drawing a line on a diagram to link the condition and related concepts
is a construction. Here is an example of how to brain storming and construct in a
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proof problem:
Problem: (Menelaus Theorem)
Aj
F
E
B
C
D
As the diagram shows, in △ABC, line DF intersects AC at point E. Proof that:
AF BD CE
∙
∙
=1
BF CD AE
Proof: we are trying to use our own method to find the solution. By the first step
we have talked about, the most significant characteristic of the statement we
need to proof is that it contains lots of ratio between segments.
When brainstorming about rations between segments, we first came up with the
idea about similar triangles. There are two basic form of using similar:
We select the second model or heuristics to approach the problem, so we are
trying to construct the second model in our diagram by drawing a parallel line
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CG.
Aj
F
G
E
B
C
D
With this construction we can get information that:
EC GF
=
AE AF
and:
BD BF
=
CD GF
And the rest is simple.
b) Simplification:
You should rewrite the original problem to a simplified version using the
information you have got on the scratch paper right now. Following in the
example of proofing Menelaus theorem, here is an example of organizing the
information to a simplified problem:
Conditions:
EC GF
=
AE AF
And:
BD BF
=
CD GF
Statement need to be proved:
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AF BD CE
∙
∙
=1
BF CD AE
By writing down what we have, we simplify the original complicated geometric
problem to a simple algebra problem. We can prove the statement by plugging in
the two known equations.
c) Recursion:
If the simplified version of problem is still not that obvious, you need to
simplify the simplified problem on scratch paper again. By repeating the
brainstorming process and the simplification process, you should simplify the
problem recursively. PLEASE DO ALL THE PROCESS ON SCRATCH
PAPER! Finishing this process, the problem should be simplified to an easy
problem.
3) Carrying out the plan
This step means write down the actual proof or the actual process
4) Check the result
In this step, just check the result of your proof or calculation.
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Appendix III: Previous Problem Set of Test
Note: this is the first version of the problem set. But because the difficult of problem 2 and 3
is then found inappropriate, I changed the problem set to another version and redid the test.
An Experiment on a New Method of Solving Math
Problems
Instruction:
This experiment is dedicated to measure the amount of problem-solving abilities of
people can be improved by a new method of using scratch paper. In this experiment, you will
be asked to solve 6 problems. Problems in this test are arranged in sequence according to
difficulty. You should start with the normal way you deal with math problems until you stuck
on a problem. PLEASE DO NOT MAKE ANY GUESSES ON ANY PROBLEMS. Also in
this test, you are not asked to write a specific solution to the problem. If you are sure you get
the idea to solve the problem, you can continue to work on the next problem. Please report
after you feel you stuck on a problem to me, get a handout explaining a new method of
thinking about math problems that theoretically can improve your ability of problem-solving,
and continue by using the new method mentioned on the handout. If you stuck again, please
report the number of problem you stuck on. Pauses during the test are allowed. You can take
breaks anytime you want. Also, all theorems and formulas needed to solve problems will be
provided.
Good luck on dealing with problems!
Problems:
7) Solve the equation:
x 2 − 16x + 63 = 0
8) In the diagram, ∠ABC = ∠ADB
C
A
D
B
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Prove that: AB 2 = AC × AD
b+c=8
9) For, a, b, c ≥ 0, find all solution of {
bc = a2 − 12a + 52
10) In the diagram below, DF=FE, ∠DGC=∠EHC=90°, ∠GCD=∠HCE. Prove that GF=HF
C
H
G
E
F
D
11) In the diagram below, AB=CD, E, F are midpoints of AD, BC. MN⊥EF. Prove that ∠AMN=∠DNM
A
E
D
N
M
B
C
F
12) In the diagram below, ABDE, ACGF, BCHI are all squares. L, J, K are the centers of three squares.
Prove that AK, BJ, CL pass through the same point
D
F
A
L
J
E
G
C
B
K
I
H
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Important Theorems and Formulas
Algebra:
3) (quadratic equation) For ax2+bx+c=0 (a is not 0), we have:
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
4) x 2 + y 2 ≥ 2xy
Geometry:
VII)
For △ABC and △DEF, if∠D=∠A, ∠E=∠B, we have
AB BC AC
=
≡
DE EF DF
VIII)
For △ABC, if M is the midpoint of AB, N is the midpoint of AC, then MN∥BC, and
1
MN = 2 BC
IX)
Two triangles are congruent if:
a) All corresponding sides are equal
b) Two corresponding angles and another pair of corresponding sides are equal
c) Two corresponding sides and the corresponding angle between the sides are equal
X)
The three heights of a triangle pass the same point called orthocenter.
XI)
For right triangle the length of hypotenuse is as twice as the distance between
midpoint of hypotenuse and the right triangle.