Student Model to Provide Appropriate Feedback in a Virtual Lesson Game:
Prompting Instructors to Teach Mathematical Ways of Thinking
Toshiki Matsuda
Graduate School of Decision Science and Technology
Tokyo Institute of Technology, Tokyo, Japan
matsuda@et4te.org
Abstract: In this paper, I discuss a student model for developing virtual lesson games for
mathematics. In this model, I emphasize how instruction affects students’ acquisition and use of
views and ways of thinking as well as its effect on learning and problem solving. This model
should provide virtual lesson games with the function of exposing gaps between a teacher’s
expectations of instructional effects and actual results. The model consists of domain knowledge
expressed by a semantic network, views, and ways of thinking, and a problem-solving script. I
supposed that views and ways of thinking play important roles in acquiring domain knowledge and
in activating its usage. The results show that encouraging students to use views and ways of
thinking increases their knowledge and makes the relationships among different elements of their
knowledge broader and stronger. In the model, Keller’s ARCS has similar roles. Therefore,
providing students with problems that they want to solve is important.
Introduction
According to the PISA 2009 survey (OECD, 2009), Japanese students took high score in the mathematic
literacy test. However, a National Institute of Educational Policy Research [NIEPR] (2007) survey revealed that 18
years students could not take sufficient score in the Mathematics-1 test consisted of problems learnt by 16 years old.
The survey also asked students’ attitudes toward each element of Mathematics I. The results show that the
percentages of students who replied “interesting” for “equation and inequality,” “quadratic function,” and
“trigonometric ratio” were 31.6%, 18.4%, and 19.8%. Moreover, the percentages of students who replied that these
respective concepts were “useful” were only 14.0%, 6.7%, and 12.0%. I consider that this low motivation caused in
tendency to forget learning outcomes easily.
As in other countries, Japanese National Course of Studies [NCoS] (Ministry of Education, Culture, Sports,
Science and Technology [MEXT,] 2009) has emphasized the importance of cultivating twenty-first century literacy.
However, textbooks authorized by MEXT have not changed much because teachers tend to choose old-style texts.
Unfortunately, teachers still believe that the problem-solving abilities they should cultivate are those required to pass
the entrance exams for upper-grade schools. Therefore, they make students memorize as much knowledge as
possible and train students to apply rules and procedures to test problems.
Although there have been many proposals to improve this situation, I believe that it is important to cultivate
utilization of mathematical views and ways of thinking explicitly included in the objectives of NCoS because they
are independent from contents topics and useful for any problem-solving. I consider that they are connected with
Bruner’s (1960) idea of “structure.” Difference between them are although “structure” is acquired as a result of
learning, ways of thinking is a tool for problem solving and learning, like cognitive strategies (Gangé et al., 2005),
and then functions to gain “structure.”
However, there are various issues to consider, including effective instruction methods, relevant examples,
and ways of encouraging students to utilize the ways of thinking. In addition, it is important that there are few
examples of lessons in which a teacher appropriately teaches students about ways of thinking. With considering
these issues, I have developed a virtual lesson game (Matsuda & Ohgami, 2011). In order to heighten the
effectiveness of this tool, I consider that it is important to provide teachers with feedback that demonstrates how
their instructions affect students’ acquisition of knowledge and use of ways of thinking. The feedback will also help
teachers understand how knowledge and ways of thinking affect processes and performances of actual problem
solving.
Purpose
In this paper, I discuss a student model for developing virtual lesson games for mathematics. In this model,
I emphasize how instruction affects students’ acquisition and use of views and ways of thinking as well as its effect
on learning and problem solving. This model should provide virtual lesson games with the function of exposing gaps
between a teacher’s expectations of instructional effects and actual results.
Mathematical Thinking
A number of studies have addressed mathematical thinking, such as Polya (1945), Katagiri (1988), and Ball
(2007). These studies emphasize the classifications of mathematical thinking and tend to show many examples of
their application. However, I consider that the relationships among different types of mathematical thinking and the
importance of a holistic view should be emphasized. Therefore, I propose Figure 1 as a depiction of mathematical
views and ways of thinking (Matsuda, 1993). People need to recognize that mathematical thinking must be
conducted as a series of utilizations of these views and ways of thinking. Moreover, to help inductive, deductive,
and generalized ways of thinking, mathematical views of thinking, such as viewing phenomena as functions,
recurrence formulas, or statistics, are useful.
Figure 1: Summary of mathematical views and ways of thinking and an example of their use
(Finding the rule to calculate the addition of positive and negative numbers)
Lesson Topic and Flow to Discuss the Student Model Concretely
Typical Flow and Contents of Textbooks
To discuss the model concretely, I focus on the topic “introduction to trigonometric ratios,” which is taught
in “Mathematics I,” a compulsory subject for tenth graders. As mentioned previously, NIEPR’s (2007) survey
revealed that students have low motivation to study this topic, and thus, it is apparent that lesson innovation for this
topic is needed.
An analysis of three textbooks (Ohya, 2012; Akiyama, 2012; Matano & Kohno, 2008) shows a similar and
typical flow to instruction on this topic (Table 1).
When teaching (a) in Table 1, teachers tell students that a rectangular triangle made from the equilateral
triangle has sides of lengths 2, 1, and √3 and angles of 30, 60, and 90 [degrees]. Some students may recall the
reasoning behind this: every length of the equilateral triangle is 2, and the sum of the angles of any triangle is 180
[degrees]. However, because it is not explained explicitly, others may not understand why equilateral triangle is
shown here.
When teaching (b), it is confirmed that any rectangular triangles that have a 30 [degree] angle are similar
because “two respective pairs of angles are equivalent” is a condition for two triangles to be similar. Moreover, it is
confirmed that the ratios of the corresponding sides (opposite-side/hypotenuse, neighbor-side/hypotenuse, and
opposite-side/neighbor-side) of two similar figures are consistent, such as 1/2, √3/2, and 1/√3. Only from this typical
case, (c) can be generally concluded. Some motivated students may examine another case or consider the proof of
this conclusion, but others unquestioningly accept the result.
Table 1: Typical Flow and Contents of Instruction on “Introduction to Trigonometric Ratios” in Textbooks
[Trigonometric Ratios]
a) Show the equilateral triangle, supposing the length of its sides is 2, and make two rectangular triangles by dividing it in
half.
b) Show two similar rectangular triangles that have a 30-degree angle and explain how the ratios of two corresponding sides in
each triangle are the same.
c) Explain how the above phenomenon is consistent even if the size of the rectangular triangles changes.
d) Suppose an angle of a rectangular triangle is θ and the lengths of the hypotenuse, neighbor-side, and opposite-side are r, x,
and y respectively.
e) Explain how the ratios of y/r, x/r, and y/x are determined by the value of an acute angle θ.
f) Define the names of these ratios as “Sei-gen,” “Yo-gen,” and “Sei-setsu” and represent them as sin θ, cos θ, and tan θ.
g) Explain that “Sei-gen,” “Yo-gen,” and “Sei-setsu” are called sine, cosine, and tangent in English.
h) Explain that “trigonometric ratio” is the general term for sine, cosine, and tangent.
i) Calculate the values of sin, cos, and tan of rectangular triangles that have the following ratios of sides: 3:4:5, 1:2:√5,
1:3:√10, and 5:12:13.
j) Calculate the values of sin, cos, and tan corresponding to 30, 45, and 60 [degrees].
[Table of Trigonometric Ratios]
k) Explain that the values of sin, cos, and tan are determined by the value of θ.
l) Show a table consisting of the values of sin, cos, and tan corresponding to every degree of θ [0 ≦ θ ≦ 90].
m) Find the value of sin 12 [degrees], cos 48 [degrees], and tan 75 [degrees] in the table.
n) Find the value of angle A if ∠C = 90 [degrees] and AB = 5, AC = 4 (case 1), AB = 5, BC = 2 (case 2), and AC = 2, BC = 1
(case 3).
[Applications of Trigonometric Ratios]
o) Derive the following formulas from the definition of trigonometric ratios: a = c sin A, b = c cos A, a = b tan A.
p) Calculate lengths of BC when ∠C = 90 [degrees] and ∠A = 36 [degrees], and the length of AB [or AC] is given.
q) Ask which term is appropriate for the underlined parts: AC = AB ×
36 [degrees], AC = BC ×
54 [degrees].
r) Calculate the height and horizontal distance if I walk 100 [m] along a slope that has a 19 [degree] pitch.
s) Calculate the height of a tree if the angle of elevation from a point 10 [m] distance from the tree is 21 [degrees].
t) Calculate the height of a tower if the angle of elevation from a point 20 [m] distance from the tree is 40 [degrees] and the
height of eye position is 1.6 [m].
u) Calculate the horizontal distance AB if the angle of elevation from a tower that has 20 [m] height at A to B is 32 [degree].
Because (c) to (e) do not include any proofs, students understand that (d) to (h) are the definitions of
trigonometric ratios and should be memorized. However, (e) might be a cue to extend the trigonometric ratio to a
trigonometric function and is related to motivation for studying this topic. In addition, (e) is ignored in (i) because
the values of trigonometric ratios are calculated from, not angles, but given lengths of sides. If (j) is taught before (i),
some students may become interested in the relationship between the values of sin and cos and between the values
of tan and its inverse. Moreover, some students may gain an interest in the trend of changing these values, and they
can attempt to create a graph from the table of trigonometric ratios at (l). In this context, I expect that whether
students have these interests or not depends on their acquisition of mathematical views and ways of thinking.
Although (k) is a repetition of (e), few students notice. Most of them understand that (k) as just an
introduction to (l). Consequently, they only memorize the procedures of (m) and (n) to solve the problems.
After learning the knowledge and skills in (a) to (n), students memorize the formulas in (o), method for
choosing a formula corresponding to the situations in (p) and (q), and method for calculating numerical solutions
using the table of trigonometric ratios in (r). The problems in (r) to (u) are presented as real-world phenomena;
however, they are presented with illustrations that include rectangular triangles, then students only consider how to
apply the procedures learned in (o) to (q). In addition, all figures of rectangular triangles used from (k) to (u) are the
same as in the left side of Figure 2. This may be a barrier to cultivating students’ ability to apply their learning
outcomes.
B
c
A
C
B
a
b
C A
B A
C
Figure 2: Examples of figures representing rectangular triangles
Flow and Contents to Promote Student’s Utilization of Views and Ways of Thinking
The problems printed in textbooks are ones that can be proved directly using the definitions of
trigonometric ratios; however, there is no need to use them. To discuss how mathematical views and ways of
thinking should be utilized, appropriate examples are needed. To this end, I designed the problem shown in Figure 3.
In this figure, a ladder truck needs to fix its scaffold in order to prevent a fall. After that, the angle and
length of the ladder should be adjusted so that it will arrive at the correct place. However, when the tip of the ladder
arrives at a position that is more than one and a half times the vehicle length horizontally from the center of the truck,
the risk of a fall becomes high. For the ladder truck to advance backward, too much time is taken to change direction.
On the other hand, a laser range finder and digital angle gauge can be used to measure distance and angles; then, the
distance to any direction can be measured directly. However, it is necessary to consider the possible occurrence of
errors and to verify the reliability of the data by two or more methods. To perform the rescue safely and quickly, it is
necessary to create a work plan that supposes various situations.
Figure 3: A problem for prompting students to construct knowledge of trigonometric ratios
This problem is made not with the goal of instilling knowledge of trigonometric ratios but with the aim of
prompting students to construct this knowledge by themselves. For this same purpose, I will develop a game-type
e-learning material in order to ask students about the reasons for their judgment in problem solving in an interactive
manner. Students will notice that if the ladder is leaned gradually from the state where it stood in the perpendicular
direction, the height of the tip position will fall but it will approach the building. Moreover, they will notice that the
height and horizontal distance become long if an angle is fixed and the ladder is extended. Finally, they might notice
that the coordinates of a certain point are decided by the length and angle of the ladder. This means that the
coordinates of arbitrary points on a plane can be expressed with (r×cos θ, r×sin θ). To help the students in their
knowledge construction, I will prepare a spreadsheet that returns the values of height and distance when a ladder is
leaned at 0 to 90 degrees.
Designing a Student Model
Domain Knowledge
Knowledge stored in the long-term memory is classified into declarative or procedural knowledge, and the
former is further classified into episodic and semantic knowledge (Anderson, 1976). As a model of semantic
memory, the hierarchical semantic network model (Collins & Quillian, 1969) is well known. This model assumes
that the knowledge accompanying a node is activated and searched by following links, and the knowledge common
to the low order is summarized and memorized at the high-order node.
In the field of artificial intelligence, nodes are sometimes expressed as frames or scripts (Barr &
Feigenbaum, 1981). This means that if a certain node is activated, its property information is activated concurrently,
and it may correspond to the chunking of knowledge. For example, good teachers have many kinds of knowledge
about factors that students misunderstand, and Higuchi and Matsuda (2004) assumed that these factors can be
classified consistently in each subject area, such as numerical examples, formula, and conditions to use. I consider
this idea is connected with the idea of “structure.” In addition, I consider that it is connected with mathematical
views and ways of thinking because students can learn numerical examples by using quantification, paying attention
to conditions when using specialization/generalization. For one to utilize knowledge in real contexts, how the
knowledge is activated becomes important. Here, views and ways of thinking might play important roles, and then
knowledge should be structured in connection with them.
Moreover, according to situation theory, the role of episodic memory is also important, as it connects views,
ways of thinking, and scripts for problem solving mentioned later with domain knowledge, and activates knowledge
in appropriate contexts as well as for certain purposes. Furthermore, I consider that Keller’s (1987) ARCS model
should be taken into consideration as a factor to activate knowledge. Although his model aims to suggest effective
design of instructions taking into account learning motivation, relevance and confidence play an important role in
explaining the priority of knowledge when it is activated.
Mathematical Views and Ways of Thinking
To cultivate students’ ability to construct knowledge by themselves, mathematical view and ways of
thinking should be taught. These views and ways of thinking can be classified as follows. For example, “thinking
through quantification” refers to substituting a numeric value for the variables in a formula in order to understand
the formula’s meaning and characteristics. Because quantification itself does not reduce an answer but is utilized for
analyzing or examining a problem/answer, it can be understood that people think reductively based on some
quantified result. Therefore, quantification is recognized as a view, and reduction as a way of thinking.
People can use numerical data about a certain phenomenon to quantify, and this method is useful for
finding a functional relationship between factors. Students need to learn various methods of quantification and to
choose an appropriate method according to each situation. Therefore, I assume that quantification is a purpose and a
condition to activate a script to perform each method.
As mentioned, the flow and contents of instruction such as I propose provide more opportunities to use
views and ways of thinking than those of textbooks. Therefore, this method of instruction may serve to increase the
amount of students’ knowledge and make the relationships among different elements of their knowledge broader and
stronger. For example, students who can utilize a functional view will notice the property that if the value of θ
changes from 0 to 90 degrees, the value of sin θ increases accordingly. Therefore, they understand that the value of θ
must be less than 30 degrees if the value of sin θ is less than 1/2. This knowledge can be utilized to detect errors in
problem solving afterward. Moreover, students who can utilize the view of visualizing may create a graph from the
table of trigonometric ratios and thereby identify the relationship between sin and cos.
two respective pairs of
angles are equivalent f or
two triangles
condition
ratios of three
respective pairs of
sides are equivalent
definition
is-a
value
sinθ=y/r
sine
Typical cases
Sei-gen
△PQR
table of trigonometric ratio
Set of : Sei-gen, Yo-gen, Sei-setsu
is-a
has
similar
Trigonometric ratio
symbol
meaning
sin
∠C=90ﾟ→ sin∠A=BC/AB
angle1=30ﾟ, angle2=60ﾟ
PQ=2,QR=1,
PR=√3
figure
読み方
angle1=angle2=45ﾟ
sin30ﾟ=1/2,
sin45ﾟ=1/√2,
sin60ﾟ=√3/2
figure
ratio of sides
figure
angle2＝
90ﾟ－angle1
isosceles right triangle
Typical cases
cos30ﾟ= √3/2,
cos45ﾟ=1/√2,
sin60ﾟ=1/2
Typical cases
Yo-gen
meaning
symbol
C
∠C=90ﾟ→ tan∠A=BC/AC
0ﾟ<angle1<90ﾟ
figure
B
relation
property
phenomena
木や塔の高さを
求める手続き
tangent
5:12:13
figure
C
x=r×cosθ
y=r×sinθ
property
3:4:5
y
x
A
∠C=90ﾟ→ cos∠A=AC/AB
Sei-setsu
B
r
figure
A
symbol meaning cosθ =x/r
tan
Typical cases
1：1：√2
property
cosine
cos
property
rectangular triangles
that have a 30ﾟ angle
cases
△ABC
1：2：√3
ratio of sides
rectangular
triangle
Pythagorean theorem
has
definition
has an
angle = 90ﾟ
side32＝side12+side22
Name
Name
similar of triangle
Condition ratios of two respective pairs of sides and a
pair of angle between them are equivalent,
two respective pairs of angles are equivalent
Definition ratios of three respective pairs of sides are
Name
Trigonometric
equivalent ratio
Type
総称
Property
相似な図形同士は、対応する辺の比が等しい
Parts
Sei-gen、Yo-gen、Sei-setsu
examples table of trigonometric ratio［method to read］
reason
(x,y)は、(ｒ，θ)で決まる
Name
Sei-gen, sine
Part-of Trigonometric ratio
Symbol sin
examples sin30ﾟ=1/2,sin45ﾟ=1/√2,sin60ﾟ=√3/2
Meaning sinθ=向かい合う辺÷斜辺，y/r[計算]
Name
Yo-gen, cosine
Formula opposite=hypotenuse×sinθ, y=r×sinθ
Part-of Trigonometric ratio
Property if θ:0→90 then sinθ:0→1(単調増加)
Symbol cos
examples cos30ﾟ= √3/2,cos45ﾟ=1/√2,sin60ﾟ=1/2
Meaning adjacent÷hypotenuse，x/r[計算手続き]
Name
Sei-setsu, tangent
Formula adjacent=hypotenuse×cosθ，x=r×cosθ
Part-of Trigonometric ratio
Property if θ:0→90 then cosθ:1→0(単調減少)
Symbol tan
examples tan30ﾟ= 1/√3,tan45ﾟ=1,tan60ﾟ=√3
Meaning opposite ÷ adjacent，y/x[計算］
Formula opposite=adjacent×tanθ，y=x×tanθ
Property if θ:0→90 then tanθ:0→∞(単調増加)
•quantification
・assign values to variables
Suppose typical cases
B
r
A
x
y
C
rectangular triangles that
have a 30ﾟ angle
Is-a
rectangular triangles
Definitionhalf of equilateral triangle
has
side1：2：√3, angle30,60,90
Figure
Name
isosceles right triangle
Is-a
rectangular triangles
and isosceles triangle
Definition正方形を対角線で２等分
has
side1：1：√2, angle45,45,90
Figure
Name
rectangular triangles
Is-a
triangles
Definition an angle = 90ﾟ
has
hypotenuse, opposite, adjacent
examples one that has a 30ﾟ angle, isosceles
right triangle,(3,4,5),(5,12,13)
Figure
Property angle2＝90ﾟ－angle1
has
Property Pythagorean theorem
Name
Pythagorean theorem
Condition rectangular triangles
Meaning square of a side=sum of squares of
the other sides
examples
Formula hypotenuse2=opposite2+adjacent 2
Proof
•functional view
・if y increases accordingly with increase of x →monotonic increase
・monotonic increase→f(x0)=y0＆y<y0→x<x0
・using quantitative data
•specialization
・if … then procedure
・if … then …..
•abstraction
・if … then procedure
・if … then …..
•reduction
・if … then procedure
・if … then …..
Figure 4: Overview of the proposed student model
•analogy
・if … then procedure
・if … then …..
Script for Making Plans to Solve Problems
In this section, I discuss the properties of script knowledge for generating and performing a plan for
problem solving. Mathematics instructors do not teach this type of knowledge explicitly in Japan. However, even so,
students gain a certain framework and procedure for problem solving from everyday life. Of course, the framework
is not necessarily efficient and adequate, but they utilize two or more problem-solving scripts according to the
situation.
For example, a trial-and-error problem-solving script that consists of following rules is supposed as the
simplest script.
- Rule-1: When you have no idea what to do, leave the problem as it is.
- Rule-2: When you have no idea what to do, look and ask what another person is doing.
- Rule-3: If any direction or clue is given, act accordingly or make a plan based on it.
- Rule-4: …
Here, although Rule-1 and Rule-2 have the same condition, I suppose that rule selection depends on the
meta-rule that determines the priority of rules. For example, if a student’s motivation to solve the problem is high,
Rule-2 is chosen preferentially, but if he/she has low confidence to follow another’s approach, Rule-1 is chosen.
Here, I notice that the ARCS model plays an important role in rule activation. On the other hand, I also notice that it
is anticipated that if the rule name is set at the action part of each meta-rule, the numbers of meta-rules become huge.
Therefore, it is better to set any properties as the action part of each rule, such as the cost of performance and height
of profitability, and to choose the rules that should satisfy these properties.
Rule-2 and Rule-3 serve an opportunity to learn a new problem-solving script. Of course, people do not
necessarily acquire a learning outcome from an experience. I consider that ARCS affects whether students want to
acquire a learning outcome or not.
On the other hand, I consider that scripts taught in lessons are more procedural, unlike those students learn
on their own. This means that because there are few competitive rules and ineffective rules in an instructed script, it
can be performed efficiently. This is why problem-solving methods should be taught explicitly in lessons. However,
in order to prompt teachers to understand this effect, it is necessary to carry out simulations that show the effects of
these different learning outcomes.
Future Perspectives
The model proposed in this paper consists of domain knowledge expressed by a semantic network, views,
and ways of thinking, and a problem-solving script. I supposed that views and ways of thinking play important roles
in acquiring domain knowledge and in activating its usage. The results show that encouraging students to use views
and ways of thinking increases their knowledge and makes the relationships among different elements of their
knowledge broader and stronger. In the model, Keller’s ARCS has similar roles. Therefore, providing students with
problems that they want to solve is important. In this case, students need to be confident to solve problems, and then
they have to acquire views and ways of thinking.
Because this paper only proposed a model, it is necessary to develop a virtual lesson game that implements
this model and to verify the effects of the game in teacher education. Further, e-learning materials should be
developed based on the model in order to verify the model’s effects on cultivating student’s ability and to confirm its
validity.
Acknowledgements
This research was supported by the Japan Society for the Promotion of Science, Grants-in-Aid for
Scientific Research (C), No. 23501137, 2011.
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