Abstract: The purpose of the study is to ascertain the veracity of the students’ level of understanding on its six facets of the basic concept of functions and relations before and after applying understanding by design as a pedagogical strategy of the study. The study uses the pretest posttest control group design to an intact class of nursing students enrolled in College Algebra at Capitol University during the first semester of the school year 2010-2011. Thirty open-ended teacher made test was administered to the class. Teacher made test was based on the six facets of understanding developed by Wiggins and McTighe, (2006), namely: explanation, interpretation, application, perspective, empathy, and self-knowledge. Data analysis revealed that most of the respondents improve their level of understanding. Understanding by design is thus one of an effective pedagogical strategy of proposing an approach to curriculum and instruction designed to engage students making connections and binding together the knowledge into something that makes sense of things.

Keywords: assessment, understanding, and understanding by design 1.0 Introduction

Relations and functions constitute a unifying theme of mathematics. Other fields of mathematics deal with concepts that constitute generalizations or outgrowths of the notion of function; like algebra considers operations and relations, and mathematical logic studies recursive functions, (Ponte, 2011). Concepts are the key building blocks for the structure of knowledge of various academic disciplines. They are considered vehicles of thought process and are the critical component of an individual’s cognitive structure of knowledge. A concept is an idea or understanding of what a thing is and it is in an ordered information about the properties of things and also related to other things. Moreover it also enhances the ability to learn subject matter content in a meaningful way. A learner who has a clearly delineated conceptual idea has much better opportunity to learn and remember particular information than one who tries to process and store incoming information without any conceptual connection on which to hang all the details (Hudgins et al., 1983).

This study aims to assess the students’ level of understanding of the basic concepts of functions and relations. Instruments and rubrics used in the study were based on the six facets of understanding developed by Wiggins and McTighe, (2006) namely: explanation, interpretation, application, perspective, empathy, and self-knowledge. Its primary goal is to analyze on how do students’ explain, interpret, apply, have perspective, empathize, and have self-knowledge on the instrument administered to them and test if there is a significant improvement.

2.0 Brief Review of Literature

Current research on intelligence and brain suggests that learning is best when it is engaged in meaningful classroom learning experiences that help discover and develop the strengths and talents (Silver, Strong and Perini, 1997). Research in cognitive psychology (Bransford, Brown, & Cocking,

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2001) challenges the notion that students must learn all the important facts and basic skills before they can address the key concepts of a subject or apply the skills in more complex and authentic ways.

Understanding is the feeling of gaining a new insight and is a visceral and vital aspect of learning. It reflects an important insight into any genuine understanding. They are not facts but ideas about facts. They are not data, but what the data suggests; they are not formulae but why the formulae matter and how they can be derived (Bransford, Brown, & Cocking, 2001). Assessment is the act of determining the extent to which the desired results are on the way to being achieved and to what extent the students have achieved. It is the giving and using of feedback against standards to enable improvement and the meeting of goals.

Most recent curriculum orientations clearly emphasize the importance of functions (National Council of Teachers of Mathematics, 1989). Depending on the dominant mathematical viewpoint, the notion of function can be regarded in a number of different ways, each with different educational implications. The concept of function is rightly considered as one of the most important notion in all of mathematics.

Understanding by design is mainly a curriculum design model that focuses on what the teachers teach. It does not focus on whom they teach, where they teach, or how they teach. However, one aspect to mention is the six facets of understanding. These include: explanation, interpretation, application, perspective, empathy, and self-knowledge. Research by learning theorist and others concludes that less than one-fourth of all students are abstract learners; most students learn best when they can connect new concepts to the real world through their own experiences and experience teachers can provide them, (Wiggins and McTighe, 2005).

3.0 Methodology

This study used pretest posttest qualitative design to ascertain the students’ level of understanding of the basic concepts of functions and relations and to analyze on how do students’

explain, interpret, apply, have perspective, empathize, and have self-knowledge through the instrument administered to them and test if there is a significant improvement after instruction.

The respondents of this study were the students enrolled in the researcher’s college algebra class in nursing department during the first semester of school year 2010-2011. There were 43 students in an intact class which were normally distributed into 5 different groups for some of the designed activities. From 43 students, only 30 were included in this study. Students who missed some of the designed activities were not included.

The researcher formulated 40-item test with particular academic prompts which are categorize into six facets of understanding. Of the 40 items formulated, only 30 were valid with an index of difficulty and discrimination between 0.20 and 0.80. The instrument reliability coefficient is 0.93. The 30–valid item test was given to the students before and after the treatment in order to obtain information about the students’ level of understanding of the basic concepts of functions and relation. Assessments on the students’ answers to the academic prompts were based from rubric for the six facets of understanding designed by Wiggins and McTighe, (2006).

Before the instruction started the instrument was given as pretest. The researcher personally handled the class where the students were exposed to understanding by design (Ubd) using the WHERETO process. The experiment lasted for 2 weeks. Topics covered during the experiment were functions and relations, particularly: finding the domain and the range of linear and quadratic functions, and their graphs. After the topics were taken, same teacher’s made test was given to the respondents as their posttest. The data were analyzed using sign test.

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4.0 Discussion of Results

The data collected were analyzed using sign test. The 30-item teacher’s made test were categorized into six facets of understanding, that is, 7 items were under explanation, 3 items were under interpretation, 5 were application, 4 were on perspective, 6 were on empathy and 5 were on self-knowledge. Students responses in each of the facets were classified into three different levels based from rubric for the six facets of understanding designed by Wiggins and McTighe, (2006).

Summary of the results of the students’ level on six facets of understanding of the basic concepts of functions and relations are shown on the tables below.

Table 1: Students’ Level of Explanation and the p-value

Problem Pretest Posttest p-value

Number Level % Level %

1 Naïve 83 Developed & Sophisticated 43 < .001*

2 Naïve 50 Sophisticated 73 < .001*

3 Naïve 53 Developed 43 0.01 *

4 Naïve 73 Sophisticated 70 < .001*

5 Naïve 73 Sophisticated 50 0.0 *

6 Naïve 77 Naïve 80 0.656

7 Naïve 83 Sophisticated 43 0.0 *

* - significant at ∝ = 0.05 level of significance

It is shown in Table 1 that most of the students’ responses were rated as naïve during the pretest in all items. The table also shows that after the treatment only item number 6 remains naïve explanation and the only item showed no significant improvement. However, most of the students’

responses on problem number 3 improves one step higher from naïve to developed which means students were making the work he own, going beyond the given but have insufficient argument.

Furthermore, the table revealed that majority of the students’ explanation in most of the items showed significant improvement from naïve to sophisticated. This implies that after the treatment students enable to give deep, broad and clear explanation beyond the given information. Samples of students’ explanations on some of the items before and after the treatment are shown in Figures 1, 2 and 3. Answers were copied verbatim from the students’ explanation. No correction of grammar or whatsoever was made.

As shown in figure 1, J2S6 had no answer during the pretest, but she got a sophisticated explanation during the posttest. During the pretest K2R5 had already an idea that the domain of a relation does not belong to the right side of an ordered pair but he does not know the meaning or the difference between (2, 3, 5) and {2, 3, 5}. However, his explanation is not sophisticated since {2, 3, 5} is not also a range, an element only on the second ordered pair. J1I4 on the other hand knows the elements in the domain in a given relation even before the treatment but she does not know how to write the set correctly. It is evident from the samples shown that there is an improvement on their explanation.

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Figure 1

Figure 2

For problem 6, both L2R4 and L1L8 have no idea before the treatment; their explanations are classified as naïve. However, after the treatment L2R4 knows that x = - 3 cannot be in the domain of the function because it is not an element in a set of an ordered pairs and is correct since if x = -3, its image cannot be found in set Y. Thus, it cannot be in an ordered pair. L2R4 had already an idea but cannot explain it well. Her explanation still not sophisticated. Similarly, L1L8 and CE8

know only that if – 3 is substituted to x in f(x), f(x) becomes 0 and is not correct.

It can be observed from figure 3, A2T8 and K1L7 had no answer and H2M had an answer but it is not clear. However, during the posttest the respondents’ explanation was clear, precise and can be understood by everybody. The result reveals that the students were able to explain with sophistication on how a set of ordered pair be defined as a function.

Problem!3:!!The!set!{2,!3,!5}!is!not!the!domain!of!the!relation!R=!{(1,!2),!(2,!3),!(3,!4),!(4,!5)}.!!Why!is!

this!so?!

J2S6!

! Pretest:!“No!answer”!

! Posttest:!is!not!the!domain!of!the!relation!R!=!{(1,!2),!(2,!3),!(3,!4),!(4,!5)}!it!is!because!all!!

the!said!real!numbers!all!belong!to!the!element!of!y.!

K2R5!

! Pretest:!(2,!3,!5)!since!it!belongs!to!the!right!side,!it!is!a!function!(!!!!!!!!,!!!!!!!!!)!

! Posttest:!It!is!because!this!set!is!a!range,!the!value!of!y.!

J1I4!

! Pretest:!because!we!should!include!the!4.!

! Posttest:!The!correct!domain!of!the!relation!is!the!1,!2,!3,!4,!because!the!domain!represent!as!

a!x.!

!

Problem!6:!!Explain!why!x!=!>3!cannot!be!in!the!domain!of!the!function!f!defined!by!f(x)!=!^!!!!!!!_!!!!!!!?!

L2R4!

! Pretest:!“No!Answer”!

! Posttest:!It!cannot!be!the!domain!of!the!function!because!x!=!>3!is!not!a!set!of!an!ordered!!

!!!!pairs.!

L1L8!

Pretest:!x!=!>3!cannot!be!in!the!domain!of!the!function!of!f,!it!is!because!the!domain!there!is!!

positive!a!and!its!already!given.!

Posttest:!x!=!>3!cannot!be!the!domain!of!the!f(x)!=!^!!!!!!!_!!!!!!,!because!if!we!put!>3!in!the!!

equation!it!becomes!0.!

CE8!

! Pretest:!Because!it!or!the!divisor!should!not!be!equal!to!zero.!

! Posttest:!Because!if!you!put!>3!the!^!!!!!!!_!!!!!!!become!0.!

!

x a

!!!

y b!

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Figure 3

Table 2: Students’ Level of Interpretation and the p - value

Problem Pretest Posttest p - value

8 Literal 67 Perspective 40 < .001*

9 Literal 63 Perspective 43 < .001*

10 Literal 80 Literal 93 .063

As reflected on table 2, most of the students’ interpretations in all statements before the treatment were literal. This means that at first students were not able to interpret the problem or have no idea about the statement. However, after the treatment most of the students’ responses on two of the three questions were already classified as having perspective which assesses it has showed significant improvement. This implies that students were able to give an analysis on the importance or meaning of the problem. None of the students were able to give an illuminating interpretation on the given statement. Unfortunately, one of the three problems remains to have still literal interpretation. And, it does not remain only to be literal but the number of respondents who had a literal interpretation increased from 80 percent in the pretest to 93 percent on the posttest. This implies that there are already many respondents who are confused on the interpretation on the equation x = 2. However, Adolescence, (2001) said that not all six of the facets of understanding necessarily be accomplished in one setting.

Results on table 2 are evident in the work of students’ interpretations as shown in Figures 4 and 6. Answers were copied verbatim from the students’ explanation. No correction of grammar was done.

As seen in figure 4 below, L2N2 at first do not know the answer. However, after the treatment L2N2 was able to relate the given set to human relationship as every person is allowed to have 1 wife or to have 1 husband. On the other hand, D1E’s pretest and posttest interpretations are “one wife is equal to one husband”, and “one husband must marry only wife”, respectively. D1E would mean that men and women are equal; they are both entitled to marry one person and these are true for Christian community. Thus, L2N2 and D1E able to have perspective in the given concept which is an important notion and principle of married life. However, G2M1 was able to give profound interpretation on the given set, the set of couples. An illuminating and insightful story of couples as far as Christian belief is concerned.

Problem!7:!!

!ℎ!!!"#$!!ℎ!!!"#!{(0,1),(0,−1),(3,2),(5,2),(−3,1)}!!"!!"#!!"#$%"!!"!!"#$%&'"!!!!"#$%&'#?!!

A2T8!

Pretest:!!“No!answer”!! ! Posttest:!Because!0!has!many!partners!such!as!1!and!>1.!

K1L7!

Pretest:!“No!answer”!

Posttest:!The!set!{(0,!1),(0,>1),(3,2),(5,2),(>3,1)}!!do!not!define!or!describe!a!function!!

!!because!0!has!two!pairs!in!the!range.!

H2M!

Pretest:!It!does!not!describe!the!function!because!it!is!only!a!set.!

Posttest:!It!does!not!describe!a!function!because!0!has!two!partners!in!the!set!of!y.!

!

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Figure 4

Figure 5

In figure 5 the students have written their answer for problem 10. None of the students’

answers have perspective in the pretest. Students’ responses are all literal, like “x = 2 means it is positive” and “x = 2 means x is to be replaced with 2”. Students are decoding with no interpretation and no sense of conceptual understanding. Some of the answers interpreted that x = 2 is a constant function. May be because the value of x is fixed, the value of x is always 2. They forgot that the lesson is on an ordered pair in the Cartesian plane, the value of x = 2 have infinitely many pairs which make the equation x = 2 not a function. The respondents failed to see the partners of x = 2 in the Cartesian plane.

Problem!9.!!Let!X!=!{Ana,!Lyn,!Rose,!Joan,!Mae}!be!the!set!of!wives!and!!

!!!!!!!!!!!!!!!!!!!!!let!Y!=!{Noli,!Joel,!Peter,!Jacob,!Lito}!be!the!set!of!corresponding!husbands.!!What!does!

the!set!!{(Ana,!Jacob),!(Rose,!Lito),!(Lyn,!Peter),!(Mae,!Noli),!(Joan,!Joel)}!illustrate!as!far!as!human!

relationship!is!concerned?!

L2N2!

! Posttest:!It!is!stated!that!every!person!is!allowed!to!have!1!wife!&!husband.!

D1E!

! Pretest:!Explains!that!one!wife!is!equal!to!one!husband.!

! Posttest:!One!husband!must!marry!only!wife.!

G2M1!

! Pretest:!The!given!set!{(Ana,!Jacob),!(Rose,!Lito),!(Lyn,!Peter),!(Mae,!Noli),!(Joan,!Joel)}!!

illustrate!the!set!of!couples.!

!!!!!!!!!! Posttest:!From!the!given!set!I!would!say!it’s!a!set!of!couples!as!far!as!human!relationship!!

!!!is!concerned.!

Problem!10.!!What!does!the!equation!x!=!2!imply?!Does!this!describe!a!function?!

M2A!

! Posttest:!I!think!this!describe!as!a!function,!a!constant!function.!

M5D!

! Pretest:!Yes!because!its!positive!

! Posttest:!Yes!it!describes!a!function!it!a!type!of!constant!function,!the!value!of!x!is!also!the!!

!!!!value!of!y.!

S1J5!

! Pretest:!This!equation!implies!that!x!is!to!be!replaced!with!2!

! Posttest:!Yes…it!implies!that!if!f(x)!=!2!then!therefore!2!is!equal!to!x!(x!=!2)!because!f(x)!=!x!!!

and!y!are!both!the!same!

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Table 3: Students’ Level of Application and the p - value

11 Novice 100 Novice 97 0.5

12 Novice 70 Novice 53 0.03 *

14 Novice 67 Novice 60 0.3036

Application is the third facet of understanding and students’ response is classified as novice, able and masterful. As reflected on table 3, most of the students’ responses in all items are just classified as novice in both pretest and posttest, which means that students can not apply the concept of function. However, during the posttest 50% of the items showed significant improvement on the students’ responses although responses are still on the lowest level. Samples of students’ answers are shown in figures 6 and 7. No correction of grammar was done.

Figure 6

Figure 7

Students’ answer for problem 11 had no response during the pretest as shown in figure 6.

However, after the treatment M2T9 and BA1 were able to cite only few elements of the range not the range itself. Only AE9 was able to give the correct answer. In the same way, only R1M9 was able to give a correct answer on problem 14 in both pretest and posttest among the three samples. From the above samples, it is evident that majority of the responses on application level are just classified as novice.

Problem!11.!!Let!y!=!x²!+!2x!+!1,!determine!the!range!of!the!function.!

M2T9!

! Pretest:!“No!Answer”! ! ! Posttest:!!{4,!9,!16}!! ! !

BA1!

! Pretest:!!“No!Answer”! ! ! Posttest:!{1,!2}! ! !

AE9!

! Pretest:!“No!Answer”! ! ! Posttest:!All!real!numbers! !

Problem!14.!!What!is!the!image!of!x!=!>5!under!the!function!f!defined!by!f(x)!=!x²!–!2?!

RC4)

! Pretest:!f(>5)!=!x²!–!2! ! ! Posttest:!x!=!>5!!!!!f(>5)!=!>5²!–!2!!

J2S6)

! Pretest:!x!=!>!5! ! ! ! Posttest:!

! ! f(x)!=!f(>5)! ! ! ! x!=!>!5!is!the!range!or!the!y!of!the!relation!

! ! f(x)!=!x²!–!2!!

! ! !!!!!!!=!>5²!–!2!!

! ! !!!!!!!=!25!–!2!! ! ! ! R1M9!

!! ! f(>5)!=!23!! ! ! ! ! Pretest:!!x!=!23!

! ! f!!!!!!=!!23!+!5!! ! ! ! ! Posttest:!F(x)!=!(>5)²!–!2!=!25!–!2!

! ! f!!!!!!=!28!!! ! ! ! ! ! !!!F(x)!=23!

! !

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Table 4: Students’ Level of having Perspective and p - value

16 uncritical 97 uncritical 67 0.002 *

17 uncritical 97 uncritical 57 < .001*

18 uncritical 83 considered 57 < .001*

19 considered 63 considered 73 0.395

There are four items which will assess if student possesses the fourth facet of understanding which is perspective. As seen on table 4, only problem number 19 had 63% of the respondents have a considered other point of view during the pretest. This implies that in problem 19, 63% of the respondents able to give reasonable, critical and comprehensive points of view in the context of one’s own view before the treatment. After the treatment applying Ubd in the classroom, 63% of the respondent becomes 73%. However, the table shows there is no significant improvement on the perspective level of the students on problem 19 at 0.05 significance level.

However, only problem 19 showed no significant improvement among the four items though two of them remain uncritical, which means they are unaware of differing points of view, which shows no perspectives, or has difficulty of imagining other ways of seeing things. However, after using Ubd as a strategy of discussing functions and relations, majority of the students are already able to give reasonable, critical and comprehensive points of view of the concept. Samples on the students’ awareness on the concept of functions and relations are shown in figures 8 and 9 below which are copied word for word.

For problem 16, MA2 is able to give an excellent answer. However, majority shows unawareness of differing points of view, like S1J5 and KB3, they show vague idea and has not give other views on the concept. On the contrary, for problem 19, only AE9 gives uncritical points of view during the pretest. BA1 gives a reasonably critical and comprehensive look at all the points of view in the context of one’s own in both pretest and posttest. Furthermore, M2T9 gives an insightful viewpoint in both pretest and posttest also.

Figure 8

Problem!16:!What!are!some!ways!to!distinguish!a!constant!function!easily?!

MA2!

! Pretest:!the!constant!function!

! Posttest:!a!constant!function!is!a!polynomial!function!which!has!a!degree!of!0.!

S1J5!

! Posttest:!n!=!0!of!the!form.!

KB3!

! Posttest:!Constant!function!can!be!distinguish!if!the!element!of!X!or!the!domain!is!not!!

!!!repeated!or!paired!only!once.!

!

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Figure 9 Table 5: Students’ Level of Empathy and the p - value

20 egocentric 93 egocentric 43 < .001*

21 egocentric 63 aware 63 < .001*

22 egocentric 80 egocentric 57 0.046 *

23 aware 47 aware 60 0.032 *

24 egocentric 73 egocentric 53 0.046 *

25 egocentric 60 aware 77 0.002 *

Empathy is the fifth facet of understanding in which many of the respondents’ responses in most of the items are classified as egocentric during the pretest. This means students’ responses have little or no empathy beyond intellectual awareness of others. After the treatment, there are only 50%

of the items where the students’ responses are considered as egocentric the rest are already classified as aware. In any case, 100% of the items on empathy level of understanding show significant improvement. Students are already able to see things through there own ideas, views, and feelings.

This implies that many of the students who are unaware of others views and feelings on the concept of functions and relations are already aware of what others knows, sees and feels. Evidences on the result of table 5 are shown in figures 10 and 11 below.

Figure 10

Problem!19:!Is!it!reasonable!to!say!that!all!functions!are!relations!but!not!all!relations!are!functions?!!

AE9)! ! ! ! ! ! ! BA1)

! Pretest:!No!not!all.!! ! ! ! ! Pretest:!Yes.! ! ! !

! Posttest:!Yes.!! ! ! ! ! ! Posttest:!Yes.! ! ! !

! M2T9)

! Pretest:!Yes,!it!is!reasonable!because!in!the!rule,!all!functions!are!considered!as!relations!

but!not!all!relations!are!functions!because!a!function!is!a!set!w/!no!two!ordered!pairs!have!the!

same!first!term!and!diff.!second!term.!

!!!!!!!!!!!!!Posttest:!Yes,!because!it!is!not!a!function!if!it!is!not!a!relation.!A!function!is!always!a!relation.!

!

Problem!22.!What!do!the!other!students!think!when!they!try!to!make!classification!of!the!functions?!

A2T8!

! Posttest:!If!they!really!know!the!concept!of!relation!and!functions,!they!will!be!fine,!they!!

!!!would!not!feel!nervous.!

J1I4!

! Pretest:!They!can!do!it.! ! ! Posttest:!Confused.!

CE8!

! Pretest:!They!think!it!is!easy.!! ! Posttest:!They!must!observe!the!given.!

!

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Figure 11

A2T8 and J1 I4 show no empathy beyond intellectual awareness of others; they see things through their ideas and feelings in both pretest and posttest. CE8 improves from egocentric during pretest to being aware in the posttest. The table 5 reveals that majority of the respondents in problem 22 remains to be egocentric but there is a significant improvement. On the other hand, problem 25 shows that majority of the students improve from egocentric to being aware and this further shows that there is a significant improvement on putting their shoes on the feet of the others.

Table 6: Students’ Level of Self – knowledge and the p - value

26 innocent 90 thoughtful 60 < .001*

27 innocent 90 innocent 80 0.254

28 innocent 93 innocent 67 0.011 *

29 innocent 93 innocent 53 < .001*

30 innocent 80 innocent 16 0.011 *

As shown in table 6, the most frequently occurring response of the students on the sixth facets of understanding which is self-knowledge are classified as innocent. This means that most of the students during the pretest are completely unaware of the bounds of one’s understanding.

However, in the posttest, only item 26 had superior response from the respondents. This implies that 60% of the students deeply aware of the boundaries of one’s own and others’ understanding on difference between functions and relations. However, 80% of the items on the self-knowledge remain innocent and 80% of the items show a significant improvement. Results are evident on some of the following samples of students’ responses as shown in figures 12 and 13. No correction of grammar was done.

Problem!25.!What!would!I!feel!if!other!students!get!the!wrong!answer!to!a!problem?!

K2R5!

! Pretest:!If!my!answer!is!correct!and!the!others!are!wrong!i!will!feel!pity!on!them.!

! Posttest:!I!will!feel!sad!

KB3!

! Pretest:!I!will!feel!sad,!but!I!know!that!getting!further!more!on!learning!about!solving!a!!

!!problem!will!make!them!right.!

! Posttest:!I!will!feel!pity!and!sad!on!them.!

LD7!

! Pretest:!ask!them!where!did!you!get!that!answer.!

! Posttest:!its!start!in!there!evaluation!on!the!problem.!

!

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Figure12

Figure 13

From the sample of the students’ responses as shown in figure 12, students are generally aware of what is and what is not understood in the posttest which shows an evidence that there is a significant improvement on the self-knowledge level. On the other hand, samples for problem 27 show unawareness of the bounds of ones understanding.

5.0 Conclusions & Recommendations

Base on the findings, the following conclusions and recommendations were drawn: 80% of the teacher’s made test showed significant improvement on the students’ level of understanding on the basic concept of functions and relations at 5% probability level as influenced by understanding by design. Ubd have helped the students develop an insight that leads to conceptual understanding with procedural skill. Hence, understanding by design is one of the effective pedagogical strategies that should be included in the curriculum and instruction designs to engage students making connections and binding together the knowledge into something that make sense. The results of this study confirms that of Canoy’s (2007) observation that poor experience on the proof writing ability and poor understanding of mathematical concepts can cause some difficulty in writing proof.

Moreover, it is recommended that mathematics should formulate questions which will assess the six Problem!26.!How!do!I!know!that!I!am!able!to!differentiate!a!relation!from!a!function?!

K1L7!

! Posttest:!the!relation!has!two!pairs!of!x!to!the!range!and!the!function!has!a!pair!of!the!!

!!!!range.!

MD5!

! Pretest:!If!I!can!already!answer!all!the!questions!correctly.!

! Posttest:!A!relation!is!a!set!of!ordered!pairs!while!the!function!is!the!classification!of!the!!

!!!relation.!

K3U0!

! Posttest:!by!determining!the!domain!and!range!and!by!looking!the!problem.!

!

Problem!27.!What!are!the!limits!of!my!understanding!about!functions!and!relations?!

H2M!

Pretest:!the!limits!of!my!understanding!about!functions!and!relations!up!to!the!things!or!!

!!lessons!I!know.!

! Posttest:!the!limits!of!my!understanding!would!be!up!to!the!concept!that!I!really!!

!!!understand.!

F2M1!

! Posttest:!I!can!only!able!to!identify!each!equation!base!on!the!difference!of!the!domain!and!!

!!!!range!of!an!equation!

R1M9!

! Pretest:!I!don’t!know,!but!there!is!no!limitation!if!you!desire!to!understand!it!

! Posttest:!There!is!no!limit.!

!

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facets of understanding in all learning activities, experiences to give students chance to develop an insightful knowledge of at least three or four of the facets of understanding. Finally, a similar study is recommended to be conducted to other intact classes or to other fields of specialization.

Insights

The use of understanding by design as a strategy in teaching algebraic concepts and theorems is becoming more acceptable in the classroom particularly for unlocking the difficulty. Increasing students’ posttest scores from the pretest scores could be one significant impact of such strategy.

Learners were given an opportunity to develop the six facets of understanding, since they were given a chance to explain, interpret, apply, have perspective, empathize, and to have self-knowledge on a given particular topic. Students will be able to go beyond what they see and to make meaning of it in order to be able to use or apply knowledge and skill wisely and effectively.

References

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Educational Psychology, F. E. Peacock Publishers, Inc., U.S.A.

Bransford, J., Brown, A., & Cocking, R. (Eds.). (2001). How people learn: Brain, mind, experience, and school. Washington, DC: National Research Council.

Canoy, S. R. Jr. (2007). Difficulty in Writing Mathematical Proofs: An Analysis. Doctoral Dissertation, Mindanao University of Science and Technology, Cagayan de Oro City.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Ponte, P. J. (2011). The History of the Concept of Function and Some Educational Implications, Volume 3 Number 2 © The Mathematics

Wiggins, G. & Mc Tighe, J. (2006). Understanding by Design. Pearson Education, Inc., Upper Saddle River; New Jersey

Wiggins, G. and McTighe, J. (2005). Understanding by Design. Alexandria: Association for Supervision and Curriculum Development.

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