MAED 314A Unit Plan Template

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Reflection on my Short practicum
I had a very nice and memorable time during those two weeks.My first day at Earl Marriott (White Rock) was very interesting. Because My SAs were unaware of the fact that they are going to meet me that day.I entered his room and he looked at me and tried to guess.............?.So it was just fun i'm introducing myself giving them a news, I have to sit in their classes to observe. Anyhow,first day passed observing different classes .Next day he was very well prepared, he gave me proper guidelines and introduced me to students.Next day I started teaching 1or 2 blocks daily.The best thing I found that my both SAs liked the way I taught students and my FA liked the stuff I taught in his presence.Both my SAs encouraged me to teach more blocks and they said I have knowledge of subject matter and the only thing that Is to observe and impliment new strategies while teaching.I tried to observe different teachers and different grades and I learnt a lot during that time.So overall it was a very important time for me to learn.I already have taught for a number of years but, now I tried to learn for a multicultured classroom.which I think was not enough time, because as a teacher, learninng will always be there during my whole life.

Doing, Assessing and Designing a Math project

Group: Rong, Darshan & Prem
The Project we chosen from "Mathematical magic is a variation of Fibnocci sequence.As we all know fibnocci sequences have amazing properties. This game is a generalisation of one of the fibnocci sequence's property.

in this game, some seemly unrelated numbers are actually related.By using mathematical analysis we can find out the reason behind the magic.The students will be surprised that we can use the one number to get the sum of ten numbers.Students will be benefit from the game by witness the of power of math.
However this game has a lot calculation of addition,which we think is both strength and a weakness.We say it is strength because it gives students a lot of practice doing addition fast and accurately. weakness is that if student makes only one small mistake , the results will not mathch to the magician's prediction.

Therefore, we think it might be better if students play in group of two.Each student will pick a number to play.Luckily, this game can be modified for two students to play at the same time, because it needs two number to start playing. When they are doing the calculation,one can do the addition and other can check the calculation results. This will reduce the addition mistakes to a minimum an densure the game to play smoothly.
Another problem to this game is that the calculation time and acuracy depends upon the addition calculation skills of the students.therefore it is better for low grade and highly struggling students to allow to use calculators if the game takes too long time.

The "Read Your Mind Game"
The project we have created is for grade 9 students.The major problem for grade 9 students is that they regard math as a hard and a boring subject.The project is intended to arouse student's interest in learning math.Math can be challenging.Math can also be fun.

In order to understand the theory behind the game they have to understand the polynomial addition, which is closely related to the topic of chapter:5 in "Math Links" grade 9.Therefore this activity is also good for students to get a deeper understanding of polynomial addition.The detailed activities are also described on the poster.

We created this game by ourselves and also refers to "Math Links 9" and "Mathematical Magic" The length of the game is 10-15 minutes.students are required to create a key from the chosen number. During the game students are required to take out their pens and papers.They need follow magician's guide to produce a key from a chosen secret number.
Because this is a fun activity for the students participate, the students will not be maked on their performance.However, for those students who are actively partcipating in the game and solving the mystry behind it,some bonus marks will be given as a reward.

Reviewing my group's work.

The solutions of number game given by my group members are really excellent.Amilia considered the options which I didn't. Min-chee's work I could not understand in detail I should discuss with her to know all methods in detail, As she has done totally in different but Excellent way. Darshan's first step looked similar to me but next step I found is out of the way as, the problem was to discuss for the first player and choices for winning with different totals.But perhaps she gave mixed options.Overall the problem solution has been found by four members in four different ways and that is a best approach on the way of problem solving . so now we have four different solutions for a single problem but, if all the solutions are correct.so now I have a lot of things to learn from the group.

Thinking Mathematically Problem Solving

It was a fun to solve this problem bcs it was like a game of numbers to win. The problem is to check a number best to start when playing in pair of two . The numbers given are 1,2,34,5
the players alternately name a number .The winner will be that person who first brings a combined total of 31.
2.Can you find a strategy.
3. what if 31 changes to another number.
4. What if numbers are 1,3,5 or 2,3,7.


I figured out total of digits 1,2,3,4,5 is 15 so I thought 15+15=30.i.e two full rounds of all digits and 30+1 is the game to win.In other words if I start announcing 1 as a first number then a total of 31 can be achieved at the end with 1.
Again if total changes to 33 then I can start with 2 and so on for 35 or 37 .........also for 29, 27.............. At the same time if digits change to 1,3,5 I was able to follow the same strategy of starting with 1 for a total of 31.Bcs every time it follows a sequence of choosing number from a fixed group of numbers. So it looks to me as a fixation of numbers.I found it is the best strategy only if the players have to select all digits differently in all rounds.If all digits can be repeated the I can go simply choosing 24+7, 26+5, 25+6 or 28+3 and so on..................... but then second player can also choose the last number to announce by figuring out the total before the last number.

Thinking Mathematically Problem Solving

So sorry to have a trouble in posting the scanned solution for my problem .
Solution for the problem is here
Game of 31

Mental Solution Strategy & Descriptions

1+2+3+4+5 Sum of digits 1to5 is 15, so if player A

1+2+3+4+5 starts with 1 will win at the end with 31.

1= 31 so announcing 1 is the best option to win

2+3+4+5 Now to find the strategy ,If the first player

1+2+3+4+5=29 or chooses to start with 2 he can only win if

1+2=32 total changes ,either 2less or 1more than 31

3+4+5 yes, it works well. now if he starts with 3 1+2+3+4+5 then to win he needs a change in total either 1 =28 or 2+3=33 3 less or 2 more than 31

4+5+ 1+2+3+4+5+1+3=31 again I check if he starts with 4 he can win if total changes to 27 or 34 i.e either 4 less or 3 more than 31.

Question: what if numbers are 1,3,5then I have 1+3+5=9x3=27+4=31, simply three full 1+3+5+1+3+5+1+3+5+1+3=31 rounds with 1,3,5 and wins for total of 31.

Hypothesis

3+5+1+3+5+1+3+5+1=27 or +3+5=35 i.e if he wants to start with 3 then for

winning condition of total should be either

27 or 35.

hence it gives me only one fixed number to start for a fixed total if I reduce or increase the total then I can have another option for first number to announce.

Timed writing

Reflection on Timed Writing
Writing activity before or after a class is really an enjoyable activity for those who love writing.It could be warm up for the students before doing a hard mathematics exercise.But at the same time some can find it a boring one.Because all students are not actually fond of writing.

Strengths:---Timed writing could be brain storming.Students can express their feelings in words and can explain their understanding of a particular topic especially in mathematics where more topics are not in approach of all students.Their struggling for some difficult topics can be recognised by the teacher through this activity.Students could bbe asked to explain their understandings for the topic disscussed during particular class times. This can provide students an opportunity to recognise difficult terms during writing for that topic.As we felt for division by zero during our timed writing exercise.

Weeknesses:---The only weekness I can see, is that some students may feel bored while writing. But a good teacher can make it enjoyable applying his/her skills. So overall this can be applied for improving standards of learning different topics in math class.

Divide by zero poem

"ZERO POEM"
It's the Zero which makes things meaningful
kepping itself meaningless.
Bank Acoounts having more zeros
10
100
.
.
.
.
10000000000 written after a digit are of great sense.
but,having no digit before million zeros make no sense.
Invented by Aryabhatta during 676
provided number theory a new direction, but
could not find a direction for itself.

Assignment #2 Micro Teaching in a Group.

I feel 15 minutes Micro teaching in a group is only a very short version of actual teaching. Our group tried well to teach the prepared lesson plan.We selected a topic from grade 8.Which we tried to explain step by step using different word problems.We tried to lead the class using detailed approach for getting solution .I feel time frame was the main issue for organising more participatory activities.The topic selected was good enough to relate with the daily activities but we could not explain its importance (Due to lack of time).But I feel that our knowledge of such things would be useful during our actual teachings.

Summary of comments we received from peers:--

Classroom Management Issue--we might need to play stern teacher's role when students do not pay attention.
Last Activity was liked by many people but they suggested for having more clear picture for making their own word problems.
-Suggested shared and equal amount of group work:Many commented that I didn't get much to speak while teaching.
-Need to consider the whole class: The group sitting by the door was feeling that we didn't involve them for most of the time during teaching.
-need to consider all students with different abilities: Our lesson might be boring for students having exceptionality.
-Information was useful: Many commented that information of methods solving equations can be useful during their daily life activities.

Lesson Plan for MIcro Teaching

Teaching in a Group of Three with Booppps.
Group Members:--Alice, Elain & Prem.
Topic: Math 8 Variables & Equations.
BRIDGE: A word problem Say this sunday you went on shooping you baught 5 video games for 110 dollers. how much you paid for one game?

Take a minute to think ..................How did you come up with your answers?

PreTest:--??????
Providing them a figure in physical to ask them for finding one side of the figure when one side & area are given.

Teaching Objective:--
*To effectively teach the concept of Linear Equations.
*To Develop students' problem solving/thinking ability.
*To Relate the mathematics of equations to real life problems.
*To encourage group learning.
*To help students to take responsibility of learning by aloowing them to ask the word problems
created by themselves.
Learning Objective:--
*Students will be able to find the unknown of linear equations at the end of the lesson.
*Students will be able to design problems using the concepts they learnt.
*Students will have an opportunity to present their learning and also to learn from each other.

Participatory:--
WE'll teach the PLOs on solving
5 word problems to teach the equations.
Eqn:1 ax=b
Eqn:2 x/a=b, a is not zero
Eqn:3 ax+b =c
Eqn:4 x/a+b=c
Eqn:5 a(x+b)=c
Asking them how they try to find their solutions.

Post Test:-- Make your own word problems (Group) and present them with their solutions.

Summary:--We will adjust the amount of teaching under time constrain.
(we might not be able to teach all types of equations in 15 minutes.)
-We want students to know the importance of the lesson and how applicable it is to our daily life.
-We want to encourage our students to find different solutions for a problem.
-We want our students to take participate actively in their learning.

Citizenship and Democracy in Mathematics

Response to--" Citizenship Education in The Context of School Mathematics" by Ellain Simmit

I felt very anxious holding the article in my hands.I was very curious to know about the role of Mathematics making good citizens.Before this I have a very strong feeling that people who learn mathematics are very critical minded and also try to find logic behind each and every task that they do.Sometime I try to see these qualities in my students and even in myself also.But after reading the article I found that really there are many facts which shows how math education connects us directly to the society.All the mathematical study is a number game and every field of today's society is based on those numbers.As we're in a society of quantifiers.I think the article is ponting about the same theme which we are learning during our educational studies that we, as a teacher have to make students able enough not only for finding right answers of their math problems but also having critical and a deep understandings.I feel every person's life go through many plus and minuses as well and one can gain terminology to balance those plus and minus by learning mathematics.We as a human being are the part of society and our every move effects the whole society aroud us. So as a teacher our first priority should be to search the methods which would serve the dual purpose of teachinng mathematics i.e for educating students to be active citizens.I liked the idea of building community in the math classroom and to focus on explanations, multiple and diverse solutions.I can think about the techniques to apply in my own classroom so that I can lead my students towards constuctiveness,and I can hope that they might be able to solve the real life problems while living in a democratic society.

The Art Of Problem Posing

"WHAT IF NOT"
The Term "What If Not" refer to the strategy of problem posing to develop a creative thinking
It's all about asking questions in a right way to develop compound ideas and concrete understanding of the concept. I like this technique to use by teachers because it is helpful for making students able to learn in a different way which actually they like and also it's a creative thinking process with a fresh perspective which enables us to expand our ability of thinking.
Teacher can select interesting questions and all correct questions have multiple answers from different learners which creates an atmosphere of rich conversations with many leading aspects for different topics.which enables the teacher to assess students learning process.
The Important thing which should be considered while using this technique is that the level of questions asked should be appropriate. one or two challenging questions which might encurage students make them curious to find the solution of problem should be framed so that all students can participate actively.
I find that it also relates us to the principles of teaching. Where we learn to make our students growth mindset instead of making them fixed mindset. Using problem posing while teaching mathematics with perfect understanding can lead students to a way of growth mindset.
I hope during our micro teaching we'll be able enough to use problem posing.The only limitation is a framed time which is of only 15 minutes and this procedure require enough time.
The only limitation for using this technique is insufficient skill. I feel the best results requires precise knowledge and also a skill. Otherwise it can create a confusion among the students.
Students having moderate level mathematics knowledge can feel difficulty grasping certain topics and can create a phobia type feeling for maths in their minds. To avoid such situation a basic knowledge of using inquiry based methods is must.

Letter from that student who never liked me

Hi, Mrs. Narang

I'm writing because I want to tell you that, what has left behind in my life due to your methodology of teaching math.I could never enjoy sound sleep during my high school because of Homework assignments.I was an intelligent student and you always assigned me a work to do,that was always different from the whole class.In the class also you always expected more and more from me that put me in pressure all the time.I was bore of your so called math quiz,But I could never tell you.Although your labour for making different assignments for me helped me making a good mathematician but still I missed somthing related leisure during my teenage.

Anyways,Thanks a lot
Actually you won a battle!

Letter from a student who liked my ways of teaching

Hi, Mrs. Narang

Today is the most memorable day of my life, I've got my masters of mathemstics.I just wanted to share this happiest moment of my life with you because, I think the Mathematics which I learnt during my high school studies made me able to get this degree today.I remember the ways you assisted me during my learning. your patience for a struglling student changed an impossible to a possible, and I could do my masters in mathematics. I remember your philosphy of making math approachable to all .I'm happy that you proved it .Your subject knowledge in addition of different teaching methods of making math easy to understand made me able to learn .An interest that you inculcated in me for learning math has proved wonderful.

I wish, I could be a person like you.

Ten Art of Problem posing: 10 Questions.

What is the right way of posing problem to make students independent of context?

How does our past experience limit our thinking?

What type of caution should we consider while posing a sensible math question?

How does common sense limit the problem posing?

How can we make problem posing more significant?

Is it benefitial for students to have brain storming questions with peers?

At what level do we able to assess their acquirement?

Should problem solving be technical or not?

How do we ensure students stay foccused or on track?

Can problem solving create a relation between known and the unknown?

Reflection on DAVE HAWITT's methods of teaching.

Reflection on DAVE HAWITT'S methods of teaching.

It was a wonderful experience of watching and learning new approach of teaching mathematics in an orchestrated way. Which shows us not only how to teach but also tells the way of classroom management. I liked the way in which teacher introduced the topic and grabed the attention of each individual sitting in the class by using loud sound of ruler.Every student was learninng attentively and everyone was anxious to see what the teacher is going to do.It was a drammatic way to teach.As first he introduced the simple subtraction and addition then introduced the unknown.He took students from a simple to the complex using patterns and assisiting them to learn by themselves.Students were not taking notes and teacher's response for not saying right or wrong for any aswere of the students, shows that teacher is not working as an "answere checker".In this way he was able to encourage students to do more and he made them able to aquire a full understanding of the topic.I feel teaching in this way could be helpful for students to be problem solver.Because students were self checkers in that class.Which is totally different way from traditional ways of teaching when teacher act as the instructer. This method makes the teacher as assistance who assists students' learning process .Which is the principal of teaching.This method was really relational and helpful for developing problem solving skills.
Overall, watching this movie gave me inspiration to create similar techniques of teaching different topics in mathematics, which I can impliment in my classroom to assist students learning mathematics.

BATTLE GROUND SCHOOLS

Summary And reflection on "BATTLEGROUND SCHOOL"

Summary:- The whole article adresses standards of math curriculum and teaching criteria followed during last sanctury.Primarily,it shows a clash betwwen prgressivists and conservatives for standards of knowledge of math from a teacher to a student.

The Progressivist form during(1910-1940) was influenced by pre media era and the only purpose was to learn basics of Mathematics and the fluency without critical knowledge of the topic.AS the society changed, the Industrialisation reformed the era of new math upto 1960s.During that period "Space race" became a national issue and Ameriacan realised a lack of qualified and educated scientists.After reforms Chicago movement unified Algebra and Geometry during early 1960s.
It was John Dewey who introduced the concept of knowing and doing. He recognised that the process of experimentation and inquiry was difficult than simple lectures.He advocated the development of high quality mental processes and a scientific attitude.Initially his ideas were not accepted across north american schooling, but lator on Deweys' methods for Programming the Environment were put into practice in some classrooms.In the post war period it was realized by Americans that USSR beating the United space in space race.

After post war period School mathematics study group (SMCG)started a movement and in colaboration of National council of Mathematics Teachers (NCTM)efforts were done for rewriting the math curriculam in United States that resulted into a rejection of Geometry where as Set Theory,Algebra,Calculus and other topics were selected to be taught throughout K-12 system.So the movement gained momentum worldwide .But there were problems regarding familarity of topics among many teachers and movement came to an end by early 1970s.

In the mid 1980s NCTM created its own standards.The development of curriculum,course for teachers and involvement of parents and educational technology were the main features.The revised eddition of this was published in 2000.Where as Recent publication of NCMTs'new focal point has pulished in 2006.

REFLECTION:- I feel serious battles for Mathematics curriculum and teaching techniques are still continue. A number of ammendments in math curricula has already been done still there is a need of critical evaluation of the contents.Future teachers can play a major role in raising standarads of mathematical learning by adopting creative techniques of teaching mathematics in their classrooms.

Reflection for Interviews.

SEP 27 , 2009.
"Reflectinon for Interviews"
There are many things which I learnt after analysing the interviews directly and also from other classmates' experiences. The first thing I feel is that current students needs teachers who are super intelligent and can give them more then one ways for solving math problems.They want more fun while learning and energetic teachers. On the part of teacher the most important thing is to recognise individual abilities and making arrangments for all of his /her students saparately.All students should be served individually according to their personal needs .Intelligent students should be assigned challenging work,where as struggling students should be provided normal works.As one of our peers' groups explained the fact that boys are always ready to do experiments but girls like structured materials.Students should be given break and more practice work But all their assignments should be marked in time .
A good teacher is that who works upto the pace of students.Students also want interaction with the teachers .Teachers shouldn't be lazy instead they should be well organised and eveready to help the students.Humour during study can play a major role for making the subject easy and also in approach of all students and a teacher should try to make math a relevant subject.Teacher should create an environment that can explain for what and why questions of their learning of a particular topic.Most important thing I feel personally is that a teacher should always be in approach of all of his/her students so that he/she can provide a quick assistance for their learning needs.

Group report for Interviews.

Sep 27,2009.
Group Report for Interviews.
We were two members.Mrs.Darshan and me.We planned to meet two different teachers and two students. we selected five questions to ask form both teachers and students.all those questions we asked during our actual meetings with our teachers and students were not only those selectedd five, but we asked them many other related questions .which gave us knowledge about high school teaching and learning process.Which we feel that is very important aspect of teaching prfession .Because teaching itself is a learning process and we always learn from two different type of experiences.One of them is our own and the other is in the forms of following answers which we get from others' experiences.
First important question which we asked from two teachers was
1 Is it possible to relate all topics in math to technology ?
The answere which we got, made us able to understand that in Mathematics we can add technology to most of the topics, but not for all .All topics can't be teach with movies,pictures or by other helping aids. Most of the time teachers feels bounded for meeting the requirements of provincial exams,and finishinng curriculam in time doesn't allow to adopt other methods of teaching except instrumental.
2. How do you compare teaching math with teaching of other subjects ?
.One of the teachers was a math specialists and other was physics specialists.So the teacher with math as a math major said that teaching math for him is very enjoyable activity and he feels that he can inspire better his students while teaching math in place of other subjects.The other who was having physics as a major explained that during teaching science he can use more audio visual aids than the aids he uses for maths.
Hence we learnt that teaching techniques mostly are concerned to subject matter to taught and also the knowledge of that particular topic.Limited knowledge can also restrict teachers for selecting different methods of teachings.
The questions which we asked from the high school students were also very informative and inspiring. One of our students' favourite subject was also math. Firstly, we asked them about their experiences with their math teachers. And their replies for this question made us aware of the things that students notice for the patience of teachers . She Said " some teachers are very good and explains the topics until we don't understand.They explains all terms and formulas step by step.Like my current teacher explains all terms by linkinng them to each other".She also told about her previous teacher that "her method was not good, she stayed busy with the board only for the whole year and never made intractions with students.
This means that students want to interact directly with the teacher, they don't like those teachers which explains whole concept using board .So they need some different methodology which gives them ways of making intractions with the whole team during their learning.
Secondly we asked about curriculum.The mixed respons of both the students was that some topics they really like to do and some topics are there in the curriculum which they don't feel easy.Which gives information of diversity in learning and special needs of different students.
Third question was for knowing the methodology, their teachers are using at present.
One of them replied that " Our math teacher first introduces the topic then she tells us what we have to learn from it, and then she explains all steps . somtimes we do disscussions in groups but not all the times . In other subjects we do more works in groups".
This reflects to us that most of the teachers using instructional methods .

Hence overall our work for getting information from different teachers and students made us aware of many things related to methodology, curriculum, diversity in learning and different strategies of teaching maths.Which I must say, is very important to be a successful learner and a teacher.

"Reflection" for Robinson's Article

I agree with the fact that learninng by doing is always more valuable.It is very important for a teacher to provide students an opportunity to use their own skills for interpretation of results by participating in strong disscussions with peers .It's a wonderful idea. I already have experienced this practice during my teaching career. I have examined the changes in students' attitudes towards learning Although it requires a lot of patience but, I feel when students are allowed to give their opinion in a group they not only try to draw conclusions for some specific topics but they develop leadership qualities among themselves.Somtimes they feel as all their feers for complexity of some particular topic have lost.I also agree with robinsons' suggestion that activities like TPS focuses on the students' thinking and promotes disscussion of mathematics.they have cerrtain task to complete, but they must synthesize their findings and develop logical conclusions rather than practice specific skills[pg.70]
I feel this must be the scenerio of my future classroom.

Two most Memorable teachers.

I believe teacher is a person who not only teach us but try to make our lives better. whenever I remember my school or college time I always feel that all those teachers who have taught me in different grades are really wonderful teachers.The one of them who taught me Math in grade eight was mrs. devinder.She was so simple and polite that everyone liked her.I think she gave me so much knowledge that I can never forget, Perhaps her methods of teaching were so impressive that were able to establish their impact deep in our minds. Other teacher I'll describe here is the one who is very special not only for me but, for all .He is such a genius personality that his achievements can't be explained in words.His Name is Mr. Gobind. An exceptional body that he got from God is just of a 3 feet height and a super brain like a computer.I think the person, I'm today is just only because of him. he taught me during whole of my university studies.Ican't forget him during my lifetime. Even today I send him an e-mail and told him all about my studies at UBC.I feel he'll be very happy to listen all about this.

Peer's Assessment summary

Sep 17 2009.
we were four in our group.
GiGi, Darshan & Sara.
Gigi, found that introduction during micro -teaching was a fun and so it made the lesson interesting.Demonstration was in a nice way.And, she suggested to do more for active participation of students.
Sara,said almost same for the introductory activity, for it was very encouraging and she praised provision of some individual knowledge of practically saving enegy by making wise use of sources of energy. she suggested for making some written notes for the students.
Darshan, said the objective of teachinng was very importantand she suggested for making some teaching Aids.

Self Assessment: Micro Teaching.

Things Went Well :- Introduction was approprate and created an interest among peer students to know about the work they"re going to do in class.

Things Need Improvement:- Use of More teaching Aids is required to do more accurate teaching.

Things reflected on based on Peer's feedback:- I tried to answere all those questions ,which were asked by me but weren't answered properly.

Sep 17,2009. Lesson Plan For Micro Teaching

Topic:-Energy,Its Resources & Uses.

Bridge:-An Activity of rubbing hands & asking questions will be organised .

Q 1.How we are able to do work?

Q2.From where do we get energy to do work? etc.....

Introduce the topic & explain all necessary information related to the topic.

Participating activity:- Questions related to the information provided earlier during teaching will be asked to create more curiosity for the topic.

then answers for all questions will be given in addition some deep knowledge of topic.

Teaching Objective:- Emphasizing students to think about wise use of non-renewable sorces of energy.

Learninng Objective:- 1.To learn about energy and its types, Need of reasonable use of available sources of energy.

Post Test:- Students will be asked to make a chart showing differances of renewable and non-renewable sources of energy.

Reflection on Skemp's Article.

Response to Richard R Skemp's "Relational Understanding And Instrumental Understanding"

As Instumental Methods provide important steps to solve a mathematical problem where as relational methods provide a deep knowledge of the terms. Both methods have some advantages and dis-advantages.One can really understand the need of earning rewards quickly. which can be done by learninng mathematics instrumentally.Where as we can not ignore the need of learning the terms deeply;which can be done using relational methods only.The Article provides us an opportunity to differenciate strengths of both methods.As Skemp decribed his own experience of exploring the map to find all shortcuts and extentions.I feel a mathematics student should also be able to navigate around mathematics independently. But being a teacher we also have to have a look on the Syllabi.The burdened syllabus has also commented in the article. students can comment, if they are able to score well by doing less; then what is the need to delve deeper into the subject matter, when the end results are same.Also they need quick rewards to earn their degrees.So in my view a balance of both methods can provide a solution of this problem. It depends on the efficiency of a teacher how he/she creates situation of learning math in the classroom.Both methods should be included during teaching mathematics. Providing formula sheet and telling students the steps of solving a math problem is in no way a justice with their creativity.So a combination of both is better.Relational understanding can cover the weeknesses of instrumental understandings and vice versa.