This assignment is due Thursday, September 11, at the start of class.

**Assignment. Ā **Choose ONE of the following two topics. Ā Write a reply to this post, responding to the topic. Ā Begin by telling us which topic you chose. (1-2 paragraphs).

**Topics.**

- Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened toĀ you, tell us about it.
- Choose an experience you had in which you suddenly understood a math concept (it could be any kind of math, from elementary school up through college). Ā Describe what happened. Ā Do you think you could explain it to others in a way that they could have the same flash of understanding?

**Extra Credit. Ā **For extra credit, write a response to one of your classmates’ comments. Ā Do you feel the same, or different? Ā Did you learn anything? Ā Did you get any ideas about teaching, or about learning?

** Why are we doing this, anyway?**Ā We are following two ideas that have come up already in class — things that mayĀ notĀ seem related to learning math, but research shows that engaging in these activities canĀ

*dramatically*Ā increase the amount that you learn, and change the way you learn it. Ā The first isĀ

**Ā ā something not typically associated with mathematics. Ā When you express your ideas in words, it forces you to think them through very carefully, detail by detail. Ā A great way to check and see if you really understand something is to try to explain it to someone else, either out loud or in writing. Ā**

*writing**Example: if you know how to add fractions, try teaching it someone who doesnāt know how. Ā*The second is calledĀ

**, or āthinking about thinking.ā Ā This happens when you think about what was going on in your head while you were working on a problem or trying to learn a new idea. Ā What train of thought did you follow? Ā Where did you get stuck, and what did you do next? Ā What were you feeling at the time? and so on. Ā Combining writing and metacognition can be a tremendously powerful tool in identifying the ways we learn best and the ways we make mistakes, and learning to improve. Ā However, like any skill, it takes practice. Ā Thatās why weāre getting started by writing a little about our past experiences with mathematics.**

*metacognition*

I remember in Jr, High School when I first learned how to use FOIL. My professor’s name was MR. Reinoso. He was a great guy and made math fun and enjoyable. There was one day when he had just finished learning about “like terms” , “polynomials”, ” monomials”, and eventually “binomials”. As we all know in math, we learn about the basics first then we move on to the hard part. So for me the hard part at that time was when he introduced “FOIL”. We used FOIL in order to multiply 2 binomials. Each letter in FOIL represents something. The “F” stands for First Terms, the “O” stands for Outer Terms, the “I” stands for inner Terms, and the “L” stands for Last Terms. In the beginning it was very difficult for me to comprehend this. I needed several examples in order to fully understand. There was a lot of algebra involved, well mostly recognizing for “like terms” and coefficients.

However towards the end of the year, I mastered FOIL and even until now I will never forget it, all thanks to my 7th grade math teacher. With the helpful drawings he made and different examples he provided, there was NO WAY I can forget it. If I were a teacher I would like to teach in the same manner my 7th grade teacher did to me and that was through Patience. Students who are willing to learn and don’t fully understand something, they must have the proper guidance and necessary steps to lead them towards the answer. This is what I believe my teacher did to me and all of my classmates and I thank him for that.

In my opinion I thought using FOIL was a brilliant technique and it was one of my favorite things to use when multiplying two binomials in my math course. When I was first introduced to it, It seemed common sense since the letters in the word FOIL each represented something, so I didn’t have trouble with it when I was first taught it. But I do agree that FOIL is something you can’t ever forget. That technique is still in use today and even I use it when I have to. But sometimes I wonder if there is another technique to multiply two binomials together instead using FOIL.

FOIL is nothing different from the distributive property between two binomials. You are still multiplying each term of the first binomial with each term of the second. The only thing it does that it leaves the terms of a resulting trinomial of at least degree 2 in a preferred organization of descending power of at least one variable. In that regard, I myself do not see it as ingenious, but rather a simple implication from a known mathematical property.

Their are several other ways to do it. I was taught by using FOIL, but then their was the box method and the vertical method. I find the vertical method to be the easiest one because it follows the same way we multiply numbers. FOIL is a good way but it has a limit with what equations or expressions can be used, which is why I liked the vertical method the most.

Choose an experience you had in which you suddenly understood a math concept (it could be any kind of math, from elementary school up through college). Describe what happened. Do you think you could explain it to others in a way that they could have the same flash of understanding?

Recently in my Calculus 3 course, my professor used a method to find the determinant of a 3×3 square matrix that I had not encountered when studying Linear Algebra. He copied the left two columns and added them to the right of the matrix, creating a 3×5 matrix. Starting from the top-left corner, he drew three diagonal lines down and to the right. Then he drew three diagonal lines down and to the left starting from the top-right corner. Each line represented the product of the three numbers each line intersected. Lines moving right meant addition and left meant subtraction.

This actually lead to a curious reaction from the few of us in the class who actually took linear algebra, as some of us actually had not learned this technique. (Feel free to imagine me as the classic confused puppy with the wide eyes and tilted head ). As the professor performed the technique once again, I spoke with one of the other former linear algebra students (and former student in this class) about the technique we did know and then the light bulb went off in my head; I instantly knew what my professor was doing.

Could I explain to other students what I saw? It was the same technique I had learned, but performed in a different order. I had been taught to pick a row and omit columns to create three 2×2 square matrices multiplied by the different values of the selected rows. I realized that if I had actually used distributive property in the method I learned as opposed to standard usage of order of operations, I would come up with the same numbers my professor was coming up with on the board.

Would I use this new technique over what I do know? Maybe. But it’s good to know both.

Actually I had quite a similar reaction provided that I haven’t taken linear algebra before. However I remember my physics professor used this same technique to solve the matrix when we studied 3D vectors. I was glad I remembered the technique that was used. And yes! It’s true that it’s good to know both ways especially when the use of i,j,k method involves sign change. This way is intriguing!

It is a good discovery. Thank you for sharing.

I like this idea of seeing more than one way of doing something – it can be very powerful to look at problems from different perspectives (and making the connection that two ways of doing something are actually the same – even better!)

I remember when I was in my 9th grade one of my favorite math teacher asked us at random to calculate the sum from 1 to 100. She also said that we must do it in about 5 minutes or less because a ten year old boy was able to it quickly. Well, none of us knew if we could do it or not. All I remember was moving my hands really quick and start adding from 1 to 100, and just as I reached 30 my time was up. At that time I was thinking to myself finding the sum up to 100 in 5 minutes is humanly impossible but how did that 10 year old do it?! I remember I felt so stupid. Moments later I saw my teacher grin like she fooled us with this problem. I also hope she did fool us because I was literally going crazy.

Anyway she told us about this young mathematician I think his name is Carl, and that he really did find the sum of 100 so quickly that he made his teacher and class mates speechless. His trick of finding the sum is what we call and use in calculus as the arithmetic series.

She wrote the sum on the board as 1+2+3+4+5+……+99+100, and then wrote the progression again in reverse order and then added them up. Having the numbers aligned properly and adding them gave 101 for every pair. For example 1+100 =101, 2+99=101..This created a pattern. Since there were 50 pairs of this, the sum is 50 times 101, and the answer is 50(101)=5050.

I can’t forget that day, because I remember relating this story to my dad that same day, and he thought i was going crazy by over thinking it. It was funny because my dad calmed me down and said” relax! that boy is only a mathematician” and everything was okay after that.

Do you think you could explain it to others in a way that they could have the same flash of understanding?

I think I love to explain things in the same way I understood for the first time, let it be math, physics or even a rare incident. Last semester in MEDU 1021 class we went through this topic of triangular number and series and I remember relating this story to my partner and she was quite convinced. I’ve explained it here too and I just realized it.

I will have you know that if you ever play Cranium, finding the sum of the first 100 numbers is the task on one of the cards.

I like this example a lot, because the explanation is such an important part of it – if you just gave the formula (50*101), I don’t think anyone would understand *why* it worked. But the explanation makes the formula very clear!

(ps. the boy in the story was Carl Friedrich Gauss – he did that about 250 years ago, in the late 1700’s)

Writing about a mathematical mind-opening moment sounds as though I have to write out something very impressive and special. However, as ordinary as I am, I am sorry that I have not encountered any moment was that enlightening. What I think is worth to writing about is how I changed my opinion about math.

Last semester, I made my class observation in a high-school called Frances Perkin. while there, I witnessed how my mentor taught factoring of binomials with theleading coefficient 1. However, she did not teach anything more than that because other cases do not fall into the requirements of the regents. Yesterday, I tutored my math 1275 workshop, none of the student notice that 3×2 -13 x +4 =0 can be further factored, and most of them do not know how to factor it. All these make me think about what math has been to us. Inevitably, math has been an unwilling obligation and responsibility for some of us because we have no right to not choose it. Thatās why oftentimes students ask the questions: why do I have to learn math if I have already know how to count? Why do I have to learn higher math if in my life I do not have to use calculus in any way? Sincerely, I do not have any answer which can convince you that math is necessary for you to be what you want to be. In that sense, I feel math is a procedure during which we learn how to overcome and endure what we do not desire, but from which we cannot escape from. Standing at the height of discussing what life is, math is just an experience. At least, it can tell you in which aspect you can make your contribution to our society and gain your life. From the other point of view, I ask myself what math is for me. Back to the case of factoring 3×2 -13 x+4 =0, I can say the answers immediately, (3x-1) (x-4) =0, then x=1/3 or 4. Sadly, I cannot tell why. I only mechanically remembered the steps, and with practice, it becomes instinct. I feel sorry for confessing that I do math but without real understanding. This separation between understanding and application makes me to think of math as a diagnostic tool. It does tell me that my way of learning is not correct, and my goal of passing a course with an A is a wrong focus.

This is a change of how I think about math. Math is no longer merely a science with numbers for me, rather now itās the compass that tells me in what direction and how I should invest my time to make real learning happen.

Hi Jian sun, i totally agree with you that Math is an experience , and i actually feel the same way you do that sometimes i can answer questions immediately but i cannot tell why i got to this answer.

Hi Jian, that was an amazing piece. You have made me realize something I think I have never wanted to admit in the past, to wanting to just pass a class without fully understanding the concepts. It’s a habit of knowing how to solve the problem without hesitation but not knowing how or why it works. I think now being in this major and being in the math education courses have helped me to use how certain math concepts work. I really like the questions you used because most students say that and I feel like they are wrong in stating those questions because in our everyday life we always are using math, even without even knowing it. Basic one is, metro cards, knowing how much many to put in, is using math. It might not be high level math but their are always something to connect math with.

Jian, your self-awareness in this piece is so admirable. I think everyone who does math does what you describe, to some extent – learns to complete the problems, without really understanding “the why” (I am guilty of this at times, too!). Our educational system has many aspects that encourage this thinking – the overemphasis on standardized testing, the importance we place on getting the right answer, and the very very little time that we set aside in our math classes to just “play around with math”. Keep working, keep trying, and don’t forget to give yourself time to play!

ps. That’s one of the reasons I love puzzles – they are a way to just play, and they always push me to understand “why”!

Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened to you, tell us about it.

I think the first topic relate to me because It happened to me many times. I got through many experience Since English is my second language so things seems to be harder for me to understand it. It always takes a while from me to understand concept. I always have to do more work than any other person who might understand things once its taught,but for me i have to go through a lot of things to be fully understood. I have to go for tutoring many times and i have to read the chapter and keep practicing so i could understand everything. One of my experience that happened to me was last fall in my calculus 1 class. The first topic that was given was derivatives. I did not understand anything in the class at this time even the formula and the definition i did not understood them. Most of the class understood the problems except me. I felt very bad at that time and i thought that it is very hard to understand it and that i will never get it. I spent almost three lectures without understanding anything just because of the derivatives. I kept trying and trying and i had a friend who was very good at Calculus. I asked him for help and he did help me a lot. He explained derivatives for me step by step and in a very easy way that i got it once he explained it. I went back to the homework that was given before and found out that i could answer every single problem and get the correct solution for it. After that, i found out that derivative is a very easy topic not as it seemed to be in the beginning . I used to solve it very quickly and laugh at myself that i did not understand a very simple and easy topic like that.

I understand your feelings because i am an ESL student also. Do no feel weak about that because learning in a different language than yours has never been easy for anyone. I remember my first year at city tech, i was wondering how i will write my math test in english or explain a concept to my fellow students. Your way of learning is a classical way that every good student should follow. I mean praticing as much as we can to in order to master a concept. In my case, i never feel comfortable with the english language. But by practicing, i did overcome this weakness and i do not feel no more any fear to write my math homework or test in english

That experience of a new topic seeming impossible, but after you master it seeming easy, or even obvious, seems to be very common in math. The ideas may be simple – but sometimes learning a simple idea can be much more challenging than learning a difficult one!

Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened to you, tell us about it.

I learned a mathematical puzzle called four fours last semester again after 12 years. The goal of four fours is to find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical symbols and the digit four (no other digit is allowed). I was very bad in math, and I didnāt like it at all. One day, I had a Substitute teacher in my seven grade math class. He taught us a math game called four fours. I was surprised that math can be so fun and different. My friends and I were work hard on that game. It was more like a co-op competition. The answer was given, and there were more than one approached in some case. The game was not only help us on math, but also given us personal achievements.

A motivation for academic success is very important. When we tell the kids if you donāt study hard, you will have trouble in your life. How many students understand what it means? Most of students are smart, but not all of them willing to study. Group study will help students learning because studentās attention only span is about 10 to 15 minutes long.

I do not like ability-grouping in class because it hurt some students feeling. What happen if the students in the worst group? They are a group of low learning ability and motivation, and with no help from group mate. In most of the case, they will be always the worst. I did have dramatically changed because I got out of worst group in eighth grade. Thanks for my Substitute teacher.

I also dislike ability grouping – often, groups with mixed skill levels are the most successful, and not just for the students who are struggling.

ps. I also learned the four fours game in the seventh grade (well, a similar game) – and I fell in love with it, too!

2:Choose an experience you had in which you suddenly understood a math concept (it could be any kind of math, from elementary school up through college). Describe what happened. Do you think you could explain it to others in a way that they could have the same flash of understanding?

A:While taking Calculus II i believe the greatest obstacle I had to overcome was understanding Integration by parts. Just the aspect of applying the formula correctly was nerve wrecking and tedious. Add on top of that remembering which part should be U and which should be V, and in cases involving Trig or Exponential functions, repeating the process over and over until I had all the terms, was just too much to handle. When my professor decided to introduce the tabular method for integration by parts I was awestruck.

Basically the tabular method involves making a table with two columns. On the left column you put the part that if you repeatedly take the derivative will eventually become 0 and list all the derivatives until you get to 0. In the right column you take the other part and repeatedly integrate until you have an equal number of terms as the right column. Next in front of every other term in the left column add a negative sign. Finish by multiplying the first term from the left with the second term from the right , second from the left and third from the right and so on until you reach the last term on the left column. The sum of all these resultant terms is the answer. Being a visual learner it took me no time to learn the process and then repeat it successfully. Moreover I found that it was a lot more organized and easy to follow than the old method.

I would always take an opportunity to teach what I learned to others but I have learned that not every one learns in the same way and that the way I find easy may just be confusing to others. It is possible that I can spark a similar sense understanding in someone else but in the end it all depends on them.

1. Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened to you, tell us about it.

I remember when I was in my 10th grade math class; I always asked myself āwhat is the point of learning this stuff anyway. Iāll never use this stuff in life.ā I guess I was one of those students that looked for a reason to hate math more than what I already did. Every time I went into that class I would enter the classroom saying ātime to learn something uselessā, but one particular day was different instead of learning some new material, the teacher showed us a video on how mathematics is connected to everyday life. In this video it showed how mathematics is applied in everything we do without us being aware of it. The video showed and explained how math helped us be where are today, without the knowledge of mathematics we wouldnāt be able to have the technology, that nowadays, we depend on. After seeing this video and how it explained the idea of math in our lives I came to realize (in my opinion) that math is the center of all that surrounded us, meaning that from math everything else was created. For example with math we were able to constructed cities, create objects that are used every day (machinery, technology), count, etc. I came to realize that even if we donāt physically do math every day, mentally we are; we calculate things every single day without paying much mind to it. For example we calculate how long will it take us to get from one destination to another and what would be the distance. This recognition help the change my aspect towards math it me realize that no matter what I do in life I will be always be doing math whether I like it or not.

I agree with Leonardo’s response to the question. I agree with him in the sense that we all use math everyday whether we like it or not. We calculate how long it takes us to get from one place to another, and we also calculate how much money we wasted today on either food, groceries, expenses (Some of us do). We use numbers everyday, math is exposed to us in our everyday lives. It takes a few people everyday to realize we use math in our daily lives. If it wasn’t for the brilliant mathematicians in the past, we wouldn’t have the machinery & Technology that has been exposed to us today. This is why I personally believe Leonardo’s Post is the best and I agree with him 100%.

Great! If you are able to track down the name of the video, post it here – I’d like to see it.

Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened to you, tell us about it.

I’ve always loved math class. I was the little kid that said math was my favorite subject and I took pride in having good grades. Everything always clicked. This continued up until trigonometry. There were now words in my equations. I needed a calculator with special buttons. The calculator was performing functions that I couldnāt check in my head. My notes became foreign to me and homework was a struggle. I failed a pop quiz for the first time ever. Through the struggle I wondered if college math was going to be this hard, would math still be my favorite subject? What if I did poorly on a test, people would know that I wasnāt good at math anymore? I started telling my dad that science was really my favorite class, there was math in science class. My math confidence was shook.

A few weeks into the unit things started to make sense. They didnāt click but I could stumble through and homework became a learning experience. Many math classes since 10th grade have been hard. I’ve even gained to confidence to say I donāt know what’s going on and to ask for help. And now the harder concepts make me appreciate the power of math more and more. My opinion of math has never been more positive, even if my mastery of the topic at hand has not been great.

I totally understood your situations and i have also once told my father that science is my favorite because there is lots of math in it.

Choose an experience you had in which you suddenly understood a math concept (it could be any kind of math, from elementary school up through college). Describe what happened. Do you think you could explain it to others in a way that they could have the same flash of understanding?

I remember when I finally understood how to use the AC method when factoring. I never used the AC method before, I was always comfortable and good in using the trial and error until I encountered the AC method when I first started college and had math 1175. I had Professor Douglas for math 1175 and the class was fairly simple until he introduced how to factor trinomials. I always was able to factor trinomials using the trial and method where I had to guess which two numbers would go into the trinomial by putting those two numbers in a binomial form and using FOIL to check the answer. When I was introduced to the AC method I thought this was a faster way to factor a trinomial especially if you have a larger trinomials numbers you have to factor out instead of guessing and checking. Now whenever I have to factor a trinomial I sometimes use the AC method if it is difficult for me to factor out.

I defiently could explain this method to other students especially if they have trouble factoring trinomials. A lot of students who I had as a group as their peer leader don’t really know how to use the AC or was never taught that method and I prefer the AC method over the trial and error just because it’s faster and easier to solve and you can solve with little steps unlike the trial and error method where you have to basically guess and check the answer. It’s always good to know that there are more then one method to solve a problem because this helps you understand the concept better and it gives you practice and opens your mind to new ideas when solving, But I always tell my group to use the method they are most comfortable with so that just in case if the AC method was useful or not useful to some students they can have at least another method they can use to solve the same problem

After guessing and checking, having a method to follow is real relief!

Topic #2

Throughout elementary school one of the worst experiences that I had with math was fractions, it was just a concept I couldnāt grasp to understand. Iām not so sure how I really did it getting pass by fractions, but guessing and checking did the work. Once I got to secondary school, my teachers werenāt profound of students showing work only by guess and check because they actually wanted to see the work on how to solve fraction problems. When I encountered this I got really frustrated with myself, because I wasnāt able to solve fraction problems.

During my third year of secondary school the best thing happened to me. A teacher of mine who I will never forget announced that she was going to hold an after school class in order for her to help students understand the concepts better. I would never forget spending the afternoon and Saturday mornings with my teacher, it was a lot of fun; she made me understand the concept so well that I was actually enjoying math and enjoying fractions, but at the same time feeling embarrassed because it was so simply to solve. All that extra work and fun activities were really helpful and useful. Since I peer lead 1175 and 1275 workshops before, my concepts or knowing and understanding fractions were put to test and I thought I did a good job on it. I followed a similar way that I was taught and I made sure I gave a lot of examples and made fractions fun to learn. I did it by playing math games and using tiles, even though it sounds kind of babyish but its a good way to learn how fractions work.

You’re lucky to have had that experience – you will find many students up through high school and beyond that still struggle with fractions (and other that can solve the problems, but don’t really understand what is going on). Games and tiles are great learning tools, too!

Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened to you, tell us about it.

I’ve loved began loving math in middle school, but, when I went to high school that changed a little. In high school I took a math course that dealt with trigonometry (shapes, sin, cos, and etc). It was horrible because there was so much things to remember and I had a hard time remembering it. Also I had to use a calculator most of the time and I hated using it because I rather show my work. I wasn’t really sure how I passed the class; I think I engraved some of the things in my head before the tests were given. Than after I passed that class, I had to take a class where I had to do proof and reason. That was worst because I still had trouble with trig. So at that time I had no choice but to go to tutoring. After that I grasp a better understanding of it and somehow passed. I still don’t like that subject till this day but overall, it taught me that learning math takes a lot practice at times. In the end I still love math.

I’m glad that negative experience didn’t change your mind altogether. And here you are, back in a “proof and reason” class — good for you!

When I was in Math 1275 we were learning trig identities and I was completely lost. I went over several class examples at home but the concept of using these words as numbers to make them equal still didn’t click. Later I read trig help on math help online and they gave me tips that really stood out. I knew I had the major trig identities memorized so that wasn’t the problem. Once I put the concepts I read online and what the professor taught into practice I began to master the problems. I ended up helping other students in my class on solving the problems and I got a 99 on the test.The professor even wrote ‘good’ on both my trig problems.

Great! Trig identities are one of the trickiest algebra topics – but the lessons you learn in them (about substituting one expression for another, simplifying complicated expressions, doing algebra with sines and cosines instead of just numbers and variables) will help you throughout your math career.

Choose an experience you had in which you suddenly understood a math concept (it could be any kind of math, from elementary school up through college). Describe what happened. Do you think you could explain it to others in a way that they could have the same flash of understanding?

When I was in middle school my math teacher taught us the rule for solving problems in combined operation. The teacher wrote āBODMASā in the white board and all of us student started to laugh because if we take this rule as a world then it sounds funny but also the meaning is naught in my native language. At first we didnāt understand what the math teacher was try to explain then she wrote BODMAS stands for : B for bracket, O for order of, D for divide, M for multiplication, A for Addition, S for subtraction. I was so amazed by this new rule and after solving few problem I felt that day like as if I was solving a puzzle using code which was the new rule which i have just learned. Later when I move to United States I got to learn from a friend that over here they say PEDMAS instead of BODMAS.