OpenLab Assignment 1: Sudden understanding

This assignment is due Thursday, September 12, at the start of class.

Assignment.  Choose ONE of the following two topics.  Write a reply to this post, responding to the topic.  Begin by telling us which topic you chose. (1-2 paragraphs).  Be sure to include your name in the post, and which section you are in (so I can give you credit).

Topics. There is a common experience that  many people describe having at various times when they are learning mathematics.  It is the experience of suddenly understanding a concept for the first time – something that might have seemed confusing, or hard, or just didn’t make sense,  all of a sudden becomes clear.  This can happen in class, because of something a teacher says, or because a friend explained it to you in a new way.  Sometimes it happens on your own, because you were thinking hard about a particular problem.  Afterwards,  the concept might even seem obvious, or simple – for some people it might even be hard to remember exactly why you found it so confusing.

  1. 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.  What was the mathematical concept?  Who was involved? How did it happen?  Finally, do you think you could explain it to others in a way that they could have the same flash of understanding?
  2. If you have never had an experience like this, then talk about your own experiences in learning math.  Choose one mathematical concept that you feel you know well – how did you learn it?  What did your teachers/friends/parents/tutors/yourself/etc. do that helped you to learn it?  What was the most important factor that contributed to your understanding?

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

Why are we doing this, anyway?  Having progressed this far in your school career, you are familiar with many of the tools for learning math:  studying, practicing by doing problems, asking questions when you need help, and so on.  I’d like to talk about two activities that may NOT seem related to learning math — but research shows that engaging in these activities can dramatically increase the amount that you learn, and change the way you learn it.  The first is writing — something not typically associated with mathematics.  When you express your ideas in words, it forces you to think them through very carefully, detail by detail.  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.  Example: if you know how to add fractions, try teaching it someone who doesn’t know how.  The second is called metacognition, or “thinking about thinking.”  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.  What train of thought did you follow?  Where did you get stuck, and what did you do next?  What were you feeling at the time? and so on.  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.  However, like any skill, it takes practice.  That’s why we’re getting started by writing a little about our past experiences with learning mathematics.

39 responses to “OpenLab Assignment 1: Sudden understanding

  1. Last semester, I learned mathematical induction. I had some initial exposure to it in the beginning of the semester and I didn’t really understand it. For some reason, I thought that the inductive step followed from the basis step. This made no sense to me– and rightfully so– as I learned later. The two are separate statements that are used in tandem. That understanding was like a bolt of lightning for me– completely instantaneous. I fixed the error in my understanding and everything simply fell into place. I have a much better understanding of it now.

    If someone had the same error in understanding, I think I could explain the concept to them. I think that I could explain induction pretty well, though I’ve only tried once or twice.

  2. Adam, thanks to you I have learned a little bit about proofs and logic during the summer research program at Rutgers University. I came across this problem as well until I got some feedback from you and other students in the program.

  3. When I was learning calculus I had some trouble understanding the concept of limits. I do not even know why I couldn’t understand it. But then as I was reading the textbook it suddenly came to me and I was almost embarrassed that a concept so simple was so hard to grasp for me. I think I could help someone else understand the concept in a non-confusing way.

  4. I chose paragraph 1 because the sudden understanding experience just happened to me on spring 2012 when I took Calculus I. It was one of the definite integrals properties. I positively got it as a property, but the first example confused me. The professor explained it to me in other ways and so did one of my classmates afterward, but it just didn’t make sense to me. when I went home, I did couple more examples. Finally it was clear and I felt stupid because it was just the other direction of the property. After having this experience, I think I can explain it even though it explains itself.

    • Hi Saloua,
      Thanks for your submission! Both you and Alan (above) talk about something that really rings true for me – sometimes, after you gain understanding of something, you feel stupid or embarrassed for not having understood it before. I think this is a very universal feeling! When you think about being a teacher yourself, keep this firmly in mind – when your students are confused, it’s not just a matter of the mathematical concepts giving them trouble, but the emotions at work are a powerful force as well. Not wanting to feel stupid is one of the number one reasons why people don’t ask questions – but sometimes that is all it takes to make the difference between confusion and sudden understanding!
      -Prof. Reitz

  5. Albina Yevdayeva

    The Eureka effect refers to the common human experience of suddenly understanding a previously incomprehensible problem or concept. It happened and happens with me all the times. Some topics don’t make any sense to me at all. As a person with English as a second language, it sometimes, takes me a while to understand the concepts. I would even say, I understand the material after I go over it myself, after I do the homework: read the chapter, and solve the problems. Only repetition reinforces the understanding.
    I always felt stupid for asking questions from people. I always felt like if I do ask them questions, then it would make me look weak and useless. I always believed that when learning something, you absolutely have to understand it on the first try, otherwise people will think you are dumb and stupid for not understanding something really obvious that everyone else gets right away.
    But I think we all have to understand that asking questions means being very brave. There is always somebody, who wanted to ask exactly same thing. And I wanted to share an expression: Human beings are like coffee beans, we don’t find out how strong we are until we’re in hot water!

    • Albina I very much agree with all that you had to say. For myself it also takes a while to comprehend the material, but by reinforcing what was done in class whether by reading the book after, or doing practice problems these are great methods to help understand the material. I also agree with your comment on asking questions. In the past I’ve been very resistant to that, but over time I have learned how it is critical in the learning process to learn to ask questions and in knowing that no question is a dumb question.

  6. I chose topic 1 because it is relevant to me. For myself I’ve had many different experiences that relate to this question. I’ve had experiences which range from elementary school to last year in Calculus class. The experience I would use is my time in Calculus 1 last semester. The concept we were learning was the definition of derivatives and originally I found that to be difficult. One primary reason it was difficult is because I was away from school for 6 years, so to come back and have to take Calculus can be quite overwhelming. The formula for definition of derivatives was originally a struggle for me to understand, but through homework and repetition from tutoring I figured it out and it all came to me. Another reason why I found it to be challenging is because I was overthinking things and not relying on my instincts. Eventually through the help of my tutor and constantly being given problems, it became second nature to solve these type of questions. It’s funny now looking back at it because I would find myself laughing at how something so simple, was so difficult to comprehend. I now think that through what I experienced I can now teach this same concept to others as well.

    • Ricky, your experience is similar to my experience with limits and derivatives in Calc 1. At first it was really hard to grasp these concept because of the ineffective way the professor taught it, but like you, with patience and practicing over and over, I also was able to finally understand limits and derivatives.

    • Your story is so similar to mine, leaving school for such a long time and coming back is a very tough thing to do. My first year math class back was Calculus as well. I did not remember anything when they did a review for 1375. As you have seen and learned, a lot of studying and hard work can take you really far.

    • Ricky, I understood how you may have had some problems with the concept because you had to be careful with everything you wrote or the mistake will throw off your whole solution. I wasn’t away from school for any time at all except breaks and vacations and I can see how transitioning back to school work especially Calculus can be overwhelming. There was no need to be down on yourself for not understanding the concept because it takes time and commitment to understand a topic. Learning is a process and it works, just differently in each case.

    • I to had an experience similar to yours Ricky. I’ve been out of school for almost 4 years and when I came back the material seemed to be a bit trickier. Thankfully I had a great Calculus II professor whom made learning the material relatively easy.

  7. One concept I had figured out on my own was that multiplication is simply addition repeated. When I was younger, my parents pushed me to learn multiplication before learning it in school by making me memorize multiplication tables. After writing the tables a couple of times, I realized I could add the previous number to the original number. For example, if I was doing the 7 times table, and I was up to 28 (7 *4), I could add 28 to 7 to get the next number.

    • I was also like that, learning the table before understanding the concept, and then realizing that you could just add on to the last answer and it’ll continue on the table. But I really enjoyed it when I saw how you can split lets say those 28 into 4 groups that holds 7 things.

  8. I chose concept 1 to write about. I remember the time when I had first learned how to divide by using long division. I was in the 3rd grade. At first, I was kind of shocked and embarrassed that I couldn’t divide. How could something so basic be so hard to do? I was shocked because I had always understood math concepts the first time they were taught. I was also embarrassed because I was one of the people who had to ask for help when I usually the one helping my classmates. When my teacher had explained step by step to me how to do it, I couldn’t understand how the concept worked to give you the quotient. Then when I asked my friend, after school, how to one of my homework problems, she explained it to me much more clearly. She showed and explained why you have to bring down this number and subtract that number. She explained the reasoning behind each step, which is what really helped me finally understand how to solve long-division. I understood that the reason you subtract the numbers is too assure the “left overs” isn’t forgotten about. I also realized the place value was very important in achieving the correct answer. I was ecstatic and the next day I had showed my teacher that I finally got it.

    • Isn’t it great how often our fellow classmates help us understand concepts? In middle school and high school, I actually found one of the most effective methods for me to help study was to help explain concepts to my friends. It’s like a great circle of learning.

  9. Did we have any homework from the textbook? if so which problem numbers


    • Hi Asia,

      We don’t have any written work (in the textbook) this week. For reference, here’s a link to the weekly assignment:

      Homework Week 2

      I’ll post something similar for each week so you can see what you need to do (I posted a new one, Week 3, for this week earlier this morning).

      -Prof. Reitz

    • Hi Asia,

      We don’t have any written work (in the textbook) this week. For reference, here’s a link to the weekly assignment:

      Homework Week 2

      I’ll post something similar for each week so you can see what you need to do (I posted a new one, Week 3, for this week earlier this morning).

      -Prof. Reitz

  10. I pride myself in understanding mathematical concepts easily. I become extremely frustrated when I don’t understand something. There have been times where I had to study or go over problems but the most difficult time I ever had was in linear algebra. I thought about changing my major because of it. It just wasn’t one concept, it was overall. It was like another language to me until I had a friend that really helped me through it. We studied together and she broke every little part so I can understand it (Thanks, Renautha). I still hate matrices but I understand them. As far as explaining it to someone else, bits and pieces not over all linear algebra.

    • Aww. Glad I was able to help! It’s funny because I understood the beginning of the semester more than I did towards the end. I have to thank you as well for explaining and helping me to understand those last few topics in Linear last semester. From this experience, I definitely realized practicing was the only way to pass this class.

  11. Jean Poyo MAT 2070- D 640

    I choose topic 1. When I took Calculus 1, the topic I had problem understanding was Related Rates. Related Rates is the application of implicit differentiation. For related rates problem, you have to analyze the information given and figure out which values you have to find. While I was learning it, I had problems analyzing the problem and that made it hard to solve it. My professor explained it many different ways but I still didn’t get it. It was while I was doing the finals review packet when I solved the related rates problem. I had helped my friend solve the problem and when I got home, I tried a whole bunch of related rates problems. I solved all of them and that’s when I realized that I finally understood the topic and I felt comfortable doing related rates problems.

    • I had that same experience last semester when I took Calculus 1. One of the many things I appreciate about Math is how one can re-do problems over and over again until one understands it. Unlike a subject like English, in Math one can pick out problems and do them until we understand. It’s great that you didn’t get discouraged, but rather chose to stick to it and do those problems until you fully understood them. Definitely a great learning experience.

    • I had a similar problem with related rates in Calculus as well. I knew that solving those problems involved using your knowledge of implicit differentiation, but after spending weeks solving purely numerical problems, I had a bit of an issue understanding what it was that the related rates problems wanted me to do. I feel as though related rates problems aren’t something a professor or teacher can fully explain since each individual problem tends to ask a different thing.

      Coincidentally, like you, I was also able to understand how to solve the word problems by working on them with a friend. Whenever he got stuck on a problem we hadn’t solved, I’d look at the problem again and think, “Hey, I think we’re supposed to be doing this based on what the problem is asking us.” Studying with someone else helped me.

  12. Loudia Desir Topic #1
    When I was younger I had a great deal of issue when fractions , mixed numbers was introduced to me . I think my greatest issue with fractions and mixed numbers was the fact that my English was very limited and the fact that I had to learn the concept behind it and also learning a new language. I struggle with the concept of mixed number and improper fractions, and how to convert a mixed number into regular fractions. I had to reteach myself the concept in my native language . I also translated important vocabularies words in my native language to further help me understand the topic.I think the fact that I was scared to ask questions in fear of being laughed at also cause me to not grasp the topic.The many hours of tutoring helped me a great deal . I can certainly explain this topic to anyone now with no difficulties.

  13. Patty Arredondo
    MAT 2070- D 640

    I chose topic 1. I find myself having a “eureka” moment often, especially lately. I haven’t been in a classroom in 5 years, and in a math class in 10 years, until this summer when I took Matrix Algebra. While some concepts and theories relating to Calculus, Algebra, etc. I still remembered, they were way back in my memory and at times took a while to bring back to the front of my mind.

    It’s quite remarkable how the human mind works. In my Matrix Algebra class this past summer, on the first exam we had a problem which involved many many steps to arrive at a final conclusion. The last step, the professor had asked that we simplify into simplest form. I knew I needed to factor out this long equation… Something as simple as factoring out an equation. I hadn’t factored out an equation since I was 14! (I’m 27). I could not remember at all how to factor. And after staring at the page for sometime, I actually closed my eyes and pictured sitting in my advanced algebra course my freshmen year of high school, and thinking of my teacher factoring on the board, and CLICK it all came back to me. It seems so silly now because factoring is so simple, but just goes to show, 10+ years later, things that were well-taught really stick back there somewhere!

    • It is truly remarkable how the brain functions. I definitely have had moments like that one Patty, so we are on the same page. It’s amazing how the brain holds onto so much information, but when information needs to be extracted it’s amazing how something as simple as closing your eyes, and concentrating can be the key.

  14. Dividing polynomials had a great deal of issue. I remember the time when I had first learned how to divide polynomials it was the most confusing concept to. I was a master at long division and I couldn’t understand why I couldn’t grasp what the professor was doing. At first, I was embarrassed because I am a math major and I couldn’t grasp something that was so easy to 80% of the class. I refused to go to tutoring or raise my hand because I felt I was going to be asking a “stupid question”. I finally pushed my pride aside and asked the professor if he could explain step by step. Then I noticed that the 80% that I thought grasped the concept sighed in relief that I was the one to ask for a better explanation. I felt better about not being the only one not being able to understand. Once he explained it, dividing polynomials was just as easy as long division when I was in 4th grade. I feel that I can explain the concept to another struggling student.

    • I always have the same experience not wanting to raise my hand or go to tutoring because I always feel like I’m going to ask a stupid question or say something stupid. I know for sure we aren’t the only one’s that feel this way, many people have encountered this type of experience. I noticed once I pushed my pride aside that everything got better from their on, going to tutoring is really helpful and useful and I’m always encouraging everyone to go to tutoring when one doesn’t understand or grasp a concept.

  15. I can’t remember when I had an experience like topic 1, I believe I have that astonishment look, every time I look at any problem. Just because I have to remember what to do, and how I have to go through the problem to get an answer. When I do see it, then I’m like ah ha! Math was always my favorite subject, it started when I was learning my multiplications as a kid. My older cousin was playing teacher and sat me down one day and taught me how to multiply by 5’s and 7’s. She at first just made me memorize the table, but it was when I asked her why 3×5 is 15 and she explained 5 for each 3 was when I understood and enjoyed the whole reason of multiplication. My love for Math basically came in algebra class, when solving for x,y or z was one of my favorite things to do.

    • Rafael,
      I love that you used an example from very early mathematics – having a real feel for the basics, including insights about how multiplication works, is absolutely essential to the math we do later. You will find in your mathematical career that there are many students who simply got by with memorizing the tables, and never really got that 3×5 is 15 because “5 for each 3”. This makes it very hard to think through multiplication problems on your own – if you forget the tables, you are lost!
      -Prof. Reitz

  16. My sudden moment of understanding a math concept happened my second semester here at City Tech. I came in as a transfer student from a state university. While retaking Calculus II , we started on the dreaded topic of series. At my prior school, I failed this section poorly and I believe it was because of the way the professor explained the concept. He rushed it and barely took the time to explain in full details the concepts behind these series. Here at City Tech, I became more comfortable with series, thanks to my professor that semester. He took the time with each student making sure they understood the problem, he also made it enjoyable and gave us enough time to practice series before the exam. I then realized it wasn’t as bad as I thought it was. I actually enjoyed learning series this time and around. It was also in that class, that I decided I wanted to teach math.

  17. That was such a powerful moment I’m sure Renautha. It’s so amazing how having a certain professor can change everything. They can either make us hate a certain subject, or they can motivate us and guide us on the path to not only enjoying the class, and the concepts but also make us want to go the route of teaching.

  18. Yanira Garcia
    MAT 2070 section D640
    Topic #1

    Throughout elementary school one of the worst experiences that I had with math was fractions, it was just a concept I couldn’t grasp to understand. I’m 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 weren’t 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 wasn’t 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. Now that I lead a workshop for 1175 and 1275 I’m capable of explaining fractions to others and feel like I can make them understand it.

  19. Yes Albina, all that you said is very true about the English language issue. I have the same problem, except for me I don’t have understanding problem but I still can’t ask questions even if I have a bunch of them because I have speaking matter. when I start talking I just feel shy and that confuses me.

  20. Darnell James MAT 2070- D 640

    I chose Topic #1. Most of my understanding of math has always come from memorizing what was supposed to be done in order to solve problems. When I was in middle school, I could tell you that 9^2 was 81, but then I would have trouble explaining why this was correct if you asked me to. Because of this, basic algebra was a pretty tough for me.. Remembering that 30 was the product of 5 and 6 didn’t tell me what I was supposed to do with an equation such as x *7x + 4 = 18. As stupid as it sounds, I was kind of frustrated that I was suddenly supposed to add letters together instead of numbers.

    Eventually, I learned to just think of the variables as if they were numbers instead. I still had no idea what “x” was supposed to be, and didn’t know how I was supposed to find out what 2 * x was. But I did know that 2*5 could be expressed as having two groups of five things each. So if I pretended that x was a number, 2*x could be expressed as having two groups of “x” objects. Looking at the unknown variables as though they were numbers helped me understand that, although I didn’t know their value, they still followed the exact same rules as “regular numbers”.

  21. Almas Hossain
    Right now i can not remember any incident to understand a math concept suddenly.that’s why i choose the topic 2. One concept of math which i enjoy always to do that is to factoring the second degree polynomial equation( Quadratic equation). I learned this topic when I was eighth grade. one of my math teacher named by Gaianandho Babu who showed up us how to easily break down the quadratic equation ( second degree polynomials equation) into more manageable way.He taught us how quickly we can find out the solutions set without using the standard formula for quadratic equation. For example consider a quadratic equation
    Now factoring this equation first multiply 6 and 6 and get 36. Now adding up any two numbers which give you middle number and give you 36 when you multiply them. Now decomposing the above equation we get
    Now organize the equation so that we can take out the greatest common factor from first and also from second two terms.

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