OpenLab #4: The importance of “Why?”

Kids love to ask “Why?”

If you’re like most people who attended public school in the U.S., most of the math you’ve done in your life has been primarily computational, or algorithmic – you learn a kind of problem, you learn a method to solve it, you practice it a few times, and you demonstrate that you’ve learned it by solving a problem on an exam.  Maybe the solution involves memorizing a formula, or a set of steps (an algorithm).  Very seldom are you asked to consider the reasons behind the method – what I call the “why” questions.  And if you ever asked these kind of questions, there is a good chance that you were not often given a very satisfying answer.

Questions you may not have spent much time on in math class:

  • Why does the formula look like this, instead of something else?
  • Why do we do these steps, in this order? How do we know this method actually works?  Why does it work?
  • Why is this fact (“theorem”) true?  I don’t believe it – can you convince me?
  • What’s the big deal about this fact (or formula)?  Why is it important?

To begin this assignment, I’d like you to spend some time thinking about the different kinds of math you’ve learned in your life.  Look at math textbooks or class notes from past and current classes.  Think about the  math skills you learned at different ages – from early experiences with counting, up through things like algebra or geometry, and on to Calculus or Linear Algebra or other recent courses.   If you still have some old math work from your high school, middle school, or elementary school years that might help too (maybe your parents kept some of your childhood work?).  If you’re in the Math Ed program, you can think about the math that you are working on in your MEDU courses.  As you consider these topics, think about the “why” questions – when you were learning this topic, did you spend time in class talking about “why?”   Did you do activities or other work that helped you understand the reasons for the formulas and methods that you learned?

Assignment (due Thursday 10/2/14). The assignment consists of two parts.  Be sure to answer each part completely (they are equally important).

1.  Describe a time that you DID learn about the “why” behind a mathematical idea (why something works, or where it came from, etc.).  Tell us about it – how old were you, what was the class (or did it happen outside of class?), what was the topic, and what did you learn?

One example that came up in a class discussion was the activity introducing the number \pi by having students measure the diameter and circumference of various circular objects, and showing that the ratio was always a little more than 3.

2.  State three questions about math that you do not know the answer to (but would like to!).  For each question, tell us why you find it interesting.  The questions…

  • …should involve “why” or “how”
  • …should be about math topics that you are already familiar with – things you’ve studied (you need not be an expert)
  • …should come from different math topics, at different levels (some examples:  arithmetic, fractions, algebra, geometry, functions, calculus, probability, … )

Some examples of “why” questions. 

  1. Why does a \times b equal b \times a?
  2. Why do we need a common denominator when we add fractions?
  3. How do we know that \sqrt{2} is irrational (not equal to a fraction)?
  4. Where does the quadratic formula come from?  How do we know it always works?
  5. The fundamental theorem of calculus gives us a way to evaluate an integral – for example, to find the area under a curve.  How do we know this actually works, and gives the correct answer?


22 thoughts on “OpenLab #4: The importance of “Why?”

  1. There was once a time in math where I was first introduced to Perimeter and Area. These are two mathematical concepts that I believe I learned in the 4th or 5th grade. My teacher had first introduced this concept through word problems. These word problems always had to deal with fences, or a quadrilateral figure, (usually a square or rectangle). The perimeter of a shape I learned is the length and width that is around the shape. In order to find the perimeter we add the 2 lengths and 2 widths of our given shape or opject. We normally use perimeter to measure the lengths of yards or gardens, as well as in real life applications we use it for carpenting, hardwood flooring and painting. Area is the amount of space taken up in the object. For perimeter we used it for measuring around the space, (think of it as the lengths that are on the outside). Now for area we are measuring inside the space or object itself and that is why the unit is squared. For example the answers can be in (cm^2, m^2, inch^2) . No matter what the unit is, it is always squared. The reason we use Area is also because in real life applications we can use it to measure the meters squared of a hardwood floor, or measure ceilings, (like how much space do you need inside the object). After realizing that perimeter and Area are important in real life applications, it makes me wonder what are other formulas that we can use to apply in our real life situations.

    Three math questions that I would like to know are :
    1. Why is “pi” go on infinetely?
    2. What is the very, very last number on the number line on both ends besides
    3. Why do we need to know the logarithm in math? Also the inverse logarithim? How can we use this in real life applications?

    The reason why I find question 1 interesting is because I am very curios into finding out the very exact final number of pi.
    The reason why I find question 2 interesting is because I also would like to know the last integer other than infinity. Numbers can go on forever and ever.
    The reason why I find question 3 interesting is because understanding log funcions and ln functions is very complicated. I would like to know why and how can we use this in real life applications. (Dealing with the number “e”)

    1. er… pi is irrational. Once it has a final decimal, pi would finally become a rational number. There is a world record for finding the accuracy of pi. The record as it stands is to the 67,890th decimal place. But to be fair, who wants to try to number crunch that many digits?

    2. This is something which i have learned in high school and hope this answers your 3rd question. The graph of all exponential function y = eËŁ follows similar pattern and it can be used to represent exponential growth and decay, which can be used as a model in real life. Examples, the decay in radioactive isotopes or the population growth or the fall in value of a car.

  2. One year ago, I learned how to solve algebra equations by using tiles in Professor Rojas MEDU 2010 class. Usually, for solving the equation x2+5x+ 6, students use the so-called AC method. I did not remember exactly how my teacher taught me to factor quadratic equations. However, after a long time of practice I developed an instinctual ability without any real understanding. Using tiles provides a visual explanation, which is more understandable. For example, X2 is the area of a square with the base x, 5x is 5 rectangle with the base x and the height 1, and 6 squares of a base 1. Since the overall area does not change, we can rearrange the above tiles and the new picture will look like this: the new figure’s base will be (x+3) and its height will be (x+2). Therefore, x2+5x+6=(x+3) (x+2)
    This is the best strategy for me to truly understand how to factor quadratic equations and the visual demonstration will make my memorization no longer stiff and senseless.
    2. Three math questions that I would like to know are the following
    A. how did Archimedes discover the arithmetic spiral?
    B. what is the difference between the Archimedean spiral (or arithmetic spiral) and logarithmic
    C. what are the applications for these two kinds of spirals?
    I came up with these three questions because of my history of math class. The textbook mentioned the Archimedean spiral as a referral of how Archimedes found the sums of series. Since there is very limited information with regard to the Archimedean spiral, I would like to ask if some of us know the historical background of it. Secondly, there is another spiral called the logarithmic spiral. I am interested in knowing the connection between each of them or if they are completely irrelevant. Lastly, I am more curious about their real applications because I can possibly guess that they are useful in the architecture field.

    1. Hi Jian,
      I often find that I remember pictures much better than formulas or algorithms — in many cases, if I just have the right picture in my head, I can use it to figure out the formula or steps I need to solve the problem. This is one of the best alternatives to just memorizing!
      -Prof. Reitz

  3. When I was little I had problems in comparing fractions i.e. part of a whole. For example back then I couldn’t tell if 1/2 was bigger than 1/8. And usually if I don’t understand a mathematical concept, I try to relate to what I know, like for instance; I could tell 1/2 is greater than 1/4 because those fractions are very commonly used in everyday life.
    I learnt about rational expressions and their representations in Prof. Rojas’s class for MEDU 3011 this semester. After years of taking math courses comparing fractions was not a problem to me, however learning the concept and understanding it was something I always wanted to know.
    My professor used manipulatives to compare fractions that developed my conceptual understanding behind this idea. At a symbolic level comparing 1/2 with 1/8, most students are tempted to see 2 in the denominator as being the smaller number than eighths. However using fraction models to represent each fraction in algorithm and see that 1/2 is greater than 1/8 gave me a clear visual understanding as well. Not only that equivalent fraction was easy to observe in multiple ways and simplification of fractions can be determined too. Using the same example 1/2 or half of a whole was the same as 2 of 1/4th and 4 of 1/8th. And simplifying 4/8 gives 1/2. This way I was less confused and I don’t have to memorize rules to remember which fraction is greater.
    The three math questions I like to know are:
    Is factorial the fastest growing function? I find this question interesting because the result of factorials grows faster than exponentials and I wonder if there is any other single function that grows faster than these.
    The second question is how do we know or why do accept that BODMAS or PEDMAS is the correct way to solve a problem? Is there any proof as to why we must follow the order of operation? This is another interesting question because the order of operation is considered important in solving a math problem and also certain problems in math you can have your own shortcuts but when it comes to algebra this is only way out.
    The third question I want to know is if there is an object with one surface and how can it be represented graphically? This question occurred to me because of my calculus III class. I learn about vectors in 3 dimensions and I would like to know if one surface objects or shapes has a function that can be shown graphically and if so how many dimensions.

    1. In my elementary school when I was in my 3rd grade in Algebra class , My math teacher gave us two fraction that we had to try to figure out which one is bigger or smaller than the other using the symbols > < . I did not understand how can i know which one is bigger than the other. So my mom tried to explain it to me in a way that i can understand it. Her way was that each time i have two fractions i have to imagine an orange in front of me. For example if i have to compare 1/2 and 1/5 so i have to image that i have to image the first orange that i split it to half, and the second orange to split it to five pieces and then from looking at the sizes of the orange i could imagine which one is bigger. i used this examples few times but i did not understand why i have to do this in order to know. Some ties the teacher gave us larger fractions that i could not symbolize it as my mom taught me. it was really hard for me to figure out why there is no other way to know these fractions. Until i got back to school and my teacher explain it to me in very easy way. For example, if we he 4/3 and 2/4 we have to multiply the numerator from one side to the denominator of the other fraction and the same process for the other fraction and we write it on the top of the fraction, so 4.4=16 for(4/3) and 2.3=6 for(2/4) so 16 is larger which means 4/3 is larger. from now on it was very easy for me to understand it and know why we use this to determine the fractions.
      Three math questions i would like to know :
      1. why we can not add variables that has different exponents like X+X^2?
      this question is interesting to me because i want to know the reason of it because i used to get really confused some times and i used to add them.
      2. why do we use the quadratic formula the way it is?
      this question is interesting because i want to know why it is in this order and what will happen if we switch any variable .
      3. Why is “pi” go on infinitely?
      this question i interesting because i always think about it and never got an answer .

    2. Yes I totally agree with you there! I’m not a huge fan of fractions but this semester I learned a lot about fractions is Rojas class as well. Which is pretty funny because all this time I thought I knew how to compare fractions when I really didn’t know about it’s background of it and now since I know how to represent it both verbally and visually it’s easy for me to understand why it works

    3. Fractions are one of the mathematical objects that can be looked at and understood in a great variety of ways – and I think many students, faced with this complexity, give up and just memorize rules. Having a clear picture in your head of fractions as “equal parts of a whole” is a great basis for understanding (I still have this picture in my head when I think about fractions!).
      -Prof. Reitz

  4. In high school, I had learned about transformations. I could grasp the ideas of reflections, translations, and dilations, but rotations always seemed weird to me. Sure I knew the formula, but one question kept popping into my head. Why is it that we choose to rotate counterclockwise? Perhaps the quadrants on the Cartesian plane may have something to do with it, but I was not buying it. Unfortunately, I would almost never work with rotations again until after my first stint in college.

    In one of my tutoring jobs, I was asked if there were any methods to figure out how to easily determine a 90 and -90 degree rotation without being forced to memorize the formula. I don’t know why, but an idea just popped into my head at the time. I told the tutee to rotate the x,y axes clockwise 90 degrees and then plot the point again as if the rotated plane were the original set of axes. Then I asked what is that new point on the old set of axes. Lo and behold, it worked. Not only did the student learn a cheap method for finding the answer, I managed to BS my way into understanding it myself. At least I hope my logic wasn’t flawed in trying to understand it.

    -Speaking of which, why are the quadrants numbered the way they are?
    I’d assume the same reasons for rotations.

    -How can a shape with an infinite boundary have a finite area?
    It sounds like a contradiction, and yet we can calculate it.

    -What is this span I keep hearing about in linear algebra?
    It was too much information for me to digest over a short period of time over the summer.

    -Why in error calculations must we use absolute value?
    Could we not determine in which direction we have erred using the sign of the resulting error?

    1. I like your observations about transformations and your strategy to determine a 90 degree rotation. However I don’t think you are restricted to only have rotations in the counter clockwise direction.

      1. of course they are not. I could have reworded it better. i meant why is going in that direction positive. Eh, I’m starting to think I have to rewrite the whole thing.

  5. I saw a multiplication table in my pen box when I was first grade, and I didn’t understand why it only had half of it. For example, it only showed 1 X 9=9, and there was no 9 X 1 equal to. I asked my teacher why multiplication table only show half of it, and he told me why.

    We can ask students to make a table, which measure the change of diameter and circumference in different circle. Students will discover the relationship between diameter and circumference little more than 3.

    Why do I need to learn geometry? How geometry help in real life if I want work in McDonald?

    Expert can spend a life studying in Laplace transform . How much do we need to study something’s that most people do not need?

    Why do we need to learn induction in this class that we learned in pre-calculus?

    1. The fact that 1×9 = 9×1 (“the commutative property”) is one of those very simple facts about multiplication that seems so natural to us now, we often forget that we had to learn it for the first time – I love that you can remember it so clearly! It is so important that it is one of the axioms of arithmetic (we’ll be spending more time on axioms later in the class).

      Regarding your question on induction, I’m curious to see what you think of it in this class (compared to precalc) – is it easier or harder, do we cover it in more detail or less, etc? Let me know.

  6. In middle school, when I was in my seven grade algebra class, my teacher introduced to the class pythagorean theorem (a^2 + b^2 = C^2). This formula is used to find any side of the right triangle. The formula itself was really easy to use but in the end I didn’t understand where the formula came from and why it alway worked until I went to my eight grade class. The teacher explained the formula by drawing a big square and than a smaller square inside of it in a way that created four congruent right triangles with the sides a,b,c (a= base b=height c= hypothenuse) as well. After that, we found the area of the smaller square c^2 was equal to the area of the big square,(a+b)^2 [it’s a+b because from the drawing we saw that the length of the sides of the big square was equal to the base of one triangle plus the height of the other triangle], minus the area of the four right triangles, 4(1/2 ab). when we simplify it, we get c^2= a^2 +2ab -2ab +c^2. After we cancel we get a^2 +b^2 =c^2, which is pythagorean theorem.

    The three questions I want answered are:
    1) what is the last number?
    I want to see what word it is and how many letters the last word will have in it

    2) why do we have to follow PEMDAS?
    Alway wonder why we have to follow this rule, why can’t i just do it in any order

    3) why is it when you multiply a negative with a negative you get a positive?
    I just wondered this because it doesn’t apply for multiplying two positives.

  7. As a kid I never learned why when you add and odd number with an odd number you get an even number. When you add an even number with an even number you get an even number. When you add an odd number with an even number you get an odd number. Taking number theory has helped me understand why this is by writing it’s proof out. Rather then using numbers to test the theory out there is a proof where you can set a as an odd number being 2n+1 and you set b as an even number being 2n and you are able to factor out the two you can see that this will always equal odd. I thought this was pretty cool which made me more interested in number theory especially dealing with prime numbers because prime numbers are mysterious and seeing that when you write out a list of prime numbers they start to spread out.
    The questions I would ask is
    1) why is the new common core math harder now for 1st and 2nd graders. What is the point of making easy math problems more complex.
    2. Why is 0 not a prime number? Why is 0 not a natural number?
    3. why are there infinetly many primes

  8. When I was in 9th or 10th I remember I was introduced to trigonometric functions. We were taught to find all angles of sin, cos and tan. For each trigonometric functions there were different positive and negative values. I was confused and were wondering why the each angles values can’t be same like positive or negative for each trigonometric functions. Later I was introduced to four quadrants, and each trigonometric functions angles is different for each quadrants.

    1. Why can’t zero be in the denominator but can be in the nominator?
    2. Why do we have infinities?
    3. Why the value of e is 2.71828?

  9. I remember when I was in the 3rd grade and we were learning how to divide. The teacher was showing us how to use long division to figure out the answers and how it tied into multiplication. By her proving that the answers were like products of the answer and another number helped me grasp the concept quickly.
    I found it interesting last year when my daughter was being taught how to divide numbers and they only show how to get the answers by grouping them together using multiples. I guess long division is a thing of the past.
    I always wondered in calculus why the derivative of e^x is e^x while everything else takes thought and remembering rules.

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