Open book learning versus rote learning

       What is rote learning?

Rote learning is the concept that you will memorize a procedure by endlessly repeating it until you “understand” it.  Some believe this is the only way to truly understand a topic in mathematics.  It is the “practice makes perfect” paradigm.  The rote learning idea is that understanding is derived from memorization and recall over time. Understanding means to comprehend or to grasp the meaning of. Students with good memories will just simply memorize the formulas and recall them on demand at test time, and the teacher will believe they understand the concept and move on.  Students who do not have great memories will struggle and get frustrated while trying to memorize the formulas. We can single them out in an attempt to give them extra help to understand, which in some cases actually embarrasses the students.

 Embarrassment even in the smallest amount begins a psychological impairment on a child’s ability to learn. No matter how we approach the situation singling out one student or even multiple students can impair learning.  Now I am sure there are some very exceptional teachers with the amazing ability to approach a student to find out why they are failing with minimal impact. Unfortunately, not all teachers can do this; it is a talent of charisma that we just simply cannot teach everyone. We are now left with the question: How do we give the children that need this extra attention help without singling them out? We first need to remove the extra pressure of rote learning exercises. We can do this by utilizing Open book learning approaches.

                           What is open book learning?

Open book learning takes away the pressure of having to cram for a test. Relieving a student’s stress will create a better atmosphere of learning. In introductory courses, having a system based on pure open book examinations can cause some problems of students not studying or reading the material before the test. To alleviate this problem you can implement both open book and closed book exams.  The exams need to be structure for whether it is open book or not. For the open book tests do not put in direct information that can be simply copied. Present the questions where the students will have to apply the knowledge rather than recall or look them up and simply copy it from the book. Keep it random as well don’t let the students know what kind of test they are going to be having, so you don’t discourage study.

                      Don’t be one sided.

               

Stay flexible and implement all of these strategies together. The mechanical rote learning approaches should stay limited to introductory courses. Rote learning should never be the pass/ fail in math education.  A final examination should be on concepts and understanding not recall of facts that anyone can look up.  

The purpose of education is career preparation.  An engineer can still build a great bridge even if they have to look up a formula. However; if they don’t have a conceptual understanding of what to look up or how to look it up they won’t be an engineer for long. 

Some Mathematical FACTS about poverty in the U.S.

   According to the Dept of health and human services, the poverty threshold in the U.S is as follows.

2012 Poverty Guidelines for the 48 Contiguous States and the District of Columbia
Persons in family/household	Poverty guideline
1	$11,170
2	15,130
3	19,090
4	23,050
5	27,010
6	30,970
7	34,930
8	38,890
For families/households with more than 8 persons, add $3,960 for each additional person.

Above Chart from http://aspe.hhs.gov/poverty/12fedreg.shtml

 

LET'S DO THE MATH

Let’s break this down a little bit.   For the family of 3 the chart indicates that to be considered in poverty you have to make less than $19,090 annually.   The National housing Conference gives the fair market average price of rent on a 2 bedroom as $960 monthly which is $11,520 annually.  The U.S. dept. of Agriculture gives the cost of food for a family of 3 as $439 monthly which is $5,268 annually. The average cost of utilities (natural gas, water, electricity) varies widely but sits around $250 monthly for most families. This is $3,000 annually. 

Now to add all of this data up we get a base cost of $19,788 that puts us close to the poverty threshold.   This base cost does not include

Average fuel costs as reported by CNN are about $368 monthly which adds another $4,416 to the annual cost of living. Our total is now $24,204 annually

Let’s stop at this number for now and see how it compares to wages.   The current minimum wage is $7.25/hour At 40 hours a week that is $290.00 weekly( $15,080 annually  before taxes)   Now if both parents were to be working 40 hours a week we double this to get $30,160 before taxes.(this puts you above the poverty level and makes public assistance unavailable)   Now this is $5,956 more than the annual cost we have above, but with this situation we have a new problem(child care) Which makes a new expense.   

USA TODAY reports that child day care ranges from $3,803 to $13,480 annually, for those lucky enough to live in an area on the low average that will still leave you with $2,153 a year. However we are talking about a family of 3 here with a child. Nowhere in our expenses did we include the other basics such as clothing and toys(yes toys they are essential for a child’s cognitive development)

If your child is in diapers and you shop wisely you will spend around $30 a week $1,560 annually on diapers (this number can vary slightly depending on special needs… sensitive skin allergies etc. )   Subtract this from the $2,153 we had left above we are now down to $593 However, with diapers we also need diaper wipes. A person could bargain shop and get away with around $12 month on those. That is only $144 annually, and when subtracted from $593 leaves us with $449 clothes, shoes, toys etc….

Another expense I left out is motor vehicle insurance which is now a legal requirement in most states. The average costs varies greatly but is around $1,500 annually leaving our money left over -$1051   There are more expenses that are “necessary expenses” That I have not included, but the fact remains It is impossible to live on minimum wage with a family of 3 and both parents working full time. This is why many people get caught up in major credit card debt, have poor credit and stay in a cycle of survival by destroying their financial future. Bad credit means higher interest and high payments on any loan they try to get, which keeps them always broke. .

For those that do manage to survive by whatever means they can are on a slippery slope. One emergency (car problems, health issues preventing work, cut in hours etc…) can completely devastate a family living off minimum wage.

The $1mil lottery winner being bashed for continuing to receive food stamps.

I recently saw the article about the women(Amanda Clayton) who won the million dollar state lottery, and kept getting food stamps.  Although I don't completely agree with it, I decided to crunch some numbers and see if this really hurts or if it still helps the economy.    She won $1,000,000 after taking the lump sum she got $700,000. Then the key portion here is she was "taxed" $200,000"  Leaving her winnings at $500,000   Now according to what  have read so far is she was receiving $200 a month in food stamps.

The key argument that everyone has is that Amanda Clayton  is "cheating" the system by using "other" peoples tax dollars to get food. Lets break that statement down a little bit. Whose tax dollars is she spending?

Lets take the $200,000 "she" paid in taxes from the lottery.  If she is receiving $200 a month in food stamps lets figure out how much of the $200,000 in taxes "paid by her" from her winnings covers with her food stamps. so we will divide the $200,000 by $200 dollars per month we get 1000 months out of the $200,000 "she paid in taxes" from her winnings. 1000 months divided by 12 months a year  gives us 83.3 years worth of $200 dollars a month in food stamps.  From this basic math you can see that she is actually going to be living off of her own tax dollars for 83.3 years.  I am not sure if there is a cut off on how long you can keep receiving benefits. So lets assume you can keep it for 18 years(because most people get it for their child)  83 years minus 18 years = 65.33 years so after she is past the limit on drawing food stamps there is 65.33 years left over of $200 a month for others . so lets divide that by the number we decided on of 18 years to see how many people can receive benefits off the remainder 65.33/ 18 = 3.629

Lets look at from the perspective of the amount of money left over instead of the time.  18 years at $200 a month is $43,200 which is how much she will use in 18 years of being on $200 dollars a month food stamps. Leaving $156,800 for the food stamp program.

Lets divide that up to see what else can be done with it.( now keep in mind we are still working with the money that Amanda Clayton paid for the taxes on her winnings.) We will take the $156,800 left after Amanda receives 18 years of food stamps out of the taxes she paid on her winnings.
 At $200 a month 1 person receives $2,400 a year in food stamps at 18 years that one person can receive $43,200 over the 18 year period in food stamps  now lets divide that number out of  the $156,800 left from the money paid in taxes by Amanda from her winnings. to see how many people can receive 18 years of benefits $156,800/ $43,200= 3.629  so basically 3 people in addition to Amanda can receive  $200 a month in food stamps for 18 years off just the money that Amanda paid in taxes from her winnings.

Now lets Recap Amanda Clayton paid $200,000 in taxes from her lottery winnings. After she Receives 18 years worth of food stamps at $200 a month there is  $156,800 left. Enough for 3 more people to receive 18 years of food stamps at $200 a month.  and a surplus left in taxes of $27,200 which is still enough for 11 additional people to receive 1 year of food stamps.
 So the question I leave you with is, "whose" tax dollars are being used be Amanda Clayton? , and Is Amanda Clayton hurting or helping the economy?
From the results above, She is only receiving her own tax dollars, and she is helping the economy by supporting additional people in the tax program from her taxes paid from her winnings.

Does poverty hurt academic success?

          While reading through the various new articles about test scores, I have seen a common argument about poverty affecting test scores.  Is this true or is it just an excuse?

I will start out with a personal note. I was a very impoverished teen in high school. I had to work two jobs while going to school. My parents were so poor I eventually had to live with my grandparents, who although were not much better off than my parents, they at least provided a steady home.

However they were unable to afford all of the things that I needed for school. My school required special clear backpacks which my family could not afford. I had to buy those and  I was responsible for all of my clothing, car payment and insurance.
School supplies was another expense I had to work for.

While most students were sitting at home doing homework or out with friends, I was at work. I became an emancipated teen so I could work the late hours I needed just to survive financially.  Unlike the students who would go out with friends I was unable to stay up late doing homework, or do it in the morning before class. I worked so late that I was already only getting a few hours of sleep.  I did however pay attention in class, and read while on my lunch breaks at work.

Luckily most of my teachers, would focus on teaching an understanding of material rather that just rote memorization. My teachers also spoke to me on their level like I was a colleague in some situations, or they would at least speak to me like an adult rather than like a third grader. Not all of them were like this, I did have some bad teachers. However, most of it was due to the student to teacher ratio being so unbalanced that inexperienced teachers had to sometimes ":fill in" the gaps.  No matter how old you are , most teachers treat their students like small children.

 I believe it was the way some of my teachers would treat me that helped me not only understand the material, but also have the desire and ability to learn on my own.
Due to school policies homework was 2/3 of the grade and I did very poorly in my classes. However all of my final exams were A's and B's, which is the only reason I was allowed to graduate.  ( thanks to an amazing teacher Mrs Janice Miller who rallied the rest of my teachers to get the school to allow me graduation and a diploma) I also was ale to achieve a 35 on the ACT test which shows that I was still able to learn despite my financial situation.

I have gotten a little off topic here, but I believe it is important to see that the way we treat students can go a long way in helping them learn. So if you have a student who is on a free lunch program or their parents can not afford all of the things they need for school, treat them like an adult who is at the same economic status that you are. They will respect you so much more for it , and will be more apt to get a lot more out of your lessons.

I will leave off with a question. Does how much money you have determine intelligence? Does a student whose parents can buy them a a brand new laptop, have an advantage over the student whose parents can barely afford  pencils and paper? Should we change how we teach students of different economic backgrounds?

The answer to all of these is no. If a course in high school is designed to require a laptop the school should provide one, or not require it. If a school is in a lower income district it should still get the same amount of funding that a school in a high income district gets.

I will just leave this open to comments and continue as I get questions. I may come back to this in a later post.

What math means to me?

   I love math, but I also realize that not everyone looks at math the same way. When I look at math, I see the beautiful mystery that can define everything in our world and beyond. I see a glimpse into unlocking the mysteries of the universe. Once you begin to delve deeper into mathematics, you begin to have an understanding of the world around you that only mathematicians get to enjoy.

    After you have gotten into to higher level mathematics, some of the problems you solve will begin to fill up you page like poetry of numbers.  I know when I am solving bigger problems that fill up a page I will take my finished product and just step back and look at the beauty of it.

I know that most people do not look at math the way I do, and I am interested in getting some feedback on What math means to you?  Everyone has their own story or way of looking at math, even if you are completely frustrated with math I would like to see the comments I get.

So please leave some comments on what math means to you so we can discuss and compare how everyone looks at it. I am very interested to see the variety of responses I get. 

An interesting pattern with derivatives

I found an interesting pattern while finding  higher order derivatives using the power rule. For readers who are not familiar with derivatives you might want  to watch the video here MIT open course ware derivatives.

I was curious what the 3rd derivative of  x⁵ was.
So I used the power rule  to find the first derivative and I got f¹(x)= 5x⁴ I applied the rule again to get the second derivative and got f²(x)= 20x³ I once again used the power rule for the third derivative and found the third derivative was f³(x) = 60x² Now by this time I started to get curious as to what the Nth derivative,fⁿ(x) would be for functions using the power rule. So I began to to work it out algebraically. I started with the general function f(x)= x^a and began taking the derivative by continuously using the power rule, and got the following results:

1. f ¹(x)= a x ^a-1
2. f ²(x)= a (a-1) x ^a-2
3. f ³(x)= a (a-1)(a-2) x ^a-3
4. f ⁴(x)= a (a-1)(a-2)(a-3) x ^a-4
5. f ⁵(x)= a (a-1)(a-2)(a-3)(a-4) x ^a-5

From these a pattern begins to emerge With the a as I took the higher order derivatives. Here is the pattern of the a throughout each of the derivitives

1. a
2. a (a-1)
3. a (a-1)(a-2)
4. a (a-1)(a-2)(a-3)
5. a (a-1)(a-2)(a-3)(a-4)

As we can see it looks like a pattern similar to a! or a factorial, but this alone is not the pattern we need to a be able to see how this relates to the nth derivative. Well lets look at the derivatives. The second derivative of  f(x)= x^a  is f²(x)= a (a-1) x ^a-2 ,and from this we see that it is not just a! it is a!/(a-2)! because a! keeps going until (a-some number) = 1  For the second derivative to make sure a! does not go past a (a-1) we use (a-2)! in the denominator because it cancels out all of the other factors past that point.  That brings us to the general form of a!/(a-n)!

Now with the a in the exponent  x^a we see that the a reduces by the the derivative we are on for example the second derivative ,  f²(x), the exponent is x ^a-2   That makes the general form for the exponent x ^a-n

To put all this together we get the formula for the nth derivative using the power rule as

f ⁿ(x)=  a!/(a-n)!  x ^a-n



 I am not sure if this pattern was already discovered by someone else or not. I have done some research and have not been able to find anything about it online.

Number Theory: Not as Scary as it Sounds

What is Number Theory?

                Number theory is simply the study of the properties of numbers. The topics in number theory range from a variety of fairly simple to extremely complex. (Most people can understand and follow along with topics in the middle of this range.) The simplest properties of numbers are still a topic of research in number theory today. A person can start becoming an amateur number theorist by simply picking a number and finding all of the ways that number can be mathematically manipulated, and paying attention to and recording any patterns that arise. While doing this you can do a little research to find other ways to manipulate the numbers you chose, and through the induction/deduction process of logic find your own theories. You might not find anything groundbreaking, but you will certainly find fun ways to learn and enjoy math on your own. I found an interesting “shortcut for multiplying by the number nine by simply making a chart of all of the multiples of nine and analyzed it until I found a pattern. I am not sure if someone else has already found this or not, but it was a very fun “trick” to use. You can find a full explanation with examples here Multiplying by 9,99,999...etc

                In high school, you were taught some of the basic properties of multiplication.  Such as: the distributive property, and  the associative property. You were most likely shown a formula to explain these similar to
a(b+c) = ab+bc, Which describes the distributive property.

Well where did we get this?  It is simply a result of number theory, or another way to define it is the logic of numbers. It is difficult to tell who to actually give credit to for discovering this property; it is one of the most widely used properties in mathematics.  It was, however; discovered by analyzing the properties and patterns found when multiplying and dividing numbers. The use of the letters, or variables, comes from what is called abstraction. Abstraction is simply taking something out of its original context and making a general form of it. This is not a great definition, but hopefully you see that when we find a pattern that is very useful, we need to make a method to apply it to all numbers.

Now some may understand the distributive property better if it is shown with actual numbers. Like this 2(3+4) =  2x3 + 2x4  but without abstraction some people would look at the example using numbers and think it only applies to those numbers.  Effective abstraction leads into using proofs to check your pattern to see if it applies to all numbers or just a certain set of numbers of numbers with specific properties.  I will leave that discussion for another post. Hopefully, this will give you a basic understanding of what number theory is and inspire you to learn more. 

Create inspiration to learn math.

               One thing the current math education policies do not address is relieving math anxiety. Many educators and policy makers talk about it, but do they really have a solid plan for relieving it.  The common discussion is all about how math is taught, but what about the kind of math that is taught.  Some educators are already implementing styles of teaching that relates to what students are interested in. This is a step in the right direction, and this does increase an interest in math education.  Many educators and even individuals have taken to the internet to provide lessons, practice, discussion boards on math education and homework. When you look at all of these education sites they all mostly just reflect the same things we teach in schools.

 Does this inspire students to learn mathematics? Does this relieve math anxiety? One way to answer these questions is to look at the students that are actively participating in these. Are they doing better on tests? Maybe…  We might be improving scores in the small groups that are actively participating in these online programs, but once again we have the question, are we relieving math anxiety?  Maybe to a small degree..

A simple way to relieve math anxiety is to introduce students to simple math “tricks”, or other ways to simplify or shorten the math calculations that most people only know one way to do.  I have already presented a couple of these methods. Such as Long multiplication in Reverse also called left to right multiplication, or multiplication without carrying. The “left-to-right” method is a good trick which helps with mental math. Students can learn to answer more difficult multiplication problems in their head with relative ease.                               

   This will make them more confident in their math skills. It will also change the way their peers look at them. They will feel smarter and it will increase their desire to be smarter by learning more about math.  The “left-to-right” method is not the only “trick” to make multiplication easy. There is also Russian Multiplication also known as binary multiplication. This method is great because you don’t have to have the multiplication table memorized to use it. You just simply have to double numbers and cut numbers in half and then add.  

All of these simple methods make math easier to students, relieving their initial anxiety. With the many ways that we can use to make math easier, we can teach a new “trick” every day.  The whole point is to increase desire for learning and relieve math anxiety.  These are the two main obstacles in helping children not only have better understanding of math, but also to do better on tests.