kitchen table math, the sequel: Should math students explain their answers in words?

Tuesday, January 21, 2014

Should math students explain their answers in words?

Complex information, such as that required for motor skills, can be learned implicitly, without awareness.

[snip]

Imagine you are riding a bicycle, and you start falling to the right. How would you avoid the impending crash? Many cyclists say they would compensate by leaning towards the left, but that action would precipitate the fall. When responding to the same situation while actually riding a bicycle, these same cyclists would turn their handlebars in the direction of the fall. The example (from Ref. 1) highlights the distinction between implicit and explicit knowledge*. Implicit learning refers to the ability to learn complex information (e.g. skills such as bicycle riding) in the absence of explicit awareness. Anecdotes such as the bicycle example offer subjectively compelling demonstrations for the existence of implicit forms of knowledge that are distinct from (and possibly in conflict with) explicit knowledge, but the existence of such learning without awareness has been difficult to prove scientifically.

Implicit learning revealed by the method of opposition
Tim Curran
TRENDS in Cognitive Sciences Vol.5 No.12 December 2001
It's an article of faith, inside schools of education, that procedural learning is dumb.

Having spent a great deal of time immersed in the literature on the basal ganglia, which handle  procedural learning, I'm pretty sure that assumption is wrong. Possibly very wrong.

The emerging research on the intelligence of nonconscious learning and the cognitive unconscious hasn't surprised me at all, mainly because, years ago, I read Arthur Reber's Implicit Learning and Tacit Knowledge: An Essay on the Cognitive Unconscious. Reber's book was so world-altering that I have kept it on my desk ever since.

As I recall (it's time to re-read), Reber opens his book with the question of expertise: how to transmit expert knowledge from one generation to the next.

In other words, he opens with the question of education.

e.g.: We have, today, people who know how to perform open-heart surgery. We will need, tomorrow, people who also know how to perform open-heart surgery. Since the people who know how to perform open-heart surgery today will grow old and die, we need to transfer their knowledge to the next generation.

How?

People's first thought (again, I haven't read Reber's book in years, so take my summary with a grain of salt)…. People's first thought was that experts should simply tell novices how they do what they do.

Simple!

But that didn't work out.

The reason it didn't work out: experts don't know how they do what they do.

That fundamental insight into the nature of expertise has never left me.

Experts don't know how they do what they do.

The fact that experts don't know how they do what they do has made me highly skeptical of "explain your answer" questions as the sine qua non of math achievement and comprehension. As far as I can tell, adults who are really good at what they do have an enormous amount of nonconscious knowledge and comprehension, so shouldn't that also be the case with children who are good at math?

I don't think the second proposition necessarily follows from the first. Perhaps math students should be able to explain, in words, why they did what they did in order to arrive at a correct answer. However, anecdotally I do see "math kids" who "just get it" -- and, anecdotally, those kids always look like the good-at-math kids to me.

In any event, I was tickled to discover that people who know how to ride a bicycle not only don't know how they do what they do but in many cases consciously believe they do exactly the opposite of what they actually do.

59 comments:

Anonymous said...

It seems to me that the only knowledge or skill domains where it's essential to be able to explain in words what you are doing and why, are the word-based domains -- the ones where you produce sentences and paragraphs as the primary output. Literature, social sciences, philosophy, etc.

In other domains, words may be used to reinforce what is being learned through other methods: mathematical problem-solving, open heart surgery, bicycle riding. But they are not essential and the main focus should be on the doing, not the describing in words.

In math, if a student is reliably using her/his math knowledge and skills to arrive at correct answers to problems, s/he should not be penalized for difficulty in describing her/his work in words. Just as we don't think students who are not good at making poster displays to illustrate their knowledge should be penalized. A good poster display may be very useful in indicating knowledge, but it's an add-on unless the class is one in graphic arts.

SteveH said...

No.

Anonymous said...

My husband, who is a PhD quant who works in a financial company, is insistent that our kids learn to explain their math answers in words.Not fuzzy words, but very precise explanations. Why? He says he spends much of his time at work explaining his algorithms and mathematical reasoning to other bright people, who need to be convinced of what he is doing. He thinks being able to precisely explain mathematical reasoning is core, and should be taught early on.

Glen said...

Unlike many skills, mastery of math requires conscious mastery. It's fine to produce beautiful art or music from a finely tuned intuition that cannot be explained.

That's not acceptable in math. Yes, mastery of math is built on a foundation of implicit intuition born of repeated experience, but on top of that foundation you must have enough explicit, conscious understanding that you can PROVE your claims.

You don't prove that art is beautiful. A variation on a piece of music won't make it wrong. Riding a bike where you want to go without falling off is good enough.

But there is no good enough in most of math. There are some fascinating exceptions, but in most cases you're not finished until you can explain why your answer is literally perfect, and you can't do that with intuition alone.

Explicit, conscious understanding of how math works---not how your own mind does math but how math works---is required for mastery.

SteveH said...

There is a huge difference between what a math expert would want for an explanation versus what a K-12 educator would want. Therein lies the crux of many misunderstandings. There is also the question of whether the explanation is used to prove that one understands something versus an explanation to convince others to do something. Those are quite different things. Public relations and salesmanship often require many things that have little to do with real understanding. The problem in education is that words that warm the cockles of pedagogues' hearts have little to do with true mathematical understanding.

Explaining mathematical reasoning might involve showing that an overall merit function or solution approach is reasonable or viable, but those "words" have little to do with what goes on in K-12 education. One has to be careful about the words that educators redefine as their own. I know a number of STEM-type parents who quickly find out that educators' words are not what they think they are. Educators love to talk in generalities to get parents to go away. If you peek behind the veil, you will be horrified. You will be questioning competence, not higher order thinking. You will see low expectations, not the understanding that comes from true mastery of mathematics.

Glen said...

When I talk about the need to prove a math result, I don't necessarily mean prove it to other people in some formal sense. Most of the time, you only need to prove it to yourself, but you still need to prove it, which you can't do if any of it is based on intuition.

I used to be a quant at a financial company, but I had no need to prove my work to others, because I was the only quant, and they always took my word for things. I still had to prove my work to myself, though, because I had to be sure I was right. This meant first finding the answer by some awkward, bumbling process, then going back and cleaning up the path from problem to solution until it was as clear, obvious, and undeniable as possible and, when possible, including alternative paths that led to the same result.

My kids want to use their intuition to guess at an answer, then ask, "Is that right?" I tell them not to give me an answer until they don't have to ask. "Just do whatever you have to do to prove to yourself that your answer is correct, then prove it to me."

"But how am I supposed to do that?"

Learning the answer to that is required for math mastery.

Anonymous said...

Glen,

How is what you are talking about different from "show your work?"


I totally understand "show your work" (and insist on it with my child ... with the explanation given that: in the real world, people need to understand how/why you got the answer you did). But "show your work" isn't the same thing as "explain the answer in words."

-Mark Roulo

Anonymous said...

The comment above about guessing was interesting to me, since I'd just found this this morning. It's about reading rather than math, but quite relevant:

http://thenewamerican.com/reviews/opinion/item/12860-nytimes-education-confab-ends-in-zero?tmpl=component&print=1

"Indeed, the most destructive philosophy that permeates primary education is the notion that accuracy is no longer essential in developing basic skills. That is why guessing is encouraged in reading. An advocate of this philosophy was Julia Palmer, founder of the American Reading Council and a believer in the whole-language approach in teaching reading. She said that it was OK if a child read the word "house" for "home," or substituted the word "pony" for "horse." "It's not very serious because she understands the meaning. Accuracy is not the name of the game." (Washington Post, 11/29/86) Ms. Palmer may have said that in 1986, but it is still the philosophy of teaching today in many primary public schools. Her Council folded in 1991.



As a tutor I discovered how destructive this philosophy is when tutoring a 14-year-old boy who thought he was stupid because he could not guess the right word in reading. He had been taught to read by the look-say method which encourages guessing. He assumed that knowledge was obtained by guessing, and that if you were a poor guesser, you were born stupid. But after I taught him to read with intensive phonics, he found out that he did not have to guess the word on the page. He could sound it out. That revelation changed his life. He discovered that gaining knowledge was not a matter of guessing, but a matter of knowing how to read phonetically. He had never learned how to use his brain. The school had taught him that learning consisted of guessing and immediate magical knowing. But after months of tutoring he realized that he was not born dumb, that learning was a matter of using his brain and phonetic skills to figure out the words on the page.



How many children grow into adulthood believing that learning is a guessing game, and that because they are poor guessers they will never become good readers? This is a question that the Times conferees would never even know how to ask. And until they know that such questions should be asked, they will never be able to improve public education."

- BL

SteveH said...

I also don't agree that math is related to knowing how to ride a bicycle but not being able to explain why. When one moves from Algebra I towards a full understanding of algebra that allows the manipulation of any expression or equation, students must be able to justify what they are doing. I remember having to put the rule or identity next to each of my steps, and most people remember how their teachers never allowed them to do two things in one step. That annoyed me, but I fully understand why. Nobody would ever call this process a description using words, and nobody would call it some sort of intrinsic knowledge. Those who can't justify what they are doing (mathematically) will probably run into many problem variations they can't handle.

Of course, this sort of thing is NOT what educators mean with they talk about using words to explain. What they want is what MathLand did when they asked students to explain why 2+2=4.


"Experts don't know how they do what they do."

I disagree with this. Some things might be fairly automatic, but I know what and why I do them. I might have to stop and think for a while to figure out the best way to explain it to someone else (knowing my audience), but that is a different skill.

Back when I started teaching math, I spent some time trying to organize all of those things that I thought would help fill in typical gaps in knowledge and skills. It didn't mean that I became better in math, but I did become a better teacher.

Glen said...

@Mark, for simple calculations, "prove your answer" can be as simple as "show your work." For more complex problems, especially those that start out as situations or objectives explained in words rather than given as equations, proving one's work (as I mean it) is more than just showing a sequence of math notation.

It means such things as showing that the approach you're taking is valid, that your mathematical representation of the problem makes sense, that the steps you take form a valid chain of logic that isn't missing any links, that your answer passes validation by one or more forms of checking, and so on.

I certainly don't require all of this for every problem, but I insist that they be prepared to provide any or all of it on demand. The older the child, the more open-ended the problem, or the more unsure the child is of his answer, the more likely I am to demand, "prove it."

Even in math proof can be a matter of degree, but I think it's important to teach a child how to go from hoping he got the right answer to making sure of it.

chinaphil said...

This debate gets very abstract very quickly unless you tie it to some concrete stuff. So here's an attempt: consider the carrying operation in adding.

To understand the carrying operation in adding, you need a very clear idea of how place value works, and how columns work.

I taught my boy (now 7) to do adding with carrying before he had a clear idea of place value. He was able to do the process mechanically, and now he is able to talk about why it works.

I think that's true of a lot of things in maths. There are useful operations one can learn to do without having a clear grasp of the big concepts behind them. Grasping the concepts is a greater knowledge, and it is something one should aim for. But teaching the operation has its place.

Allison said...

Yes, Chinaphil, teaching the algorithms has a place, but there is no reason a student can't learn to understand place value first. Doing so makes learning the algorithm a piece of cake, because it is learning a "nice shortcut for keeping track of what you know."

The main problem with teaching the algorithm first is it is very easy for an expert to mistakenly think the novice understands when they don't, because we naturally "fill in" their poor explanation and knowledge with our own. Once mastered, it is difficult for us to "forget" what we know and "not see" what they don't see.

So we meet a child who can do pages of 3 digit by 3 digit addition with renaming who when asked to add horizontally
8 + 5 =

says "13, but the answer can't be bigger than 9."

Allison said...

the riding-the-bicycle example is interesting in how little light it adds to the issue.

When adults are given lessons in some motor skill work, they are told things about what they are doing wrong. Consider tennis:
" bend your elbow more; spread your feet; check the angle of the racket"--constant coaching that is verbal and conscious.

I learned to ice skate as an adult. Again, words, lots of words, coupled with my practice.

As a young adult, I learned to snowboard. Not very well, but well enough to make it more than 5 feet without falling down the very first time by having been told in words what I am supposed to do, and what to do better.

Explicit instruction works. This is consider inauthentic by constructivists.

The irony is that the authentic action of experts is a "forgetting"- Husserl talked about all of his in his phenomenology work. Experts don't know what they are doing: Pilots can't tell you how they land a plane, race car drivers can't tell you how they know to pass. Teachers are truly special because the authentic ones accurately learn what is actually done, rather than assume they can trust their judgment, and then explicitly force themselves to disclose those truths. Ironic, no?

chinaphil said...

Allison:
"...there is no reason a student can't learn to understand place value first."
No, I don't think that's right. Doing things with numbers is how you come to understand them. Apart from anything, you literally can't understand place value without understanding addition, because place value *is* addition. 34 means three tens plus four units. You can't get place value without getting addition; you can't get addition without getting place value.
So I don't think it's right to say students "should" learn one before the other. And I don't think there's any reason to consider the abstract concept of place value to be superior to or prior to the concrete skill of addition.

Karen W said...

Absolutely kids can learn place value first--and Alison is right that it makes the addition algorithm easy to teach as just a way to record what you know.

The Montessori golden bead materials (single beads, ten bars, hundred squares, and thousand cubes) give kids a concrete understanding of place value and work with physically exchanging/renaming when used to perform four digit addition and subtraction problems. When they get it (and get tired of manipulating all of those materials) they move onto using just the abstract addition and subtraction algorithms.

chinaphil said...

No, using beads doesn't get you out of this quandry. In order to use the beads properly, children have to be taught what to do - algorithms. Those algorithms of how to manipulate the beads are no more "basic" than the algorithm of the carry when doing addition on paper.

There is no way to argue your way out of the fundamental interlinking of arithmetic operations and place value. Focusing entirely on abstract knowledge of place value at first seems unlikely to be helpful to me as a strategy; focusing entirely on the procedural knowledge of arithmetic also seems to be limiting.

But it's worth remembering that learning more maths is often about abstracting and generalising what you learned before. Therefore to someone who has learned a lot of maths, it's easy to see any subject as a particular instance of general rules. To the learner, though, those rules are invisible. Learners often grasp the general by doing the particular.

Catherine Johnson said...

Fascinating comments!

Mark wrote:

But "show your work" isn't the same thing as "explain the answer in words.

That is exactly the way I've always seen it (though I have a semi-open mind on the explain-in-words issue).

I strongly favor students showing their work; I don't, personally, see the need for explanations in words.

For what it's worth, my operating assumption is that of Anonymous above: being able to explain things in words is important in word-based fields, not in math (math-based fields?).

Catherine Johnson said...

Glen - what kind of consciousness are you talking about?

Mathematicians seem to have eureka moments the same way scientists do (or is that wrong?)

Eureka moments happen after unconscious problem-solving/thinking/creating….whatever you call it.

It seems to me that an awful lot of 'explanations' are post hoc: we create them after we've had the eureka moment.

Catherine Johnson said...

Teachers are truly special because the authentic ones accurately learn what is actually done, rather than assume they can trust their judgment, and then explicitly force themselves to disclose those truths. Ironic, no?

This is SOOOO true.

I know I've said this a number of times, but when I started teaching composition I was chronically confronted with the fact that while I knew how to write I had no idea how to explain what I did to my students. Crazy!

I had a fabulous experience along these lines this semester.

I had brought to class an exercise on restrictive & nonrestrictive clauses to see whether everyone could read them accurately.

Actually….I should make this into a post...

gasstationwithoutpumps said...

Doing mathematics and writing about mathematics are separate skills. Both are important.

Generally math problems are solved by fairly complicated searches through the space of mathematical transformations. Sometimes there is an "aha!" moment that connects together ideas and simplifies something that previously seemed very difficult.

But written mathematics prunes away almost all the real work and reorders the work that remains, leaving only a clean path from problem statement to resolution.
This writeup is a done almost entirely after the mathematics work is done. The result is an elegant mathematical statement that usually contains more sentences than equations.

The only time I've seen K–12 students taught anything reasonable about mathematical writing in in the "Art of Problem Solving" on-line classes. And the teaching about math writing was more in the feedback on the homework than in the class, pointing out where there were missing assumptions or steps in the proof that did not follow.

Glen said...

Catherine, the kind of consciousness I'm talking about is explicit and declarative, as opposed to implicit and procedural. Mathematicians are like painters and bike riders in that all of them use unspoken intuition for part of what they do. The difference is that painters and bike riders can leave it at that. They don't have to understand what they do; they just have to produce good results.

Not so with a mathematical result. You may use intuition to guide your search for a solution, but you're not done until you understand what you have found well enough to PROVE it explicitly.

I'm not talking about understanding the human mind. I'm talking about understanding painting and math. If you implicitly understand painting, you can produce great art, even if you don't explicitly understand it. If you implicitly understand math but don't explicitly understand it, that's not good enough. If you ask me for the capacity of a hemispherical punchbowl, and I say that my intuition says that it's something like (2*pi*r^3)/3, you could reasonably ask me to come back when I have an answer I can prove. It's not as if the formula is going to speak for itself, like a painting or graceful bike maneuver.

I asked my son a few days ago for the sqrt(a^2 + b^2). He said it was (a+b). I asked him to prove it to me. He found it surprisingly difficult to prove for something so obviously true. He eventually managed to prove that it WASN'T true---not cast doubt on it but PROVE that it wasn't true.

For simpler items, showing your work can be sufficient. But the more challenging the problem, the more there is to explain. Articles in math journals are full of words as well as math notation. In that respect they resemble journal articles in economics, physics, neuroscience, etc. They have to explain their reasoning and prove their math.

Anonymous said...

Here's an interesting example. I think that mathematical modeling is a very important topic and that simple Algebra I/II word problems are a gateway to modeling.

I gave a project recently based on mixture word problems - Sally mixes 50ml of 10% HCl solution with some 40% solution to make a 20% solution. How much of the 40% solution is needed?

The way this is often taught is:

.10(50)+.40(x)=.20(x+50)

Very few students understand WHY this equation represents the situation described in the problem. They learn (and are taught) that this is the equation to use to get the answer and they move on.

One of the most important things I ask for when I assign a written project is that any equation introduced to solve the problem must be explained and justified. At the same time, I provide an enormous amount of support in the form of oral explanations.

The project ends up showing one example (a 25% solution that requires 100ml), asking the students to solve two others (40%/25ml and 70%/10ml) and then has them make the percentage a variable (x) and the amount of the solution added (y) and graph the resulting relationship so that all the answers appear in the graph.

I think that justifying their work is helpful to both the students and to me as I sometimes see explanations for a project on logs and exponents like:

"multiplying on both sides by ln"

which demonstrate that, although the student can solve the problem, they don't really understand the concept.

Anonymous said...

This was discussed on a physics teacher list a while back. One of the participants, a guy with lots of "real world" experience, put it roughly this way:

We hope that one day, some of you will actually do something with some of this. We have to teach you in a way that assumes that it's not all just an exercise. In your future, no one will trust you to do anything important if you can't present your work in a way that can be followed and checked.

In that spirit, note that it is almost time for this year's Moody's Mega Math Challenge. You can see more about it at

http://m3challenge.siam.org/

Phil

allison said...

--There is no way to argue your way out of the fundamental interlinking of arithmetic operations and place value. Focusing entirely on abstract knowledge of place value at first seems unlikely to be helpful to me as a strategy; focusing entirely on the procedural knowledge of arithmetic also seems to be limiting.

I didn't day this. I said that teaching the standard algorithms-the vertical algorithms-0 can come second.
Which is what Singapore's Primary Math does, among others.

But you can decouple place value from base 19 addition, by teaching other base systems, and showing it is still place and value. Yes, a student still needs to understand what 56 steps from 0 is, however you write it.

The problems are 1) too many of our students and teachers work the vertical algorithms and have no idea what the places are saying (literally don't know why the tens place is 1/10 of the hundreds place) and 2) too many others are never taught fluency with the algorithms.


but it is a lot easier to teach fluency with the vertical algorithm if they already know how to add by grouping into tens and ones.

Ironically, someone defended me using Montessori, which is the program used by the girl who told her mother
:8+5 is 13, but the answer can't be bigger than 9." Montessori school had taught her the vertical algorithm, and in doing so confused the idea of the value of a sum and the way you write it in base 10.

allison said...

Our students are taught to pattern match. Pattern matching is not math.

They can do this well enoug to take a standardized test, and to convince the unobservant they are competent but like a trained seal, they don't understand the math they do.

Forcing them to explain their work in *some* manner, not necessarily
verbally, is necessary to force them to see math is more than pattern matching; it is about using known truths to derive others.

Precise words are very good. So are presentations at the board. It is all immensely time consuming, and can only be done by someone who is truly a mathematical person, who thinks mathematically and knows a lot of math. Most engineers don;t even have this trait.

Cassandra Turner said...

I'm going to second Allison here: "The problems are 1) too many of our students and teachers work the vertical algorithms and have no idea what the places are saying (literally don't know why the tens place is 1/10 of the hundreds place)"

I've worked with thousands of teachers over the last 7 years. I am constantly astounded by how many do not understand place value and can do not a lick of mental math. They line up 45+7 vertically.

Once had a 7th grade math teacher actually say, "Oh, this is the first time I've realized what those little 'ones' we write above the numbers mean."

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