The "common core check guy"

In the news recently, we see the story of Doug Herrmann. Doug is a father from Ohio who out of frustration with his child's math homework, wrote a check using common core methods. He was frustrated because he was unable to understand what the school was teaching his son, and therefore was unable to help his son with his math homework.

The ten card system that his “viral” check picture was intended to highlight is not a terrible method of mental math. One exception is that for some it abstracts the idea so far that it seems to become separate from the actual “math”. Utilizing grouping techniques is in no way a “bad” technique to understand mental math. However, it becomes a problem if too much emphasis is placed on a single rigid method.

Using the ten card system is one of many useful techniques for understanding grouping. We only run into problems when the standardized testing systems force children to “believe” there is only one right way. The testing should only test for the ability to solve problems and not for the strict adherence to a single “correct” way of getting there. The practice tests I have looked at appear as though they are actually structured that way. The second grade tests do emphasize place value, but do not seem to indicate a single method. Now as far as the rigidity of the teachers, I have not seen how this comes into play. I do know that in many instances students fall behind due to teachers being too strict in the techniques that you use.

Please understand that I am well aware that there has been a rising trend in math illiteracy. I am not completely blasting common core. I , along with other parents, have some questions about the implementation and flexibility of these standards.

Some questions I have include:

• How flexible are the methods used ?
  • Are students being taught that the ten card system is the “only correct” method ?
  • Will students be penalized for utilizing other techniques to arrive at their answers ?

• How are conflicts between common core, and how the parents teach their children handled ?
  • Do teachers penalize their students for the way their parents teach them to solve problems ?
• Is there an awareness of the different ways people understand concepts ?
  • Is there an accommodation in place for students who understand place value through other means than just grouping visualizations? ( e.g. positional notation instead of ten cards.)

These questions go beyond just common core. The rigid adherence to single techniques has long been the culprit behind the fall in mathematics education.

On a personal note, I have been a victim to the rigidity of teachers. When I was in high school(many years before common core) I failed a math class simply due to my teacher not approving of my mental math techniques. To be fair, I would like to emphasize the math teacher was not actually a math teacher, she was a soccer coach moonlighting as a math teacher due to a poor student teacher ratio.

The conflict arose when I was forced to show my work(and by show my work I mean that my teacher wanted to see the remedial addition, subtraction, multiplication, and division) This was an algebra class, where showing your work meant to show the steps taken to simplify the expressions. However, this instructor insisted we not just show that we had to multiply she wanted to see the actual steps we took to multiply. I thought this was odd in an algebra class, because learning to multiply was second grade. However, I did comply and attempted to be verbose with “showing my work”. However, my method of “long hand” multiplication was different from what she was taught. Most people are taught the “only” way to multiply is to start in the one column and carry and borrow and all these other concepts. I used a more “short hand” method which lends itself well to mental math. I would start with the highest power of ten column and work left to right instead of right to left. For example if I were to multiply two digit numbers I would start in the tens column and multiply no need for carrying and borrowing. You are just simply adding up zeros. This method isn't understood by all, but it was how I understood numbers and place value. Unfortunately, my math teacher was strict and rigid in how she wanted it done and failed me on all of my quizzes. I understood the algebra part and understood the basic calculations part, and arrived at the correct answer. I just applied a different method for the basic calculations.

Now my story is sadly similar to many other students. For most, these situations reinforce their frustrations with mathematics. It can quickly lead students to make snap judgments about the efficacy of learning and understanding math. They soon start to see math as a foolish endeavor with rigid methodologies not worth their time. Or they could simply just give up believing they will never understand and struggle to merely pass.

 I don't exclusively place the blame on teachers for the problem of rigidity. The parents also hold a share of the blame. As with the “common core check guy” Doug Hermann and some of the comments on his posting, they also find themselves rigid in their understanding. Simply not understanding a technique of solving a problem does not condemn it. Rather, it simply means you should probably learn more about it before condemnation.  

Propositional calculus: an introduction

Propositional calculus is the branch of mathematics that deals with the rules of logic and evaluation of statements or propositions. It is sometimes referred to as sentential calculus for its use of the sentential conjunctions. It is primarily concerned generating laws for evaluating conjugated statements. Propositional calculus is part of a broader science of logic and proof.

Some of the symbols used in propositional calculus:

Symbol	Meaning
¬	Not or negation
^	and
v	or
→	implies
↔	If and only if
P	Sentential Statement
Q	Sentential Statement
T	True
F	False

In propositional calculus the above symbols are used in constructing truth tables for evaluating compound or conjugated sentences. The truth tables provide a short hand tool for deriving new laws from the simple compound sentential statements. Truth tables like the one below shows some of the simple rules and the new rules derived from those.

P	Q	P→Q	P^Q	PvQ	(P^Q)→P	P ↔ Q
T	T	T	T	T	T	T
F	T	T	F	T	T	F
T	F	F	F	T	T	F
F	F	T	F	F	T	T

This is just a simple introduction to propositional calculus, it is part of a much broader branch of propositional logic . It is useful in the formulation of logic rules, and methods of proof.

The "Reality" of math and logic

Logic has been used as a methodology even before its formalization. In order to survive, our progenitors had to be able to deduce whether a particular place or event was dangerous. These more earlier versions of ourselves, needed to determine what was safe to eat and where food was located. In this sense they were using logic intuitively.

As societies evolved, we became aware of a need for a system of accounting. We developed symbols to serve as abstract models for real world objects. Empirically , we can demonstrate the need for a modeling system. We can also demonstrate that these models of real world objects work. The models functionality means they can be used for future discoveries, and aids in civilization building.

Enumeration permeates our entire civilization and history on multiple levels. It observably exists separate from language. From the simplest counting systems, patterns began to emerge. The laws of mathematics became more complex, and a method of proof was needed that was empirically verifiable. The earliest proofs were purely empirical as used by the Pythagoreans and Thales of Miletus. Over time, mathematical proofs slowly became less heuristic and a formalized system of logic was becoming more prominent. Logic was largely an emergent aspect of these methods of empirical verification. The methods of logical proof slowly grew over the next two and half centuries from Thales around 550 BCE to Euclid around 300 BCE. Euclid is credited with formulation of the axiomatic method of proof.

The Euclidean style of proof through axioms, appears assumptive, or only justifiable a priori.

The axioms served as descriptions or definitions of geometric objects. The definitions could be demonstrated empirically in general, but they did rely on assumptions. For example, take the statement 2 points determine a line. This is a postulate proposed as self-evident, and deduced from it is many other theorems. If you were to take this as meaning all statements derived rely on a non-empirical premise you would be incorrect. The statement can be demonstrated as empirically true. You would simply have to draw a line through only one point. The line drawn, would be a point and not a line. To construct a line you would have no choice but to have it pass through multiple points. So, we can see the statement is both self-evident and empirically demonstrable.

The evolution of mathematical systems of proof allowed for a more formalized system of logic. All of which finds its origins in our innate pattern recognition. The ability to recognize patterns and natural desire to seek them out led to the mathematical modeling of real world objects. The modeling techniques made a complete system of logic and mathematics more easily constructed and shared. We formulated logical and mathematically complete laws and theorems from some of the simplest models.

We see that these models are intuitive,empirical, and justified in their use.

Positional Notation

Positional Notation

Our number system is base ten, as many learned in elementary school. Which means that there is only ten symbols to represent numbers. 0,1,2,3,4,5,6,7,8,9 . To represent ten we use 1 and 0 (10). Which when we learned place value, we recognized this as 1 in the tens place and 0 in the ones place. What this means is that the position of the symbol will determine the value of the number. For example when we see the symbol 34 we recognize it as 3 tens and 4 ones. We would often see something like:

Tens	ones
3	4

 We are mostly taught this and just accept it as is and move on. To dig a little deeper into to place value I will introduce a new way of looking at this (Positional Notation) . To rewrite our example ,34, into positional notation we get this:

3 x 10¹ + 4 x 10º

The reason I write it this way is that is represents a number as powers of the base it is in. In this case it is base 10. Positional notation gives us a way of looking at numbers relevant to their base. The reason this is useful to learn, is that it gives us a quick way to convert numbers from other bases to base ten. For example, if you need to convert the number 101 in binary to base 10 you can rewrite 101 in positional notation and convert it easily. Since 101 is in binary we will use powers of 2 instead of 10 and we get

1 x 2² + 0 x 2¹ + 1 x  2º

1 x 4 + 0 x 2 + 1 x 1

4 + 0 + 1

5

As you can see from the table above just by rewriting the number into positional notation relevant to the base it is in you add it all up and you get the number in base 10. So when we convert 101 from binary to base ten we get 5.


 This is just a neat little trick I use to quickly convert numbers from other bases to base 10. With a little practice, you can convert numbers from most bases back to base ten almost effortlessly. There is other ways of converting number to other bases, I just really like this one.