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How to make change when handed a $20... and help democracy

If someone owes you $5.35 and hands you a $20 bill, every reader of this blog can easily make change. You have a calculator, a cash register, or you do it in your head.

However there is a faster way that I learned when I was 12.

Today it is rare to get home delivery of a newspaper, but if you do, you probably pay by credit card directly to the newspaper company. It wasn't always like that. When I was 12 years old I delivered newspapers for The Daily Record. Back then payments were collected by visiting each house every other week. While I did eventually switch to leaving envelopes for people to leave payments for me, there was a year or so where I visited each house and collected payment directly.

Let's suppose someone owed me $5.35 and handed me a $20 bill. Doing math in real time is slow and error prone, especially if you are 12 years old and tired from lugging newspapers around.

Instead of thinking in terms of $20 minus $5.35, think in terms of equilibrium. They are handing you $20 and you need to hand back $20... the $5.35 in newspapers they've received plus the change that will total $20 and reach equilibrium.

So you basically count starting at $5.35. You say outloud, "5.35" then hand them a nickel and say "plus 5 makes 5.40". Next you hand them a dime and say "plus 10 makes 5.50". Now you can hand them 50 cents, and say "plus 50 cents makes 6". Getting from 6 to 20 is a matter of handing them 4 singles and counting out loud "7, 8, 9, and 10" as you hand them each single. Next you hand them 10 and say "and 10 makes 20".

Notice that the complexity of subtraction has been replaced by counting, which is much easier. This technique is less prone to error, and makes it easier for the customer to verify what you are doing in real time because they see what you are doing along the way. It is more transparent.

Buy a hotdog from a street vendor and you'll see them do the same thing. It may cost $3, and they'll count starting at 3 as they hand you bills, "3..., 4, 5, and 5 is 10, and 10 is 20."

I'm sure that a lot of people reading this blog are thinking, "But subtraction is so easy!" Well, it is but this is easiER and less error prone. There are plenty of things you could do the hard way and I hope you don't.

It is an important life skill to be able to do math without a calculator and this is one of the most useful tricks I know.

So why is this so important that I'm writing about it on my blog?

There are a number of memes going around right now that claim the Common Core curriculum standards in the U.S. are teaching math "wrong". They generally show a math homework assignment like 20-5.35 as being marked "wrong" because the student wrote 14.65 instead of .05+.10+.50+4+10.

What these memes aren't telling you is they are based on a misunderstanding of the Common Core requirements. The requirement is that students are to be taught both ways and that the "new way" is such that that they can do math without a calculator. It is important that, at a young age, children learn that there are multiple equivalent ways of getting the same answer in math. The multi-connectedness of mathematics is an important concept, much more important than the rote memorization of addition and multiplication tables.

If you've ever mocked the way people are being trained to "stop thinking and just press buttons on a cash register" then you should look at this "new math" as a way to turn that around. If not, what do you propose? Not teaching them to think about math in higher terms?

In the 1960s there was the "new math" movement, which was mocked extensively. However if you look at what "new math" was trying to do: it was trying to prepare students for the mathematics required for the space age where engineering and computer science would be primary occupations. I think readers of this blog should agree that is a good goal.

One of the 1960s "new math" ideas that was mocked was that it tried to teach Base 8 math in addition to normal Base 10. This was called "crazy" at the time. It wasn't crazy at all. It was recognized by educators that computers were going to be a big deal in the future (correct) and to be a software developer you needed to understand binary and octal (mostly correct) or at least have an appreciation for them (absolutely correct). History has proven they naysayers to be wrong.

When I was in 5th grade (1978-9) my teacher taught us base 8, 2 and 12. He told us this was not part of the curriculum but he felt it was important. He was basically teaching us "new math" even though it was no longer part of the curriculum. Later when I was learning about computers the concept of binary and hexadecimal didn't phase me because I had already been exposed to other bases. While other computer science students were struggling, I had an advantage because I had been exposed to these strange base systems.

One of these anti-Common Core memes includes note from a father who claims he has a Bachelor of Science Degree in Electronics Engineering which included an extensive study of differential equations and even he is unable to explain the Common Core. Well, he must be a terrible engineer since the question was not about doing the math, but to find the off-by-one error in the diagram. To quote someone on G+, "The supposed engineer must suck at his work if he can't follow the process, debug each step, and find the off-by-one error."

Beyond the educational value or non-value of Common Core, what really burns my butt is the fact that all these memes come from one of 3 sources:

  • Organizations that criticize anything related to public education while at the same time they criticize any attempt to improve it. You can't have it both ways.
  • Organizations who just criticise anything Obama is for, to the extent that if Obama changes his mind they flip and reverse their position too.
  • Organizations backed by companies that either benefit from ignorance, or profit from the privatization of education. This is blatant and cynical.

Respected computer scientist, security guru, and social commentator Gene "Spaf" Spafford recently blogged "There is an undeniable, politically-supported growth of denial -- and even hatred -- of learning, facts, and the educated. Greed (and, most likely, fear of minorities) feeds demagoguery. Demagoguery can lead to harmful policies and thereafter to mob actions."

These math memes are part of that problem.

A democracy only works if the populace is educated. Education makes democracy work. Ignorance robs us of freedom because it permits us to be controlled by fear. Education gives us economic opportunities and jobs, which permit us to maintain our freedom to move up in social strata. Ignorance robs people of the freedom to have economic mobility. The best way we can show our love for our fellow citizens, and all people, is to ensure that everyone receives the education they need to do well today and in the future. However it is not just about love. There is nothing more greedy you can do than to make sure everyone is highly educated because it grows the economy and protects your own freedom too.

Sadly, Snopes and can only do so much. Fundamentally we need much bigger solution.

Posted by Tom Limoncelli in Rants

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4 Comments | Leave a comment

Side A does not like Side B and rejects their ideas.
Side B does not like Side A and rejects their ideas.

Neither side has gotten past their disdain for the other to even talk about the technical merits of the proposal.

You are will not get the other side to listen because you are not addressing their issue.

On the technical side at least in regards to Common-core:math

Common-Core:math is to brute force password cracking
Classical math is to rainbow tables.

Both have advantages and disadvantages.

Personally I like the Montessori education idea better for schooling then current system. I would couple it with Mastery based advancement instead of yearly timebox and fixed age groups. Some kids are just not ready to be at the same spot as other kids and we are forcing them to be compared to their peers.

Thanks for sharing your thoughts DMonTech. I do respectfully disagree with the "both sides do it" argument you have made. There is only one party who has embedded anti-science views into their (written) platform and only one party that has, as a platform, rejected anti-science views.

"Side A does not like Side B and rejects their ideas.
Side B does not like Side A and rejects their ideas."
Politically we have tried understanding, reason and all manner of civilised approaches. As Tom says: "There is only one party who has embedded anti-science views into their (written) platform and only one party that has, as a platform, rejected anti-science views." and they should be held to ridicule.

As for the wars that Teachers get into with curriculum design. Well this can be a very hot topic. Teachers being rational people can gradually approach change, provided it is backed up with understanding and intelligence. Sadly often curriculum and pedagogy changes are brought about by Politicians and lobby groups from either side. This leads to some very bad outcomes for students (see basic skills testing, ie $20 is too difficult a note to work with).

If Common Core means teaching multiple ways to come to the answer, I am all for it. I was one of those kids who would have benefited by learning that "counting up" method of giving change. It wasn't till I worked with cash at a family store that I learned that trick from my Dad. Afterwards, I could hand back change as quickly as any Math Whiz.

My problem with Common Core comes in the testing/implementation. Just like the "counting up" method worked for me, simple arithmetic is all that is needed for the Math Whiz. Why should that person be penalized for being able to come to the answer in fewer steps (arithmetic) while not being able to (or not seeing the point of) counting it out the long way?

The final answer is the same, and if it is correct. It should be marked correct. Period.

If you can do $20- $3.65= $16.35 in your head, you should get just as much credit as if you write it out counting up-
3.65+.10= $3.75+.25= $4+1=$5+5=$10+10=$20

That is not the case in most schools. Students are not just benefiting from learning in a way that makes sense to them, they are being punished for not learning every way that is taught. When in life are you required to show ALL of the ways you could possibly arrive at a correct answer?

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