The danger of the Number Sentence

This is outside the normal for my blog, but I had something to say.  This is commentary on American education.

When I was young, I went to school and had teachers fill my head with all sorts of things. Most of the things I learned when I was very young have been replaced by newer things. The process has really been an evolution of ideas that is not unlike the evolution that has occurred on this planet. I want to discuss with you the dangerous new idea that has evolved and I want you to consider it in evolutionary terms. I want you to do this, because I think it is important that we select what kinds of ideas reproduce and spread in future generations.

There has been an outcry against the “Common Core” and really it terrifies me. The research I have done has really been very cursory, but as I write this I am digging a little deeper. First let me give you my basic qualifications in broad terms, so you can assign some value to my words, but I will not give you the details and specifics of my experience, so please research on your own as well. I worked for an educational technology company for several years as a software engineer and eventually as the Development Team Lead. One of the products I was intimately involved with was designed to track and analyze students’ success towards meeting standards and to gauge the alignment of curricula to maximize the learning plan for students in order for them to be provided the tools to meet the standards over the course of their education.   This exposure made me very wary of “Academic Standards” and weary of the “No Child Left Behind” mindset.

Under “No Child Left Behind,” federal funding was linked to student success at standards. This lead to my being tasked to create tools to identify which children were closest to making the “next step” so that academic resources could be targeted at them. To clarify, the goal was not to help all students, not to help the worst and not to help the best. The goal became to help students who would increase the schools federal compensation by targeting the students who were closest to reaching the next goal. I understand fully that by doing so and increasing or maintaining the federal funding, all students in the school could be enriched by the additional funding. I also am passing no moral judgment on the educators and administrators who set us to this task, because I understand the need to follow the dollar.

I became a huge cynic about the government because of this. This is what shattered my “Great America” illusion.

At the center of it, Common Core is a new approach to standards that corrects what I see as the heartbreaking failures of NCLB. As of now, it is not linked to federal funding. Tools and testing are much more available than under NCLB. There are great things happening here. This is one of the great things about evolution. Things get refined.

However, evolution is powered by random mutations that are not always beneficial, favorable, useful or attractive. One of the things that has mutated is the standards themselves. Again, my looking at the standards is cursory and you should research the actual standards yourself. I will post one standard here so that you can really decide if you’ve “looked at the standards” and so that I can discuss it a little more closely. This one is from the high school level in the realm of “Statistics and Probability”

Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

First, I must say that I have mixed feelings on the change to the tone of the standards. I appreciate the absence of multiple “Education is a science” words, however the tone is oversimplified. I think it is great that the standard feels approachable for a larger portion of the population, however the most vocal proponents and opponents of any educational change will have a higher level of education and therefore the concrete examples of “state lottery” and “fast food restaurant” seem too specific and low brow. This can easily be corrected by formatting and defining the standards a little better(i.e. separating the examples from the text of the standards).

Second, I really appreciate the fact that the standard is a clear and quantifiably testable objective. If I give you 10 questions that require you to calculate anticipated payout based on reward values and probabilities, I can clearly say, “You got 8 right, so you understand the idea, but we need to work on details.” This is perfect when it comes to standards. There have always been and always will be “standards” that require too much interpretation for people to measure success in any standard way. It appears the group responsible for Common Core Standards understood this.

Now, that being said, I want to explain the “Number Sentence” and where things are going wrong. I have been lightly exposed to the shift in primary school mathematics education and it has troubled me greatly.   I am very upset by the teaching of shortcuts as methods. In 6th grade, I was pretty good at math and I would help other kids in the class and Mrs. A pulled me aside and told me that I can’t teach people the shortcuts until they learned the full long way. I got so angry at her. Then she explained that the rest of everything they learn will be built on understanding the long way. It did nothing to lessen my anger, but it is the most important thing she ever taught. I almost said it was the most important thing she ever taught me, but it is unequivocally the most important thing she ever taught. Period.

Number sentences are a small problem in primary school education sequencing. Ultimately, children need to be taught to convert English language sentence into mathematical expressions. However, the English vocabulary they have at the time is not sufficient to use the word “expression” in a mathematical context. However, as it is in primary school mathematics and has an easy word, it is used to classify disdain for the actual methods that are being taught.

The methods being taught represent mathematical shortcuts that waiters and cashiers use all the time. The idea of breaking up more complicated addition and subtraction into multiple operations is used every day. If I ask you to make $0.67 in change, most people would subtract $0.50, then subtract $0.10, then subtract $0.05 and realize that it’s got $0.02 left over. Ultimately, this is the easiest way for our minds to perform simple operations is to break down the numbers. We do it often enough that it becomes trained habit. I didn’t realize that this is what my brain was doing until I stopped to think about it.

This is where the danger lies.

I can teach a 6 year old how to make change and it is a valuable skill. Even if they never work a register in their entire lives, at least they’ll be able to make sure that whoever is giving them change is doing it right. However, we should teach them to “do math.” We should teach them to love to learn. We should teach them that when they learn long subtraction with carries that it will become long division. We should encourage them to find shorter ways as long as they work, but first we must teach them the long way.

In 11th grade, I had a math teacher that I will never forget. I finished our first pre-calculus exam and was playing with my graphing calculator.

9×9 = 81
99×99 = 9801
999×999 = 998001
9999X9999 = 99980001

When the bell rang, I walked up to the teacher all sorts of proud and wrote out:

999,999,999,999,999,999,999×999,999,999,999,999,999,999 = 999,999,999,999,999,999,998,000,000,000,000,000,000,001

Which was well beyond the limits of even the most powerful graphing calculator in the room. He smiled and said only two words to me, “Prove it.”

I did. He got it published in “Mathematics Teacher,” for what it’s worth. But really he gave me a lifetime’s worth of education in two words.  He didn’t teach me the shortcut. He probably didn’t know the answer that day when I walked up to him, but I bet he proved it before I did. He gave me the encouragement to find the shortcut. He supported me and told me I was wrong when I was wrong. He told me to dig deeper when I wasn’t necessarily wrong.  Guidance is education.

When I have children, I will teach them the long way home and make sure they know it. I will show them other ways home and make sure they know those ways too. I will hope, every day that they will come home 15 minutes early and tell me about this great shortcut they found. I assure you that if you teach them the shortcut, they will never learn the long way or the other ways. I assure you that if they are comfortable with only the shortcut, that if they are faced with the long way, it will be intimidating and they will just feel lost. These are simple lessons. So is long hand subtraction with carries.

So, let us not just cry about the failures of Common Core. Let us raise our voices in praise of intellectual evolution. Let us embrace the good that has come and fight against the bad that is the price. If ideas go untested, unjudged and unattempted they may as well be bad ideas. If bad ideas go unchallenged everything starts to go bad. Challenge the bad pieces but embrace the good ones. We do not need to start over every generation; we can keep some of the good stuff.


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