(not) Explaining “Half”

“If you can’t explain it to a 6-year-old, you don’t understand it yourself.” – Albert Einstein

 

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“I’ve drank half my water”

 

My daughter isn’t a 6-year-old… yet. I get a couple more months before Dr. Einstein’s quote applies. Regardless, today I found myself at the breakfast table trying to make sense of one of those topics that exists in the intersection of math and common language.

It’s good to make mathematics common place, but the problem with making technically-specific terms commonplace is that it often leads to the usage of the word becoming a bit less technical. ELA teachers will tell you that this has happened with the word “literally.” (For more on “literally”, this post will literally blow your mind.)

In that same spirit, today’s breakfast was abuzz with the word “half”. My daughter is comfortable with the word “half.” She will often take a couple of gulps from her cup leaving ~5 ounces in a 6-ounce cup. Then she’ll say, “I drank half my water.” To her that means that she has consumed a some portion of her water, more than “just a little”, and less than “all” or even “a lot”. It seems the developing spectrum looks something like this:

Water Drinking Spectrum

Water Drinking Spectrum

It’s clear that spectra like this make clear the need for fractions, but that’s a discussion for another time.

Well, today, she asked about what “half” means. Which was a question that I should be qualified to answer, but you wouldn’t know it from how the conversation went.

Her: “Dad, what does ‘half’ mean?”

Me: “Well, it’s like broken into two pieces that are the same size.”

Her: “No, like, I drank half my water”

[See? My explanation didn’t work because I was thinking the two pieces were the water in the cup and the empty space. She only sees water. That’s only one piece.]

Me: “That means that the amount of water that you have already drank is the same as the amount of water you have left.”

Sage: “I don’t get it.”

Me: [holding up clear cylindrical glass that is “about half-full”] “See? This glass has water up to here [I point at the water line], and then the rest of the glass is empty. It’s like there’s the same amount of water and empty.”

Sage: “Um… Maybe we should talk about this when I’m older.” [Sage continues eating oatmeal and the topic changes]

 

Good thing she’s only five. I have a few months to prepare for Dr. Einstein’s test of whether or not I really understand the idea of “half.”

 

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“Ugg.. I don’t get ANY of this…”

I like working with teenagers for a lot of reasons. One such reason is because it is usually quite easy to get bold, absolute statements out of them.

For example: “I don’t get ANY of this.” (emphasis not mine…).

It’s tricky business handling that statement because most of the time, the author of that quote genuinely feels that way. That doesn’t make the it true, but if he or she is already feeling like all the attempts at math are turning up wrong, combating their perception by simply saying it’s false is probably not the best approach.

I enter into evidence the following photo:

 

See it? He made the same mistake twice.

See it? He made the same mistake twice.

I gave these two problems as part of a short formative assessment. I saw the answers shown above a bunch. Most of my classes were lamenting the assessment because, “They don’t get any of this.”

Which, as I said above, couldn’t be farther from the truth. Those two answers show quite a lot of understanding, actually. There is a single mistake that led to incorrect response in both cases. That student was unable to distinguish the shorter leg from the longer leg. That’s it. Fix that and the answers get better.

And when I show them the answer to those two, many of those bold, absolute thinkers are likely going to think, “Yup, I knew it. It’s wrong. I knew I didn’t get any of this.”

At least I know that going into the discussion. Now, if I can just convince them that understanding is a spectrum more than a light switch.

Perception and Reality – (Lean not unto thine own understanding…)

In Basic Economics, Thomas Sowell tells a story about a decision made by a New York politician who was attempting to address the homeless problem in New York City. The politician noticed that most of the people who were homeless were also not very wealthy. The politician moved forward with the idea that the apartment rent prices were simply too high for these people to afford a place to stay.

So, he decided to cap the rent prices… and the homeless problem got worse. How could this possibly be?

Well, according to Dr. Sowell, lowering rent prices, while making the apartments more affordable for those in need, did the same for everyone else. The suddenly cheaper rent prices decreased the rates of young folks sharing apartments. Also, people who have several places they call home throughout the year might not have found it reasonable to pay a high rent price to keep a NYC apartment that they might only stay in a few times throughout the year. Lower rent prices made that seem more reasonable.

Evidence also suggested that there was an increase in apartments being condemned. Lowering rent costs meant that landlords found themselves with fewer resources to maintain buildings, repair damages, pay for inspections, etc.

While the decision made the apartments more affordable, it also made them more scarce. There was a disconnect between a decision-maker’s perception of a situation and the reality. That disconnect led to a decision that ended-up being counterproductive.

I may have just done the same thing… maybe.

Perception-Reality

 

Sometimes things make so much sense. If we did this, it would HAVE to produce that. It make so much sense. How could it possibly not work?

This perception was in place among some in my community. It led me to decide to try The 70-70 Trial, which I’ve been at for about 10 weeks now. The perception in place goes like this:

a. Formative assessments prepare students for summative assessments.

b. Students who struggle on formative assessments are more likely to struggle on summative assessments (and the inverse is also true.)

It’s with these two perceptions in mind that we assume that the if we can ensure a student achieves success on each of the formative assessments (regardless of the timeline or the number of tries), we improve their chances of success on the summative assessment.

The 70-70 trial did what it could to ensure that at least 70% of the class achieved 70% or higher proficiency on each formative assessment. (There were four.) This included in-class reteach sessions and offering second (and in some cases third) versions of each assessment. With all of those students making “C-” or better on each formative assessment, how could they possibly struggle on the unit test? That was the perception.

50% of the students scored under 50% on the summative assessment. That was the reality.

Now, I am not an alarmist. I understand that one struggling class in one unit doesn’t discredit an entire education theory. But it sure was perplexing. I’ve never seen a test where, after 8 weeks of instruction on a single unit (Unit 4 from Geometry), half of an entire class unable to successfully complete even half of the unit test.

And when you consider that this class was the one class I had put the most effort into defeating just that kind of struggling, well it seems like the intersection of my perception and the reality wasn’t nearly big enough. I just got a better view.

And I’m having a hard time making sense of what I’m seeing.