Table of Contents
1. Pappas Page of the Day
2. The First Summer Post?
3. Problems Adapting to Change
4. Classroom Management
5. Neutering the No Talking Rule
6. Neutering the Discipline Hierarchy
Pappas Page of the Day
This is what Theoni Pappas writes on page 152 of her Magic of Mathematics:
"Finding experts who are able to translate ancient Chinese writing is difficult. Finding experts able to translate manuscripts that deal with mathematical ideas is difficult. This explains why Chinese ideas of mathematical themes are scarce."
This section is called "The Chinese Method of Drawing Squares." In this lesson Pappas demonstrates an ancient Chinese proof of the Pythagorean Theorem. It's similar to the cut-out proof that I tried to show my eighth graders at my old school. Pappas states that the proof could be as old as 1200 BC -- which would be before Pythagoras.
Rather than try to describe the image here, let me link to the Cut the Knot page devoted to proofs of the Pythagorean Theorem:
The proof in Pappas is given as Proof #4 on the Cut the Knot list. Notice that according to Cut the Knot, the Chinese only proved the case for the 3-4-5 triangle, with the general proof given by a Hindi mathematician Bhaskara about 1500 years after Pythagoras (so no, this wouldn't actually qualify as a pre-Pythagorean proof).
The First Summer Post?
Notice that I'm not posting any more SBAC Prep worksheets. I'm not going to keep up the pretense of posting SBAC Prep for a school where I no longer work, for students who already finished testing.
I'll be using these posts to reflect upon the past year -- especially where I went wrong. And I think that the first thing I failed to do is mentally prepare for my first year as viewed in my blog posts. In fact, we see that when I was hired for my first job at the end of April 2016, most of the following blog posts were still about substitute teaching. Not until mid-to-late June did I start writing (and thinking) about the upcoming year.
In May 2016, I continued to work as a substitute -- and of course being in the classroom did help to prepare me for my new teaching job. I had two weeklong assignments that month -- the first week I was in a high school math class, and the last week I was in a middle school science class. Ironically, the science class would be more relevant to my upcoming teaching job since I'd have to cover science, but I didn't know this at the time. There was a blogging challenge that month, MTBoS30, and my weeks as a sub gave me something to write about, though I did squeeze in discussion of the upcoming year as well.
Problems Adapting to Change
In fact, this is one reason why I struggled in my first year -- problems adapting to change. In fact, I would be given one schedule upon being hired or at the start of summer, and I'd even post about it here on the blog -- and then I found out something else when school started last August.
When summer began last year, I was handed a copy of two Illinois State texts -- the STEM workbook and a traditional text. I blogged about how I'd divide the year into two-week units, with the first week devoted to STEM projects and the second to traditional lessons.
As it turned out, this fit well with the requirement to submit photos of STEM projects to Illinois State every two weeks. The problem is that the plan left out other parts of the Illinois State curriculum -- most notably Learning Centers and arts projects. I never truly incorporated these into my lessons.
Moreover, I was thinking of each STEM project as a unit. But on the official Illinois State pacing guide, each unit should be a Common Core standard, not a STEM project. There are so many standards that each unit needed to be covered in one week, not two weeks. I should have changed my plans completely once I returned to school and learned more about the Illinois State curriculum.
Most importantly, the part of my curriculum that was missing was science. When I was hired, I had no idea that I'd need to teach science. The previous year, my old school had a separate science teacher, but she left when I arrived. Most likely, the administration was hoping that they'd be able to find a new science teacher, but by August this attempt failed. Again, upon finding out that I'd be teaching science, I should have changed my plans rather than stick to what I posted here on a blog when I didn't have all the information.
But of course, my classroom management in my first year was less than ideal. I can't begin to discuss how I can improve as a teacher unless I address this elephant in the room.
I don't write about management much on this blog, preferring to stick to the math. If you are an experienced teacher, I recommend you skip these posts on classroom management, because many of these posts will be about discovering things you've probably known all along. If you're a novice teacher, then take the rest of this post as a warning.
The most I wrote about management was during the "Day in the Life" posts, since becoming the ideal classroom manager was my goal for the year. Also, last summer I wrote four rules I wanted to implement in my classroom. These rules were:
1. The Teacher Respects You
2. Respect Your Honesty
3. Respect Yourself and Others
4. Respect Your Class Equipment
Keep this in mind as we revisit some of the issues I had with classroom management. Let's begin with a school rule -- no chewing gum in the classroom. This was a school rule which was not negotiable, and so I was expected to enforce this rule.
The first time I saw a student chewing gum, I told her to stop chewing it. (I'm sorry to use gendered pronouns here, but this tended to be a girl problem.) Her response would be, "I'm not chewing gum," as she hid the wad of gum under her tongue. Of course, later she'd resume chewing the gum -- otherwise she wouldn't have placed it under her tongue.
This would happen several times with different students. Every time I tried to give a consequence for chewing gum, the student claimed that there was no gum, and so it was unfair for me to give any consequence for gum unless I could prove beyond a shadow of a doubt that there was gum. The implication was that I was a power-hungry teacher who wanted to punish innocent kids for no reason.
One day, I told a student that I saw her chewing gum, so I had the proof I need. The girl then removed the object from her mouth, which turned out to be a wad -- of paper. She then told me that nowhere in the rules did it say not to chew paper. The implication was that I could never be certain that the student is chewing gum, and so it would be unfair to punish anyone for chewing gum. The result, of course, was that the rest of the year, students openly chewed gum, and I was powerless to do anything about it.
This is what I call neutering a rule. The students come up with tricks to convince me that enforcing a rule was unfair. In this case, the tricks were hiding the gum under the tongue and chewing paper. I know that people in general don't chew paper -- the girl did so for the sole purpose of neutering the no gum rule. Soon the no gum rule was rendered unenforceable -- by me, at least.
Let's contrast this with the actions of a better classroom manager -- my support staff member. She would tell the student to spit out the gum -- and usually, the gum was spit out within five seconds. It never occurs to the students to hide the gum under their tongue or start chewing paper if my support staff member is the one telling them to spit out the gum.
Also, notice that I had a much better time enforcing the no food rule. There's no room under the tongue to hide food or make paper look like food. The no food rule was never neutered.
Neutering the No Talking Rule
Notice that breaking the no gum rule is mostly harmless. Sure, the gum made the room messy, but it didn't affect student learning. But having found success in neutering the no gum rule, the students went on to neuter a rule that did affect education -- the no talking rule.
On the second day of school, my support staff member came up with a simple call-and-response to keep the class quiet: "When I say 'Listen,' you say 'Up!'" I thought it was a great idea, and so I incorporated it into my own management.
The first two times I tried it, it worked. But the third time, a student continued to talk. When I called him out on it, he claimed, "I wasn't talking!" And you can see what this led to -- students claimed that it was unfair to punish anyone for talking, and the no talking rule was about to be neutered.
When the class became loud, it could be heard all the way in the next classroom -- the middle school English class, where another grade was trying to learn. So when that class reached my room, they claimed that it was unfair to punish them because I let the first class talk whenever they wanted.
It was difficult for the students to learn anything. And of course, the talking didn't stop even when it was time to take a test. As usual, only my support staff member could quiet the class, and so hardly any work was accomplished when she wasn't in the room. This included IXL time, which turned into SBAC Prep time later on.
The students figured what to say to me to get a rule neutered. For example, students tried to say "You're weird!" when I tried to enforce an unpopular rule. Actually, this phrase didn't work -- I replied that I didn't mind being called "weird," and the students stopped using this one. As we've seen, the phrase "You're unfair!" was successful for neutering the rules.
Another phrase that students used effectively was "That's juvenile!" The first time I used it was when I decided to give the classes a pop quiz because -- why else -- they were too loud. It was my oldest students, the eighth graders, who claimed that only elementary teachers give pop quizzes. They were making two implications there -- the first is that there's no point in giving them pop quizzes because they'll just leave it blank, and the second is that if I don't already know any "non-juvenile" methods of management, then the only fair thing is to let them talk as much and loudly as they pleased.
The "juvenile" excuse came up a few months later in the class. I was assigned an instructional aide in order to help me with management. She came up with several methods, such as turning off the lights when she needs the class to pay attention. As soon as she left the room, one students then tells me that turning off the lights is juvenile. But get this -- I never turned off the lights! The aide who came up with turning off the lights had already left the room. In other words, the students who told me that turning off the lights was juvenile wouldn't dare say it to the instructional aide.
By the way, my students didn't neuter the "no fidget spinner rule" -- if only because fidget spinners didn't become popular until right after I left the classroom!
So now we see the pattern -- I give an unpopular rule, and the students come up with all sorts of excuses not to follow the rule. They quickly learn that the phrases "that's unfair" and "that's juvenile" are effective at neutering rules to render them unenforceable -- by me at list. They say things to me that they don't even consider saying to other adults.
Neutering the Discipline Hierarchy
When the instructional aide was in my room, we came up with the following hierarchy:
1st Infraction: Warning
2nd Infraction: Write a 150 word essay telling what did and how you will improve your behavior.
3rd Infraction: Call parents
4th Infraction: Send to Principal
The students reacted to this the same way they responded to all my management attempts -- they neutered it. The second step in this hierarchy was to write a 150 word essay -- but sometimes students said that they didn't know what to write for 150 words. Some students wrote "I will not do that" thirty times, while others just wrote a few sentences and didn't even try to reach 150. So the second level was neutered to basically a slap on the wrist.
Furthermore, students insisted that I give them a clean slate once they finished writing. So if they commit a third infraction, I should count it as a first infraction rather than call the parents. (Part of this was due to the fact that if a student refused to right standards, I counted that as the third infraction, thus if they do write the standards, I should give them a clean slate.) Therefore the implication is that I should allow the students to commit many infractions and give them very little punishment for committing all those infractions -- anything else would be unfair.
And this is why I often yelled in class. Sometimes I absolutely had to get the students quiet, but they wouldn't be quiet merely because I told them to. I yelled at students who talked when the ideal classroom manager would merely give them a consequence from the hierarchy -- but I couldn't because the rules and consequences had been neutered.
This is a summer post, so I do want to get back to the traditionalist debate. Here is a link to Barry Garelick's most recent post:
In working with a group of fifth graders in need of math remediation at my school, I had them do the exercises in their book. It involved multiplication of fractions, and it used the area model of a square as the means to illustrate what multiplication of fractions represents, and why one multiplies numerators and denominators. A problem like 3/4 x 2/3 then is demonstrated by dividing a square into three columns, and shading two of them, thus representing the 2/3. Then the square is divided into four rows, with three of them shaded–the 3/4. Thus 3/4 of the 2/3 have common shading, and the intersection of the shading of the 2/3 and 3/4 portion yields 6 little boxes shaded out of a total of 12 little boxes which is 6/12 or 1/2 of the whole square. The students see what 3/4 of 2/3 means in this model in terms of area and the reasoning behind multiplying numerators and denominators.
The purpose of this post is to criticize the Common Core practice of making the students draw diagrams every time they want to multiply fractions. He writes that he was tired of forcing them to draw, and the students were tired of drawing them.
The co-author of that blog, SteveH, has much to add:
My explanation is that understanding comes in levels, abstraction, and variations. There is no such thing as understanding first if one uses understanding in a generic sort of way. Conceptual (low level) understanding may motivate one a little bit, but that’s always happened forever. Every unit in a proper textbook introduces new concepts built on top of old ones. All of my traditional teachers from decades before did that, even for simple arithmetic skills. That’s just the start. What is missing these days (even with CCSS so-called emphasis on skills) is practice on all variations and depth. K-6 teachers, and therefore students, do not value any proper level of practice. They think repetition is just about speed and not understanding. That’s completely false. All proper textbooks provide problem sets with variations and depth, and all competent teachers assign and expect individual completion of non-trivial sets of homework problems. Each problem is really NOT the same. There are subtle differences that challenge one’s understanding. Individual practice still does not happen properly in K-6. Educators seem to only concern themselves with what goes on in class – often still in mixed ability groups where many kids fall though the cracks. Words are not understanding. Understanding comes from doing problem sets individually. I had so many nightly “discoveries” from that process. Modeling that process with a group in class does NOT transfer to many other problems. It has to be done individually and in quantity.
By the way, tonight was the Scripps National Spelling Bee. The winner turned out to be a sixth grade girl from California. Of course, I'll be watching one of my favorite movies, Akeelah and the Bee, which is also about a sixth grade girl from California who competes in the National Spelling Bee.
Last year around the time of the Bee, I mentioned the contest in a traditionalist post. Naturally, the Bee represents everything a traditionalist stands for -- long hours of practice memorizing long lists of spelling words. Consider how much time Ananya Vinay needed to prepare for her victory, and in the movie, we see Akeelah training for the Bee with a professor. The traditionalists like SteveH like to point out that hard work and effort is the only way that anybody learns anything -- and that applies to math as much as spelling. This is why he emphasizes doing individual homework problems, as he does in the above quote.
When the year began, I wanted to think of the students in my class as mathematical Akeelahs -- especially the young sixth grade girls who may show some promise in math. But these girls often wondered, why should they work hard in math?
In many ways, the traditionalist debate is related to classroom management. The traditionalists emphasize that students need to do the homework in order to learn, but I, as a teacher, have no control over what goes on at the students' homes. On the other hand, classroom management is just that -- management of the classroom. It doesn't matter whether it's a group project day (which traditionalists despise) or the traditional lesson day, students won't learn if they are talking the whole time.
Before this year started, I wrote that while traditionalism works for younger elementary students, middle school students aren't motivated by repetitive drill and practice, which is why I favored a more progressive pedagogy in the middle years. But I saw that many of my students just talked right through the projects (having neutered the requirement that they work on them), while Garelick writes that his students enjoyed his traditional assignments and learned much from them.
The difference, of course, is management. Garelick likely has strong classroom management skills -- when he gives the students a traditional assignment, they work on it even if it's boring. I, on the other hand, need to improve my management skills -- so even when I gave my students a project, many refuse to work on it even if it's interesting.
I began this post with a discussion of the Pythagorean Theorem. Earlier this year at my old school, I wrote about my eighth grade Pythagorean Theorem lesson. It's true that confusion over my implementation of the Illinois State curriculum is to blame for the students' failure to understand it, but equally to blame is my management and the students refusing to do traditional practice problems on the theorem.
The conclusion is that maybe I am not qualified enough to say whether or traditionalist or progressive pedagogy is superior until I have the management necessary to implement them effectively. That's the only way I can bring out the Akeelahs and Ananyas in my classes. For the rest of this summer, many of my posts labeled "traditionalists" will actually be about management.
I'm returning to a summer blogging schedule, so I'll be posting only once or twice a week. As I promised earlier, I'll be returning to spherical geometry in my next post.