But this class appears to be different from the February visit. The teacher for this class is different (but the teacher from February is no longer listed on the school website.) Moreover, in second period, the students create the announcements video (to air during third period), while in February, no video was created for morning announcements.
This is the same school where I subbed for art two weeks ago, and the schedules for both the art and film teachers are similar. Both teachers have their most advanced class during second period, then the next advanced class during third period, and then the regular classes from fourth period on. Of course I won't do "A Day in the Life" for today, but I will describe some of the classes.
Naturally the two best classes of the day are the advanced classes. I actually put third period a little ahead of second period. In this class, one senior girl takes over as a student TA. Sometimes, even having a senior TA is as helpful as having another teacher in charge. She knows exactly what the students should be working on in third period, and so she redirects them back on task.
This reminds me much of the project that the coding teacher tried to do with the -- oops, ixnay on the arterchay odingcay! (Wow, so many days of subbing remind me of the year that I promised not to mention during the Eugenia Cheng reading!)
Meanwhile, In the regular classes, the students must study quietly for 15 minutes, and then I get to show them a movie of my choosing. Three of the four regular classes have TA's, but in this situation they are mostly irrelevant. This marks my only real classroom management of the day. I decide that sixth period is the best class of the day. This class is right after lunch, and so I must enforce my no restroom rule until midway through the period. (I send my attendance roster with the TA, so there's no loophole here.) But other than that, these students do work silently for the 15 minutes. A few students do try to whisper or talk in the other classes.
In seventh period two girls decide to study Geometry. They are completing a worksheet based on Lesson 2-4 of the Glencoe text, "Deductive Reasoning." This lesson fits with the logic that we're currently reading about in Eugenia Cheng's book.
The regular teacher directs me to access his iTunes account and choose a movie -- the students aren't allowed to choose. I decide to go to the Action/Adventure section and choose a random number (on my TI-83) between 1 and 91, since there are 91 movies listed there. If I choose an R-rated movie, I make another selection so that the rating is PG-13 or lower.
Here are the movies I randomly select today:
4th period: Dawn of the Planet of the Apes (PG-13)
5th period: Now You See Me (PG-13)
6th period: Chronicles of Narnia: The Lion, the Witch, and the Wardrobe (PG)
7th period: Kong: Skull Island (PG-13)
Chapter 10 of Eugenia Cheng's The Art of Logic in an Illogical World, "Where Logic Can't Help Us," begins as follows:
"Cardiac surgeon Stephen Westaby writes in Fragile Lives about the fact that if the heart stops, the brain and nervous system will be damaged in less than five minutes. So he often had five minutes or less to decide how to perform surgery."
As the title implies, there are just some situations where logic just can't help us. Clearly, surgeons don't always have enough time to do a full logical analysis. Cheng writes:
"It's important to understand how far logic gets us and where emotions have to help, rather than pretend that logic can get us all the way there. But we'll start by thinking about where logic can start to kick in, which is only after some help with finding starting points."
Cheng's first example involves language. She points out that there's no logical reason, for example, why the word for "cat" is, well, "cat." Indeed, she contrasts this with her native language:
"'Cat' in Cantonese is a high-pitched 'mow' (rhyming with cow), which does sound quite like the sound a cat makes. More than 'cat' does, anyway."
Of course, English speakers would render the sound a cat makes as "meow." Anyway, young infants learning how to speak the language must figure out for themselves what a "cat" is, since for most words there's no logical connection between the word and the object -- again, except for certain imitative words like English "cuckoo" and Chinese "mow."
The author moves on to discuss flashes of inspiration. She writes:
"You might argue whether they really exist or not, but I have definitely had moments that I would describe it like that. Perhaps it would be less melodramatic to call them an 'idea.'"
Cheng tells us that art and music are all about such flashes of inspiration. Only occasionally would a musician use "logic" to compose a song:
"Some composers, famously Bach and Schoenberg, use symmetry to transform parts of their composition into new but related music."
And indeed some Common Core Geometry texts mention this in the transformations unit -- in particular, this appears in Lesson 4-8 of the U of Chicago text (Third Edition).
According to the author, even math often begins with flashes of inspiration:
"Once we have had the idea we proceed using logic, but that part comes afterwards, when we test and exhibit the robustness of our idea."
Having shown us where logic begins, Cheng now proceeds to tell us where logic ends. She presents us the following menu:
Marinated Roast Breast of Chicken -- $18.50
Pan Fried Ostrich -- $21.00
Char Grilled Fillet Steak -- $26.95
Smoked Haddock Fish Cake -- $16.95
Roasted Summer Vegetable Tart -- $17.95
Cheng explains:
"Perhaps you've decided that you can't spend more than $20, and also you don't like fish. This logically narrows down your options to the chicken and vegetable tart, but beyond that, logic can't tell you anything."
And of course, there isn't always time to think logically in an emergency:
"If there is a fire you will, I hope, have an instinctive reaction 'I must get out!' If this is an instantaneous reaction, it probably wasn't exactly processed logically."
But Cheng tries to write it out logically anyway, as follows:
A: There is a fire.
X: I must get out.
It might go like this:
A is true (there is a fire).
A implies X (If there is a fire I must get out)
Therefore X (I must get out) by modus ponens.
\
The author also inserts additional steps in case we need to explain this to a child:
Let A = There is a fire.
Let B = I stay here.
Let C = I burn.
Then we have:
A is true.
A and B implies C.
C implies bad.
Therefore I must make sure B is false, i.e. I must get out.
On the other hand I think we can all agree that this would not be a logical deduction:
There is a fire.
I will stay here.
(But replace "fire" with "hurricane" and consider recent news events. Google: refuse to evacuate.)
Cheng continues to write about how insufficient time leads to insufficient information to make a logical decision:
"This can happen in an emergency but it can also happen in sport, where the trajectory of a ball is in principle entirely governed by physics, but we can't take all the necessary measurements in time to do the calculation before needing to hit the ball."
And she adds:
"Any consequences involving human reactions to things are almost certain to be guesses about human behavior rather than logical conclusions."
This applies to both economics and voting.
In fact, there's one very famous conundrum in which insufficient information leads directly to people making an illogical decision -- the prisoner's dilemma. David Kung, in his thirteenth lecture, discusses the prisoner's dilemma, but I only wrote a little about it in January 2016 (only because my post that day was jam-packed with other information).
So I'll try to explain it better using Cheng's book today. Alex and Sam are two prisoners, and the prosecutor asks each of them separately to testify against the other:
- If neither testifies, each get a year in jail.
- If both testify, each gets five years.
- If exactly one testifies, the snitch goes free and the other gets ten years.
Imagine you are Alex. At first you might considering staying quiet:
"What if Sam actually lands you in it and goes free? Then it would be safer for you to testify as well, to protect against that possibility."
Of course, Sam is thinking the same thing. Thus both confess and each gets five years.
Cheng points out that this is counterintuitive. And so she comes up with a clarifying example by assuming that there are more than two people involved. (I've done the same in the past with the Monty Hall paradox -- imagine that there are more than three doors.)
Cheng points out that this is counterintuitive. And so she comes up with a clarifying example by assuming that there are more than two people involved. (I've done the same in the past with the Monty Hall paradox -- imagine that there are more than three doors.)
- If no one betrays the group, each gets $500.
- If more than one betrays the group, each traitor gets $0 and everyone else loses $1000.
- If exactly one betrays the group, that traitor gets $1000 and everyone else loses $1000.
The author explains:
"If the only other person involved is your best friend then hopefully you know and trust each other enough to know that you won't denounce each other and you'll go home with $500 each. However, imagine doing this with a group of 100 strangers."
Of course, most people will be afraid that someone will betray the group, and hence they'll betray the group as well. She writes:
"This shows us that in all scenarios for other people's behavior, you get a better outcome if you betray. In game theory this is called a dominant strategy and the logic says that this is the strategy that you should take for the best outcome in either scenario."
(Kung uses another name for "dominant strategy" -- the "Nash equilibrium," named after the mathematician featured in the film A Beautiful Mind, John Nash.)
"This shows us that in all scenarios for other people's behavior, you get a better outcome if you betray. In game theory this is called a dominant strategy and the logic says that this is the strategy that you should take for the best outcome in either scenario."
(Kung uses another name for "dominant strategy" -- the "Nash equilibrium," named after the mathematician featured in the film A Beautiful Mind, John Nash.)
Cheng explains that climate agreements, such as the Paris Climate Agreement, work similarly to the Prisoner's Dilemma:
- If no one defects, there is some cost but the benefit is global.
- If everyone defects, the effect on the world could be drastic.
- If exactly one country defects, then that country benefits the most -- they reap the benefits of conservation without any of the costs.
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next two weeks and skip all posts that have the "Eugenia Cheng" label.
(This is even though Cheng never names the country which would defect from the agreement.)
"Now according to the logic of the prisoner's dilemma, we should expect everyone to defect. It is perhaps heartening that this isn't universally the case."
As the author explains, it's all about trust:
"I think what this is actually saying is that if a community is infused with enough trust to act as a coherent whole rather than as a collection of selfish individuals, then the logic of the situation changes, and becomes one that can benefit everyone rather than everyone suffering as a result of a few selfish individuals."
This is what little I wrote in January 2016 about the Prisoner's Dilemma (including a link):
Prisoner's Dilemma
I claim that the Prisoner's Dilemma is one of the most commonly played games by the teenagers in our classes -- except it's the Lover's Dilemma:
And Cheng concludes the chapter similarly:
"We have seen that logic cannot explain and decide everything in the world, so we are going to have to do something when it runs out. We should not pretend that those non-logical things are logical, but we should also not assume that those non-logical things are bad."
I've decided to add the "traditionalists" label to this thread. This is because traditionalist Barry Garelick posted today. Even though this post isn't directly about the Barbara Oakley article, Garelick mentions Oakley in passing, so this post can be considered part of the extended Oakley discussion:
https://traditionalmath.wordpress.com/2018/09/29/3099/
(Again, notice that this post is dated September 29th, yet it was visible on my computer in time for my own post dated the 28th. And this is despite Garelick's living right here in California. I'm wondering whether the Wordpress clock is now suddenly defaulting to Greenwich.)
And Garelick responds to Emily Parnell (the KC Star columnist) by citing Oakley (and linking to her infamous article):
Barbara Oakley, a professor of engineering at Oakland University in Michigan, wrote a book called “Learning how to Learn” in which she describes techniques one can use to succeed in difficult subjects. She does not hold memorization in disdain, nor learning facts, nor practice. Recently she wrote an op-ed that appeared in the New York Times about the value of practice and memorization in becoming proficient in math–and was castigated in comments that followed as well as in blogs for what was characterized as narrow-mindedness and resistance to innovation in education. The criticisms bore a resemblance to the Kansas City Star columnist’s view of having to deal with the great unwashed.
Pot, kettle, black -- to me the ones who don't want to deal with "the great unwashed" are, of course, the traditionalists. Which side, after all, is promoting tracking to keep "the great unwashed" out of the highest tracks?
Once again, I agree that traditional memorization and p-sets can be beneficial, provided that the students actually do the memorization or the p-sets. Even professional mathematician Eugenia Cheng doesn't like to memorize -- how much less, then, do the students in our classes like to memorize?
I'm glad that I labeled this post "traditionalists," because today's Lesson 3-2 will contain an activity that's definitely an alternative to traditional p-sets.
Lesson 3-2 of the U of Chicago text is called "Types of Angles." In this chapter, students learn about zero, acute, right, obtuse, straight, complementary, supplementary, adjacent, and vertical angles.
In the new Third Edition of the text, this actually corresponds to Lesson 3-3. But the definitions of acute, right, and obtuse are actually combined with yesterday's Lesson 3-1. Only the last four definitions (mainly adjacent and vertical angles) remain in the new Lesson 3-3.
(By the way, Lesson 1-4 of the Glencoe text, which I covered in the special ed class that I subbed for last Friday, is most analogous to Lesson 3-1 of the U of Chicago Third Edition. Last week's students learned all about angles, including acute, right, and obtuse angles as well as bisectors, in that lesson.)
In between these, in the new Lesson 3-2, are rotations. I've mentioned before how strange is this that both the old and new editions define a rotation as the composite of two reflections in intersecting lines, yet the new edition has a section on rotations before defining reflections! The U of Chicago most likely placed this section here so that in introducing rotations, students become more familiar with angles. (Again, I point out that Hung-Hsi Wu of Berkeley, in his recommendations for Common Core Geometry, teaches rotations before reflections, but he defines rotations differently. His lessons have nothing to do with the new Lesson 3-2.)
In fact, Jackie Stone -- a Blaugust participant last month -- also introduces rotations when teaching her students about angles, just like the U of Chicago text:
https://mathedjax.wordpress.com/2017/08/23/what-is-that-how-do-you-use-it-blaugust/
What was intended to be a five minute “review” of these skills to launch into the real lesson activity of the day turned into a much more in depth “teaching” of how to use this tool. Although they might NEVER use a protractor outside of my class again I do find the task of measuring something using a tool useful. The task also spoke to the CCSS Math Practice Standards of attending to precision and using tools strategically. It is so challenging (especially at the beginning of the year) to determine what are appropriate scaffolds to help students work on a task. Moving forward, I plan to assume less which is actually a good thing because then we can talk about refined meanings of things. For instance, because of their lack of background we were able to really talk about that the measurement in degrees was actually a measurement of a rotation. I think next year my approach might be different.
This is what I wrote last year on this lesson:
Section 3-2 of the U of Chicago text discusses the various types of angles. It covers both the classification of angles by their measures -- acute, right, and obtuse -- as well as related angles such as vertical angles and those that form a linear pair. Complementary and supplementary angles also appear in this lesson.
As usual, I like to look for other online resources by other geometry teachers. And I found the following blog post from Lisa Bejarano, a high school geometry teacher from Colorado who calls herself the "crazy math teacher lady":
http://crazymathteacherlady.wordpress.com/2014/08/23/
Bejarano writes that in her geometry class, she "starts with students defining many key terms so that we can use this vocabulary as we work through the content." And this current lesson, Section 3-2, contains many vocabulary terms. So this lesson is the perfect time to follow Bejarano's suggestion.
Now two textbooks are mentioned at the above link. One is published by Kagan, and unfortunately I'm not familiar with this book. But I definitely know about the other one -- indeed, I actually mentioned it in my very last post -- Michael Serra's Discovering Geometry.
The cornerstone to Bejarano's lesson is the concept of a Frayer model. Named after the late 20th-century Wisconsin educator Dorothy Frayer, the model directs students to distinguish between the examples and the non-examples of a vocabulary word.
Strictly speaking, I included a Frayer-like model in last week's Section 2-7. This is because the U of Chicago text places a strong emphasis on what is and isn't a polygon. But Serra's text uses examples and non-examples for many terms in its Section 2-3, which corresponds roughly to Section 3-2 in the U of Chicago text.
Here are steps used in Bejarano's implementation of this lesson:
1. I used the widget example from Discovering Geometry (chapter 1). It shows strange blobs and says “these are widgets”, then there is another group of strange blobs and it says “these are not widgets”. I have students define widgets in their groups. Then they read their definition and we try to draw a counterexample. Then we discussed what makes a good definition and we were ready to go!
(Note: Bejarano writes that she found the "widget" example in Chapter 1 of Serra's text, but I found it in Section 2-3 of my copy. Of course, I don't know how old my text is compared to Bejarano's -- mine is the Second Edition of the text, dated 1997. Hers could be the Third Edition or later.)
Notice that "widget" here is a non-mathematical (and indeed, a hypothetical) example that Bejarano uses as an Anticipatory Set. As we've seen throughout Chapter 2, many texts use non-mathematical examples to motivate the students.
3. Students worked in small groups with their 3 terms copying the examples & non-examples, then writing good definitions for each term. I set a timer for 10 minutes. (Yes, I'm skipping her #2.)
Now I don't necessarily want to include this as a group activity the way Bejarano does here. After all, I included the Daffynition Game in my last post as a group project [...]
And let's stop right here, because today's an activity day, and I haven't posted that Daffynition game yet this year -- so let me post it today! This is what I wrote last year about the activity. (Oh, and if you thought we were done with Serra's text after finishing Chapter 0 last month, think again!):
It's tough trying to find activities that fit this chapter. One source that I like to use for activities is Michael Serra's textbook, Discovering Geometry. Just like most other geometry texts, in Chapter 2 he discusses the concept of definition (Section 2-3, "What is a Widget?") Then the text introduces a project, "the Daffynition Game," where the students take turns making up definitions to real words.
A few comments I'd like to make about the game as introduced by Serra. Step 3 reads:
3. To begin a round, the selector finds a strange new word in the dictionary. It must be a word that nobody in the group knows. (If you know the word, you should say so. The selector should then pick a new word.)
The problem is that this depends on the honor system -- how do we know that a student who knows the word will actually admit it? Rather than depend on the students' honesty, why not make knowing the word an actual strategic move? That student will then earn a point for knowing the word -- and the student can still make up a fake definition in order to earn even more points? This means that the selector must be very careful to choose a word that isn't in the dictionary.
Another question is, what affect would this project have on the English learners? I'd say that this would be a great project for them, since they can learn both English and math in this lesson. But English learners might be at a disadvantage in this game, since if they choose a word that a native English speaker knows, the English speaker would earn a point (since I'm not counting on the honor system here). One debate that always comes up in a group activity is whether to group homogeneously or heterogeneously. For this project, it may be best to group homogeneously, but by English, rather than mathematical, ability.
Finally, this project requires students to look up words in a dictionary -- but what dictionary? I threw out the problems in earlier sections that depended on the availability of a dictionary. Perhaps the night before this activity, part of the homework assignment could be to look up a word in the dictionary and write down its definition -- but that assumes that the students will actually do the homework, and besides, there's no guarantee that the students have access to a dictionary at home (or online) either.
My solution is for the teacher to have enough index cards with words and definitions on them. Therefore the selector chooses an index card, not a word from the dictionary. Indeed, the teacher can give an index card to each student even before dividing the class into groups! But the selector should still follow the other steps as originally written in the Serra text.
OK, so let me post the worksheets. I decided to post only the first page of Lesson 3-2 (Lisa Bejarano's lesson) and then go directly to the Daffynition Game.
In cutting Bejarano's second page, I'm dropping some terms that don't appear until later in Chapter 3, but I also dropped "vertical angles" and "angle bisector," which do appear in Lesson 3-2. Teachers can either make sure to write those two dropped terms on index cards in the Daffynition Game, or else go full Bejarano and use the Frayer models as a full group project, just as the Colorado teacher originally intended.
[2018 update: Jackie Stone still posts weekly on her blog. Lisa Bejarano hasn't posted since April.]
(This is even though Cheng never names the country which would defect from the agreement.)
"Now according to the logic of the prisoner's dilemma, we should expect everyone to defect. It is perhaps heartening that this isn't universally the case."
As the author explains, it's all about trust:
"I think what this is actually saying is that if a community is infused with enough trust to act as a coherent whole rather than as a collection of selfish individuals, then the logic of the situation changes, and becomes one that can benefit everyone rather than everyone suffering as a result of a few selfish individuals."
This is what little I wrote in January 2016 about the Prisoner's Dilemma (including a link):
Prisoner's Dilemma
I claim that the Prisoner's Dilemma is one of the most commonly played games by the teenagers in our classes -- except it's the Lover's Dilemma:
- If I like my friend, and my friend likes me back, then we can start a relationship.
- If I don't like my friend that way, nor does my friend like me, then we don't start a relationship, but at least we can remain friends.
- If I like my friend, but my friend doesn't like me back, then our friendship is ruined. My (former) friend gets to laugh at me -- the best of all situations for my (ex-)friend, the worst for me.
And Cheng concludes the chapter similarly:
"We have seen that logic cannot explain and decide everything in the world, so we are going to have to do something when it runs out. We should not pretend that those non-logical things are logical, but we should also not assume that those non-logical things are bad."
I've decided to add the "traditionalists" label to this thread. This is because traditionalist Barry Garelick posted today. Even though this post isn't directly about the Barbara Oakley article, Garelick mentions Oakley in passing, so this post can be considered part of the extended Oakley discussion:
https://traditionalmath.wordpress.com/2018/09/29/3099/
(Again, notice that this post is dated September 29th, yet it was visible on my computer in time for my own post dated the 28th. And this is despite Garelick's living right here in California. I'm wondering whether the Wordpress clock is now suddenly defaulting to Greenwich.)
The Kansas City Star published a column in defense of Common Core’s math standards, containing the usual rhetoric–to wit:
I was recently part of a conversation about education. It was a social media conversation, intended to bash the alternative strategies of teaching math.The strategies have been caught up in the term “common core,” but are actually teaching methods designed to help kids reach common standards.I offered an alternative viewpoint to the woman’s outrage. I was once told by the faculty at my kids’ school that these learning strategies aren’t designed just to teach the material, they teach the kids to learn. How to analyze. How to understand why math works, not just how to solve a problem.
And Garelick responds to Emily Parnell (the KC Star columnist) by citing Oakley (and linking to her infamous article):
Barbara Oakley, a professor of engineering at Oakland University in Michigan, wrote a book called “Learning how to Learn” in which she describes techniques one can use to succeed in difficult subjects. She does not hold memorization in disdain, nor learning facts, nor practice. Recently she wrote an op-ed that appeared in the New York Times about the value of practice and memorization in becoming proficient in math–and was castigated in comments that followed as well as in blogs for what was characterized as narrow-mindedness and resistance to innovation in education. The criticisms bore a resemblance to the Kansas City Star columnist’s view of having to deal with the great unwashed.
Pot, kettle, black -- to me the ones who don't want to deal with "the great unwashed" are, of course, the traditionalists. Which side, after all, is promoting tracking to keep "the great unwashed" out of the highest tracks?
Once again, I agree that traditional memorization and p-sets can be beneficial, provided that the students actually do the memorization or the p-sets. Even professional mathematician Eugenia Cheng doesn't like to memorize -- how much less, then, do the students in our classes like to memorize?
I'm glad that I labeled this post "traditionalists," because today's Lesson 3-2 will contain an activity that's definitely an alternative to traditional p-sets.
Lesson 3-2 of the U of Chicago text is called "Types of Angles." In this chapter, students learn about zero, acute, right, obtuse, straight, complementary, supplementary, adjacent, and vertical angles.
In the new Third Edition of the text, this actually corresponds to Lesson 3-3. But the definitions of acute, right, and obtuse are actually combined with yesterday's Lesson 3-1. Only the last four definitions (mainly adjacent and vertical angles) remain in the new Lesson 3-3.
(By the way, Lesson 1-4 of the Glencoe text, which I covered in the special ed class that I subbed for last Friday, is most analogous to Lesson 3-1 of the U of Chicago Third Edition. Last week's students learned all about angles, including acute, right, and obtuse angles as well as bisectors, in that lesson.)
In between these, in the new Lesson 3-2, are rotations. I've mentioned before how strange is this that both the old and new editions define a rotation as the composite of two reflections in intersecting lines, yet the new edition has a section on rotations before defining reflections! The U of Chicago most likely placed this section here so that in introducing rotations, students become more familiar with angles. (Again, I point out that Hung-Hsi Wu of Berkeley, in his recommendations for Common Core Geometry, teaches rotations before reflections, but he defines rotations differently. His lessons have nothing to do with the new Lesson 3-2.)
In fact, Jackie Stone -- a Blaugust participant last month -- also introduces rotations when teaching her students about angles, just like the U of Chicago text:
https://mathedjax.wordpress.com/2017/08/23/what-is-that-how-do-you-use-it-blaugust/
What was intended to be a five minute “review” of these skills to launch into the real lesson activity of the day turned into a much more in depth “teaching” of how to use this tool. Although they might NEVER use a protractor outside of my class again I do find the task of measuring something using a tool useful. The task also spoke to the CCSS Math Practice Standards of attending to precision and using tools strategically. It is so challenging (especially at the beginning of the year) to determine what are appropriate scaffolds to help students work on a task. Moving forward, I plan to assume less which is actually a good thing because then we can talk about refined meanings of things. For instance, because of their lack of background we were able to really talk about that the measurement in degrees was actually a measurement of a rotation. I think next year my approach might be different.
This is what I wrote last year on this lesson:
Section 3-2 of the U of Chicago text discusses the various types of angles. It covers both the classification of angles by their measures -- acute, right, and obtuse -- as well as related angles such as vertical angles and those that form a linear pair. Complementary and supplementary angles also appear in this lesson.
As usual, I like to look for other online resources by other geometry teachers. And I found the following blog post from Lisa Bejarano, a high school geometry teacher from Colorado who calls herself the "crazy math teacher lady":
http://crazymathteacherlady.wordpress.com/2014/08/23/
Bejarano writes that in her geometry class, she "starts with students defining many key terms so that we can use this vocabulary as we work through the content." And this current lesson, Section 3-2, contains many vocabulary terms. So this lesson is the perfect time to follow Bejarano's suggestion.
Now two textbooks are mentioned at the above link. One is published by Kagan, and unfortunately I'm not familiar with this book. But I definitely know about the other one -- indeed, I actually mentioned it in my very last post -- Michael Serra's Discovering Geometry.
The cornerstone to Bejarano's lesson is the concept of a Frayer model. Named after the late 20th-century Wisconsin educator Dorothy Frayer, the model directs students to distinguish between the examples and the non-examples of a vocabulary word.
Strictly speaking, I included a Frayer-like model in last week's Section 2-7. This is because the U of Chicago text places a strong emphasis on what is and isn't a polygon. But Serra's text uses examples and non-examples for many terms in its Section 2-3, which corresponds roughly to Section 3-2 in the U of Chicago text.
Here are steps used in Bejarano's implementation of this lesson:
1. I used the widget example from Discovering Geometry (chapter 1). It shows strange blobs and says “these are widgets”, then there is another group of strange blobs and it says “these are not widgets”. I have students define widgets in their groups. Then they read their definition and we try to draw a counterexample. Then we discussed what makes a good definition and we were ready to go!
(Note: Bejarano writes that she found the "widget" example in Chapter 1 of Serra's text, but I found it in Section 2-3 of my copy. Of course, I don't know how old my text is compared to Bejarano's -- mine is the Second Edition of the text, dated 1997. Hers could be the Third Edition or later.)
Notice that "widget" here is a non-mathematical (and indeed, a hypothetical) example that Bejarano uses as an Anticipatory Set. As we've seen throughout Chapter 2, many texts use non-mathematical examples to motivate the students.
3. Students worked in small groups with their 3 terms copying the examples & non-examples, then writing good definitions for each term. I set a timer for 10 minutes. (Yes, I'm skipping her #2.)
Now I don't necessarily want to include this as a group activity the way Bejarano does here. After all, I included the Daffynition Game in my last post as a group project [...]
And let's stop right here, because today's an activity day, and I haven't posted that Daffynition game yet this year -- so let me post it today! This is what I wrote last year about the activity. (Oh, and if you thought we were done with Serra's text after finishing Chapter 0 last month, think again!):
It's tough trying to find activities that fit this chapter. One source that I like to use for activities is Michael Serra's textbook, Discovering Geometry. Just like most other geometry texts, in Chapter 2 he discusses the concept of definition (Section 2-3, "What is a Widget?") Then the text introduces a project, "the Daffynition Game," where the students take turns making up definitions to real words.
A few comments I'd like to make about the game as introduced by Serra. Step 3 reads:
3. To begin a round, the selector finds a strange new word in the dictionary. It must be a word that nobody in the group knows. (If you know the word, you should say so. The selector should then pick a new word.)
The problem is that this depends on the honor system -- how do we know that a student who knows the word will actually admit it? Rather than depend on the students' honesty, why not make knowing the word an actual strategic move? That student will then earn a point for knowing the word -- and the student can still make up a fake definition in order to earn even more points? This means that the selector must be very careful to choose a word that isn't in the dictionary.
Another question is, what affect would this project have on the English learners? I'd say that this would be a great project for them, since they can learn both English and math in this lesson. But English learners might be at a disadvantage in this game, since if they choose a word that a native English speaker knows, the English speaker would earn a point (since I'm not counting on the honor system here). One debate that always comes up in a group activity is whether to group homogeneously or heterogeneously. For this project, it may be best to group homogeneously, but by English, rather than mathematical, ability.
Finally, this project requires students to look up words in a dictionary -- but what dictionary? I threw out the problems in earlier sections that depended on the availability of a dictionary. Perhaps the night before this activity, part of the homework assignment could be to look up a word in the dictionary and write down its definition -- but that assumes that the students will actually do the homework, and besides, there's no guarantee that the students have access to a dictionary at home (or online) either.
My solution is for the teacher to have enough index cards with words and definitions on them. Therefore the selector chooses an index card, not a word from the dictionary. Indeed, the teacher can give an index card to each student even before dividing the class into groups! But the selector should still follow the other steps as originally written in the Serra text.
OK, so let me post the worksheets. I decided to post only the first page of Lesson 3-2 (Lisa Bejarano's lesson) and then go directly to the Daffynition Game.
In cutting Bejarano's second page, I'm dropping some terms that don't appear until later in Chapter 3, but I also dropped "vertical angles" and "angle bisector," which do appear in Lesson 3-2. Teachers can either make sure to write those two dropped terms on index cards in the Daffynition Game, or else go full Bejarano and use the Frayer models as a full group project, just as the Colorado teacher originally intended.
[2018 update: Jackie Stone still posts weekly on her blog. Lisa Bejarano hasn't posted since April.]
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