I. The concept of the derivative can be generalized to apply to any two dependent quantities.
II. The derivative can be defined and interpreted in many different settings.
III. Suppose a belt is put around the equator of the earth (about 25,000 miles). Suppose we increase the belt by 6 feet so it hovers above the Earth. How far away from Earth will it be?
IV. We have now seen the importance of the derivative.
Starbird begins by showing us various figures and their area functions. In each case, the area of the figure is dependent on its dimension.
area of a square = x^2
area of a circle = pi r^2
In the case of the square, he considers changing its side length from x to x + delta-x. The rate of change of its area is thus ((x + delta-x)^2 - x^2)/delta-x. The professor reminds us that this rate of change is just like a derivative, and that we can show geometrically that it approaches 2x, the length of two sides of the square, as delta-x approaches zero.
For the circle, Starbird gives specific values for its radius -- he shows what happens when we change its radius from 3 to 3.01. The rate of change of its area is thus (A(3.01) - A(3))/0.01. He tells us that A(3.01) - A(3) is approximately the circumference * 0.01, and so (A(3.01) - A(3))/0.01 is almost exactly the circumference of the circle, or 6pi. In fact, A'(r) = d/dr(pi r^2) = 2pi. For those who have wondered whether the derivative of the area being the circumference is a coincidence, the professor has shown us that it's not a coincidence, because we can prove it!
Starbird now jumps to examples from economics. Every product has its supply and demand curves, which map the quantity in terms of price. If the derivative of the demand curve is steep (negative), or if the derivative of the supply curve is steep (positive), then the market is more volatile and slight price changes will make huge differences in the quantity demanded or supplied.
The professor now moves on to the belt question listed in the outline above -- this is, as it turns out, a classic riddle. He uses the derivative of the circumference function to solve it -- since C(r) = 2pi r, its derivative is just the slope of the line, C'(r) = 2pi. Thus a 1-foot change in the radius leads to a 2pi or 6.28-foot change in the circumference -- and this is true independently of the radius. Thus adding 6 feet to the length of the belt surprisingly causes it to hover nearly an entire foot above the earth!
Today is the Chapter 8 Test. It is also Day 90, the mathematical midpoint of the year. As we already know, most Early Start schools don't actually begin early enough in August to have a true semester of 90 days before winter break.
Three years ago at the old charter school, the mathematical second semester was when SBAC Prep began. Indeed, the bell schedule was changed so that SBAC Prep would replace most of P.E. time. Since we'd lacked a real conference period anyway, things truly became tough for me once SBAC Prep began.
Oh, and before I write about the test, you might point out that today's a lousy day for the test -- especially since I state that my calendar is based on the district where I sub today. Not only is today Monday, but at least one district school has a minimum day today. In other words, not only am I expecting students to study over the weekend for the test, but now they have only 35 minutes per class to finish the 20-question test.
You might ask, why was today chosen for the "once a month" minimum day schedule? Well, there are only two Mondays in January to choose from -- last week the students didn't return from winter break until Tuesday, and next Monday is MLK Day. Usually, the minimum day is as early in the month as possible, and so today's it.
And so I can't help it that blindly following the digit pattern unfortunately makes the test land on the worst possible day, a minimum day Monday.
When this happened last year, I wondered whether this would a good day for the test after all, and so I considered making changes for this year -- perhaps reducing today's test to a "quiz" instead. At first, I thought that I should obviously make the change -- but then I realized something -- that minimum day only occurs at one of the district schools.
I also mentioned that most of the schools in this district are on a block schedule, so a class like Geometry would meet only every other day. Yet for the entire history of this blog, I posted everyday as if Geometry meets everyday, even though I claim to be following this district calendar. You might argue, if I were truly adhering to my district calendar, I should only be posting thrice a week.
Well, there is one loophole here -- one of the schools in the district doesn't have a block schedule. So I can actually claim that this blog does not merely follow the calendar for the entire district, but only the calendar for a single school -- the one where class meets everyday. And lo and behold -- at that particular school, today is not a minimum day. (That school does have late days, but they work completely differently from the rest of the district.) Thus I'd be justified in keeping this a Chapter 8 Test, since the schedule is based on the school where today isn't a minimum day.
(Note: All of this schedule business refers to my old district. The blog calendar has nothing to do with my new district, even though it's the district where I receive most of my subbing calls.)
Therefore, what's my final decision -- quiz or test? Well, recall that on Friday, I sub for a teacher with similar dilemma -- she couldn't decide whether to call her assessment a "quiz" or a "test." In the end, she compromised -- tomorrow, her students will be taking the "Chapter 5 Quest."
And voila -- the solution falls neatly into my lap. Today I'm posting the "Chapter 8 Quest."
Here's how today's "quest" will work. It will contain ten questions. It consists of last year's Questions 11-20, mainly because last year, I explained that this years Questions 9-10 are no longer valid. (So this year's "quest" contains more than half of the 18 valid questions from the test. If I had wanted this to be a "quiz," I would have included just eight questions, or fewer than half of last year's.)
This is what I wrote last year about today's quest:
11. Choice (a). The triangles have the same base and height, therefore the same area.Let's worry about the Chapter 8 Quest that I'm posting today on the blog. Here are the answers:
12. 3s^2.
13. 13 feet.
14. 6 minutes.
15. 780,000 square feet.
16. 133 square feet.
17. 1/4 or .25. Probability is a tricky topic -- the U of Chicago assumes that the students already know something about probability. Then again, it should be obvious that the smaller square is 1/4 of the larger square.
18. 4,000 square units.
19. 13.5 square units.
20. a(b - c) or ab - ac square units.
I've decided to add the "music" label to today's post and discuss the Mocha music conversion that I promised you last week. Let's convert "Under the Sea," from Disney's The Little Mermaid.
Let's begin by noting the rhythm of this song. The song is in common 4/4 time and there are no sixteenth notes -- just eighth notes, quarter notes, dotted quarter notes, half notes, and whole notes.
One important part of this song is syncopation -- in many bars, there is a quarter note on beat 2 (or an eighth note on the second half of beat 2) tied to an eighth note on beat 3. Of course, these could be written as a dotted quarter note (respectively a quarter note), but the third beat in 4/4 is so important that most scores show the tied notes instead. In Mocha, we'll just represent these notes as if they were indeed a dotted quarter note of length 6 (respectively a quarter note of length 4), since Mocha doesn't care about syncopation.
My old random music generator from October 2018 doesn't allow for syncopation. This is because the random generator would sometimes create rhythms that don't occur in real music -- for example, it might take a dotted half note and divide it into four equal notes. Each of these would technically be a dotted eighth note (and would truly count as syncopation), but four consecutive dotted eighth notes never appears in a real 4/4 song. Indeed, a dotted eighth note must always be adjacent to a sixteenth note or rest. It might be possible for four straight dotted eighth notes to appear in a song with a strange rhythm such as 6/16 (or even 12/16), but not 4/4. Thus I simply dropped syncopation from the generator altogether.
But this isn't a randomly generated song -- it's an actual song from a movie. Therefore we're free to include all the syncopated notes we want -- and we will. (It might be possible to modify the random generator to make some syncopated rhythms -- in exchange, we'd drop its ability to include sixteenth and dotted eighth notes so that the generated rhythms will be similar to today's song.)
The general pattern in this song is a quarter rest, then a syncopated bar, then a non-syncopated bar, before the pattern repeats with another quarter rest.
OK, so that's enough about the rhythm -- let's move on to the scale. My score has the song written in the key of C major. But in the actual movie, Sebastian the crab begins the song in Bb major before modulating up to C for the last verse. I wonder whether my score keeps the entire song in C or does it modulate up to a higher key (such as D) for the last verse. I'll never know, because in the classroom last Thursday, I found only the first page of the score. It ends just after the second verse -- we never reach the key change (nor the bridge, when Sebastian invites the other sea creatures to sing).
We've generally used 18EDL for converting songs in major keys, even though in the past, I once suggested that either 18EDL or 20EDL will work. The difference is that, while 20EDL contains a just major third (yellow 5/4), it lacks a usable fifth. On the other hand, 18EDL has a perfect fifth (white 3/2), while its third is a passable supermajor third (red 9/7). The fundamental key for 18EDL in Mocha is the key of white D (N=1 or N=8) -- we can convert this to either white C (N=9) to match the score, or green Bb (N=10) to match Sebastian.
Since we're discussing three different keys here, let's avoid mentioning specific notes such as D, since this can be either the root (key of D), second (key of C), or third (key of Bb). Instead, we'll only refer to the interval names (second, third, fourth, and so on).
We recall that 18EDL has, in addition to the tonic note, just (white) seconds and fifths, while its third is supermajor (red). On the other hand, its fourths, sixths, and sevenths differ significantly from a just major scale -- the fourth is about a quarter-tone sharp, the sixth is about a quarter-tone flat, and the seventh is about a semitone flat.
But one thing about "Under the Sea" is that fourths and sevenths aren't emphasized -- in fact, the tune is almost pentatonic (1-2-3-5-6-1). There are a few sevenths, but they aren't stressed. Meanwhile, the fourth doesn't appear in the melody at all. (There is a fourth in the intro before Sebastian sings, and the chord line does show a IV chord, but no fourth in the melody that the crab sings.) Thus only the sixth will sound significantly different from the just major scale.
The lowest note in the melody is the second, while the highest is the third of the next octave (which we may call the tenth for simplicity). Here are all the notes used in our melody:
Degree Ratio Interval (where root=18)
16 9/8 white 2nd
14 9/7 red 3rd
12 3/2 white 5th
11 18/11 lavender 6th
10 9/5 green 7th
9 2/1 white octave
8 9/4 white 9th
7 18/7 red 10th
Since the root at 18 doesn't appear, we could arguably label this as a song in 16EDL rather than 18EDL, but it's ultimately based on our 18EDL major scale. Since Degree 13 doesn't appear, this is officially an 11-limit melody, rather than 13-limit.
In coding this in Mocha, let's keep this song as simple as possible. I decided to have an empty FOR loop of length 400 represent the quarter rest every two bars. The code is set up so that whenever it encounters either eight consecutive notes (which adds up to seven beats) or a long note (half or whole), then the next bar will begin with the quarter rest.
But there are two instances during the refrain (chorus) where there is no quarter rest, but an actual quarter note instead. Yet I chose to keep the code as it is -- in both cases, the correct note listed in the score is the sixth. Since our lavender 6th can be quite dissonant in this position, I decided just to keep a rest in that position (but the performer should still sing a syllable during that beat).
Here's the resulting song:
https://www.haplessgenius.com/mocha/
10 FOR V=1 TO 3
20 IF V=3 THEN N=9 ELSE N=10
30 FOR L=1 TO 16
40 FOR I=1 TO 400:NEXT I
50 FOR X=1 TO 8
60 READ A,T
70 SOUND 261-N*A,T
80 IF T>=8 THEN X=8
90 NEXT X,L
100 RESTORE
110 NEXT V
120 DATA 9,2,9,4,9,4,9,2,10,4,8,4,9,4,12,4
130 DATA 14,2,12,4,12,4,14,2,16,4,12,4,14,8
140 DATA 9,2,9,4,9,4,9,2,10,4,8,4,9,4,12,4
150 DATA 14,2,12,4,12,4,14,2,16,4,12,4,14,8
160 DATA 11,2,9,4,9,4,11,2,12,4,9,4,12,4,9,4
170 DATA 10,2,8,4,8,4,9,2,7,4,8,4,9,8
180 DATA 11,2,9,4,9,4,11,2,12,4,9,4,12,4,9,4
190 DATA 10,2,8,4,8,4,9,2,7,4,8,4,9,8
200 DATA 7,6,8,2,9,4,11,16
210 DATA 7,6,8,2,9,4,8,16
220 DATA 7,6,8,2,9,4,11,4,12,2,7,4,8,2,9,4
230 DATA 12,2,7,4,8,2,9,4,7,16
240 DATA 7,4,8,6,9,2,11,4,9,2,9,4,11,2,9,4
250 DATA 8,4,8,6,9,2,7,4,8,2,9,4,7,2,9,4
260 DATA 7,6,8,2,9,4,11,4,12,2,7,4,8,2,9,4
270 DATA 12,2,7,4,8,2,9,4,9,16
As usual, click on the Sound button before you RUN the program.
This song will repeat three times. The first two times it plays N=10 (key of green Bb) and modulates up to N=9 (key of white C) for the third verse, to simulate the way that Sebastian sings it. Actually, when the crab moves up to C major, he doesn't repeat the "verse" part but only the "refrain/chorus" part (starting with the lyrics "Under the Sea"), but we repeat the whole song in C anyway (to make up for the lack of a bridge).
Again, this would be one of those times that I wish that Mocha BASIC resembled Atari BASIC so that we can RESTORE different DATA lines at different points of the song. This will allow us to simulate the song more closely. For example, after the second chorus, the last line "Under the Sea" leads to a repeat of the chorus -- that is, the last note in Line 270 should be Degree 11 instead of Degree 9, and then we repeat Lines 210-270 before moving on to the bridge. Oh, and the final chorus, Sebastian adds several bars before singing the final line "Under the Sea." Again, this would be easier if we could RESTORE different DATA lines to reflect this, but we can't.
Also, sometimes I wish we could just use a special Degree such as 0 to represent a rest. If the value of A read in Line 60 is 0, then the formula 261 - N * A becomes 261, the special bridge degree. There is no Sound 261 (as there shouldn't, as this would be like strumming a string of length 0), and so I'd like the notation SOUND 261,4 to produce a rest of the correct length. But this doesn't work -- Sound 261 would produce an error message. Instead, we must use the empty FOR loop to generate a rest -- and while length 400 is approximately the quarter rest, we need it to be exact if we wish to add harmony to this song.
How can we add an accompaniment (in a second window, of course)? My score is written in four parts, presumably to represent the soprano, alto, tenor, and bass parts. It's traditional for the melody to be written in the highest part or soprano (even though the crab Sebastian is at best a tenor). So we could use one of the lower parts for our accompaniment here.
But I won't. The bass part as written in the score is complex in some parts. We already had to write 27 lines of code to represent the melody, and I don't want to make you enter 27 more lines just to add harmony here.
Instead, I'll use the chord line for our harmony -- we just use the root of the chord. For example, our chord line begins with the tonic and dominant chords, C-G7-C-G7-C-G7-C-G7-C. Since we're writing in three different keys, let's use Roman numerals instead -- I-V7-I-V7-I-V7-I-V7-I. In our accompaniment, we'll represent these by the tonic and fifth notes.
The F chord (IV) appears often, as do the Am (vi) and D7 chords once each in the refrain. Officially, we should write D7 in Roman numerals as V7/V, but as far as we care here, it's just a second. (There is also a C7 chord for V7/IV, but again we ignore the 7 here.) So the only notes we need in our accompaniment here are the tonic, second, fourth, fifth, and sixth.
Since these are bass notes, we need to use Degrees that are below the ones used in the melody. We can use Degree 18 for the tonic since the lowest note of the melody is Degree 16. For the fifth, we go one octave lower, Degree 24. The second can be played at Degree 16, since the only notes played over D7 in the score are in the higher octave.
The fourth is an interesting note here. The fourth of 18EDL is at Degree 13, which we can drop one octave to Degree 26. But the thu fourth is dissonant here. Fortunately, in the lower octave, we also have Degree 27, which produces a white fourth that's more consonant with the other white notes in the melody (most notably the white octave). Indeed, we'll often see that there are extra notes in the bass octave (of odd Degree) that aren't available in the melody octave.
But there's just one problem here -- in the key of Bb (N=10), Degree 27 would be below the lowest playable note in Mocha (Sound 1 = Degree 260). There are two possibilities here -- one is to keep Degree 26 (which is Sound 1 = Degree 260) even though this is the dissonant thu fourth. The other is to change Line 20 so that we modulate from C to D rather than from Bb to C. This means that it no longer matches Sebastian's keys, but at least the consonant Degree 27 is available for N=9 and N=8.
In either case, we should use Degree 21 instead of Degree 22 for the sixth in the bass line. Degree 21 is the red sixth, which is more consonant with the red tenth in the melody.
So let's write the bass line now. We open up a second window and type in the following:
5 SOUND 261-18*10,8:SOUND 261-24*10,8
10 FOR V=1 TO 3
20 IF V=3 THEN N=9 ELSE N=10
50 FOR X=1 TO 8
60 READ A,T
70 SOUND 261-N*A,T
90 NEXT X
100 RESTORE
110 NEXT V
120 DATA 18,16,24,8,18,24,24,8
130 DATA 18,24,24,8,18,24,24,8,18,8
140 DATA 26,16,18,16,24,16,18,16
150 DATA 26,16,18,16,24,16,18,32
160 DATA 26,16,18,16,24,16,18,16
170 DATA 26,16,24,16,18,32
180 DATA 26,16,24,16,21,16,16,16
190 DATA 26,16,24,16,18,16
Don't forget to click the Sound box in this second window. Just as I did about a week ago, we can have the word RUN already typed in both windows. In this code, Line 5 is there to play two introductory notes (the tonic and 5th) at a half note each. The start of the third note is when the melody needs to begin, so we switch to the other melody window and press ENTER just as the third note is about to begin. Don't be alarmed that the melody doesn't start right away -- it does begin with a quarter rest after all.
Here are some adjustments to be made. First, if the quarter rests aren't timed properly, we can adjust 400 (Line 40 in the melody window) so that it's timed better. If this is too difficult, we can replace Line 40 so that a quarter note (timed exactly) is played there instead of a quarter rest:
40 SOUND 261-N*12,4
Here I chose Degree 12, the fifth, to play here. I could have played the sixth instead, since this is the actual correct note in some of the bars in the chorus. (Remember?) But that sixth is dissonant, especially when played against the bass note (which is Degree 24 in those bars). Thus by playing Degree 12, we replace the dissonant 24/11 (an octave plus 12/11) with 24/12 (a perfect octave).
Speaking of dissonance, if Degree 26 is too dissonant for the bass fourth, we can change it to Degree 27 provided we change the key to C-D (rather than Bb-C). We change all the 26's in the DATA Lines 140-190 to 27's, then change:
5 SOUND 261-18*9,8:SOUND 261-24*9,8
20 IF V=3 THEN N=8 ELSE N=9
Line 20 must be changed in both the melody and bass windows, while Lines 5 and 140-190 are changed only in the bass window.
Today is also a "traditionalists" post. It's fitting that I've been writing about Michael Starbird's Calculus course in these traditionalists' posts, since after all, Calculus is the traditionalist-approved capstone high school course. But recent traditionalist activity doesn't concern Calculus this time, but with something a lot more basic -- multiplication.
Barry Garelick, our main traditionalist, made the following post yesterday:
https://traditionalmath.wordpress.com/2020/01/12/dont-tell-jo-boaler-dept/
A group of students in Lake Charles, Louisiana is promoting knowing the multiplication facts.
Our first commenter in this thread is Michael M.:
Michael M.
I coach an extracurricular math program for elementary school students, and I’ve decided to give multiplication tables to all my first and second graders, as a bit of a subversive prize. I never thought I’d see the day when giving students multiplication tables would be considered an act of rebellion, but here we are.
Do first and second graders really need multiplication tables? I've always considered multiplication to be a third grade topic. On the old California Standards (the ones that led to Algebra I in eighth grade), a few basic multiplication facts (2's, 5's, and 10's) are briefly mentioned under second grade, with the rest not taught until third grade. No multiplication is listed under first grade.
Then again, Michael M. mentions that he coaches an extracurricular math program, most likely for above average students. I hope that he wouldn't give times tables to a typical first grade class.
As usual, the commenter who takes the cake is SteveH:
SteveH:
I don’t pay a whole lot of attention to Jo Boaler, but is she promoting NOT knowing the times table or that everyone doesn’t need immediate recall of 7*8?
Notice that both Garelick and SteveH invoke the name "Jo Boaler" as an example of a progressive or anti-traditionalist teacher.
Recall that I have a copy of Making Number Talks Matter. Even though Boaler isn't the author of this book, she does write the Foreword. And the actual author (Ruth Parker) often quotes Boaler throughout the book.
So let's answer SteveH's question and find out what Boaler believes about the times table. We might as well go to Chapter 5, since that is the multiplication chapter. Parker writes:
"Flash cards and timed tests have continued to make early appearances in US classrooms as early as the second grade despite decades of evidence that, at best, they don't work very well -- as any middle and high school teacher knows. Timed tests, in particular, which cause many children to dislike and avoid math, have long been associated with math anxiety."
Presumably, flashcards and timed tests are what the traditionalists prefer. Here Parker says that these traditionalist methods lead only to math anxiety, so now we must ask, why do the traditionalists prefer to promote math anxiety in young students?
At this point, Parker does indeed quote Jo Boaler:
"Occurring in students from an early age, math anxiety and its effects are exacerbated over time, leading to low achievement, math avoidance, and negative experiences of math throughout life."
Parker compares traditionalist flashcards to a stack of cards with unrelated math facts like "b * g = z," an overwhelming task to memorize the entire stack. On the other hand, the progressive method of Number Talks shows that these math facts are related.
SteveH claims that according to Boaler, not everyone needs immediate recall of 7 * 8. Hey, that's a coincidence -- the exact problem that Parker gives as an example on page 61 is indeed 7 * 8. In this example, Parker quotes three students who explain how they get the answer 7 * 8:
Marta: I know 7 times 7 is 49, so I added one more 7 and got 56.
Jacob: Well, 4 times 7 is 28, so if you add 28 and 28, that would be the same.
Teresa: 10 times 7 is 70, and you could take away two 7's, or 14, and that's 56.
SteveH:
I thought it was their idea that your don’t memorize facts out of context, but that they magically happen as part of their process.
So far, the only ones promoting magic are the traditionalists. Before students learn the table, they know 0 out of 100 facts, and after they've learned it, they know 100 out of 100 facts. But the traditionalists assume that students magically go directly from knowing 0 to knowing 100 without any intermediate process where they know some, but not all, of the 100 facts.
On the other hand, Parker's method fully accepts the idea that at some point, students know some but not all of their times tables. In the above given example, Marta knows 7 * 7, Jacob knows 4 * 7, and Teresa knows 10 * 7. They use all of these previously known facts to learn the new one, 8 * 7.
Parker's method breaks down 7 * 8 into known facts, just as phonics breaks words down into their known sounds. On the other hand, the traditionalists promote the whole language of learning the whole times table at once.
SteveH:
How many eggs are in 8 dozen? How many older people can do that in their heads (as 8*10 + 2*8 with understanding!) and not their grandkids in K-6 now?
No, Parker's next example isn't 12 * 8 -- but it is 12 * 18:
Ruth [Parker, the author]: Who is willing to explain how you got one of these answers and why it makes sense?
Keanon: I did 10 times 18 ... Well ... I broke the 12 into 10 plus 2, and then I did 10 times 18 and got 180. Then I did 2 times 18.
Just change 18 to 8 and we get SteveH's problem solved with the same process. I'm hoping that by "K-6," SteveH doesn't literally mean K. I don't expect kindergartners to be able to do this at all.
Before we leave Parker's book, notice that she fears that timed tests send the wrong message about math -- that being "good at math" is equivalent to being "fast at math." Recall that I have no problem with the idea that students should know basic math -- that's what my "Dren Quizzes" were for. But I gave plenty of time for students to complete the Dren Quizzes.
Oh, and speaking of quizzes:
SteveH:
When I was young, if you didn’t get good report cards based on specific quizzes and tests, kids and parents knew it quarterly and you ran the risk of summer school, or worst of all, staying back a year. Now, individual problems fester until it’s too late.
Presumably, these "specific quizzes and tests" are the "timed tests" that lead to math anxiety. And the research surrounding staying back a year ("grade retention") is mixed. Many sources state that grade retention doesn't work.
I'm not quite sure whether grade retention works or not, but it does remind me of tracking -- a topic that I've mentioned in many traditionalists' posts. And of course, almost any tracking discussion leads to a discussion of race and politics.
I can't help but notice that the teacher and student mentioned in the news story (linked from the original Garelick article) are black. It's almost as if Garelick wants to taunt us -- see, the progressives like to claim that they're helping other races, but only the traditionalists are actually helping them.
Once again, I have no problem with anyone of any race learning the times tables. I just disagree with the traditionalists regarding the process.
Looking at some other recent Garelick posts, I see that he brings up the big Barbara Oakley debate from a year and a half ago.
And I already wrote about his anti-PBL post from last week. The day after I linked to it here, SteveH added a comment, so let's look at this here:
https://traditionalmath.wordpress.com/2020/01/04/pbl-a-guide-to-the-hype/
SteveH:
As if these students have a clue to what the needs are for “all learners.” What if a student said that “all learners” should not be in the same classroom? What if a student says that PBL does not meet the needs of her learning style?
What if a student says that traditional math does not meet the needs of her learning style. (Two can play at that game, SteveH!)
SteveH:
Who gets to decide what’s “ultimate” – schools or students?
...or traditionalists, for that matter? (Two can play at that game, SteveH!)
SteveH:
I want to see her PBL examples for AP Calculus.
It would really be interesting if Michael Starbird, in his Great Courses video, suggested something that leads to a PBL example for Calculus. We haven't reached such a project yet, but there's still time, considering that we're only a quarter of the way through his course.
(Then again, I could imagine converting Starbird's lecture today on circles, squares, and belts into a project, perhaps to be taught before students learn how to differentiate polynomials. The students might enjoy the belt riddle!)
SteveH:
School bands and orchestras can do interesting stuff when they know that most of the students have been taking private lessons for years, paid for by parents. That’s apparently now the model desired by all teachers. Do the fun stuff in class and assume that engagement gets the rest of the job done. At least the music teachers know that’s wrong.
To me, "engagement" means "not leaving things blank." I admit it -- yes, I assume that not leaving things blank gets the rest of the job done, or at least more than leaving things blank does. PBL is all about doing things that students don't want to leave blank, as opposed to say, traditional p-sets, which they do want to leave blank.
OK, here is the Chapter 8 Quest. Let's hope that our students know enough Geometry not to leave it all blank:
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