"Swiss mathematician Leonhard Euler (1707-1783) had learned of the Konigsberg bridge problem in St. Petersburg while in the service of Catherine the Great of Russia."
This is the second page of the Konigsberg subsection. And as I promised earlier this week, Euler's name comes up in association with this problem. Unlike the Knight's Tour, Euler's connection to this problem appears to be genuine.
Again, this is one of my favorite problems to discuss and begin the school year with. So I definitely want to look at what Pappas writes about the Konigsberg bridge problem. Here are some excerpts:
"[Euler] set into motion ideas that launched the field of topology. For example, if a triangle is distorted into a square or circle, topology studies which of the object's properties remain unchanged. For each part of the city to which the bridges led he assigned a vertex point and illustrated each bridge with an arc. Euler identified each vertex as either an even or odd point."
And of course, as soon as we find the even and odd vertices (or nodes), we see that we can't have too many odd vertices. This is what one of my eighth graders was able to figure out so quickly last year after I told her to count them. As Pappas explains:
"Thus any graph that was traceable (no doubling back) could only have at most 2 odd vertices -- i.e. 0 if they were all even or 2 if one was a beginning point and one an ending point."
There is only one picture on this page. It's a very simplified example of a network with just three vertices, in order to illustrate what networks and nodes are. By the way, notice that Pappas usually calls the sets of vertices and paths a network, but she also uses the word graph. This graph is similar in many ways to the graphs in the graph theory of Erdos, but there are differences. In Konigsberg there may be two bridges linking two islands, but in Erdos there is at most one path directly between two nodes.
I've decided to label this as another traditionalists post, since it is test day. It's been a while since we've looked at our favorite traditionalist, Barry Garelick. He wrote his most recent post on Monday, so I could have made Monday my only traditionalists post this week, But lately I've been falling off of schedule, so let me return to labeling all posts on test days as "traditionalists."
Today's post isn't about the Bridges of Konisberg, but some different "bridges" altogether:
“(Bridges) focuses on developing the students’ understanding of math concepts,” Davis said. “It is not about how students can memorize certain skills, but really around their ability to problem solve and look at math in more complex ways…and explain their reasoning to their teachers and peers.”
The "Bridges" mentioned here is a elementary math text, Bridges in Mathematics. As you might expect, Garelick criticizes Bridges because they focus too much on problem solving and not enough on rote memorization.
This post has drawn two comments, and both of them are by -- who else? -- SteveH, our other favorite traditionalist. Moreover, Garelick even quotes SteveH at the end of the post itself. That is to say, there are more words here from SteveH than from Garelick.
Let's look at SteveH's second comment, where he critiques one particular grade level in Bridges:
Bridges in Mathematics – 3rd grade questions –
OK, so we're looking at the third grade text here.
telling time to the minute.
SteveH doesn't state the reason that this is questionable content. Here is the relevant standard:
CCSS.MATH.CONTENT.3.MD.A.1
Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.
But most likely, SteveH would prefer that this standard, like many Common Core Standards, be taught at a lower grade level. For example, telling time was a second grade standard under the old California Standards. Of course, we know that many students in higher grades have decided that telling time on an analog clock is obsolete and refuse to learn it (but as I've seen before, students sometimes just claim they can't tell time as an excuse to take their phones out and do other things besides tell time).
Then again, let's compare this to another third grade skill -- single-digit multiplication -- that many eighth graders lack (the "drens"). Yes, traditionalists lament the fact that many eighth graders haven't mastered multiplication. But to them, this isn't a problem for a textbook to rectify. To them, eighth grade is the year for Algebra I, and so an acceptable eighth grade text must cover Algebra I standards only -- that is, they must assume that all K-7 standards are already mastered. And so likewise, an acceptable third grade text must assume that telling time -- to them, a second grade standard -- is already mastered.
Add 86 cents to $1.23 and show your work. What if you do it in your head?
To me, "doing it in your head" is problematic, because students sometimes cheat by copying a neighbor, then claiming that there's no work because they did it mentally. Surely traditionalists want the students to do the math, not copy the answers from a neighbor.
But most likely, SteveH is contrasting "doing it in your head" with using the Common Core-approved "strategies" for solving math problems. We know that SteveH would prefer that students use the standard algorithm for solving the problems rather than the Common Core "strategies." but the standard algorithm isn't taught until a higher grade.
Let's compromise here and allow students to use the standard algorithm without penalty. But even use of the standard algorithm requires students to show work -- enough work to prove that they didn't copy or cheat. Students should set up the standard algorithm by converting "86 cents" to "$0.86" and then line up the decimal points. Teachers may also demand that the student writes out carrying the "1" from the tenths place to the dollars place. This is enough to prove mastery rather than cheating.
“Story” multiplication problems of the difficulty of 6 X 2.
15 – 8 = ?
3 X 9 = ?
"Story problems" or "word problems" are another double-edged sword. Traditionalists often find "story problems" to be a waste of time. They may say that in the time it takes to do the three word problems that lead to the examples above, a student can complete 30 or more traditional problems.
Then again, what good is it to be able to solve any single-digit addition, subtraction, multiplication, or division problem in one second each, without knowing when to add, subtract, and so on? Perhaps a compromise could be that the only word problems that appear are those in which three numbers are involved in the computation, perhaps with two different operations involved.
And this is a classic:
“Keiko has 7 coins in her pocket that add up to 48 cents. What coins does she have. Show your work.
OK, for this one I sort of agree with SteveH. The problem itself is interesting. If we use a greedy algorithm and attempt to reach 48 cents with as few coins as possible, then we would use a quarter, two dimes, and three pennies. But this is only six coins, not seven. The easiest way to add an extra coin is to replace one dime with two nickels. So the is that Keiko has a quarter, a dime, two nickels, and three pennies.
But SteveH is probably arguing that third grade questions should be algorithmic, rather than depend on a trial-and-error process. Not until, say, factoring in Algebra I should students be required to use any trial-and-error.
Do their methods work? How do they know?
We know that to traditionalists like SteveH, success in math entails passing eighth grade Algebra I and senior year Calculus. So when he asks "Do their methods work?" he's really asking "Do students who take Bridges in elementary school go on to pass eighth grade Algebra I or AP Calculus?" (And of course, he means the students who learn all their math from Bridges, not outside tutoring.) And if the answer is "no" -- as SteveH already suspects is the case -- then Bridges should be thrown out and immediately replaced with a curriculum that does lead to Algebra I and AP Calculus. And naturally, he believes that such a curriculum is likely to be a traditional math curriculum.
Let's wrap up this traditionalist post with a revisit to my subbing from earlier this week. I mentioned that I actually taught math to a pair of students who spoke mostly Spanish and very little English. I recall back when I was a student teacher, and I found that I actually had slightly more success helping out the English learners than native English speakers. I wonder, why is this the case?
I suspect it might be because sometimes when I teach math I speak too fast, and students who have gaps in their prior knowledge are not able to keep up. But when I'm helping English learners, I already know they have gaps in their knowledge -- most notably English skills -- and so I force myself to slow down for them. In other words, math is, to many students, itself very much like a foreign language, but I don't respect that view -- except when the students themselves view English as a foreign language.
But it's also possible that the opposite is true. I treat all students like my English learners -- so I slow down and repeat myself too often. Native English speakers then tune out my lessons, and so they end up not learning.
Whichever is the case, I need to improve my teaching skills to English natives -- where there are no gaps in language to cover up deficiencies in math. This was arguably a factor in my first year as a teacher was not as successful as I'd like.
This is what I wrote two years ago about the answers to today's test:
1. C' is the same as C, but D' goes up diagonally to the left. This is tricky because the line of reflection is not perfectly vertical.
2. I' goes up diagonally to the right.
3. There are two symmetry lines -- the segment joining the two points and its perpendicular bisector.
4. The angle measures 62 degrees.
5. The angle measures 2x degrees.
6. The reflection image over line AD of ray AB is ray AC. This is tricky because it's been a while since we've seen the Side-Switching Theorem.
7. This is officially the Figure Reflection Theorem -- just make the right vertices correspond.
8. Reflections preserve distance.
9. The orientation is clockwise.
10. The orientation is counterclockwise, because reflections switch orientation.
11. There are three pairs: angles B and C, angles BAD and CAD, angles ADB and CDB.
12. There is one line of symmetry -- the line containing the angle bisector. This follows from the Angle Symmetry Theorem.
13. F' = E follows from the Flip-Flop Theorem. FG = EH is because reflections preserve distance.
14. Proof:
Statements Reasons
1. MO = MN 1. Given
2. M' = N 2. Given
3. MO = NO 3. Reflections preserve distance
4. MNO is equil. 4. Definition of equilateral
It's possible to add more details, such as O' = O, Transitive Property, etc.
15. The rectangle has two lines of symmetry, one horizontal, one vertical.
16. The isosceles triangle has one line of symmetry, and it's horizontal.
17. The images of the vertices are (1,3), (7,1), and (6,-2).
18. The image is (c, -d).
19. The angle measures 140 degrees.
20. The shortest distance is the perpendicular distance.
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