But it's rare when Google features a mathematician in its Doodle. But today, Monday, March 23rd, Google celebrates the 133rd birthday of Emmy Noether, who was not only a mathematician, but a

*German*mathematician. And so I couldn't resist making my planning about the German education system on the day that one of the most famous products of that education system was born. I wrote this post at the last minute, and so even though I'm posting this late in the evening (Pacific Time) of March 23rd, by the time most of you read this it will be March 24th and no longer Noether's birthday.

But who, exactly, was Emmy Noether? As usual, whenever I want a biography about a mathematician, I link to the University of St. Andrews website:

http://www-history.mcs.st-and.ac.uk/Biographies/Noether_Emmy.html

This page describes her as an abstract algebraist:

"At Göttingen, after 1919, Noether moved away from invariant theory to work on ideal theory, producing an abstract theory which helped develop ring theory into a major mathematical topic.

*Idealtheorie in Ringbereichen*(1921) was of fundamental importance in the development of modern algebra."

Here "algebra" refers not to high school Algebra I or II, but

*Abstract A*

*lgebra*, a college class that's far beyond Calculus. Nonetheless, this "ring theory" actually appears, believe it or not, in the Common Core Standards!

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Now, as it turns out, this system that's analogous to the integers -- a system where one can freely add, subtract, and multiply -- is formally called a

*ring*! The word "ring" is the hidden word behind this Common Core Standard. The standard could easily have been written:
CCSS.MATH.CONTENT.HSA.APR.A.1

Understand that polynomials form a

Understand that polynomials form a

*ring*, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Similarly, we notice the following standard:

CCSS.MATH.CONTENT.HSA.APR.D.7

(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

The formal term for such a system that's analogous to the rational numbers is a

*field*:
CCSS.MATH.CONTENT.HSA.APR.D.7

(+) Understand that rational expressions form a

(+) Understand that rational expressions form a

*field*, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Notice that there are four different fields mentioned in the Common Core Standards -- the field

**Q**of rational numbers, the field**R**of real numbers, the field**C**of complex numbers, and the field of rational expressions mentioned in HSA.APR.D.7, often written as qf(**R**[*x*]). There are six rings mentioned in the standards -- all fields are rings, so the four mentioned fields are rings. The two additional rings mentioned in Common Core are the ring**Z**of integers and the polynomial ring mentioned in HSA.APR.A.1, often written as**R**[*x*]. But the standards avoid the words "ring" and "field" in order to avoid confusing the math*teachers*who never took Abstract Algebra. So instead of "ring" we have the verbose phrase "system analogous to the integers."
Not only did Emmy Noether study ring theory, but there is a category of rings that is named after Noether herself. A

*Noetherian*ring is a ring that is simple -- in a precise sense that can only truly be defined in an Abstract Algebra course. Suffice it to say that all of the rings that appear in the Common Core standards are Noetherian -- as it turns out, all fields are Noetherian, and the two extra rings mentioned in the standards,**Z**and**R**[*x*], are also Noetherian. One must study advanced college-level math before ever encountering a ring that isn't Noetherian.
But we can appreciate Noether's impact on mathematics without knowing anything about Abstract Algebra, but instead sticking to Physics and even Geometry! Here is another link to an article about Emmy Noether:

This article tells us that Noether studied "invariants in algebra," and provides us with the following example of what "invariants" exactly are:

Her early work focused on invariants in algebra, looking at which aspects of mathematical functions stay unchanged if you apply certain transformations to them.(To give a very

*basic example of an invariant, the ratio of a circle's circumference to its diameter is always the same — it's always pi — no matter how big or small you make the circle.)*
And of course, we discussed and celebrated the invariance of that ratio last week. Here's another example of an invariant: the article states that an invariant is something that stays unchanged when when certain

*transformations*are applied to them. So let's consider a certain type of transformation that appears throughout Common Core Geometry -- the*isometry*. What are the invariants of a figure when we apply an isometry? That's easy -- angle measure, betweenness, collinearity, and distance -- and there's even a theorem that states that these four are invariants, the A-B-C-D Theorem found in Section 6-5 of the U of Chicago text. Of course, if we replaced "isometry" with "similarity transformation," then there are only three invariants, as distance is no longer invariant. This is implied by the Size Change (Dilation) Theorem of Section 12-5.
But the importance of Noether's work is revealed when we apply these invariants to physics. As it turns out, these invariants in geometry correspond to Conservation Laws in physics. (Recall that Stephen Hawking was also looking for invariants in searching for his Theory of Everything.)

These correspondences are known as Noether's Theorems. Let's return to the article for a description of Noether's Theorems:

Here's an example: Let's say we conduct a scientific experiment today. If we then conduct the exact same experiment tomorrow, we'd expect the laws of physics to behave in exactly the same way. This is "time symmetry." Noether showed that if a system has time symmetry, then energy can't be created or destroyed in that system — we get the law of conservation of energy.

Likewise, if we do an experiment, and then do the exact same experiment again 20 miles to the east, that shouldn't make any difference — the laws of physics should work the exact same way in both places. This is known as "translation symmetry." Noether showed that translation symmetry leads to the law of conservation of momentum.

Finally, if we put our experiment on a table and rotate the table 90 degrees, that shouldn't affect the laws of physics, either. This is known as "rotational symmetry." But if rotational symmetry holds in a system, then angular momentum is always conserved. (That is, if you have a spinning bicycle wheel, it should spin in the same direction forever unless friction slows it down.)

Aha! Now the phrases "translation symmetry" and "rotational symmetry" should sound

*very*familiar to anyone who reads this blog, because translations and rotations appear all over the Common Core Geometry standards. And so we see that the Common Core's focus on transformational geometry -- derided by its critics as "fluff" -- is in fact the key to understanding the laws of physics. I can only imagine someone trying to tell Noether that all of her work is based on "fluff"!
So far, this post that I wanted to devote to the German education system discusses precious little about German education system per se, but about a particular German mathematician. It's now time to move on to a discussion of the education system.

Here's finally what I wanted to discuss. There are problems that our current education system under Common Core -- and considering that there are so many walkouts at schools due to the Common Core tests, that fact alone is evidence that the system is flawed. What I want to know is, what would happen if the U.S. were to adopt certain facets of the German education system?

There are some parts of German education that would be universally unpopular if they were to be adopted here. For example, many German high schools go up to the 13th grade. I can't see anyone, no matter the political persuasion, who would consider 13th grade a good idea to have here. Indeed, many

*German*schools are dropping the 13th grade, so it would make no sense to bring the extra year here.
I wish to compromise and accept parts of the German education system that liberals or Democrats would support as well as parts that conservatives or Republicans would like. Yes, it's difficult to avoid making this into another political topic -- which is why I buried it here in a spring break post rather than add politics to the posts that are supposed to focus on geometry.

But before I give any proposals based on the German education system, let's recall that the German model is

*not*my preferred model for education. I've stated my own preferred ideas -- for example, replacing grade-levels with "paths," and using the full potential of computer-based tests like the PARCC and SBAC to meet student needs. These were mentioned in previous posts. Instead, I mention the German model because I was*intrigued*by it when I subbed in a German class, and I wanted to consider whether adopting parts of the German model would solve some of the issues that parents, students, and teachers have with Common Core.
In Germany, many universities are tuition-free, and there was a backlash when the universities tried to charge tuition. It is probably impossible to make the colleges free in the U.S., but the closest we can get to Germany would be to adopt the president's proposal that community college tuition be free.

But of course, this isn't the most controversial part of the German education system. That, of course, is the rigid tracking system -- the division of all students into three educational paths. Here is a link to an article that discusses the three-track German system:

*As the 19th century came to a close and the 20th was opening, Germany had uniform elementary schooling with compulsory elementary education for all children aged 6 through 10, providing four years of basic education. This demonstrated Germany’s commitment to a state-run system of basic education. Following completion of elementary school, students were streamed into one of three types of school:*

*• Volksschule – The students thought to be of low ability (the majority), were streamed into the Volksschule (People’s School, later call the Main School or Hauptschule) where they would get a few more years of education, and receive a qualification entitling them to apply for training leading to working-class jobs in Germany.*

*• Realschule – Students thought to be of higher ability were streamed into the Realschule, where they would prepare for a qualification entitling them to apply for more training that would lead to more prestigious jobs such as clerks, technicians and lower-level civil servants.*

*• Gymnasium – Those thought to be of the highest ability were streamed into the Gymnasium, where they would be given a broad preparation in the humanities and prepared to take examinations for the Abitur, which was the sole gateway to the professions, teaching and the upper levels of the civil service.*

Again, the word "Gymnasium" clearly doesn't mean the same in English that it does in German. We notice that, as is common in European countries, the entire Gymnaisum is geared towards a single goal -- passage of a big test known as the

*Abitur*. And we see how important the*Abitur*was, as university entrance or a professional career were impossible without it. Certainly there were never huge walkouts at the*Abitur*the same way there are walkouts at PARCC or SBAC now.
According to the article, the three educational racks began around the end of the 19th or beginning of the 20th century. That is, they were formed around the time that Emmy Noether was a student. Let's go back to the St. Andrews website and learn more about Noether's secondary education:

"After elementary school, Emmy Noether attended the Städtische Höhere Töchter Schule on Friedrichstrasse in Erlangen from 1889 until 1897. She had been born in the family home at Hauptstrasse 23 and lived there until, in the middle of her time at high school, in 1892, the family moved to a larger apartment at Nürnberger Strasse 32. At the high school she studied German, English, French, arithmetic and was given piano lessons. She loved dancing and looked forward to parties with children of her father's university colleagues. At this stage her aim was to become a language teacher and after further study of English and French she took the examinations of the State of Bavaria and, in 1900, became a certificated teacher of English and French in Bavarian girls schools. She was awarded the grade of "very good" in the examinations, the weakest part being her classroom teaching."

Notice that Noether was born in 1882, so she was clearly a prodigy -- she was admitted to secondary school at age 7 and left at age 15.

The Pearson Foundation article tells us that students are tracked around age 10 -- that is, as they enter 5th grade. But -- since once again, the education system is run by the states -- individual states may choose different ages to enter secondary school. Some states don't begin tracking until the students enter 7th, rather than 5th, grade.

If we were to adopt this division in the U.S., it would make the most sense to track as the students make the transition from elementary to middle school -- but this is state-dependent, as many states have middle schools start at different grades. Here in California, middle school generally starts with the 6th grade, although I know of at least three local school districts where middle school doesn't begin until the 7th grade. Beginning middle school with 5th grade is rare in California, though I do know of at least one charter school network that includes 5th grader in middle school. So most likely, a German tracking system applied to California would track entering 6th graders. In states where 5th graders in middle school is more common, tracking may occur with students entering 5th grade, just as in most German states.

Of course, the early age at which tracking occurs is only part of the controversy. The real problem occurs when the tracks line up with other demographics, such as race/ethnicity or social class -- or even to a lesser extent, gender. We can observe the inequalities that were present in the German system by looking back at Emmy Noether's education.

Undoubtedly, Noether faced gender discrimination. But notice that according to the St. Andrews link, Noether attended a "Tochter Schule." As "Tochter" means

*daughter*, this implies that Noether attended an all-girls school. The discrimination that she would face due to her gender would occur after she left secondary school and headed for the university:
"Instead she decided to take the difficult route for a woman of that time and study mathematics at university. Women were allowed to study at German universities unofficially and each professor had to give permission for his course. Noether obtained permission to sit in on courses at the University of Erlangen during 1900 to 1902. She was one of only two female students sitting in on courses at Erlangen and, in addition to mathematics courses, she continued her interest in languages being taught by the professor of Roman Studies and by an historian. At the same time she was preparing to take the examinations which allowed a student to enter any university. Having taken and passed this matriculation examination in Nürnberg on 14 July 1903, she went to the University of Göttingen. During 1903-04 she attended lectures by Karl Schwarzschild, Otto Blumenthal, David Hilbert, Felix Klein and Hermann Minkowski. Again she was not allowed to be a properly matriculated student but was only allowed to sit in on lectures. After one semester at Göttingen she returned to Erlangen."

Eventually, Noether would meet the prominent mathematician David Hilbert. I mentioned back around the first day of school, and again in November, that Hilbert was the one who wrote a rigorous formulation of Euclid's original axioms of geometry. As it turned out, Hilbert allowed Noether to lecture with him, and the link above shows an advertisement for a Noether lecture:

*"Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr E Noether, Mondays from*4

*-*6

*, no tuition."*

Now Hilbert was sharply criticized for allowing a woman to lecture with him. Hilbert's response to his critics was, "We are a university, not a bath house."

So we see the uphill battle that Emmy faced as a woman. But it is noticeable that nowadays, it is the

*male*students who are struggling to get into Gymnasium, not the female students. This phenomenon occurs here in the U.S., where there are more female college students than males. Sometimes I refer to this as the "Bart and Lisa Problem," referring to two children on*The Simpsons*. Notice that it's not the son Bart, but the daughter Lisa who is more academically minded, while Bart couldn't care less about getting an education at all. (I point out the character Gaston in Disney's*Beauty and the Beast*, which takes place in the 19th century. Gaston notices that Belle is always reading books and tells her that it's improper for women to read. Compare this with the attitude of Bart and many boys in the 21st century, who believe that it's improper for*men*to read!)
Sometimes I wonder how we got from the 19th century, when women like Noether had trouble getting an education, to the 21st century, when

*men*have trouble getting an education. I suspect that it has something to do with traditional gender roles -- males, of course, have always preferred to be the more active of the two genders. In the 19th century, when women were expected to work in the home, getting an education was an*active*thing to do, and so men preferred that they themselves be the only ones entitled to an education. But in the 21st century, women are no longer assumed to be housewives and technology has increased to the point that there are so many other things to do in the world besides get an education. So reading and education were no longer considered*active*things to do, and so males like Bart reject them!
On the other hand, Emmy Noether escaped the class divisions associated with German tracking, for her own father Max Noether was a mathematician in his own right. From Max's biography, also from the St. Andrews site:

"Max attended school in Mannheim but his studies at the gymnasium were interrupted in 1858. He suffered an attack of polio when he was 14 years old and it left him with a handicap for the rest of his life. For two years he was unable to walk and was unable to attend the Gymnasium. However, his parents arranged for him to receive lessons at home and so he was able to complete the Gymnasium curriculum without returning to school. At this stage Noether was interested in astronomy, so before beginning his university studies he spent a short period at Mannheim Observatory."

Notice that Max's family were iron wholesalers, they so most likely had a vocational education. But see that Max, fortunately, was able to enter Gymnasium -- his main problem was his infection with polio and his inability to walk (similar to FDR).

The last problem associated with tracking is, of course, race and ethnicity. We return to the Noether family and notice that they were Jewish. This wasn't a problem for Emmy until, of course, the rise of the Third Reich in the 1930's:

"Further recognition of her outstanding mathematical contributions came with invitations to address the International Congress of Mathematicians at Bologna in September 1928 and again at Zurich in September 1932. Her address to the 1932 Congress was entitled

*Hyperkomplexe Systeme in ihren Beziehungen zur kommutativen Algebra und zur Zahlentheorie*. In 1932 she also received, jointly with Emil Artin, the Alfred Ackermann-Teubner Memorial Prize for the Advancement of Mathematical Knowledge. In April 1933 her mathematical achievements counted for nothing when the Nazis caused her dismissal from the University of Göttingen because she was Jewish."
Of course, the Third Reich no longer exists in Germany. But we can see that many immigrant groups are being placed on the lowest track:

"In this way the old tripartite system was quietly transformed. In the past, most of the graduates of the Hauptschule
went on to apprenticeships and had a good shot at a decent job and a good career. When the transition was
complete, the Hauptschule had become, in some schools and in some parts of the country, a dumping ground for
students who would find it hard to get a qualification of any kind – these included immigrants and native Germans
from lower class families alike."

The article tells us that many immigrants and other low-class students are locked onto a dead-end track that doesn't lead to success. This is why many people oppose such rigid tracking systems, in both Germany and the U.S.

Is it possible to have tracking without such significant racial and class-based inequalities? I say that it's possible, but politically difficult. It's hard for me to avoid politics when discussing the possibilities here.

Someone on the liberal or Democratic side of the political spectrum might say that if only certain tracks lead to success, then there should be some sort of Affirmative Action to help those groups that are less represented on the tracks that lead to success. I'm not sure whether hard quotas or extra points would increase access to success for the underrepresented groups. It might depend on the degree to which these groups are underrepresented, as well as the disparity between the levels of success between the highest and lowest tracks. Notice that the Common Core already contains a longitudinal data system, P20, which may be used to determine which tracks lead to success and which groups are represented on these tracks.

But someone on the conservative or Republican side of the spectrum generally opposes Affirmative Action and so would be against this plan. Instead, one might point to the preceding paragraph:

"As more students went to Gymnasium, and passed the Abitur, more decided to then enter the dual system and get
a qualification, as a form of insurance against unemployment, before proceeding to university. More students who
would formerly have gone to the Hauptschule put in the extra effort needed to get into the Realschule, which improved
their chance of getting an apprenticeship from a good employer. But employers offering the best apprenticeships,
who had earlier taken students only from Realschule, began to see a steady stream of Gymnasium students, some of
whom already had an Abitur, knocking on their doors. Others who had formerly recruited only from the Hauptschule,
found that they could get better candidates from the Realschule. Increasingly, the Hauptschule became a giant
storage locker for the students who had no future, a road to nowhere for those students."

This is often known as "credential inflation." It's said to happen in the U.S. as well -- so people talk about college-educated taxi drivers and bellhops.

So a conservative might say that there's nothing wrong with certain groups being underrepresented on various tracks as long as all tracks lead to success. Instead, the goal should be to stop and reverse credential inflation so that all tracks lead to success. My main problem with this idea is that many articles discuss the

*existence*of credential inflation, rather than a*solution*to credential inflation.
Thus a solution to track without inequality is politically difficult -- but then again, it would be politically difficult to implement

*any*part of the German education system. In conclusion, emulating the German system may be a good idea in theory, but not in practice, and so this is why doing so is not my preference for how to improve the American education system.
END

What possible value is there in teaching ring theory to high school students, particularly, if by doing so, topics vital to the preparation for calculus are abandoned?

ReplyDeleteBased on other sources, I see that Common Core doesn't actually require anyone to teach ring theory, nor does it appear in the standards. The standard just as well could have just said "add, subtract, and multiply polynomials" without mentioning the "system analogous to the integers."

Delete