Effective Teaching of Abstract Algebra
Abstract Algebra is one of the important bodies of knowledge that the mathematically educated person should know at least at the introductory level. Indeed, a degree in mathematics always contains a course covering these concepts. Unfortunately, abstract algebra is also seen as an extremely difficult body of knowledge to learn since it is so abstract. Leron and Dubinsky, in their paper ¡§An Abstract Algebra Story¡¨, penned the following two statements, summarizing comments that are often heard from both teacher and student alike.
1."The teaching of abstract algebra is a disaster, and this remains true almost independently of the quality of the lectures." (Leron and Dubinsky agree with this statement.)
2."There's little the conscientious math professor can do about it. The stuff is simply too hard for most students. Students are not well-prepared and they are unwilling to make the effort to learn this very difficult material." (Leron and Dubinsky disagree with this statement.)(Leron and Dubinsky, p. 227)
Thus the question is raised: if there is something the "conscientious math professor can do about" the seemingly disastrous results in the learning of algebra, what is it that we can do? As a teacher of undergraduate mathematics, I want and need to know what these effective methods of teaching abstract algebra are.
Leron and Dubinsky's paper referred to above and papers resulting from their research contain the bulk of literature that I reviewed. In this paper, they summarize their
experimental, constructivist approach to teaching abstract algebra. Among the classroom activities are computer activities, work in teams, individual work, class discussion, and sometimes a mini-lecture summarizing the results of student work (which by this time is familiar to them), providing definitions, theorems, and proofs in their abstract forms.
The computer activities use the ISETL programming language. As an example of its use, students write a program implementing the group axioms. They then can enter what they consider to be a group, and the computer will give as output a true or false response. They can use the same process to determine whether their proposed group is closed, has an identity, etc. They choose their answer and then let the computer respond. In this way, students ¡§construct¡¨ the group process, with the view that they will also have a ¡§parallel construction¡¨ occurring in their minds. Students have an experience on which to base their learning of group theory.
The method proposed here by Leron and Dubinsky certainly seems patterned after Dubinsky's theoretical foundation for student learning laid out in his work ¡§Reflective Abstraction In Advanced Mathematical Thinking.
The title from this book is referring to the story ‘Moby Dick’. Where on the first page, on the first line, it quotes “Call me Ishmael”. Once the reader reads this, all who have read ‘Moby Dick’ will recognise the name. I myself have not yet read ‘Moby Dick’ so the title did not mean too much to me but Michael Gerard Bauer told any readers who hadn’t yet read Moby Dick where and when the name is mentioned. People who go to the library having read Moby Dick may glance over the book and immediately see ‘Ishmael’ on the front cover and may influence their decision on reading the book.
A new leader arose by the late 1980s, a Serbian named Slobodan Milosevic, a former Communist who had turned to nationalism and religious hatred to gain power. He began by inflaming long-standing tensions between Serbs and Muslims in the independent provence of Kosovo. Orthodox Christian Serbs in Kosovo were in the minority and claimed they were being mistreated by the Albanian Muslim majority. Serbian-backed political unrest in Kosovo eventually led to its loss of independence and domination by Milosevic. In June 1991, Slovenia and Croatia both declared their independence from Yugoslavia soon resulting in civil war. The national army of Yugoslavia, now made up of Serbs controlled by Milosevic, stormed into Slovenia but failed to subdue the separatists there and withdrew after only ten days of fighting. Milosevic quickly lost interest in Slovenia, a country with almost no Serbs. Instead, he turned his attention to Croatia, a Catholic country where Orthodox Serbs made up 12 percent of the population. During World War II, Croatia had been a pro-Nazi state led by Ante Pavelic and his fascist Ustasha Party. Serbs living in Croatia as well as Jews had been the targets of widespread Ustasha massacres.
Yugoslavia was a very diverse, ethnic, and peaceful place under communist rule ("Genocide in Bosnia--1992-1995"). For 40 years it stayed this way ("Genocide in Bosnia--1992-1995"). Provinces declared...
The reason this genocide happened is because Bosnia declared its independence from Yugoslavia, because of this the Serb military killed thousands of Bosnian civilians. This was the worst act of genocide since the Nazis while 6 million jews were killed.
“Although many different ethnic and religious groups had resided together for 40 years under Yugoslavia’s repressive communist government, this changed when the country began to collapse during the fall of communism in the early 1990s” (Holocaust Museum Houston). This quote is a very truthful one. Many people in this world probably have never even heard of the Bosnian Genocide. It was a very tragic time in history that could have been prevented if people didn’t have so much pride and arrogance like Slobodan Milosevic did.
Bosnia is a country in Europe and its capital is a city called Sarajevo. Bosnia is bordered by Serbia and Croatia. In this genocide, between 1992 and 1995, the Serbians wanted to pursue genocide against the Muslims of Bosnia. After World War 1, a country called Yugoslavia was created in 1918. It was created out of the Austria-Hungary empire that lost the war and lost its land. Serbians, Croatians, Slovenes, and Bosnian Serbians and Muslims lived all in one country. The problem was the people didn't get along and each republic wanted to take control of the country. This went on until after WWII, when the Soviet Union took power and control over the country. Joseph Broz (Tito) was leader of Yugoslavia until the 6 republics separated. Then in 1992, Bosnia declared its independence from Yugoslavia. Soon after, the Bosnian
I remember how mathematics was incredibly difficult for me and because of this I can relate to the struggles students have with math. For a teacher to be successful they need to create relevance for the students. I understand how to relate the various topics of mathematics to topics of the world, which for most students is difficult to do, For example, I remember at the CREC School I was observing at, there was a student of Bosnian decent who was having trouble understanding how to read a map of the United States. So I showed her a map of Bosnia with the same map key, and we discerned what everything meant (where the capital was, where the ocean was, major port cities were, etc…). She caught on quickly as she already had an understanding of Bosnia and it quickly transferred over to the map of the thirteen colonies. This skill is easily transferrable to mathematics by using relevant, real-world examples of concepts learned by
"There is one front and one battle where everyone in the United States—every man, woman, and child—is in action, and will be privileged to remain in action throughout this war. That front is right here at home, in our daily lives, and in our daily tasks."- President Franklin D. Roosevelt, 28 April 1942.Women at this time were the only hope for filling the empty jobs left by the men going to war. Everyone began to realize that this war could not be won without the help of everyone, whether they were a male or female. Many people pushed and supported women being able to join the workforce...
This first encounter with God serves to identify the trouble that man has with obeying God and ignoring ones self. Even in a simple time with no worries at all, it was impossible for Adam to resist the self and obey God. Throughout the novel Moby Dick Melville addresses the relationship between man and the Judeo-Christian God. Melville demonstrates many of the shortcomings of western religion and its in ability to reconcile the benefits of the darker side of humanity. Ishmael, through his journeys finds himself in the midst of several situations that exemplify this dichotomy between the ideal relationship with the Judeo-Christian God and the practical nature in which man typically relates to God.
Gagnon, V. P. (2004). The myth of ethnic war: Serbia and Croatia in the 1990s. Ithaca, N.Y.: Cornell University Press.
Paper 205. http://digitalcommons.ric.edu/facultypublications/205 4. Williams, Heather. Andrea.
These tensions, only highlighted by the war, are an unfortunate but large part of Bosnian culture as a whole. The three main ethnic groups of Bosnia are Bosniaks, Serbs, and Croats, with 48.4%, 32.7%, and 14.6% populations respectively (CIA World Factbook). Their intense nationalistic attitudes and vastly different religious heritages cause animosities between the groups that go back beyond the times of nations. Bosniaks are generally Muslim while Croats are Roman Catholic and Serbs are Christian
The conflict between the Albanians and Serbs has been a continual issue since the fourteenth century. Ethnic conflicts rose again after the death of Tito who was the leader of Yugoslavia. Tito set up a national Yugoslav government and let the five Slavic nationalities (Serb, Croat, Slovene, Montenegrin, and Macedonian) govern their own part of Yugoslavia which suppressed any ethnic fighting (Andryszewski 14). After the death of Tito in 1980, ethnic conflicts began to come to surface again. Slobodan Milosevic gave a speech to the Serbs in Kosovo saying that “No one will dare to beat you again” (Andryszweski 18). In 1991, Croatia and Slovenia declared their independence from Yugoslavia which led to the outbreak of war since the Serb-dominated central government wanted to preserve the state. In 1995, the Dayton Peace Settlement was signed to end the war and Yugoslavia broke apart ove...
As a secondary subject, society often views mathematics a critical subject for students to learn in order to be successful. Often times, mathematics serves as a gatekeeper for higher learning and certain specific careers. Since the times of Plato, “mathematics was virtually the first thing everyone has to learn…common to all arts, science, and forms of thought” (Stinson, 2004). Plato argued that all students should learn arithmetic; the advanced mathematics was reserved for those that would serve as the “philosopher guardians” of the city (Stinson, 2004). By the 1900s in the United States, mathematics found itself as a cornerstone of curriculum for students. National reports throughout the 20th Century solidified the importance of mathematics in the success of our nation and its students (Stinson, 2004). As a mathematics teacher, my role to educate all students in mathematics is an important one. My personal philosophy of mathematics education – including the optimal learning environment and best practices teaching strategies – motivates my teaching strategies in my personal classroom.
A somewhat underused strategy for teaching mathematics is that of guided discovery. With this strategy, the student arrives at an understanding of a new mathematical concept on his or her own. An activity is given in which "students sequentially uncover layers of mathematical information one step at a time and learn new mathematics" (Gerver & Sgroi, 2003). This way, instead of simply being told the procedure for solving a problem, the student can develop the steps mainly on his own with only a little guidance from the teacher.