The midterm exam took place on Nov 9–11, 2017.
A note about writing
If you read any textbook or scientific article, you will see that there is a standard way of writing mathematics, in which equations form part of full sentences and occur naturally with the flow of the text. While completely optional, you may wish to practice this in writing your assignments. In general, the solutions and notes posted here try to adhere to this standard.
One way to write mathematics like this is to make use of the free software package LATEX, which can produce very high quality scientific documents, and is used extensively by mathematicians, physicists, and engineers. There is an excellent guide, The Not So Short Introduction to LATEX2ε, which can be used to get started. The homework assignments and solutions are all written in LATEX.
Last year's midterm exam from 2016
Last year's midterm is available for practice:
Solutions are available:
Last year the New York City school system adopted a policy requiring nightly homework assignments for every pupil, with minimums ranging from 20 minutes for first graders to two hours for high school seniors. School districts in Connecticut are now in the process of complying with a new requirement, adopted in June, that they develop a districtwide policy regarding homework.
The difficulty of developing an effective homework policy is taken up in the January issue of the Harvard newsletter, which is sponsored by the Harvard Graduate School of Education and published by the Harvard University Press.
One basic problem, the newsletter notes, is that while the topic of homework has drawn the attention of hundreds of researchers, their conclusions are ''far from uniform.''
Joe D. Austin, a professor of education at Rice University, examined eight decades of research and found 16 studies concluding that homework improved academic performance and 13 showing ''no differences between students who were assigned homework and those who were not.'' On the other hand, Herbert Walberg, a professor of education at the University of Illinois-Chicago Circle, recently pooled data from 15 studies and concluded that ''homework boosts achievement dramatically - especially if teachers grade or comment on it.''
The newsletter also notes a discrepancy between the research and the justification usually given for assigning homework in the first place.
Surveys consistently find that while teachers, principals and parents ''hope homework will strengthen skills,'' the newsletter said, they usually ''praise self-discipline and independent study habits as its most important fruits.'' By contrast, ''research on homework has concentrated on academic achievement, not on character changes.''
''This means that there is virtually a vacuum of information on whether homework achieves what we say it is for - character building,'' Mrs. Featherstone said.
Homework in Private Schools
Systematic data do exist on the amount of time students devote to school work outside class. Last month the Census Bureau, in its annual survey of school enrollment, reported that American children spent 5.4 hours a week, or just over one hour a school night, doing homework.
The bureau found some significant patterns within the overall total, including the fact that students in private schools do far more homework, 14.2 hours a week, than those in public schools, who do 6.5 hours. The study also found that elementary school students spent almost as much time on homework as their high school counterparts.
On the basis of the research she surveyed, Mrs. Featherstone suggested that young students might be spending too much time on homework.
''A lot of elementary school homework assignments are really busywork, and young children have strong developmental needs to do things other than school work,'' she said. ''They need time to play outdoors, and with the budget cuts in schools reducing instruction music and the arts, they need instruction of this sort. It's important for students to have time to read books that they enjoy, but how often do you see an assignment: 'Read a book you enjoy for an hour.' ''
Difficulties for Young Pupils
The newsletter found evidence that mathematics homework creates ''substantial difficulties'' for younger children. ''When the Philadelphia public schools set up a homework hot line for confused children and their parents, they were deluged with telephone calls - and elementary math led the list of troubles,'' it said.
''Elementary schools surely need the regular experience of puzzling alone over math problems,'' it continued. ''But perhaps they ought to do most of this at schools so that they can get informed help when they need it. One study shows that even eighth graders can learn as much from short assignments completed in the classroom as from longer problem sets taken home.''
A major benefit of homework, the newsletter found, is that it leads to communication between parents and the school. ''With students below the high school years, homework involves more than just a child, a book and a pencil,'' it said. ''Whole households participate in one way or another.''
One study of 26 blue-collar parents by Jean Chandler and Catherine Snow of Harvard found that the parents assisted their children in ''appropriate'' ways, providing encouragement and help that the children, who were interviewed separately, described as helpful.
''Homework gave these parents a window on their children's schoolwork and sometimes led them to talk to the teacher,'' said the researchers, adding that such contacts may have improved the children's chances for success at school because ''teachers expected more of boys and girls whose parents sought them out.''
Student Attitudes Vary Widely
Not surprisingly, research on homework suggests that attitudes of students toward different types of homework vary widely. A study of Australian schools by Frank Coulter found that students completed more of their assignments when teachers made it central to the work of the course, collected it routinely, spent time reviewing it in class and wrote it down rather than announced it orally at the end of class.
The researcher also found that high school students were greatly influenced by whether they regarded the homework as useful and interesting. Social studies assignments, for example, tend to be completed more regularly than those in English.
''Instead of asking how much homework is enough,'' the newsletter suggested, ''we might ask what sorts of work will convince students that if they study they will master important material. Unless the rewards are clear and attractive, many teen- agers simply will not participate.''
Mrs. Featherstone said the effectiveness of homework could be improved if teachers could find ways to ''individualize'' the assignments. ''Most homework is the same for everyone,'' she observed. ''This means that it often bores the most able kids and frustrates the least able one. It's not realistic to say that a teacher should design 30 different assignments, but a good teacher might suggest four or five ways of doing something.''
Value Seen for Slower Students
At the high school level, research suggests that homework can offer slower students a chance to compensate for deficiencies. The newsletter noted evidence from one study of 20,000 high school seniors that ''low- ability students who did one to three hours of homework a week reported grades as high as those of average- ability students who did no homework.''
The problem, though, is that teachers of less successful students tend to demand little outside work. ''If schools could convince low-achievement students and their teachers of the value of homework, they would take a step toward equalizing opportunity,'' the newsletter concluded.
Among the members of the Editorial board of The Harvard Education Letter are Patricia Albjerg Graham and Jeanne S. Chall of the education school, Albert Shanker, president of the American Federation of Teachers, and John Brademas, president of New York University.Continue reading the main story