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Assigning More Meaningful Math Homework

A small set of problems or even one substantial problem can be enough to supplement classroom instruction.

As a math teacher of more than 23 years, I have had the habit, almost like a muscle memory repetition, of assigning daily math homework for my middle school students. It wasn’t until recently that I paused to reflect, “Why am I assigning this?” The easy answer is, “My students need to practice to develop their skills.”

If I dig a bit deeper into the “why,” I wonder, “Are all of my students benefiting from this assignment? Did I assign an appropriate amount and level of problems? What resources do my students have or not have to be successful with this assignment? Is the assignment meaningful or busywork?”

Consider the following suggestions for making math homework more meaningful.

3 Ways to Create More Meaningful Math Homework

1. Think quality over quantity. The National Council for Teachers of Mathematics Homework page of tips for teachers suggests, “Only assign what’s necessary to augment instruction. If you can get sufficient information by assigning only five problems, then don’t assign fifty.”

Worksheets and problem sets from textbook publishers might contain dozens of problems that repeat the same concepts. It is OK to assign part of a page, such as “p. 34 #s 3, 5, and 17.” I tell my middle school students, “I handpicked these particular problems because they capture the objective of today’s lesson.” When students know that their teacher carefully “handpicked” a problem set, they might pay more attention to the condensed assignment because it was tailored for them.

Even one problem can be sufficient. Robert Kaplinsky, cofounder of Open Middle , routinely shares on X (formerly Twitter) single problems that are really engaging and give students a good chance to practice skills.

The depth and exploration that can come from one single problem can be richer than 20 routine problems. You might be surprised by how much depth can be inspired by a single problem.

2. Consider choice and variety. It’s unrealistic to create a personalized daily homework assignment for each student in your class. Student voice and choice can be applied to your preexisting assignments without your having to re-create the homework wheel.

Traditional assignments can be modified by offering students choice. This might look like “ Choose any five of these problems ,” or take this tip from educator Peter Liljedahl and designate problems as “mild, medium, or spicy” and let students pick their level for that assignment.

Offering homework level choice also promotes a culture of growth mindset through messaging like “You might choose mild problems for this lesson; however, tomorrow you might feel you’re ready for a medium level.” Level choice can vary day to day—your math level is not fixed.

Daily homework can also be spiced up by offering a variety of types of assignments. Consider assigning problems that go beyond providing a single number answer. Here are a few examples to get students thinking beyond just getting a particular problem right:

• When simplifying (4 + 5) x 5 - 3, what is the first step?
• When simplifying (4 + 5) x 5 - 3, Ali got the answer 18. What advice do you have for her?
• Write your own order of operations problem with a solution of 42.

Check out these websites for even more creative ways to vary homework:

• Three-Acts Math Tasks
• Open Middle
• Would You Rather Math

3. Remember, accountability doesn’t have to result in a grade. There is a major difference between assigning homework for a grade and assigning homework purely for practice. When a grade is the result of an assignment, the stakes get higher for the student.

In the February 2023 Washington Post article “ A deep dive into whether—and how—homework should be graded ,“ former teacher Rick Wormeli wrote, “When early attempts at mastery are not used against them, and accountability comes in the form of actually learning content, adolescents flourish.” If homework is truly for practice, this is an opportunity for students to make mistakes and take risks without the fear of a penalty.

Even if homework is graded as a completion grade, there are considerations of equity and meaningfulness of the practice.

Consider the following questions when deciding to give a completion grade for a homework assignment: Do all students have a home environment that is supportive of homework? Do some students have additional support, such as tutors or parents, to help them get the homework completed? Would students copy homework assignments from each other just to earn the completion grade?

If not grades, then how do we hold students accountable for practicing outside of class?

Student presentations and discussions are a way to check for understanding of an assignment and to let students know you expect them to attempt the problems. This might look like a debate in which students take sides on how to approach a problem . Alternatively, students could post their work on the board to share their strategies with the class or discuss their solutions in small groups. Communicating their mathematical thinking deepens their understanding .

Education consultants Ashley Marlow and Katie Novak write in their Edutopia article “ Making Math Accessible for All Students ” (July 2022), “When students have opportunities to think, reason, explain, and model their thinking, they are more readily able to develop a deep understanding of mathematics beyond rote memorization. The goal is for all students to experience success in higher learning of mathematics—requiring those reasoning and sense-making skills and increasing engagement.”

The next time you’re planning your lessons and assignments, pause and reflect on the meaningfulness of the homework assignment. Could it be shorter but more in-depth? Can students have a choice in their work? Will students find value in doing the work even if it is not for a grade? You might find that students take more ownership and care in their homework if it’s more meaningful to them.

Introduction to Math 444 Assignments

This sheet is about the regular individual assignments. there may be other written work stemming from group activity in class, etc., that is not covered by this description., what is the purpose of the written assignments.

As in most subjects, one masters math by doing. This means that all one's studying should involve drawing, working examples on paper, etc. But the written assignments provide the impetus to work out in polished form problems and constructions and proofs that develop understanding of geometry. They also provide a running assessment for student and instructor of what is going on in the class as to levels of understanding, areas of success, areas that need work, etc.

How often are assignments given?

In the beginning part of the course, shorter assignments will be due two or three times a week. This will gradually move to longer and less freqent assignments of about one per week. Students who don't like daily assignments are free to work ahead and treat assignments 1a, 1B, 1C, etc. as a single weekly assignment.

What kind of written assignments?

Assignment items to turn in will be of several forms.

• Fairly routine exercises or constructions.
• Problems to solve - some challenging and some less so. These can be construction problems, numerical or ratio problems, or existence of special geometrical relationships such as similarity, concurrence, etc.
• Major or minor theorems to prove.
• Physical models and drawings will sometines be assigned.
• Investigations and open-ended problems. As the course moves on, such problems will appear on the assignments.

When are assignments due?

• Assignments are due at the beginning of class on the due date .
• No further writing is allowed once you walk into the room. Experience has shown that this leads to poor quality presentation, and sometimes to copying from other students.
• Students may discuss problems outside of clas s, so long as each student writes up her/his work independently. However, "sharing" answers in the 10 minutes before class is too late for group reflection.
• Late Assignments cause problem, so they may not be accepted. Often questions on an assignment will be answered during class. Also, the grading will sometimes be scheduled on the day the assignments are due. We do realize that nobody is perfect (least of all us) and that crises occur. If it happens that a student does not have an assignment on time once or even twice and asks to turn it in late, we will probably say OK, at least for partial credit. Once or twice also will not make or break one's grade. But don't make a habit of it.

Grading of Assignments will Follow several Strategies

We feel it is important that assignments be graded and feedback provided. However, some of the problems assigned in this course are not quickly graded. A bit of calculation will show that any problem that just takes 10 minutes per student will add up to about 5 or 6 hours of grading time. Thus the instructors will adopt a number of strategies to make grading possible. In some cases, students will self-report and then groups will discuss answers.

Quality of Presentation Counts

Since Math 444 is a course for students who intend to make a career explaining mathematics, the assignments turned in should be well-written as to language and thought. Answer in complete sentences. Use white space. Break up long explanations or proofs into paragraphs. Write neatly or word process. Presentation can definitely be part of the grade (plus or minus). Egregious examples of illegibility or bad writing will not be graded

What this means is that one should often write down a draft page for brainstorming and then write a completely new page to turn in.

Good and Bad News about Assignment Grades

This section gives two contradictory messages. The good news is that in Math 444 one does not have to get 100% to get a high grade. The problems are not routine, so perfection is not required, though a high standard is expected. The bad news (for some) is that skipping the homework and trying to get a high grade by being brilliant on the tests will not work because of the weight put on assignments. Only part of the course experience shows up on the tests, so high test grades alone do not ensure a high course grade in the absence of strong assignment grades.

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Statistics and probability

Unit 1: analyzing categorical data, unit 2: displaying and comparing quantitative data, unit 3: summarizing quantitative data, unit 4: modeling data distributions, unit 5: exploring bivariate numerical data, unit 6: study design, unit 7: probability, unit 8: counting, permutations, and combinations, unit 9: random variables, unit 10: sampling distributions, unit 11: confidence intervals, unit 12: significance tests (hypothesis testing), unit 13: two-sample inference for the difference between groups, unit 14: inference for categorical data (chi-square tests), unit 15: advanced regression (inference and transforming), unit 16: analysis of variance (anova).

Assignments on writing

Examples of short assignments, term papers, designing assignments that enable students to write well.

Writing well requires mastery of writing principles at a variety of different scales, from the sentence and paragraph scale (e.g., ordering information within sentences so content flows logically ) to the section and paper scale (e.g., larger-scale structure ). To simplify teaching, you can begin the term with shorter assignments to address the smaller-scale issues so you can more easily focus on the larger-scale issues when you assign longer assignments later in the term. At all scales, students best learn to communicate as mathematicians if the assignments are as authentic as possible: if the genre and rhetorical context are as similar as possible to those encountered by mathematicians.

Many of the following ideas are currently implemented in M.I.T.’s communication-intensive offerings of Real Analysis and Principles of Applied Mathematics .

• Require that at least one question on each problem set be typed up and written in the style of an expository paper (rather than the usually terse and sometimes scattered style of a homework solution).
• Assign short exposition tasks such as summarizing the proof of a theorem done in class or filling in the gaps in an explanation given briefly in class.
• To help students learn LaTeX or how to use equation editors, have an assignment requiring at least basic math formatting due early in the semester so students aren’t required to learn it as they’re researching and writing their term papers. Begin with simple math formatting exercises, building to more complex: e.g., see the assignments for M.I.T.’s Real Analysis recitations 1 (text with math) , 2 (table and figure) and 13 (slides containing a figure with LaTeXed labels) .
• Begin with communicating simple arguments, building to more complex (e.g., having students explain the heapsort algorithm and then revise the explanation based on feedback provides a rich opportunity for teaching about writing clear definitions, giving conceptual explanations as well as rigorous details, and presenting information in an order that is helpful to readers.) See the sequence of assignments from M.I.T.’s Principles of Applied Mathematics .
• Have students revise part of a concise textbook such as Rudin’s, Principles of Mathematical Analysis in the style of a more-thorough lecture note.
• Before an exam, have students formulate and submit to you a list of 2+ questions they have about the material. Students have a hard time formulating precise questions, yet this is an important communication and learning skill. Some students may feel they understand the course material, so permit questions that go beyond the scope of the course. You can use the questions to focus a review session. More detail about this assignment is given in this lesson plan from M.I.T.’s communication-intensive offering of Real Analysis.

The following books, articles, and websites contain short writing assignments.

• Stephen Maurer’s Undergraduate Guide to Writing Mathematics has an extensive appendix of writing exercises designed to target various aspects of writing mathematics.
• Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go , by A Crannell et al . [link goes to MAA review] This 119 page book from the MAA contains “writing projects suitable for use in a wide range of undergraduate mathematics courses, from a survey of mathematics to differential equations.” Each prompt is written in the form of an (often amusing) letter from someone who needs help with a “real-world” problem that requires math expertise. Students must solve the problem and write a letter of response. On his website, Tommy Ratliff (one of the co-authors) gives a brief account of using such projects in his calculus course.
• Annalisa Crannell’s Writing in Mathematics website has writing assignment for Calculus I, II, and III as well as links to colleagues’ websites that have further writing assignments.
• Quantitative Writing from Pedagogy in Action, the SERC Portal for Educators, has many examples of short and long writing assignments based on “ill-structured problems,” which are “open-ended, ambiguous, data-rich problems requiring the thinker to understand principles and concepts rather than simply applying formulae. Assignments ask students to produce a claim with supporting reasons and evidence rather than ‘the answer.'”
• The Nuts and Bolts of Proofs by Antonella Cupillari includes exercises for an introductory proof-writing course. Proof topics include calculus and linear algebra.
• Platt, M. L.. (1993). Short essay topics for calculus. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies 03.1 , 42-46.

Additional information about journal-writing assignments and other writing-to-learn assignments can be found on the page about using writing to help students learn math .

For each assignment, indicate your expectations about audience and length, so students know how much explanation to include. An appropriate audience is often other students in the class who are unfamiliar with the specific topic of the assignment, or other math majors not in the class.

Term papers enable students to pursue areas of their own interest and so can be among the most rewarding assignments for students. To help students succeed, give students guidance for choosing a sufficiently focused topic, for finding helpful sources, and for using sources appropriately. See this assignment for proposing a term paper topic , from M.I.T.’s Principles of Applied Mathematics –it includes guidance for how to choose a good paper topic.

One of the (interesting) challenges of assigning a term paper is generating a list of possible paper topics. Ideally, each topic should have well-defined scope and have at least two or three available resources accessible to students in the course. You may want to emphasize to the students that they are not expected to do original mathematics research. However, the paper must be their own — they cannot paraphrase and closely follow a published survey paper.

One of your institution’s librarians may be happy to collaborate with you to show students how to find useful sources.

To provide students with an authentic rhetorical context for their term papers, consider showing them samples of expository papers and suggesting that they write for a journal that publishes expository papers (e.g., The American Mathematical Monthly , Math Horizons , Mathematics Magazine , and The College Mathematics Journal .

Don’t assign a term paper unless a variety of topics exist at an appropriate level. For example, a term paper may not be appropriate for an introductory class in analysis.

Be aware that plagiarism may be an issue particularly in large classes on subjects for which a wealth of material is available online. In such classes, you may find it to be helpful to tightly specify the paper topics or to supply a specific slant to the papers (e.g., apply such-and-such method to an application of your choice). Vary the assignments from year to year. These precautions may be less important in small classes.

In some classes (e.g., applied mathematics classes), it may be necessary to carefully guide students to choose topics that contain sufficient mathematical content. For that reason, using caution when approving unfamiliar topics.

A poorly focused assignment will leave students confused about what is expected of them and is likely to result in poor writing. Students are likely to write their best if the assignment is interesting and if students are told (or are able to confidently identify for themselves) the following:

• educational objectives of the assignment
• audience knowledge and interest, and author’s relationship to the audience
• purpose of the text to be written (e.g., to convince, to entertain mathematically, to teach, to spark interest)
• content to be addressed
• details of the genre ( proof ? research paper? funding proposal?)
• how the writing will be graded
• an effective writing process (you can provide support by assigning intermediate due dates or revision )

The following resources explain these points and give further guidance for designing effective assignments:

• Bahls, P., Student Writing in the Quantitative Disciplines: A Guide for College Faculty , Jossy-Bass 2012, pp. 36-46, contains sections on structuring writing assignments (includes sample prompts), sequencing assignments throughout a course, and sequencing writing from course to course.

General resources (not specific to mathematics)

• How can I avoid getting lousy student writing?
• What makes a good writing assignment?
• The webpage Integrating Writing and Speaking Into Your Subject , provided by MIT’s Writing Across the Curriculum, has several subpages about writing assignments.
• Creating Writing Assignments , MIT’s Writing Center

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Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

• This page was last edited on 5 April 2020, at 18:48.
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Mathematics > Analysis of PDEs

Title: critical mass phenomena and blow-up behavior of ground states in stationary second order mean-field games systems with decreasing cost.

Abstract: This paper is devoted to the study of Mean-field Games (MFG) systems in the mass critical exponent case. We firstly establish the optimal Gagliardo-Nirenberg type inequality associated with the potential-free MFG system. Then, under some mild assumptions on the potential function, we show that there exists a critical mass $M^*$ such that the MFG system admits a least energy solution if and only if the total mass of population density $M$ satisfies $M<M^*$. Moreover, the blow-up behavior of energy minimizers are captured as $M\nearrow M^*$. In particular, given the precise asymptotic expansions of the potential, we establish the refined blow-up behavior of ground states as $M\nearrow M^*.$ While studying the existence of least energy solutions, we establish new local $W^{2,p}$ estimates of solutions to Hamilton-Jacobi equations with superlinear gradient terms.

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