The Technical Memo

A technical memo is a document that enables a student to explain how they solved a problem. These problems are not your usual textbook problems. They are problems typically correspond to projects and require several steps They have multiple possible solution strategies. Often the problems are not well stated so much effort goes into understanding what is being asked for. Technical memos are useful when a problem requires a significant amount of interpretation, strategy, and calculation.

During the first project, I provide the students with samples of technical memos. Using these samples, I work with the students to help them understand the components of the document. Let’s look at the components of a technical memo.

Title and Authorship: Every document needs a title that reflects what is being discussed and/or solved. A technical memo should not have a cute title or an incredibly complicated title. The title should attract the reader to go farther and read the next part of your memo. THe authors name and affiliation should be placed under the title. For my purposes, I also require students to include the date on which the memo is submitted (due date).

Introduction: The first section of your memo is the Introduction. It should be named as the Introduction to make it easy for the reader to identify. In this section the student needs to demonstrate that they understand the problems as it was given to them. A good way to do this is to restate the problem in their own words. If any part of the problem is unclear, the student should clarify them and state the assumptions they used. A diagram may help them to clarify the problem. The student should identify and show any data or quantities that were provided in the project letter. The student should say exactly what they are looking for in the problem and the general direction of their solution. If the student is analyzing how the student to teacher ratio is changing in a state, they should include the data they used, what is the student to teacher ratio, the basic technique used to solve the problem, and the basic answer. You never hide the solution from your reader. By supplying an answer, you rely upon your readercuriosity to draw them into your memo to learn more. If they are an untrusting soul, they will want to scrutinize the entire solution you supply in the rest of the memo.

Materials and Methods: The name of this section can be a bit misleading. The word Materials is often left out in technical memos on mathematical solutions. It comes from the fact that this type of memo forms the foundation for lab reports in the sciences like biology and chemistry. In these technical memos or lab reports, it is appropriate to talk about the materials used in an experiment. Usually in a mathematical technical memo we don’t use materials like chemicals so you may leave out the term Materials and simply call this section Methods.

In this section, students lay out their strategy for solving the problem in general terms. Without actually solving the problem, they want to explain what they are going to do and what mathematical tools they will need to do this. they may need to explain how these tools work because your reader may be unfamiliar with them. For instance, if they will be solving a quadratic equation as a part of your strategy, they will want to show the quadratic formula and how it applies to your specific instance.

Students should use mathematical terminology in their explanation. They may even need to explain the terminology so that the reader can follow what they are saying. The tough part for students is determining how much detail they should go into. They need to give enough detail so the reader can follow the calculations, but they don’t want to do the calculations in the Methods section. Save the calculations for the Results section. The only exception to this general rule is if the strategy involves a repetitive calculation. If the student to teacher ratio must be calculated several times from data, it is perfectly acceptable to show a sample calculation in the Methods section.

Results: In this section students carry out the strategy that they laid out in the Methods section. This may include solving equations, recognizing patterns, interpreting graphs. Don’t expect the equations and graphs to speak for themselves. Lead your reader through your calculations using your strategy from the Methods section as a guide. They should use words and complete sentences in the Results section.

Discussion: In the Discussion section the student should summarize your results and check your solution if possible. Does the solution make sense with reality? How does the solution relate to the real world or the big picture? Can they generalize your solution? I often ask question that require them to compare models and the Discussion section is an excellent place for them to do this. In some projects, it is not possible to solve the problem as stated. In the Discussion section, the student then explains how the problem can be modified to allow a reasonable solution.

In grading a technical memo, I consider a number of questions:

Conceptual Understanding
Key Question: Does the student’s interpretation of the problem using mathematical representations and procedures accurately reflect the important mathematics in the problem?

Strategies and Reasoning
Key Question: Is there evidence that the student proceeded from a plan, applied appropriate strategies, and followed a logical and verifiable process toward a solution?

Computation & Execution
Key Question: Given the approach taken by the student, is the solution performed in an accurate and complete manner?

Key Question: Was I able to easily understand the student’s thinking or did I have to make inferences and guesses about what they were trying to do?

Key Question: Does the student grasp the deeper structure of the problem and see how the process used to solve this problem connects it to other problems or real-world applications?

If the student has followed the format described above (Intro, Methods, Results Discussion), I can easily determine the answer these questions and rate their performance. In the Problem Solving Scoring Guide below (adapted from the Northwest Regional Educational Laboratory), you can see the criteria I use rate a student’s performance on each of these questions.

Problem Solving Scoring Guide DOC | PDF

The Project Letter

To be as realistic as possible, I give the problem to the students as a business letter. The format is fairly standard and is designed to give them some background on the problem, identify the data they will use, and what the expectation for the solution is. To help them write the any solution strategies, the writer of the letter and their basic mathematical background are disclosed. A good letter will pique the student’s curiosity. The PDF below includes the letter with annotations identifying the purpose of the parts of the letter.

Project Letter Annotated PDF | PDF

Mathematics Resources

I use a textbook throughout the course to give students a reference source for the mathematics they need to know. I am not too picky about the text, but I do need the textbook have some type of algorithmically generated homework and quizzes. I use MyMathLab, but many other systems would work.

Online homework and quizzes are the vehicle students use to actively learn the content in the class. Students complete a homework assignment and quiz on each section in the textbook that I cover. After completing the assignments, students have learned the basic procedure required in the course. However, a deeper understanding occurs when a student must apply this information in a realistic context like the context in the project.

I find that students are not very good at reading textbooks, so I have produced short videos for all of the content in the textbook. Each video is no more than 10 to 15 minutes long (a typical students attention span and covers very specific objectives. This allows me to utilize them in a just-in-time manner. As the need occurs, I can email video links to students when they are stumped on a particular question for a homework assignment or quiz. Below are several videos appropriate to the student to teacher ratio project.

If the video for your textbook is in smaller segments that you can link to, you can probably utilize those videos to help your students learn the calculus concepts.

Technology Resources

calc_smMost students enter the class with a good sense of how to use their graphing calculator. However, the graphs that a graphing calculator produces are not suitable for a professional report. The graphing calculator is an excellent tool for students to use, but Excel produces publication quality graphs and is frequently used in a business environment. To complete a project and the accompanying technical memo (or other assessment), student need to use a graphing calculator and Excel to analyze the data. Additionally, I require students to use Mathtype to produce their technical memos.

The technology skills are learned via a series of assignments designed to give them the appropriate skills with a graphing calculator and Excel. Each week students complete these assignments and accomplish several steps in the solution strategy. In addition to keeping the student on task, it also allows the instructor to check the student’s progress on their project.

For this project, students complete four technology assignments:

Technology Assignment: Scatter Plot DOC | PDF – Students make a scatter plot on a graphing calculator and in Excel of the student to teacher ratio in the state they have been assigned.

Technology Assignment: Regression Model DOC | PDF – Student use the scatter plot from the first technology assignment and find linear and quadratic models of the student to teacher ratio data.

Technology Assignment: Rational Model DOC | PDF – To account for the asymptotic behavior that is expected from the student to teacher ratio data, students model the number of students and teachers separately. Then the functions are combined to create a ratio function.

Technology Assignment: Limits at Infinity DOC | PDF – This tech assignment helps students to compute the asymptotic behavior of the ratio function they have produced.

I grade the documents students produce for these assignments and point them in directions appropriate for their data. I try to redirect students who are on wild goose chases and congratulate students making good progress. Throughout these assignments I want them to think about what needs to be on a good graph and how to incorporate the work they are completing in their final technical memo.

Solution Strategy

Most students begin graphing the student to teacher ratio in each year. They must calculate these ratios from the NCES data. As an example, the scatter plot of the student to teacher ratio for the entire United States from 2000 to 2006 looks like the graph below.


A superficial approach to this question is to model the student to teacher ratio with a linear or quadratic function. In the case of the entire US, this leads to functions that are decreasing.


Both of these functions predict that the student to teacher ratio will eventually reach zero. This is not likely to happen. What is more likely is that the ratio will begin to level off. To get that type of behavior, the students turn to modeling the students and teachers separately. The resulting functions are divided to give a ratio function.


Finally, students take the limit as the input approaches infinity to find the level the student to teacher ratio approaches in the long term.

As outlined with the US data, the strategy does not seem difficult to follow. However, each student is only using these graphs as a guide to complete a similar analysis on the state they have been assigned to analyze. Their data may or may not behave the same way as the US data. For instance, in many states the number of teachers is decreasing. Modeling the data with a polynomial leads to a function that will equal zero. The resulting ratio function then has a vertical asymptote. Students must model teachers with a function that is non-negative to eliminate this type of behavior. Most students examine several different options for their ratio to find one that behaves as they expect it should behave based on the data.