# Math Instructional Practices

**Mathematics Instructional Practices (MIPs)**

What follows are ten mathematics instructional practices (MIPs) that are meant to capture the nuances of the majority of mathematics instruction for the first two years of college mathematics. The use of technology for instruction is not its own category. Technology is illustrated as part of the implementation of various instructional strategies, but emphasis on technology itself is not an instructional strategy that can be easily summarized and is left to future exploration. The practices are also meant to describe the nuances of teaching face-to-face. The nuances of mathematics instructional practices for online courses will be left for future research. It is important to first establish a baseline for MIP that occurs in face-to-face instruction.

Remember,** no instructor will fall distinctly into any one category**. All of us will use a mix of each of these practices, to varying degrees. In the final instrument to measure instructional practice, you would be asked to what extent you use each strategy in your courses (at a specific level of mathematics) and you would respond on a scale. Likewise, many activities that you do in the classroom may span more than one category of instruction. The intent of the categories is to lend description to different nuances of instructional practice in mathematics.

You can download the detailed MIPs and the research that went into them here.

If you cite these elsewhere, please reference: Andersen, M. H. (2009) Mathematics Instructional Practices, www.teachingcollegemath.com and the version date (April 30, 2009).

**Lecture**

*Teaching by giving a presentation on some subject for a time period longer than 20 minutes. This instructional method includes the exchange of questions and answers between the instructor and students. The key characteristic is that the students rarely interact with each other during this learning process.*

Examples:

• The instructor presents a logical narrative on exponential functions using a whiteboard. The narrative includes definitions, example problems, and application problems. The instructor periodically asks if there are any questions about the material.

• The instructor presents a lesson on graphing lines using an overhead graphing calculator viewscreen to show students how changes to the algebraic function result in changes to the graph. The students follow along, each using their own graphing calculator and occasionally interject questions when they have a problem with the technology.

• The instructor uses PowerPoint and video from the Internet to present a lesson showing students how the path followed by a cannonball is modeled by a quadratic equation, and how to find that equation. Students with laptops click through the slides as they listen and watch the presentation.

**Collaborative Lecture**

*Teaching by giving a series of short, focused lessons intermixed with student-centered activities that solidify the concepts of the lessons or serve to introduce the next short lesson (DeLong and Winter, 2002). The interaction during the activities is primarily between students.*

Examples:

• The instructor begins class by showing the class a concept question that asks the students to choose which of four sets of data represent linear functions. After the students work to solve the problem in pairs, and share their results by responding with clickers, the instructor discusses the solution with the class and moves on to present a 10-minute lesson about one of the non-linear data sets, which can be represented with a quadratic function. This cycle of using a concept question followed by 10 minutes of lecturing continues for the whole class period.

• After giving a 15-minute lesson focused on solving a triangle using the Law of Sines, the instructor puts the students into groups and asks the students to solve two problems on their own. It turns out the instructor has purposely selected the second problem to have two possible answers and craftily pushes two of the groups towards the non-standard answer. When the groups present their findings to the rest of the class, this possibility of two solutions provides fodder to motivate the next 15-minute lesson on the possibility of more than one triangle for a set of data.

**Cooperative Learning**

*Including class time for learning that engages students in working and learning together in small groups, typically with two to five members. Cooperative learning strategies are designed to engage students actively in the learning process through inquiry and discussions with their classmates (Rogers et al., 2001).*

Examples:

• All of the students in the class find a partner and a spot at the whiteboards in the classroom. The instructor reads a factoring problem aloud and the students work together to solve the problem at the board. The students help each other within pairs and between pairs, asking questions and providing hints to each other. The instructor occasionally provides hints to pairs of students, but it is primarily students who are answering each others’ questions. Every few minutes, the instructor directs one person from each pair to move to the right, and reads a new question for the new pair of students to solve together.

• The instructor poses the following question to an algebra class, “How do you find the least common denominator for any set of fractions?” Students are given two minutes to think about the problem on their own, and then they join a group to solve the problem. After 8 minutes, each group presents their solution to the rest of the class.

• Class is held in a room with eight computer stations. Students work together in groups of three to complete an activity about inverses using a spreadsheet program. One student is designated as the computer-specialist, one student has responsibility for writing the responses to turn in, and the third student will present the results of their experimentation to the rest of the class.

**Inquiry-based Learning**

*Designing and using activities where students learn new concepts by actively doing and reflecting on what they have done. The guiding principle is that instructors try not to talk in depth about a concept until students have had an opportunity to think about it first (Hastings, 2006).*

Examples:

• Students use colored red and black counters to represent negative and positive integers. Students model the additions of signed numbers by matching up and removing pairs of red & black tiles until there are no more pairs. After several problems, each student proposes a “rule” for how to add integers of various types.

• Students to use spreadsheets or the data table on a graphing calculators to explore how a change in the function equation affects the data it produces. Students propose an explanation for what they see and then devise and conduct tests of their hypotheses.

• Students use the slider bars on an interactive online model to experiment with the effect of changing a coefficient on the graph of the function. Students work in teams to come up with a precise definition for how the coefficient affects the graph.

**Emphasis on Application Problems**

*Including class time for students to solve problems based on data from real-world situations (present or past) problems that come from the partner disciplines of mathematics (e.g. Engineering, Chemistry, Biology, Physics, Economics, Business). *

Examples:

• During class, students set up two equations to model the protein and carbohydrate content for two different foods, and then they solve the system of equations.

• Students download data from the instructor’s calculator on the kinetics of a chemical reaction, find a curve to fit the data, and then make predictions based on the model they constructed.

• Students write mathematical equations to model the costs, revenues, and profits for a proposed business venture. Using the model, they find the break-even point for the venture and then present their findings at the end of class.

**Emphasis on Project-based Learning**

*Designing and assigning project work that requires students to solve a non-standard problem that requires a longer period of time than problems that would typically be assigned for homework or in class. There is often a research component where students must actively seek data, background knowledge, or formulas. Often the students work on projects in pairs or small groups. The final result of a project might include a written paper or a presentation on the findings.*

Examples:

• In an assignment that takes several weeks to complete, students are required to compile data from three different government databases, find functions to model the data, and then use the functions to make predictions about the future. The instructor reserves two days of class time for the students to present their findings to the class.

• Students in an algebra class are asked to design an experiment to test the assertion that the circumference of a circle is pi times the diameter. After writing a paragraph to describe how they are going to test the rule, and getting approval from the instructor, the student carries out several trials, records the data, analyzes it, and then writes up the results.

• Students work in groups to analyze statistics on the energy efficiency of one of the buildings on campus. Students have to determine (on their own) how to measure energy efficiency, gather the data, and analyze it. To present their findings, each group prepares a poster and then presents the results to the Supervisor of Buildings & Grounds.

**Emphasis on Multiple Representations**

*Teaching by including multiple ways (e.g. graphs, diagrams, algebra, words, data, manipulatives) to represent mathematical ideas whenever possible. The rule-of-four (representing a function visually, algebraically, numerically, or with words) is an example of multiple representations. *

Examples:

• The instructor describes the model of U.S. Postage rates (a nice piecewise function) in words. The students draw a graph of the function, the class contributes to constructing an algebraic formula for the piecewise function, and the instructor creates a blank table with inputs to which the students add the output results.

• The instructor shows the students how to multiply two binomials and the class works through several examples. Then the instructor uses a diagram of rectangles and squares to demonstrate the idea of multiplying two binomials and has the students each construct their own geometric example.

• The instructor puts the students in groups and gives each group a table of data. The students graph the data, find a function that fits the data, and then describe a situation that could be illustrated by the data.

**Emphasis on Communication Skills**

*Providing opportunities for students to practice their ability to communicate mathematical and quantitative ideas using both written and oral communications.*

Examples:

• At the beginning of each class, the instructor chooses two students to each present a selected homework problem. Each student presents their answer to the class, explaining each step to the class as they go.

• Ten minutes before the end of class, the instructor has the students write one paragraph about what they have learned in class that day. The writing assignments are turned in to the instructor, who chooses several paragraphs to copy (without student names), and then corrects the mathematical language, spelling, and grammar. The next day in class, each student receives a copy of the uncorrected student paragraphs to correct themselves. Then they compare their corrections with the instructors’ corrections to learn from what they missed.

• In an algebra class, students are required to answer every application problem in a complete sentence that summarizes both the problem statement and solution.

• Students in a trigonometry class have to write their own application problem (and solution) based on some situation in their own life. The assignment is graded on the clarity and quality of the writing, and the accuracy of the mathematics.

• Students take a mathematical version of a “spelling test” where the instructor reads five problems aloud and the student writes the problems down. Then the instructor shows the students the answers, and students correct their own work to learn from their mistakes.

**Mastery Learning **

*Designing summative assessment check-points into the instructional program where the student is tested on their mastery of a single topic (or subtopic). The instructor may coach students during class time or outside of class to help students who struggle with understanding the concepts while they are intensely focused on learning. Note that the student s do not receive partial credit for partially correct responses on mastery-based assessments.*

Examples:

• The course is taught in a self-paced lab format. Students cannot continue to the next module in the course until they pass the previous module with an 80% on the end-of-module assessment. The instructor is present during lab time to answer questions, provide coaching, and motivate students to stay on track.

• Using the course management system Blackboard, an instructor sets up check point quizzes for students to complete after class each day. The quizzes are algorithmically generated, and students may take the quizzes multiple times, but they must answer 80% of the questions correctly before the score is recorded.

• At the end of the semester, students are each given an oral exam to assess their mastery of the subject. Each question the instructor asks is graded on a pass/fail basis and the oral exam is a major component of the course grade.

• Midway through the semester, students take a Gateway exam on logarithmic and exponential functions. Problems are graded correct or incorrect. The students must get at least an 80% to have a score recorded for the exam. They may retake the exam five times over a period of two weeks.

**Emphasis on Formative Assessment**

*Making use of instructional strategies in the learning environment that assess where students are having problems so that students can learn more and learn better. (Gold, 1999)*

Examples:

• During class, the instructor pauses in a lesson to project a concept-oriented true/false question on the board. The students answer the question by holding up one of two index card “bookmarks” that says either True or False. The instructor quickly judges the number of responses, then after a quick discussion (and certainty that the concept has been understood by most students), they proceed with the lesson.

• At the end of a class, the instructor asks students to write down their muddiest point from class on a scrap of paper, and collects the responses. The next day of class, the instructor begins by addressing a few of the issues that were collected on the previous day.

• Before each class, the students log in to the online course management system to answer three open-ended questions about the textbook readings assigned for that day. During the next class, the instructor uses a projector to show the class selected answers as a way to guide the lessons each class period.

Version date: April 30, 2009