The habits of mind embodied in this practice should enrich the problem solving experiences of math learners. Some of those habits can be described as looking for patterns, generalizing results, monitoring the solving process, checking the reasonableness of answers, making discoveries, and devising new avenues to explore.
Research has shown that early work on these habits, such as looking for patterns, can have a positive impact on later mathematical and language ability. Pedagogical methods that put these habits into practice are illustrated in the “Designing Math Curricula” webinar, in the video “Cluster Problems,” and in the discussion of extending several problems in the resource called “Opening Out.”
Strategies for Implementing Practice Standard 8
The raw material for implementation comes from the cultivation of good questioning about good problems. The videos from Inside Mathematics demonstrate practice standard 8 in action in classrooms. The slide presentation called “Making Bags of Apples” shows both the teacher-student interaction as well as the explanations about the ways students are looking for and expressing regularity in repeated reasoning.
Sources of Problems
You will find resources that contain rich problems including “Grade 1 Practice Problems” (part of a vast problem bank from Port Angeles, WA); “Calling All Students” (from Exemplars); the extensive Noyce collection of “Problems of the Month;” and the problems in “Guess-Check-Improve Strategy” from Victoria, NSW. These resources will assist teachers as they search for problems for a variety of grade levels and covering many content standards.
This web page provides the individual Problems of the Month (POM) designed to be used as a school wide investigation to promote a problem-solving theme and developed by the Noyce Foundation. Each problem is divided into five levels, Level A through Level E, to allow access and scaffolding for the students into different aspects of the problem and to stretch students to go deeper into mathematical complexity. POM are organized by progression and listed for specific grades for an alternate search route. An overview of this bold initiative is presented by video (12minute, 43 seconds) along with a classroom excerpt (4 minute, 36 seconds). Other text resources: Guidelines, POM Scoring Guidelines and an index of the mathematical concepts addressed by each POM are included in PDF downloads.
This page provides videos that support educators in understanding and implementing the eighth Mathematical Practice Standard of the CCSSM. Classroom scenes show teachers monitoring and elaborating on what students notice as they work through a calculation or a problem and strive for a general understanding.
This webpage from the state of Victoria, Australia, discusses the problem solving strategy of guess-check-improve. It illustrates the importance of working systematically, record-keeping, and using information gained from initial guesses to improve subsequent guesses. The page includes teaching strategies, sample problems, and a link to an article about problem solving strategies.
This 5-minute video presents a number trick that offers an opportunity for students to practice basic addition. The presenter explains how the trick works and suggests how students can customize it to use with friends and family. The page includes links to written instructions and a student activity sheet (both pdf), and a spreadsheet (xls) which helps explain the trick.
In this 1.5-minute video, mathematician Dr Ria Symonds describes a characteristic of numbers such as 7, which after repeated squaring of digits and summing of the squares, result in the number 1, the first natural number. This introduction invites an investigation to find other "happy" numbers, and to find out what happens to "sad" numbers.
In this 4-minute video, educators can watch a teacher deliver a fourth grade lesson on using number sense, arrays, and simpler calculations to solve a more complex problem, called a cluster problem. Watch this teacher check for understanding of multiplicative distribution and the use of arrays.
In this lesson from Illuminations, students explore and discover linear relationships. Linear patterns are identified, extended and described verbally, numerically and algebraically through three investigations. Using manipulatives and the linked applet, "Chairs", learners determine the number of chairs needed when the number of tables is known, and vice versa. Instructional plan, questions for the students, assessment options, extensions and teacher reflections are provided.
This sample constructed response problem for grade 3-5 students provides an opportunity to analyze patterns and develop algebraic thinking in the context of a telephone chain. It includes suggestions for adjusting the challenge level and a task-specific rubric with annotated samples of student work at each level. Free registration allows users to download a 9-page print-friendly version (pdf).
In this 47-minute webinar Grant Wiggins discusses the difficulty students traditionally have in transferring their knowledge to problem solving, especially in standardized testing situations, and how standard math curriculum affects students' perception of mathematics. He describes how real problems differ from typical exercises, and advocates using them as the foundation of a math curriculum to ensure that Standards are met and that students develop into independent problem solvers. His slides are included as a pdf download.
This interactive Flash activity gives students an opportunity to visually model and calculate six different types of fraction applications, all in the context of solving word problems. Types of problems include finding a fraction of a number, using a known part to find the whole or other part, and problems using the four operations. A video demonstration introduces the method, and then students work on problems. Teachers can track a student's progress throughout the problem sets.
In this 39-slide slideshare presentation students are guided through the process of using repeated reasoning to discover the number of apples in a specific number of bags. This slideshare presentation was developed by the Illustrative Mathematics Project to illustrate CCSS-SMP 8. Throughout the slideshare student and teacher dialogue is presented along with commentary by two observers.
The author of this article uses four NRICH investigations to illustrate how a teacher might extend the results of students' work as a springboard for developing deeper mathematical understanding. He models good prompting, e.g., "Tell me what you notice about the result" and "I wonder what would happen if ...", and warns against trying to lead students' thinking. [The problems discussed are cataloged separately.]
Kindergartners and first-grade students listen excitedly to a modified storybook to guide their geometry activities. Using pattern blocks to create first simple patterns based on the story, they go on to create more complex patterns and ultimately to look for numeric patterns related to the geometric patterns.
This article describes the teacher's role in promoting mathematical thinking and problem solving in the classroom—identifying critical teacher actions and decisions; considering how beliefs influence the teacher’s actions and decisions; and suggesting implications for teachers and students. The author contrasts two different approaches to the presentation of a problem and points out strategies that foster understanding, perseverance, and confidence in students.
This article presents the use of a problem-based instructional task in an elementary classroom. After estimating the number of blades of grass on a football field, students write letters to explain the results of their research. Teachers will learn how to implement this task and similar problem-solving tasks in the elementary mathematics classroom.