The Math Challenge
Research-Based Interventions: Increase Communication in the Math Classroom
Adult learning theories encompass student-centered learning, increased student autonomy, collaboration among students, strategy development, and increased metacognitive practices by learners. Increased communication among students and between instructor and students is one instructional technique that incorporates all of these principles and facilitates math learning.
Building on Foundations for Success: Guidelines for Improving Adult Math Instruction identifies communication as a critical instructional and learning practice in the math classroom, which helps students develop problem-solving skills and integrate new information into previous knowledge. It also allows students to expand on their own thinking and the ideas of others by engaging in discourse about problem-solving methods and math concepts. The National Council on Teachers of Mathematics contends that communication and collaboration among students and the instructor in the math classroom foster learning from one another, increase active participation by learners, and facilitate better understanding of concepts as students integrate others' ideas into their own understanding.
The following narrative illustrates how communication fosters learning in the classroom.
[Source: ANI -1.2, About How Many]
Instructor description: One of my students went to a yard sale last weekend. She saw several pieces of furniture she wanted to buy, but she didn't have a measuring tape with her and wasn't sure whether the items would fit in her living room. She decided to pass on purchasing them. In class today, we completed an activity to help us think in different ways about what we as a group know about numbers and measurement.
I cut up strings of different lengths and placed a bunch at each of the four activity station tables. In groups of three, the learners were asked to go to a station and figure out how long each string was --without using a ruler, measuring tape, yard stick, or any other standard measuring tool. I asked for at least one person in each group to document their thinking and calculations while they worked through the problems together, and for the whole group to be able to explain their answers to the rest of the class when we were finished.
Several students seemed stumped at first, but I asked the group if anyone had ideas on how to approach solving these problems. One student offered that her husband uses his foot size to measure the length of a room because his shoe size is about 12 inches. This got the students thinking about other ways they could approximate length.
Students presented their work as follows:
James: Janice held the string in front of Kent so that it dangled down to the ground. She said that it was about half as long as he was tall. Kent said his height was 5 feet 10 inches.
Instructor: What did you do with that information?
James: We agreed that we needed to find out what half of Kent's height was to get a sense of how long the string was.
Instructor: Why did you want to find half of Kent's height?
Janice: Because in estimating the length of the string, we compared it to Kent's height and it seemed like the string length could be held up to his height about two times. We had him lie down on the ground and the string was almost two times his height exactly.
Instructor: Okay, so how did you approach that calculation?
Janice: I suggested dividing the 5 feet of his height by 2, and then the 10 inches of his height by 2.
Instructor: Why did you decide to do it that way?
Janice: I thought this would be an easy way to do it because it was small quantities, and it's the same as when I take a large number like 86 and break it down. I think, “What's 80 divided by 2?” and “What's 6 divided by 2?” and then I add the two answers together. For the string, I figured it was 2 ½ feet plus 5 inches. That is 24 inches + 6 inches + 5 inches = 35 inches.
Instructor: Are there other ways to figure out half of Kent's height?
Kent: I suggested converting my height into inches first, and then dividing by 2. This seemed less confusing to me because it was all the same unit. So, I figured 5 feet is 60 inches, plus 10 inches, is 70 inches. I divided 70 by 2 and got 35 inches.
Instructor: What did you discover when you tried different approaches?
James: We did both of the calculations and found that both methods resulted in the same answer.
Instructor: Why do you think that you got the same answers using different methods?
Group 2 measured a string of a different length.
Margaret: I was thinking about things in the room I could use to compare with the string. I remembered that the paper used in the computer printer is 8½ inches wide by 11 inches long.
Nick: We discussed which length would be easier to work with, and decided to use the long end of the paper to avoid dealing with a fraction.
Margaret: Next, we laid a sheet down along the string and counted the number of paper lengths that fit along the string.
Instructor: So, it seems you had several units you were working with. What decisions did you make next?
Shawn: I wondered if we should answer in the units of paper lengths or inches. We decided we could give the answer both ways, as long as everyone was clear about what the unit size was and we used the same one.
Margaret: The paper could be laid along the string four times. So, taking the length of the paper --11 inches --we then multiplied that number by 4 and got 44 inches.
Instructor: Does this approach make sense? Can you explain how you knew this would work?
Shawn: It does make sense because we used the paper as a unit of measure, kind of like a ruler.
Instructor: Oh, I see. Did anyone think of other ways to get an answer?
Nick: After we did all these calculations, I realized that since 11 inches is pretty close to 12 inches, which is a foot, we could have considered each paper length about a foot, and just subtracted a few inches.
Instructor: Group A, does that make sense? Does it come close to their other answer of 44 inches?
Janice: Yes, because four feet would be 48 inches, and if you take away an inch from each foot, you take away four inches, which leaves you with 44 inches. That's a good way to calculate it, Nick.
After the students determined the approximate length of different strings, they started using the strings to measure other items in the room.
Instructor: So you came up with many different ways to figure out the length of the strings. How would these informal measuring strategies work when you're at the store and you don't have access to standard measuring tools? What other situations have you found yourself in where you have needed to measure items and these approaches would be useful?
Check for Understanding
Identify where and how the different instructional strategies and math principles are reflected in the example. Use the questions below to guide your thinking. After you answer these questions on your own, refer to the analysis below the questions to check for understanding.
- What did the teacher do to encourage communication and collaboration in the classroom?
- What principles of adult learning theory, student-centered learning, and self-regulated learning are embedded in this activity?
- Which strands of mathematical proficiency are reflected? (See Appendix A.)
- Which Common Core State Standards for Mathematical Practice are reflected? (See Appendix A.)
- The teacher encouraged communication in the classroom in the following ways: Provided a small-group activity allowing for more participation by each student
The example reflects adult learning theory, student-centered learning, and self-regulated learning in the following ways:
- Created an interactive environment in which each participant could contribute based on his/her knowledge of measurement or understanding of converting from one unit to another
- Used “why” to ask students to justify and explain their thinking
- Encouraged students to provide guidance to one another on how to make comparisons and calculate the answers
- Provided opportunities to make decisions together as to how to approach solving the problem or presenting the answer
- Allowed students to share their strategies and judge the rationale of others
- Encouraged students to explain their reasoning and justify their answers
The following Strands of Mathematical Proficiency are reflected, as follows:
- The activity addresses a need that students have to understand size and measurement so they can operate in daily life more effectively; they learned that they could find alternative ways to measure objects and space when they do not have access to the standard measurement tools they would normally use.
- The task elicits knowledge the students already have and connects it to the current context; by using benchmarks of known measurements, the students were able to problem solve to find new measurements.
- The activity fosters student autonomy and encourages them to think together and ask questions of one another rather than rely on the instructor for all the answers.
- The activity provides the opportunity to work collaboratively and learn from their peers.
- The activity supports different roles and contributions by various students.
The following CCSS Standards for Mathematical Practice are reflected, as follows:
- Conceptual understanding
- Students further developed their understanding of linear measurement, and they learned that there are many ways to measure using different “units” of measure, whether standard units such as inches and feet or nonstandard units such as paper length or a person's height.
- Students connected previous knowledge to new situations when they used their knowledge of some units of measure along with their familiarity with personal or everyday objects.
- Students acquired knowledge from one another that will generate more knowledge and understanding as they repeat the task.
- Students further developed their sense of “benchmarks,” which is key to being able to think critically about the reasonableness of answers.
- Strategic competence
- Students represented the problem both verbally by describing how the measurements are similar or different to known quantities, and numerically by computing the answers.
- Students showed flexibility in the approaches they used by drawing on known measurements for comparison as well as estimation techniques.
- Students gained skills in estimation by gauging their answers, based on the lengths of known quantities.
- Adaptive reasoning
- Students thought logically about the relationships and concepts they were faced with, connecting knowledge of some measurements and spatial benchmarks with which they were familiar.
- Students explained and justified their conclusions to one another as they figured out the answers in small groups and presented their arguments to the class.
- Make sense of problems and persevere in solving them (MP1).
- Students found entry points for finding the length of the strings by relying on their prior knowledge and experiences.
- Students assessed their answers and determined whether they made sense.
- Students identified multiple ways to state the solution to the problem, using different unit measurements and estimation techniques.
- Construct viable arguments and critique the reasoning of others (MP3):
- Students explained their approaches to one another (such as how to break down larger numbers, estimation techniques, how to convert units), and they reached consensus.
- Students responded to questions from the instructor, explaining why they proceeded as they did and the math they used to find the answers.
- Students listened to the reasoning of their peers and were able to evaluate their thinking.
- Use fewer pieces of string and shorter lengths, cutting them closer to familiar benchmarks (such as 6 inches or 10 inches rather than 3.5 inches).
- Instead of using the strings, begin by asking the class to estimate what they think the length of the room is, and put the estimates on the board. Then ask students to estimate how long their own foot is. Ask them to consider whether their own foot length would be more or less than the actual 1-foot measurement of 12 inches. Using rulers, have them measure their own feet. Record these measurements on the board. Then ask students which foot would be best to use to measure the length of the room.
- Limit the resources available for measuring; for example, tell students that they cannot use sheets of paper.
- Provide a wide variety of lengths of string that are less easy to estimate, such as 17 inches or 32.5 inches.
- Ask students to determine lengths using metric measurements (such as meters or centimeters).
- Challenge students to estimate the height of the classroom or the height of the building.
Excerpt from U.S. Department of Education, Office of Career, Technical, and Adult Education. (2014). TEAL Math Works! Guide. Washington, DC: Author.
Find the complete report on the department's web site.