By Clayton Edwards, Posted November 24, 2014 – 

Getting your students to write and speak mathematically isn’t as simple as it may sound. This post will explore how to make the process a little easier. In part 1, I wrote about modeling these traits and listed activities that I use each week to make mathematical understanding a classroom staple. In this post, we will look at the time needed for student success in writing and speaking mathematically, connections to the Common Core Standards for Mathematical Practice, and some final thoughts on what can be expected after implementation.

3. Provide Significant Amounts of Thinking, Writing, and Discussion Time 

Time always factors into instructional decisions. Many students cannot develop and record a complete explanation in a short amount of time. Instead of asking a question and expecting an answer in 5 seconds (probably alienating 90 percent of the class), give students an abundance of time so that you can have greater participation by students who have actually thought about the question. Students will be less likely to shut down if they know they have time to think and work. Here is how much time I give for each of the activities discussed in part 1.

A. Two Problems 

Give written explanations, discuss with two students, and save until the end of class for a whole- group discussion. 

Time allowed: No set time. Some students may take 2 minutes, whereas others may take 15 minutes. End-of-class discussion generally takes 3 to 5 minutes.

B. Task 

Time allowed: One full 60-minute class period for students to work in groups and discuss and submit answers and explanations. They are also given a full week outside class to finish anything that wasn’t finished during the class period.

C. Daily Question 

Time allowed: 10 to 15 minutes to brainstorm, write out an explanation, and select 3 students to discuss the daily question. Allow 5 to 10 minutes to discuss as a whole class.

D. Estimation 180 Questions 

Time allowed: 5 to 10 minutes to brainstorm and write a clear understanding of thinking, with an additional 5 to 10 minutes of discussion as a whole group.

4. Connections to the Common Core’s Standards for Mathematical Practice 

In all honesty, having students write about and discuss mathematics is an essential piece of what should occur in a mathematics classroom, according to the Common Core Standards for Mathematical Practice and the NCTM process standards. Let’s consider a few of the standards that I found to be the most related to the mathematical activities I have described. Since all my activities involve the same standards, I will refer to Dan Meyer’s Leaky Faucet task in each case. Note: In this problem, I changed the amount per gallon to $0.02, so be advised that the price will not match the original problem.

CCSS.MATH.PRACTICE.MP1 

Make sense of problems and persevere in solving them. 

When students begin the problem, they do not try to get an answer in a few seconds, but instead look for a starting point, perhaps an idea of what would make sense. In the students’ minds, they have formulated an idea of what does and does not make sense before the actual solving begins. This idea of “making sense” also occurs throughout the problem-solving process. One student multiplied the number of milliliters in 10 minutes by 10 to get 1 hour instead of 6. This student was skilled in making sense of the problem and catching her mistake. A second student divided by $0.02 instead of multiplying and conveyed that $4800 did not make sense if the price per gallon is less than $1.

CCSS.MATH.PRACTICE.MP3

Construct viable arguments and critique the reasoning of others. 

 Students often work together on these tasks and are asked to write justifications for each statement. Many times, group members will devise their own lines of logic and discuss the plausibility of each. This Leaky Faucet problem has many approaches, so communication flourishes. The first student’s approach was to find how many milliliters it would take to fill 1 gallon and then to multiply by 3 for the entire sink. Her partner took a different approach and found the capacity of the entire sink first, then converted to time. Both approaches were acceptable, but each student had to convince the other one that their process made sense and do it in an understandable manner. You cannot have an argument until you fully understand your own logic, and both students were able to prove their answers to themselves first.

5. Additional Thoughts 

Developing good written and verbal explanations does not happen in a day. Most students, including higher achieving and struggling students, are going to be blown away by these expectations. Anticipate a bumpy road initially. Expectations need to be modeled multiple times, and student frustration will occur. My suggestion is to stay out of the way as much as possible. Ask questions to help students clarify their thinking instead of directly answering them. Walk away from a group when it is clear that the students need more time to think through the problem. Direct those in need to other students, so that they have additional opportunities to explain. These techniques will help students to persevere, which is one of the greatest gifts that a teacher can instill in students, math related or not. You will be surprised at what students can accomplish when written and verbal explanations are expected!

Comment or question? Join the discussion by responding below.

  Clayton Edwards, @doctor_math and cedwards@spartanpride.net, is a middle level mathematics instructor at Grundy Center Middle School in Grundy Center, Iowa. He is interested in the mathematical learning of all students of varying ability levels through self-pacing, task-based instruction, and other methods.