What’s in a Name? Think-a-Loud Protocol

By Dawn Woods, RME Elementary Mathematics Coordinator, and Marilea Jungman, RME Project Specialist

In January, Research in Mathematics (RME) conducted think-aloud protocols with second, third and fourth grade students at Nebbie Williams Elementary School in Rockwall ISD. The school volunteered to participate in the think-aloud protocol study that was designed to enrich the development of the ESTAR Universal Screener.

Background: What is the ESTAR Universal Screener?

The Elementary Students in Texas Algebra Ready, or ESTAR, is the latest intitative within the Texas Algebra Ready (TXAR framework) to support elementary students in the state of Texas to achieve a high level of preparedness in mathematics. The ESTAR Universal Screener is being designed to help educators identify students who may need additional support in becoming algebra-ready in the elementary grades and will be aligned with algebra-readiness knowledge and skills articulated in the revised Texas Response to the Curriculum Focal Points.

This document, based on the revised TEKS adopted in April 2012, identified critical areas of mathematics instruction in a framework for sequencing and developing curricula at each grade level. This document provides the content of the ESTAR Universal Screener and will be organized around foundational, bridging, and target knowledge and skill levels and simultaneously includes items written to target four levels of cognitive complexity - research indicates 4 areas critical for mathematics success: procedural understanding, conceptual understanding, strategic competence, and adaptive reasoning. Data generated from the screener will be reported in a format that helps teachers make informed decisions about the content and structure of mathematics instruction in the classroom.

Why is the ESTAR Universal Screener important?
Although performance standards are in process of being established for the State of Texas Assessments of Academic Readiness (STAAR), data from 2012 indicate that 3rd grade students responded, on average, to only 30 of 49 mathematics items correctly (61%) while 4th grade students responded, on average, to only 32 of 48 (or 66%) of mathematics items correctly (Texas Education Agency, 2012). These data speak to a need for early identification of students who may be struggling to learn critical mathematics content. One of the research steps in developing the Universal Screener is to conduct student interviews, also known as think-aloud protocols.

What is a Think-Aloud Protocol?
The purpose of a student interview or think-aloud protocol is to transform a student’s covert thinking process into an observable behavior so that the thinking process can be documented and analyzed (van Someren, Barnard, & Sandberg, 1994). Basically, we ask the student to work through a small number of math items appropriate to their grade level, and to “think-aloud” as they work. This concurrent data capture is maximized through the notes and reflections of the interviewer, the use of audio/visual, and a field observer dedicated to recording the student’s thoughts, hesitations, and gestures verbatim.

Once the student solves a math item, the interviewer asks the student to reflect on his or her thinking process after the task is completed. This is called retrospective data collection. Here, the interviewer uses questioning, prompting or dialogues to encourage the student to talk about his or her thoughts about the math item. The repetitive nature in the questioning allows student’s initial thoughts to be repositioned, and in many instances, a student alights on the correct answer after a first-round wrong choice.

We use the student interviews to verify and provide validity evidence of misconceptions captured in item designs as well as learn how metacognition (Flavell, 1979) plays a role in the planning and strategies students use in mathematical problem solving. Furthermore, student interviews also provide valuable information about students’ sense of self-efficacy, which may be correlated to student’s academic achievement (Hackett & Betz, 1989) and predicts later success for elementary school students (Bandura, 1997; Joet, Usher, & Bressoux, 2011).

Summing It Up
Think-aloud protocol is a valuable qualitative research method that enables researchers to uncover and map thinking processes. As we begin to analyze our data from our project at Nebbie Williams Elementary School, we hope to not only provide validity to our items for the ESTAR project, but gain valuable insight into how students think about math and how metacognition and self-efficacy play a role in students mathematics achievement.

Thank you again to the wonderful teachers that helped us with our project!

3rd Grade
Ms. Jennifer McCurry and Ms. Melody Carrilo

4th Grade
Ms. Christine Gregory and Ms. Lana Edwards

2nd Grade
Dr. Marcella J. Hodges and Ms. Kathleen Elam

Bandura, A. (1997). Self-efficacy: The exercise of control. U.S.A.: Macmillan.  

Flavell, J. H. (1979) Metacognition and cognitive monitoring: A new area of cognitive-developmental inquiry. American Psychologist 34(10). 906-911.  

Hackett, G. & Betz, N. E. (1989) An exploration of the mathematics self-efficacy/mathematics performance correspondence. Journal for Research in Mathematics Education 20(3). 261-273.  

van Someren, M. W., Barnard, Y. F., & Sandberg, J. A.C. (1994). The think aloud method: A practical guide to modeling cognitive processes. London: University of Amsterdam, Department of Social Science Informatics. 

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Strategies for Adding and Subtracting Decimals

By Cassandra Hatfield, RME Assessment Coordinator

Why is it that many kids struggle when adding and subtracting decimals? After working as a math specialist in elementary schools, I have some theories about why students that I taught in middle school often struggled with this concept.

When an elementary student is asked to solve 36 + 4, some strategies I have seen include:

Counting On

Making 10

The US Standard Algorithm

How would a child’s strategy change if we asked them to solve 3.6 + 0.4? The truth is, it should not:

• Counting on: If the student understands that tenths increase just as the ones place increases, they can still do this strategy.

• Making 1: Have you ever thought to relate making 10 with whole numbers to making 1 with tenths? Or making 100 with whole numbers and making 1 with hundredths? Consider teaching a lesson comparing the ways to make 10 with the ways to make 1 using tenths and the ways to make 100 using hundredths, or even tenths and hundredths. This will support students in solving with mental math instead of the standard algorithm.

• The standard algorithm: Let’s be truthful, when solving the whole number problem 36 + 4 with the standard algorithm, would you “line up the decimals?” Technically, yes. However, you weren't aware because the decimal was not visible. If students are taught to add whole numbers with the standard algorithm by “lining up the place values,” we can teach the same principal as it applies with decimals. The standard algorithm was invented to create an efficient uniform way of computing. The common theme in using the standard algorithm in addition and subtraction is that the place values are lined up. This ensures that the computation is accurate. This same principle applies when students add 3.6 + 0.4 - without a deep understanding of place value many students misplace the decimal point.
• In early elementary, students practice counting around the class or by multiples of whole numbers. Have you ever thought about counting around the class by increments of decimals?

Assessing Decimal Addition and Subtraction
In Teaching Elementary and Middle School Mathematics (2013), a suggested activity for formative assessment is to ask students to compute the sum of a problem involving different numbers of decimals places.
For Example:

75.35 + 4.7 + 0.671

For this assessment, interview students estimating the sum and then computing the exact answer. The goal of this assessment is to record “whether they are showing evidence of having an understanding of decimal concepts and the role of the decimal point. Note whether students get the correct sum by using a rule they learned in an earlier grade but have difficulty with their explanations. Rather than continue to focus on how to add or subtract decimals, struggling students should shift their attention to basic decimal concepts.”

Summing It All Up
If we place more value on mental math strategies, as well as lining up the place values when computing with the standard algorithm,  students may develop a deeper understanding for the skill.

Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally. Upper Saddle River, NJ: Pearson Education, Inc.

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Planning in the Problem-Based Classroom

By Saler Axel, RME Research Assistant

Problem-based classrooms provide natural learning opportunities for students by giving them the latitude to explore at their own pace. This type of student centered learning helps encourage exploration and in turn enhance comprehension. Many teachers though struggle with successfully implementing problem-based learning into their teaching because the approach is often very different from what they were taught.

Implementing a problem-based curriculum takes time and patience. When teachers plan lessons within this framework, it is imperative to remember that pre-planned lessons may not always follow a formal time table. Lessons need to be tailored to the students’ needs and fulfill the curriculum objectives, which doesn’t always allow for the rigidity of a schedule.

John Van de Walle offers teachers nine steps to successfully planning problem-based lessons in his book Elementary and Middle School Mathematics: Teaching Developmentally (2013).

Step 1: Begin with the Math! Consider what you want your students to learn by thinking in terms of mathematical concepts instead of skills. Students will better comprehend and retain new information when you approach your teaching in this manner.

Step 2: Consider Your Students. Begin by thinking about what your students already know. Consider what background knowledge they need and whether they have enough to begin or whether they will require a review. What do you expect may cause your students to struggle? How can you best present mathematical concepts to match your students’ prior knowledge base?

Step 3: Decide on a Task. Use Van de Walle’s book to help you compose a task that will best match the lesson and concept you plan to teach. Remember, not all tasks need to be complex or elaborate; simple can be better!

Step 4: Predict What Will Happen. Predict what your students will do with the presented task. Make sure that each student has the opportunity to participate and benefit from your lesson. Students may approach tasks differently, but it’s important that each student learns new skills. If you feel unsure about whether your task will benefit everyone, reconsider. Does the task help accomplish teaching the concepts you set out to teach?

Step 5: Articulate Student Responsibilities. For almost all tasks, students should be able to tell you:

• What they did to get the answer. 
• Why they did it that way. 
• Why they think the solution is correct. 

Consider how you expect students to share the information above. Consider asking students to answer in several different formats throughout the year. Be clear that everyone will be expected to provide this information when their tasks are complete.

Step 6: Plan the Before Portion of the Lesson. It is important to prepare students for the task at hand by first encouraging them to quickly work through easier, related tasks. This can better familiarize students with your expectations of each task and refresh their memories of past-presented information.

Step 7: Think about the During Portion of the Lesson. Consider your predictions. What types of accommodations or modifications can you provide in advance for students that will likely need extra help? What types of extensions or challenges can you offer students who finish before their peers?

Step 8: Think about the After Portion of the Lesson. Determine how your students will present their. Consider the best way to assess your students’ learning. How will you be assured of their comprehension and ability to retain any new material?

Step 9: Write Your Lesson Plan. Now that you have considered your lesson in such detail, this step should come easily! Below is a possible lesson plan outline format:

• The mathematics or goals. 
• The task and expectations. 
• Materials needed and necessary preparation. 
• The before activities. 
• The during hints and extensions for early finishers. 
• The after-lesson discussion format. 
• Assessment notes (whom you want to assess and how)

Summing It All Up
When planning lessons in a student-centered, problem-based classroom, remember that your students’ needs and learning styles should heavily influence what tasks you implement in the classroom. Take some time this week to plan a mathematics lesson using these nine steps.

What tasks will best meet your students’ needs?

Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally. Upper Saddle River, NJ: Pearson Education, Inc.

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Don’t Tell the Answers, Just Ask the Right Questions

By Sharri Zachary, RME Mathematics Research Coordinator

I recently came across an interesting newsletter that highlighted 8 tips for asking effective questions in the mathematics classroom. In this newsletter, teachers are encouraged to never say what a student can say. Rather than tell students what to do, the practice should be to stimulate thinking and deepen students’ conceptual understanding of mathematical concepts. To be able to effectively do this, teachers have to make a commitment to continually develop their mathematics knowledge so that they can ask questions that will help their students make connections between concepts.

In order to improve at asking effective questions, it is suggested that math teachers consider the following tips:

1. Anticipate Student Thinking. Consider different ways or approaches students may take to solve problems, including possible errors or misconceptions, and form questions that will prompt students to be reflective about their problem solving.

2. Link to Learning Goals. Ask questions that connect back to previously taught concepts. Consider the following example:

Learning goal –Understand that a unit fraction can represent a point on the number line, a distance between two points, and magnitude.

Student problem: Represent the fraction ⅕ on a number line. 
Possible questions:

• Why is it possible for ⅕ to be represented as both a point and a distance between two numbers?
• How would you describe the unit interval?

3. Ask Open Questions. Open questions allow for differentiation because student responses may vary depending on the level of understanding a student has about the topic.

Closed question: Is ²/₇+²/₇+²/₇ equivalent to ⁶/₇ ?
Open question: What other equivalent expressions can be written to represent ⁶/₇ ?

4. Ask Questions that Actually Need to be Answered. Avoid asking rhetorical questions because they provide students with an answer without allowing them to engage in reasoning about their answer (“Asking Effective Questions,” 2011).

5. Incorporate Verbs that Elicit Higher Levels of Bloom’s Taxonomy. Include verbs from Bloom’s list in your questions because they require students to tap into specific cognitive processes that engage thinking (“Asking Effective Questions,” 2011).

6. Ask Questions that Lead to Conversations with Others. Students often benefit from mathematical conversations held with their peers. However, it may be difficult for students to initiate this conversation. This allows for students to discuss the “big ideas” within a given topic.

7. Keep Questions Neutral. Avoid prefacing questions with qualifiers such as easy or hard, or offering verbal and non-verbal clues, facial expressions, or gestures. Qualifying questions may prevent a student from thinking through the question or may deter them from answering.

8. Provide Wait Time. Wait time encourages thinking and provides students with an opportunity to formulate their thoughts into words.

Summing it All Up
The goal, in asking effective questions, should be to help students focus their thinking about problems rather than lead to a solution. By stimulating their thinking, they will gain a deeper understanding about a concept that will lead them to make connections to other mathematical concepts. Don’t tell students what to do but ask questions that will lead them to the right answer.

To read the entire article:
Asking effective questions. (2011, July). Capacity Building Series. Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_AskingEffectiveQuestions.pdf

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Students Managing Money

By Saler Axel, RME Research Assistant

What are some helpful, creative ways to clarify your students’ understanding of money? What activities can you implement into your teaching to help teach the value of money and instill an appreciation and respect for it?

Several years ago, I read Check Out These Checkbooks: Real Banking for the Classroom in a 1999 edition of Teaching Children Mathematics by Abby Tuch. She established a classroom store filled with goodies that her third grade students could save for and purchase with play-money. They students learned to write checks and account for how much money they had by balancing their checkbooks. As a second grade teacher, I incorporated many of Tuch’s ideas into my own lessons and created a unit that promoted the discovery of money management through on-going life skills and mathematics exploration. Here is another similar unit plan from Scholastic called Creating a Classroom Economy Unit Plan.

By implementing a similar unit in your own classroom, your students can become more familiar with various denominations and learn how to save and spend responsibly. They will become familiar with words such as ledger, deposit, debit, balance, account, saving, spending, and banking. Students will use play-money, learn to keep an accurate and updated balance sheet, receive a salary to work classroom jobs, and frequently shop at the classroom store.

Before beginning this unit, your students should be familiar with dollars, coins, and multi-digit addition and subtraction. The Common Core State Standards (CCSS) for Mathematics in second grade states that students should be able to “solve word problems involving dollar bills, quarters, dimes, nickels, and pennies using $ and ¢ symbols appropriately.” The activities in this unit will help you as an educator pose real-life, project-based word problems to which your students can relate.

Image from Scholastic: My Classroom Economy

Ready to begin? Introduce money-management to your students. Ask them to contribute ideas and discus why it’s important to both save and spend responsibly. Talk to them about what people use money for (buy groceries, buy a home, buy toys, etc.) and what may happen if someone spends all the money they have.

Pass out the budget worksheets that your students will use throughout this unit’s duration. Teach them how to fill it in appropriately and revisit how to correctly calculate money problems through multi-digit addition and subtraction. Students can practice filling in their budget sheets with answers they calculate from presented word problems, as discussed in the CCSS. Spend several lessons teaching word problems and how to correctly use the budget sheets so you can assess their readiness for the next step … the classroom store!

The store is this unit’s highlight! Your students will love shopping for new trinkets that they can take home. The opportunity to spend at the store encourages students to evaluate how much money they have and determine whether they should shop now or save for later! When students make purchases, they must correctly count out the appropriate tender. To challenge your shoppers, you may require them to pay using specific tender (only using quarters or using the least amount of coins possible, etc.).

Image from Scholastic: My Classroom Economy

To earn spending money, all students work at classroom jobs for a set salary. Jobs may include routine “classroom jobs” such as black-board eraser and line leader, or more unit specific jobs like the store cashier. Explain to your students that everyone needs to work responsibly and carefully, just like in the real workforce. At your discretion, jobs may rotate every couple of weeks. Students should be paid (receiving play money) on a weekly basis, document their earnings using the budget sheets, and store their money in a designated place. A great partner activity: ask students to review a partner’s budget sheet and double check that all savings match the sheet’s total line.

Just because this unit runs smoothly for a few weeks doesn’t mean it has to end. I ran this unit year after year from Thanksgiving to summer vacation by increasing the mathematical expectations. I began introducing store coupons (ex: 25% off) which students could earn for good behavior (a great classroom management tool). These coupons were fun to earn and required computation with percentages later in the school year. Store sales are also ways to have students calculate adjusted prices before purchasing items.

Where can you locate enough merchandise to stock your store? Don’t be afraid to ask parents, administrators, or companies in your community for help! Ask parents to donate items with company logos or keep your unit in mind when they’re at the dollar store. Great store merchandise includes pencils, erasers, stickers, children's books, pencil sharpeners, bracelets, small notepads, coloring books, puzzle books, hats, or t-shirts.

Summing It All Up
By implementing a fun monetary unit into your classroom, you can challenge your students’ mathematics skills and knowledge of money while also teaching important life skills such as saving and spending responsibly. What are ways you can add even more educational excitement to this unit? 

Tuch, A. (1999). Check out these checkbooks: Real-life banking for the classroom. Teaching Children Mathematics, 5(7) 422-429.

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