Invented Strategies

By Saler Axel, RME Research Assistant 

In traditional classrooms, students are often taught one or two strategies for whole-number computation. They memorize rules to compute different operations. At first glance, teachers may mistakenly think their students “get” how to compute. What is often the case though is that students may be able to compute using a tried and true method, but cannot explain why it works. When students attempt cousin items that do not read exactly the same as what they are used to seeing, they may struggle or may not correctly calculate an answer.

When students learn how to compute algorithms, but do not learn the concepts behind them, they miss important stepping stones. By teaching students how to invent strategies, they learn what methods work best for them and which will better serve them in the “real-world.”

According to John Van de Walle in Elementary and Middle School Mathematics: Teaching Developmentally, invented strategies positively impact students’ academic success.

1. Students make fewer errors. When students compute with strategies they understand, they make fewer errors. When students make errors and do not understand the concepts behind their actions, they may have a far more difficult time fixing their efforts.
2. Less re-teaching is required. Teaching conceptual understanding is time consuming, but worth the effort! Not only can students gain the strategies necessary to be more successful in mathematics, the time spent teaching them is meaningful. When students know the “how” of computation but not the “why,” more re-teaching is necessary to help students develop computational skills.
3. Students develop number sense. Students’ development and use of algorithms provide a deeper understanding of the number system.
4. Invented strategies are the basis for mental computation and estimation. Mental computations are invented strategies. When students are taught how to use invented strategies, they are being taught mental computation. There is therefore little need to provide direct lessons in other computational formats or how to do mental math.
5. Flexible methods are often faster than the traditional algorithms. Van de Walle provides the following example to clarify: Consider the product 64 x 8. An invented strategy may be to calculate 60 x 8 = 480 and 8 x 4 = 32. Then find the sum of 480 + 32 which is 500 + 12 which equals 512. A student that uses a traditional algorithm will likely spend more time than someone that uses an invented strategy such as the one above.
6. Algorithm invention is itself a significantly important process of “doing mathematics.” When students invent successful computation strategies, their confidence in mathematics is strengthened. Younger students have been traditionally taught to compute algorithms without understanding why they work or being given the latitude to create their own methodologies. Van de Walle suggests that by opening the door to invented strategies, elementary students gain a valuable view of “doing mathematics.”

Van de Walle gives some examples of invented strategies with multiplication, such as useful visual representations, complete-number strategies (23 x 6 = 23 + 23 + 23 + 23 + 23 + 23 = 138), partitioning strategies, compensation strategies, and using multiples of 10 and 100.

By shifting your practice from teaching students traditional methods to increasing students’ awareness of how computation works, you can provide a solid foundation to enabling the use of invented strategies in mathematics.  As you teach, remember that more math drills is not the answer. Find which facts the student is struggling with and what current strategies they are using on the facts they do know. Break students into teams and challenge them to come up with multiple ways to solve a problem while always explaining how they got the answer. Being able to explain how students came to their answer is essential.

Consider your students. What strategies will you use to best encourage their mathematical thinking and “doing?”

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. 

Pi Day Activities!

By Cassandra Hatfield, RME Assessment Coordinator and Dawn Woords, RME Elementary Math Coordinator

Mathematician: Pi r squared 

Baker: No! Pie are round, cake are square!

Are you looking for a meaningful way to celebrate Pi Day in your classroom? If so, here is an assortment of resources, visuals, and activities that incorporate pi.

Pi in a Musical Sequence
With student selected musical notes, Pi in a Musical Sequence converts the first 10,000 digits of pi into a musical sequence, allowing for students to “hear” that the value of pi does not have a pattern. If you have students who are visual learners, you could adapt this activity by having 10 students sit in chairs at the front of the class with a paper labeled with the digits in order from 0-9. As you (or the class) recite the digits of pi, have each child stand up when they hear their digit called.

A Visual of Pi
For a great discussion opener, show students this animation of pi that illustrates the relationship between circumference and diameter without directly showing the diameter.

Taken from Wikipedia Commons  - Pi Unrolled Slow

Discovering Pi 
This NCTM Illuminations Lesson, geared for middle school and up, has students measure the circumference and diameter of circular objects. Then students calculate the ratio of circumference to diameter and find the average in attempt to discover the value of pi. This activity can be extended to include a technology application by having students calculate the ratio of circumference to diameter then create a scatter plot and find the line of best fit.

Discover the Formula for the Area of Pi
This NCTM Illuminations Lesson encourages students to develop the formula for the area of a circle. You can use the activity sheets in the lesson or paper plates work well too!

We also found a great video from Yummy Math where the area of a circle is explored. You can find that video here. 

Hats Off to Pi
Here is a lesson we wrote up to use for Pi Day with Grades 5-8. How are hat sizes determined? Once students have discovered pi we’ve written an extension activity to have them discover that hat sizes are actually the diameter of their head!

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Screener vs. Diagnostic

By Savannah Hill, RME Professional Development Coordinator

One project we are involved with at RME is an initiative with the Texas Education Agency and Education Service Center, Region 13 called Middle School Students in Texas Algebra Ready (MSTAR). It began in the summer of 2010 with the goals of (1) improving overall mathematics instruction, and (2) impacting student achievement. MSTAR is comprised of three lead components structured and integrated to support students and teachers in grades five through eight to achieve mathematics success: the MSTAR Universal Screener, MSTAR Diagnostic Assessment, and MSTAR Professional Development.

After talking with many teachers, we have found there is some confusion on the different ways to utilize the MSTAR Universal Screener and the MSTAR Diagnostic Assessment. The intent of this blog is provide a short description of each of these components and how they should be implemented.

MSTAR Universal Screener 
The MSTAR Universal Screener is designed to be administered to all students and identifies studentsʼ level of risk for not being ready for algebra.  The Universal Screener helps teachers make two important decisions within the Response to Intervention (RTI) process:

• Identify students on-track or at-risk for meeting expectations in algebra and algebra-readiness.
• Determine the degree of intensity of instructional support or supplemental intervention needed for students who are at-risk for not meeting expectations in algebra.

Teachers monitor studentsʼ risk status by administering comparable forms of the MSTAR Universal Screener in fall, winter, and early spring.

MSTAR Diagnostic Assessment
The MSTAR Diagnostic Assessment is designed to address those students identified as struggling in Tiers 2 and 3. The diagnostic assessment is given after the MSTAR Universal Screener to those students in Tiers 2 and 3. Its purpose is to:

• Inform educators where a student is on a learning progression.
• Identify the underlying misconception(s) that caused the student to answer incorrectly.
• Identify students current understanding of algebra-related content.

None of the diagnostic assessments are tied to a particular grade level because there may be a 7th grade student who is struggling from misconceptions about 5th grade content. However, when the teacher decides which assessment the student will take, there will be some direction about which assessment may be better for each grade. The reports given will provide information that can be used to plan supplemental instruction. This assessment is not intended to provide screening information.

MSTAR Professional Development
The MSTAR Professional Development provides tools for delivering instruction to all students in achieving algebra readiness and supports informed decision-making based on the results of the MSTAR assessments. The MSTAR Professional Development academies were created to support teachings in preparing students for success in algebra. Trainings are available in face-to-face sessions and/or online. RME researchers, along with TEA, delivered Professional Development in three training sessions for the MSTAR project for the Texas Education Agency in spring and summer 2011 and 2012. The trainings were replicated across the state by certified trainers.

The MSTAR Universal Screener can be accessed through the Project Share Gateway at www.projectsharetexas.org. It can also be accessed directly at http://mstar.epsilen.com. This option will allow you to bypass the Project Share site entirely. Users will see an MSTAR icon after logging in. The same username and password is used for either option. For more information, you can also contact your local Educational Service Center.