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Tuesday / June 18

Opening the Door to Math Screeners to Inform Instruction

Did you hear about the guy who doesn’t like going through screen doors?
He said he does not want to strain himself!

I found that using screeners in math takes some of the strain out of teaching and learning. The terms screener, diagnostic test, and inventory test are used to describe a tool that is generally used prior to instruction to identify strengths and weaknesses (misconceptions) in student understanding of specific skills and concepts. Test items similar to those on a screener may also be used during or after instruction to provide formative assessment. The purpose of using screeners is to gather evidence that will actually be used to inform targeted teaching and learning.

Have you ever begun a math class and quickly discovered that some or many students were not quite ready for the lesson? When was the last time you realized students gave incorrect responses on a test because they evidently misunderstood some basic concepts — even though you thought they were ready for the test? The use of short math screeners to probe key areas — before, during, or after instruction — coupled with targeted interventions, is one way to keep out such instructional bugs.

Fractions and Long Division

The study of fractions and division are key areas where teachers have expressed a need for screeners at all grade levels. Because fluency with fractions and division are critical in grade school and beyond, and because students of all ability levels often need a refresher to keep from being left behind, it makes sense to have short diagnostic tools readily available for such topics. It should be noted that according to a Carnegie Mellon-led study, “elementary students’ knowledge of fractions and of division uniquely predicts those students’ knowledge of algebra and overall mathematics achievement in high school, five or six years later.”[1]

My Experience with Diagnostic Testing, Error Analysis, and Intervention

I began my mathematics teaching career during the 1970s in an individualized setting in a middle school. Working with students one-on-one taught me the importance of using diagnostic testing and error analysis to pinpoint instruction and intervention. In that setting I formed my belief that simply marking a math problem correct or incorrect provides limited information to both the teacher and the student. The teacher needs to know the nature of the error to properly inform instructional activities. The student needs to know what he/she understands as well as know of any specific misconceptions he/she may have. Why would you “reteach” an entire algorithm when you can focus on just that portion of the algorithm that is misunderstood?

It is important to differentiate a “random” error from a misconception. Teachers should not be overly concerned about errors that occur due to carelessness, inattention, or incorrect recall of facts (so long as those errors are not excessive). But when students consistently make the same type of error, say, two or more times, then teachers should consider addressing the need for possible intervention. Misconceptions generally include errors related to concepts (such as not understanding the meaning of the denominator of a fraction) and errors related to procedures (such as not correctly performing the steps of a division algorithm). This is where screeners tied to error-analysis tools come into play. Such tools save teachers and students valuable time in teaching and learning for success. As we know, errors do not automatically correct themselves. If not addressed, some errors will repeat themselves during grade school, high school, and even college.

It should be noted that many misconceptions arise due to an incorrect use of algorithms — algorithms that often were engrained through rote memorization rather than through conceptual understanding. In such cases, providing (additional) meaningless drill is not the answer. Rather, a targeted approach — based on teaching for understanding — that specifically addresses the misconception is a more efficient and long-lasting solution. Often, engaging students in an alternative algorithm or approach is the way to go.

Young Adults Often Make the Same Errors as Young Students

After my middle-school career, I taught developmental mathematics in a learning laboratory at the community college level. There I discovered that many of the systematic errors made in basic math (whole number computation, fractions, early algebra) at this level were similar to those I had found during my early teaching experience. A community college student who computes ½ + ½ and obtains 2/4 probably has had the same misconception throughout grade school and high school.

Sample Diagnostic Tools

Over the years, I have recorded student error patterns that I uncovered. I also examined a wide body of academic research to study the nature of additional error patterns.

Here are some links to sample diagnostic tests with error analysis and pinpointed interventions you can use with your students. If you have any questions about how to use these diagnostic/intervention tools, feel free to contact me.

The above material is from Strategies for Teaching Fractions: Using Error Analysis for Intervention and Assessment and Strategies for Teaching Whole Number Computation: Using Error Analysis for Intervention and Assessment.

Both sets of diagnostic tests are organized into easy-to-implement parts. A teacher may implement just one small part — or an entire test. Each Diagnostic Test is followed by an Item Analysis Table that keys each incorrect student response to a specific (numbered) error pattern. This is followed by a section that describes each error pattern in detail — directly related to the foils — and then suggests hands-on, research-based Intervention Activities that focus on teaching for conceptual understanding. The interventions include short learning snippets, mathematical discourse questions, hands-on activities, alternative algorithms, estimation, instructional games, language development, student error searches, and more.

In conclusion, efficiently and effectively reaching all students often hinges on opening the door and swinging into action with screeners tied to powerful intervention tools.

David will be facilitating workshops this summer on teaching whole number concepts and computation (July 19) and on teaching fractions (July 20). The workshops will be in Buffalo Grove, IL and are based on his Corwin titles. Details are at

[1] Siegler, R.S., Duncan, G.J., Davis-Kean, P.E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M.I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science published online June 14, 2012, Retrieved from

Written by

David B. Spangler is the recipient of the 2014 Lee E. Yunker Mathematics Leadership Award from the Illinois Council of Teachers of Mathematics. The award honors an Illinois teacher for providing outstanding resources to mathematics teachers.

David has devoted his professional career of more than 40 years to mathematics education. He began as a middle-school mathematics teacher in an individualized setting. Later he taught at Triton Community College, where he gained direct experience interacting with struggling students in a developmental math laboratory. Currently he teaches mathematics methods courses through National-Louis University and ActiveMath Workshops, a professional development company he co-founded in 1994 ( Some of the methods courses and workshops he facilitates address special needs students, intervention, computation, and error analysis at the elementary level. David has literally worked with thousands of students and teachers during his career. He has authored several books and has written numerous articles for mathematics journals, such as the popular “Cartoon Corner” for Mathematics Teaching in the Middle School.

As an educator, David’s goal has always been to teach mathematics for meaning rather than in a way that promotes rote memorization. His Corwin titles, Strategies for Teaching Whole Number Computation and Strategies for Teaching Fractions were written to help teachers achieve that goal.

David lives with his wife, Bonnie, in Northbrook, Illinois. They have three grown children, Ben, Jamie, and Joey.

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