Remedying Problems Understanding the Base-10 Number System: A Short Intervention Rodger Armistead Acadia University, EDU 50G3 February 21 st, 2010

Summary of Problem  I teach Year 1 in a private, co-educational school in the United Arab Emirates. My students are between five and seven years old.  I perform periodic benchmark tests in numeracy. In December 2009, I administered a simple counting task involving student-created base-10 manipulatives.  Proficiency in base-10 activities is a precursor to proficiency in place-value and number operations.  Four children in particular had difficulty completing the task.  Their stated totals were highly inaccurate. I decided to investigate their results further.

My notes from the initial test 7 objects (seven units) 13 objects (one ten, and three units) 38 objects (three tens and eight units) Almudhaffar7 (counted 1:1)40 (counted 1:1)19 (counted 1:1, random) Maryam7 (counted 1:1)13 (counted 1:1)(declined) Saif7 (subitized)40 (counted 1:1)(declined) Sara7 (counted 1:1)40 (counted 1:1)100 During the test, quantities were arranged to allow subitization.(the quick identification of quantities without counting.) All students were able to count quantities containing only units accurately. Four students were unable to count quantities containing groups of 10. Of these, two students showed a marked lack of confidence when counting larger quantities.

Subject Description “Maryam” Female Emirati 65 Months Old 1.5 Years of School Student file is missing. “Almudhaffar” Male Iraqi 73 Months Old 2.5 Years of School Past teachers noted some difficulties communicating in English “Saif” Male Emirati 72 Months Old 2.5 Years of School Past teachers noted difficulties counting and communicating in English. “ Sara” Female Lebanese 72 Months Old 1.5 Years of School Past teachers noted some difficulties communicating in English I chose four research subjects according to the degree of inaccuracy on their benchmark tests.

Research Question  I collected several forms of baseline data, including a performance task, past anecdotal notes, and notes from past teachers.  The task was designed to eliminate the possibility of errors in manipulation and errors related to language.  I was able to establish that the problem persisted when counting in their native tongue, and that all students were able to manipulate the materials appropriately.  The problem appears to be related to a deficiency in Mathematics skills. Therefore, my research question became:  What are some ways I can help children having ongoing problems understanding the base-10 number system?

Relevant Literature  Many people have researched counting skills in young children.  Saxton and Cakir (2006) performed a large scale research study treating my research question specifically.  They proposed three key skills that have a positive correlation with success in base-10 tasks:  Trading  Partitioning  Counting-on  Van de Walle and Folk’s textbook (2001) proposes a wide range of games and activities targeting specific mathematical skills.  Alsuwaie (2001) provides a useful taxonomy for categorizing students with mathematical deficiencies.

Plan of Action  Based on the skills identified by Saxton & Cakir, I organised a programme of short numeracy games conducted during centres time and lunch time.  The games were modeled on those proposed by Van de Walle & Folk.  I conducted these activities over three weeks, and then repeated my initial benchmark test.  I then used Alsuwaie’s taxonomy to label the students according to the difficulties they were experiencing.

Data Collected during Partitioning Activity “Mariam”“Almudhaffar”“Saif”“Sara” First Session Re-counts; states total. States total correctly without re-counting. Re-counts, states (incorrect) total. Second Session Re-counts; states total. States total correctly without re-counting. Third Session States total correctly without re-counting. Re-counts; states total.States total correctly without re-counting. Students counted a quantity of objects, which was then separated into parts. They were asked “how many” again. They were deemed successful if they were able to state the correct quantity without counting.

Data collected during Trading Activity “Mariam”“Almudhaffar”“Saif”“Sara” First Session 3/3 successful attempts1/3 successful attempts Hesitant 0/3 successful attempts1/3 successful attempts Very confident Second Session 3/3 successful attempts 0/3 successful attempts Third Session 3/3 successful attempts 1/3 successful attempts2/3 successful attempts Students were asked to count a variety of objects, and trade them for a larger object when they reached a certain threshold. They continued until the first quantity was exhausted.

Data Collected during Counting-On Activity “Mariam”“Almudhaffar”“Saif”“Sara” First Session 0/3 correct attempts2/3 correct attempts 0/3 correct attempts Second Session 0/3 correct attempts3/3 correct attempts 1/3 correct attempts Third Session 1/3 correct attempts3/3 correct attempts A quantity of Unifix cubes was counted together with the students and placed in an opaque bag. Several more cubes were added, one at a time, and students were asked to state the new total of objects in the bag.

Re-test  After the intervention, I performed the benchmark test again with a quantity of base-10 units.  The quantity included two tens and two units.  Results: “Maryam” “10, 20, 21, 22” (correct) Alsuwaie’s taxonomy: Mixed Strategist “Almudhaffar” “10, 20, 21, 22” (correct) Alsuwaie’s taxonomy: Mixed Strategist “Saif” “10, 20, 21, 23” (nearly correct) Alsuwaie’s taxonomy: Mixed Strategist “Sara” “10, 20, 30, 40 “(incorrect) Alsuwaie’s taxonomy: Block Strategist

Student Profiles  Most students found a correct total during the re-test.  Some students showed deficiencies in some skills and strengths in others:  “Maryam”  Understood the principle of partitioning after the third session.  Highly confident during trading activity  Marginal understanding of counting-on activities  Accurate count during re-test  “Almudhaffar”  Never grasped the concept of number conservation (partitioning activity)  Confidence and accuracy in trading activities increased over time  Understood the concept of counting-on  Accurate count during re-test

Student Profiles (continued)  “ Saif”  Accurately stated total during all partitioning activities  Did not understand the concept of trading for larger quantities  Confident and accurate during counting-on activities  Nearly accurate count during re-test  “Sara”  Increasing understanding during partitioning activities  Inconsistent performance during trading activities  Inconsistent performance during counting-on activities, but increasing accuracy  Incorrect count during re-test

Conclusions  Most students seemed to benefit from the intervention.  The targeting of individual sub-skills as identified by Saxton and Cakir increased performance on the base-10 task.  Each student experienced difficulty in different sub-skills.  Mastery of all skills was not required to accomplish the larger task.  One student may benefit from further intervention; alternatively, she may be lacking in even more basic skills.  More assessment may be required to ascertain the nature of her learning deficiency.  Our year level may benefit from adopting a larger-scale intervention scheme if we want to help all the students having difficulty in base-10 performance tasks.

References  Alsawaie, O.N. (2001). Linguistic Relativity and Place Value Concept: The Case of Arabic. Paper Presented at the International Conference of New Ideas in Mathematical Education. Palm Cove, Australia, August 19 th – 24th,  Saxton, M. & Cakir, K. (2006). Counting-On, Trading and Partitioning: Effects of Training and Prior Knowledge on Performance on Base-10 Tasks Child Development, May/June 2006, 77(3), 767 – 785.  Van de Walle, J. & Folk, S. (2001). Elementary and Middle School Mathematics: Teaching Developmentally: Canadian Edition. Toronto: Pearson Education Canada.