The Demographics

Campus Type:


Grade Span:


Enrollment by Race/Ethnicity:

  • African American 5.2%
  • Hispanic 20.0 %
  • White 61.2%
  • American Indian 0.3%
  • Asian 10.0%
  • Pacific Islander 0.0%
  • Two or More Races 3.3%

Enrollment by Student Group:

  • Economically Disadvantaged 14.8%
  • English Language Learners 2.4%
  • Special Education 8.9%

Mobility Rate:


Total Student Enrollment:



6th Grade Mathematics and English as a Second Language (ESL) Mathematics

The Challenge

Academic Language in Mathematics. Central to the discipline of mathematics education, the academic language of mathematics is essential in the acquisition of new knowledge and numeral literacy (Schleppegrell, 2007; Purpura & Ganley, 2014). Academic language is “the language that is used by teachers and students for the purpose of acquiring new knowledge and skills imparting new information, describing abstract ideas, and developing students’ conceptual understandings” (Chamot & O’Malley, 1994, p. 40). Academic language is a fundamental component of learning and is one of the most critical determinants of academic success (Francis, Rivera, Lesaux, Kieffer, & Rivera, 2006). “Teaching and learning the language of mathematics is vital for the development of mathematical proficiency” (Riccomini, Smith, Hughes, & Fries, 2015, p. 235) and a strong predictor of early numeracy knowledge (LeFevre et al., 2010; Purpura et al., 2011; Purpura, Hume, Sims, & Lonigan, 2011). Students’ understanding of mathematics concepts are inextricably bound to word identification and vocabulary (Capraro, Capraro, & Rupley 2010; Piccolo, Harbaugh, Carter, Capraro, & Capraro, 2008; Capraro & Joffrion 2006; Kotsopoulos, 2007). Acquisition of the vocabulary of mathematics should have the same emphasis as the academic language in other content areas (Pierce & Fontaine, 2009; Schleppegrell, 2010). Sufficient language development is requisite for the comprehension of word problems and the understanding that vocabulary may have multiple meanings across content areas especially in mathematics.

The Case

The evidence of this case study builds on the substantiation of the influence of increased vocabulary acquisition and academic language of mathematics for 5th grade mathematic students through the implementation of FASL’s Principles of Practice and Design Principles. The mathematics teacher in this study served students in intermediate 6th grade students as well as an English Learners’ math class (N=75). The teacher’s target objective was to raise the level of academic achievement through building the students’ academic language in mathematics. “Developing mathematics vocabulary knowledge allows adolescents to expand their abstract reasoning ability and move beyond operations to problem solving” (Dunston & Tyminski, 2013, p. 40).

The Solution

Repetition through variation and spacing. Herman Ebbinghaus (1885), a German psychologist, pioneered the experimental study of memory in 1885. He discovered that our brains are wired to forget and the importance of the spacing effect. The spacing effect is the phenomenon of spreading learning over time as opposed to learning the content in one sitting. Ebbinghaus found that learning is greater when spaced over time. During this study, the teacher spaced practice over several days. She limited the use of direct instruction and learning activities to short periods of time ranging from 10-15 minutes, and then repeated the same content in a variety of activities each day during the same week. She designed instruction with relevant connections and elaborated content by tying memories together by connecting new knowledge with prior knowledge. She used both repetition of content and variation of the learning activities. Repetition serves as a critical function in storing memory in the brain and constructing pathways to long-term retention of vocabulary. Variation deepens the understanding of new concepts and aids in the prevention of lackluster repetitions of the same learning activity. Variation can begin informally with a matching activity and move incrementally at a faster speed into a more rigorous activity such as a word problem in math. In this case study, the teacher had students participate in a variety learning activities using paraphrasing, summarizing, simplifying, and abstracting the knowledge to connect learning and increase long-term memory recall of vocabulary words in mathematics.

The first learning assignment began with the Talking Stick Activity. The teacher divided the students into collaborative groups and provide each group mathematics vocabulary words on note cards. Each table received a talking stick. The student who held the talking stick shared the word and definition with the classmates in their group. Each student had an opportunity to hold the talking stick and share a vocabulary word with their classmates allowing each student to become an expert on that word.

The next variation in the design of learning occurred by having students work collaboratively in groups to match math vocabulary using the Matching Card Activity. Student groups were given two sets of cards; one set of cards displayed the vocabulary words, while another set of cards contained the definitions. Students worked together to determine which vocabulary word matched the definition.

The teacher designed a Think-Pair-Share Activity in which students shared with a partner how to add and subtract integers, then partner responded. Students switched and shared how to multiply and divide integers and then their partner responded. Students then changed partners, repeated the process, and changed partners again once more. Student partners then explained how to multiply and divide integers to entire class, afterwards the class provided feedback.

The next task progressed in rigor where students participated in the Inner-Outer Circle Activity. This learning activity dividing the students into two groups. Group 1 where placed in the inner circle and Group 2 was in the outer circle. Each student in the inner circle had a different vocabulary word and definition and were instructed to share with their classmate facing them in the outer circle. After a set period of time, the teacher signaled the inner circle to rotate and pair-up with a new student in the outer circle. After the group completed a full rotation, the inside circle and the outside circle traded positions and then repeated the steps above. This activity helped student engagement and cognitive processing skills by allowing students to practice active speaking and listening skills. This variety kept students actively engaged and on task.

Students continued to practice the vocabulary words as the teacher increased the complexity of the learning activities to deepen the level of rigor. The teacher designed a Fill-In-The-Blank Word Problem Story. Students used the vocabulary learned to complete the story. At the end of the class, students completed an Exit Ticket Activity and described how to add, subtract multiply and divide integers.

In the next unit of study, the teacher posted vocabulary words around the room and out in the hallways where the students participated in a Scavenger Hunt. The students were prompted to search for vocabulary words and match them with the definitions on their paper. Afterwards, the students returned to class and they completed a Crossword Puzzle utilizing the words and definitions.

The acquisition of vocabulary progressed through students creating custom flash cards using an interactive online program from the vocabulary words and definitions. The interactive online program provided a variety of activities and other games in addition to the flashcards that students could use in class, after-school, and at home to study. This tool offered an opportunity for students to master the vocabulary through repetition and variation and an opportunity of choice in the learning. She increased the level of difficulty by challenging her students to deeper levels of understanding using varied learning tasks through the repetitive exposure to vocabulary and building the academic language of mathematics.

The Outcome and Progress Measure

The teacher’s intentional design of instruction contributed to the students’ overall academic progress in this study. Following the implementation of the FASL’s Principles of Practice and Design Principles of repetition with variation and spacing of the mathematics vocabulary activities, the overall student learning growth increased 6-12% during the time of the study and found through analysis of the scores that the English as a Second Language class outperformed the regular math classes on their weekly Friday vocabulary post-assessments. At the end of the study, student academic language in math was improved through deliberate practice using repetition with variation and spacing to expand the complexity of the learning task.


Capraro, R.M., Capraro, M.M., & Rupley, W.H. (2010). Semantics and syntax: A theoretical model for how students may build mathematical misunderstandings. Journal of Mathematics Education3 (2), 58-66. doi:

Capraro, M.M., & Joffrion, H. (April-June 2006). Algebraic equations: Can middle-school students meaningfully translate from words to mathematical symbols? Reading Psychology 27 (2–3) 147-64. EJ737448.

Chamot, A.U. & O’Malley, M.J. (1994). The CALLA Handbook: Implementing the Cognitive Academic Language Learning Approach. Reading, MA: Adison-Wesley.

Dunston, P., Tyminski, A. (2013). What’s the big deal about vocabulary? Mathematics Teaching in the Middle School, 19, 38–45. Retrieved from…/mtms2013-08-38a_pdf/

Ebbinghaus, H. (1885). Über das Gedächtnis. Untersuchungen zur experimentellen Psychologie [Memory: A Contribution to Experimental Psychology] (in German). Trans. Henry A. Ruger & Clara E. Bussenius. Leipzig, Germany: Duncker & Humblot.

Francis, D. J., Rivera, M., Lesaux, N., Kieffer, M., & Rivera, H. (2006). Research-based recommendations for instruction and academic interventions. Portsmouth, NH: Center on Instruction. Retrieved from http://www.centeroninstruction. org/files/ELL1-Interventions.pdf

Kotsopoulos, D. (November, 2007). Mathematics discourse: “It’s like hearing a foreign Language”. Mathematics Teacher 101 (4), 301-305.

LeFevre, J., Fast, L., Skwarchuk, S., Smith-Chant, B.L., Bisanz, J., & Kamawar, D., Penner-Wilger, M. (2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81, 1753-1767.

LeFevre, J., Berrigan, L., Vendetti, C., Kamawar, D., Bisanz, J. & Skwarchuk, S., & Smith-Chant, B.L. (2013). The role of executive attention in the acquisition of mathematical skills for children in grades 2 through 4. Journal of Experimental Child Psychology, 114, 243-261.

Piccolo, D. L, Harbaugh, A. P., Carter, T. A., Capraro, M. M., & Capraro, R. M. (2008). Quality of instruction: Examining discourse in middle school mathematics instruction. Journal of Advanced Academics, 19, 376-410.

Pierce, M. E. & Fontaine, L. M. (2009).Designing vocabulary instruction in mathematics. The Reading Teacher, 63(3), 239-243.

Purpura, D. J. (2010). Informal number-related mathematics skills: An examination of the structure of and relations between these skills in preschool. Doctoral dissertation. Retrieved from ProQuest Dissertations and Theses, Florida State University (Accession Order No. AAT 3462344).

Purpura, D.J., Baroody, A.J., & Lonigan, C.J. (2013).The transition from informal to formal mathematical knowledge: Mediation by numeral knowledge. Journal of Educational Psychology, 105, 453-464.

Purpura, D.J., & Ganley, C.M. (2014). Working memory and language: Skill-specific or domain-general relations to mathematics? Journal of Experimental Child Psychology 12, 104-121.

Purpura, D.J., Hume, L., Sims, D., Lonigan, C.J. (2011). Emergent literacy and mathematics: The value of including emergent literacy skills in the prediction of mathematics development. Journal of Experimental Child Psychology, 110, 647-658.

Riccomini, P. J., Smith, G. W., Hughes, E. M., & Fries, K. M. (2015). The language of mathematics: The importance of teaching and learning mathematical vocabulary. Reading and Writing Quarterly, 31, 235–252. doi: 10.1080/10573569.2015.1030995.

Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading and Writing Quarterly, 23, 139–159. doi: 10.1080/10573560601158461.

Schleppegrell, M. J. (2010). Language in mathematics teaching and learning: A research review. In Language and mathematics education: Multiple perspectives and directions for research. Charlotte, NC: Information Age, 73-112.