1936

Margaret Bourke-White

The Journey is the Destination

For most of the twentieth century, the study of algebra was reserved for those students who were entering high school and were considered to be mathematically inclined (Van de Walle, 2007). The accompanying pedagogy was focused primarily on the manipulation of symbols and the solving of equations. Today, in many school systems, instruction in algebra begins as early as kindergarten and focuses mainly on the development of concepts such as identifying patterns through the use of concrete materials, the insertion of relevant representations to stand for unknown quantities, the comprehension of numerical relationships, the connection of these relationships to generalized functions, and the interpretation of equal signs as indicating equality on both sides of the equation (Van de Walle, 2007). Here are some further issues to consider.

- Not coincidently, many mathematical educators have now broadened their interpretation of algebraic inclusivity. This new wave thinking has been styled Equity and Access, and it embraces the belief that all children are capable of learning algebra (National Council of Teachers of Mathematics, 2005). As such, Equity and Access accommodates not only the very young, but minorities, the learning and intellectually disabled, English language learners, and girls (Van de Walle, 2007). As a result, many school systems now support the idea that every child should be presented the opportunity to engage in the formal, rigorous study of algebra, under the tutelage of highly qualified teachers (NCTM, 2005). In addition, a wide sample of the illuminati endorses tapping appropriate resources to undergird this more encompassing agenda (Jamar, Clark, Cooper & Curcio, n.d.).

- The challenges of implementing such an approach are imposing. The arduousness resulting from the variance in student abilities and disabilities, as well as other cultural, linguistic, ethnic, racial and socioeconomic considerations can prove daunting to the classroom teacher (Van de Walle, 2007). Not surprisingly, the instructor may be further imposed upon to provide modifications and accommodations to the material, while still affording each student the opportunity to reason mathematically (Laureate Education, 2007). On the other hand, the rewards of such an undertaking can be particularly gratifying in light of the doors it may open for the underrepresented students.

- My own plan of action for addressing the issue of diversity through equal access to algebra, (the gatekeeper of the curriculum and the gateway to higher order thinking), includes the following transformational ideas for English as Second Language learners.

- As we strive to increase the academic achievement of these students, we should reflect on their particular circumstances. “English language learners enter the classroom from homes where English is not the primary language” (Van de Walle, 2007, p.100). In consequence, although it may take these children a relatively short time to develop conversational communication skills, it takes up to seven years to learn the academic language used in algebra and other subject areas (Van de Walle, 2007). The Teachers of English to Speakers of Other Languages (TESOL) endorses the approach of having these students use English in their content courses, while still encouraging the practice of their native language for cultural as well as clarification purposes(Van de Walle, 2007). Accordingly, the teacher should initiate and maintain certain conventions that take into account the cultural and language differences of these learners. First, he/she should face the students when addressing them, speak slowly, use simple sentences and gestures that reflect the vocabulary, and be willing to repeat these instructions when requested. Non verbal answers, in the form of flash cards, the nodding of heads, and pointing to object should be encouraged. In addition, the teacher needs to be aware of differing cultural norms, such as the meaning of eye contact, possible reticence in volunteering answers, and becoming sensitized to what constitutes personal space (Williams, 2009).

- As part of the curriculum, the instructor can explicitly teach vocabulary with concept-mapping and algebraic word walls (Williams, 2009). The children can include their own definitions, pictures, and translations. The educator can also further a central tenet of bilingual education by pairing students with volunteers, who speak the same language, and who can help the child bridge the language barrier.

- Moreover, story problem language, which encapsulates the algebraic concept(s) should be kept simple, with the content-specific vocabulary commensurate with the child’s skill level. However, the difficulty of the numbers used in the problem should be maintained (Williams, 2009).

- Subsequently, after conferring with individual parents, the teacher can also incorporate the child’s background experiences into the lessons; formulate the accompanying activities with multi-sensory modalities that take advantage of the child’s individual learning style, and incorporate whatever accommodations (responses to the needs of the environment or the learner that do not alter the task) or modifications (alters the task by making it more accessible to the student) that are deemed necessary. Instructors can also use frequent mini-lessons to model problem-solving algebraic strategies such as drawing pictures, using manipulatives, incorporating symbols, tables, graphs, or number lines. The Virginia Department of Education suggests providing opportunities for students with common language differences to partner up for clarification purposes (2004). Heterogeneous groupings, which include two ESL students (with matching native languages) and one English speaking child, has been found to have a positive impact on the learning (Williams, 2009). Still others encourage students to use bilingual and picture dictionaries (Williams, 2009).

- Algebra inculcates the critical thinking skills students will need in order to become informed adults, capable of weighing options and making wise decisions (National Council of Teachers of Mathematics, 2000). Many educators agree that algebra is mastered through hard work rather than innate ability (Jamar, Clark, Cooper & Curcio, n.d.). Hence, all students should have equity and access to this rewarding domain.

- To sum up, the success of any plan of inclusion depends primarily on a number of related factors. First and foremost are the high expectations of thoroughly trained teachers, and the perception of these expectations by the learners. In addition, the following strategies will prove salutary for the nascent learner of algebra: early, conceptually-based learning with concrete manipulatives, heterogeneous groupings, attention to individual learning styles, multi-sensory dynamics, as well as the identification and affirmation of cultural, linguistic and ability distinctions, with the resulting accommodation and modifications to relevant lessons. Finally, underpinning these practices with sound financial, parental and communal support, will provide the means for all students, through their own diligence, to attain algebraic literacy.

** **

**Gender** (Van de Walle, 2007)

The belief that mathematics is a male domain persists in our society and is held by both sexes. Since many of the differences in learning between boys and girls are socially constructed, and not biological, awareness and appropriate reactions by the teacher will be salutary.

- Girls tend to be more social learners. Therefore, cooperative learning groups are valuable.
- Integrate constructionist lessons where all students talk and listen. Here there is less teacher intervention. Authority is allied with the students and the value of their reasoning. This results in more equal footing for girls.
- Make sure that you are providing both boys and girls with equal attention, as well as the opportunity to implement higher order thinking.

**The Gifted** (Van de Walle, 2007)

The term "mathematically talented" is a function of ability, motivation, experience and opportunity.

- Create open ended, depth creating programs for the gifted that include multiple ways of solving the problem, multiple possible answers and the creation of many other related problems by the student.
- Present lessons that enhance depth, enrichment and acceleration.

**Students with Intellectual Disabilities** (Van de Walle, 2007)

Children with moderate or severe intellectual disabilities (IQ scores between 50 and 75) will have circumscribed mathematical reasoning.

- Limited cogitative abilities do not alter the way children think but rather, alter the way these children acquire their learning.
- Take more time; inculcate repetitiveness in the lessons and use and concrete manipulatives.
- Have these students interact with their peer groups in cooperative learning settings by taking less demanding roles.
- Partner the student with limited abilities with a number of different students. This creates the opportunity for repeated exposure and over learning.
- Do not become obsessed with having these students attain mastery of facts and computational skills.
- Let them use a calculator at all times.
- Focus on those areas of math that are going to be of most use to them as adults.

**Students with Learning Disabilities ** (Van de Walle, 2007)

These students are mentally capable, and not retarded. Often there have perceptual (auditory and visual impairments) and cognitive difficulties. This may affect memory, or the ability to speak or express themselves in writing, perceive auditory or written instructions, or integrate abstract ideas.

- These limitations should be compensated for by emphasizing the students’ strengths.
- Never ask students to do things that rely on their deficits. In short, avoid weaknesses and capitalize on strengths.
- Seat the child near teacher or board.
- Repeat main ideas. Show only one main idea at a time.
- Have only one person in the class talking at a time.
- Face the students and enunciate clearly.
- Design tests or worksheets specifically for the child.
- Use centimeter grid paper for computations.
- Use tape recorder or video with instructions.
- Provide geometric models.
- Assign a buddy to help read, explain or repeat directions.

**Students with Memory Deficits** (Van de Walle, 2007)

These can be short or long term impairments.

- Short term: break tasks and directions into small steps and provide a buddy to help with recall.
- Long term: require over learning, frequent practice and associations with other ideas.
- Ask one or more students to put the directions in their own words to assure repetitiveness.
- Write instructions on the board.
- When working in an oral capacity allow children the option of a written format.
- Allow calculators at all times.
- Utilize frequent, brief reviews.
- Stress number relationships for memorization.

**Students with Integrative Deficits** (Van de Walle, 2007)

Children who have difficulties with abstract ideas and conceptualization have integrative deficits.

- Use experiences and ideas most familiar to them.
- Utilize invented procedures.
- Have them put ideas in their own words in either oral or written form.
- Use physical models for longer than usual.
- Have students frequently verbalize what they do, using pictures words and numbers.
- Require frequent explanations and justifications to assist them in making connections to new ideas.
- Allow repetition and practice of new ideas.
- Use multiple representations of abstract concepts.

**Students with Attention Deficits** (Van de Walle, 2007)

These children have chronic problems with attention span, impulse control, and sometimes hyperactivity.

- Establish simple predictable routines and discuss these with the child.
- Make expectations and consequences clear.
- Integrate learning activities that are active and involving.
- Have the children do independent work in an environment free of distractions.
- Use highlighters.
- Give short assignments, and pair them with a buddy to help stay on task.

References:

Bay-Williams, J. M. (2001). What is algebra in elementary school? *Teaching
Children Mathematics,*

Jamar, I., Clarke, C., Cooper, D., & Curcio, F. (n.d.). *Algebra position paper.* Retrieved August

24, 2005, from the Benjamin Banneker Association Web site:

http://www.bannekermath.org/algebraposition.html

Laureate Education, Inc. (Producer). (2007). Program Eight: Equity and Access [Motion picture].

Elementary Mathematics: Algebra, Grades K-5. Baltimore: Norwood, K.

National Council of Teachers of Mathematics. (2005, April). *Closing the achievement gap: A
position of the National Council of Teachers of Mathematics.* Reston, VA:

Author.

National Council of Teachers of Mathematics, (2000*). Principals and standards for
school mathematics. *Reston VA: Author.

Van de Walle, J. A. (2007).

developmentally

Virginia Department of Education. (2004). Mathematics: strategies for teaching students

Retrieved August 13, 2009 from Web site

http://www.doe.virginia.gov/VDOE/Instruction/ESL/LEPmathResource.pdf

Williams, M. (2009). Strategies to support esl students in math. Retrieved August

13, 2009 from Web site: http://esl-programs-

lessons.suite101.com/article.cfm/strategies_to_support_esl_students_in_math