Multicultural Mathematics

The Importance of Multicultural Mathematics

When you look at architecture, geography, computers, landscapes or even a sports field right triangles are not the first thing that might come to your mind but if you are looking you should be able to see right triangles all around your environment.  Right triangles and trigonometry is a lesson learned in high school Geometry that can reflect world diversity by showing a common knowledge and application through mathematics.  As a teacher I have the ability to engage the class with not only how to apply trigonometric ratios to solve for a missing side or acute angle of a right triangle but to show the rich history and world use of the math the students are going to learn.  Trigonometry was developed for use by astronomers and surveyors.  Trigonometry dates back to early Egypt and was further developed by the Greeks.  Indian mathematician’s updated trigonometry based on the sine function and Muslim astronomers compiled the findings of the Greeks and Indian’s.  In the 13^th century Germans defined the trigonometric ratios and then Isaac Newton continued the study through calculus and differential equations.

It is important to introduce students to multicultural content so they can see how the world has shaped what they are learning mathematically and how mathematics has shaped the world they live in.  Students often see mathematics, especially geometry, as definitions, formulas and problems that have no purpose.  Students may show more interest if they have some cultural connection to the material and see how application of something as simple as a right triangle is used and applied in different cultures around the world.  From the great Pyramids to the Eiffel tower students can engage in a world view application of mathematics.

Developing cultural competence in the classroom starts with me, the teacher.  Getting to know my students as individuals, their backgrounds and cultures and building my knowledge to engage my students.  Having applied problems and projects where students research and incorporate a culture(s)into the mathematics is another way to develop cultural competence in the classroom.  Knowing your school demographic, community socioeconomics, language(s) spoken and learning and growing along with your students. Students demonstrate cultural competence through their project work, problem presentation and interactions in the classroom. 

 

Reference

Honlyn Limited (2004). The History of Trigonometry. Retrieved http://www.trigonometry-help.net/history-of-trigonometry.php

Miller R. (2011). The Importance of Culturally Competent Teachers. Retrieved https://www.huffingtonpost.com/randy-miller/the-importance-of-cultura_b_787876.html

 

Differentiated Instruction for SLI student and Readiness

Students Whom Differentiated Instruction Would be Appropriate

 

What is differentiated instruction?  Differentiated instruction is when a teacher (like I am hoping to become) attempts to meet the needs of every student in the classroom through formative assessments, adjusting lessons, offering additional materials, and so forth.  Students have different learning styles, different abilities, and unique personalities that differentiated instruction enable a teacher to elevate each student’s requirements and preferences.  This sounds like a tall order for the teacher!

“Keep in mind that differentiation shouldn’t be something that complicates your day or life. Although additional work and effort are required up front, the payoff comes later in the lesson of study or even in the school year. The payoff comes when students achieve more in your classrooms, become more involved in classroom discussions, smile more during their school days, and, yes, even score higher on various assessments.” (Preszler, 2006).

A disability, listed by the IDEA (Individual’s with Disabilities Education Act), Speech or Language Impairment (SLI) is a communication disorder such as stuttering, impaired articulation, a language impairment, or a voice impairment that adversely affects a child’s educational performance. It would be appropriate to differentiate learning in a High School Geometry class for a student with such a disability.  Formative assessments I plan to use in my classroom are: Homework/Self-Assessment, Oral Question and Answer, Pair-Think-Share, White Board magic, and Exit Ticket. 
The Formative assessment that would need modification based on the students’ needs would be the Pair-Think-Share.  Pairing of students for this assessment would be a high priority.  I would reach out to specific students that are grasping the content at a high level and have the maturity to partner with the disabled student.  This modification differentiates the learning environment and helps the disabled student through a peer tutoring type experience.  I would also give additional time and attention to this pairing during this part of the lesson so that the content is being learned at a high level from both students.  I can also modify the problem given to the pairing, differentiating the content, for the student with the disability into specific steps and include key words and add graphics or pictures when applicable.  The learning environment of the classroom would be a high priority for the learning needs of a student with SLI. As a teacher I would need to be sensitive to calling on a student with SLI during an Oral Question and Answer Formative Assessment, instead I could set up a daily text/email with the student still assessing their learning while meeting their specific needs.   The Learning environment of the classroom should also be supportive and productive for all students - a place where learning math will be fun, interesting and productive.

Three resources that will support the SLI student(s) in the class : 

• Strategies for Speech or Language Impairment http://do2learn.com/disabilities/CharacteristicsAndStrategies/SpeechLanguageImpairment_Strategies.html
• LD Online: Educators Guide to Learning Disabilities and ADHD http://www.ldonline.org/spearswerling/Specific_Language_Impairment
• Project Ideal http://www.projectidealonline.org/v/speech-language-impairments/

Based on the above modifications of the Formative Assessment I would be consistently checking for understanding, making sure the seating position in the class allows for clear access to visual and verbal cues, and using co-operative learning such as peer tutoring.  Proper assessment is key for all students, especially those with disabilities, to be successful and competent with the content they are to master. “The assessment process is multi-tiered, multidisciplinary and occurs in a continuous cycle—from planning through to final assessment and evaluation. The assessment process begins at the classroom level, with the teacher using informal techniques such as observation, reading inventories and other diagnostic tools to explore how the student is learning and to identify areas of strength and concern.” (Speech and Language Disorders, nd).

Early detection, tracking attendance, social interaction and grades are all tools for identifying and addressing struggling students in the classroom.  The readiness level of a student refers to capability to learn and apply new concepts. Formative Assessments are a means to gauge the readiness level of each student .  Differentiating instruction for students at different readiness levels requires a full breakdown of the content.  Tiered instruction is a strategy where a lesson is organized from least complex to most complex.  A great way to organize the levels is from Bloom’s Taxonomy knowledge dimensions: Remember, Understand, Apply, Analyze, Evaluate, and Create.  For example, in the High School Geometry class the current lesson it about the Pythagorean Theorem.  The lesson would start with a very simple application of the theorem, a simple triangle.  As the lesson progresses students could be grouped or paired based on their readiness level and given a set of problems tiered to meet their need(s). 

In my lesson on the Pythagorean Theorem for example - some students may struggle with the algebra while other students may be ready for application.  Having a lesson prepared by readiness levels will allow me to group the students - and work with each groups needs.  The students who are ready for application could work on a project incorporating technology and writing skills.  These students need little instruction which allows my time to work with students needing algebra review and more explanation.  Never wanting students to feel in the low group or high group I feel it would be important to be discrete in the groupings and assignment work. 

 

References

Weselby, C. (2017). What is Differentiated Instruction? Examples of How to Differentiate Instruction in the Classroom. Retrieved https://education.cu-portland.edu/blog/classroom-resources/examples-of-differentiated-instruction/

Preszley, J. (2006). Strategies That Differentiate Instruction. Retrieved https://education.ky.gov/educational/diff/documents/strategiesthatdifferentiateinstruction4.12.pdf

Do2Learn. (2017). Disabilities. Retrieved http://do2learn.com/disabilities/Overview.html

Dr. Bloom, B. (2015). Bloom’s Taxonomy of Learning Domains. Retrieved http://www.nwlink.com/~donclark/hrd/bloom.html

n.a. (2017). Speech and Language Disorders. Retrieved https://speechandlanguagedisabilities.weebly.com/classroom-implications.html

 

Formative and Summative Assessments for High School Geometry Standard

Formative and Summative Assessments for a Geometry Standard

The standard I have chosen to write assessments for is a high school geometry standard: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 

 A formative assessment is an evaluation of learning during a planned lesson while a summative assessment is an evaluation of learning at the end of a planned lesson.  A formative assessment is for both the student and the teacher and should reveal the students level of understanding.  A formative assessment enables a teacher to refine and or revise a lesson to ensure the standard is meet by each student in the classroom.  Formative assessments can be for a grade or not for a grade.  A summative assessment tests how much a student has learned at the end of the planned lesson.  A summative assessment could be a standardized chapter test or cumulative test, for a grade that demonstrates a student’s knowledge of a subject, and application of the knowledge learned.  Only future curriculum is affected with the data received from the summative assessment.

Formative Assessment #1:  For all five of my objective I would assign daily homework as a formative assessment to reinforce the concepts learned each day of class.  My homework assignments will be unique: 

Day 1 Objective 1 = 5 homework problems from the specific objective

Day 2 Objective 2 = 5 homework problems from Objective 2 and 3 homework problems from Objective 1

Day 3 Objective 3 = 5 homework problems from Objective 3 followed by 3 problems from Objective 2 and 2 problems from Objective 1; and so forth. 

Some high schools work on A/B 90 minute schedules and I would adjust my objectives and lesson and homework accordingly.  The goal of the formative assessment is the homework assignments will gradually build with each new objective to revisit and reinforce concepts learned until the standard and total lesson is completed.  As the amount of problems decrease when new objectives are introduced the complexity and critical thinking skills will increase challenging students to apply concepts, knowledge and work towards an excellent understanding of content, thoroughly identifying how topics are applied.

Formative Assessment #2:  Randomly choose students throughout the class time to state the Pythagorean Theorem. Every class during the week I would randomly choose 5 students (or total students in class divided by five or class days) – like a verbal pop quiz.  This would be a good verbal assessment of Objective 1 and have students commit to memory the Pythagorean Theorem which is used and applied in upper level mathematics classes. 

Performance Based Summative assessment #1: (5-10 minutes) Have students answer the following question to demonstrate their depth of understanding and coherence after all objectives have been covered: “How do you use trigonometric ratios to solve for a missing side or angle of a right triangle?” 

Performance Based Summative assessment #2: (5-10 minutes) Have students answer the following question to demonstrate their depth of understanding and coherence after all objectives have been covered: “Use the given image, angle C is a right angle, to find the value of sin A and cos B. What relationship does the ratios of sin A and cos B share? What is the value of tan A and tan B? "

 

Performance Based Summative assessments will be graded on a point system to show students level of learning and understanding:

(3) Superior – Shows thorough understanding of the concepts. Uses appropriate strategies to solve problems. Computations are correct. Written explanations are exemplary. Diagrams are accurate and appropriate. Goes beyond requirements of problem.

(2) Satisfactory with minor flaws – Shows understanding of the concepts. Uses appropriate strategies to solve problems. Computations are mostly correct. Written explanations are effective. Diagrams are mostly accurate and appropriate. Satisfies all requirements of problem.

(1) Nearly Satisfactory with serious flaws – Shows understanding of most of the concepts. May not use appropriate strategies to solve problems. Computations are mostly correct. Written explanations are satisfactory. Diagrams are mostly accurate and appropriate. Satisfies most requirements of the problem.

(0) Unsatisfactory – Shows little or no understanding of the concepts. May not use appropriate strategies to solve problems. Computations are incorrect. Written explanations are not satisfactory. Diagrams are not accurate or appropriate. Does not satisfy requirements of the problem.

The following are five objectives from the above standard:

Objective 1: Students in my geometry class will be able to define, show and solve the Pythagorean Theorem by the end of the class.

 Objective 2: Students in my geometry class will be able to recognize, comprehend and compute the converse of the Pythagorean Theorem and related theorems about obtuse and acute triangles by the end of class.

Objective 3: By the end of class my geometry students will be able to distinguish, identify and interpret the lengths of two sides of a 45-45-90 and 30-60-90 triangle when the length of the third side is known (given).

Objective 4: Student in my geometry class will be able to identify and explain tangent, sine and cosine ratios for an acute angle of a right triangle by the end of class time.

Objective 5. By the end of class, students in my geometry class will be able to solve right triangle problems by correct selection and use of the tangent, sine and cosine ratios.