The first theorem is shown similarly; one can divide the random string into nonoverlapping blocks matching the size of the desired text, and make Ek the event where the kth block equals the desired string. If there were as many monkeys as there are atoms in the observable universe typing extremely fast for trillions of times the life of the universe, the probability of the monkeys replicating even a single page of Shakespeare is unfathomably small. Ignoring punctuation, spacing, and capitalization, a monkey typing letters uniformly at random has a chance of one in 26 of correctly typing the first letter of Hamlet. In the case of the entire text of Hamlet, the probabilities are so vanishingly small as to be inconceivable.
My dad gave me one dollar bill 'Cause I'm his smartest son, And I swapped it for two shiny quarters 'Cause two is more than one! I didn't understand why it came so naturally to some students, but not to me.
Looking back, however, I realize that I had an advantage that I wasn't even aware of — I understood the language in which the problems were written, even if I didn't understand how to solve them! Although it is easy to assume that many English language learners ELLs will excel in math because math is a "universal language" and students may have had prior educational experience that included mathematical instruction, that assumption can lead educators astray.
As I spoke with teachers and did research for this article, it became very clear that making sure that students understand math vocabulary and have ample opportunities to use it are very important.
Solving word problems, following instructions, understanding and using mathematical vocabulary correctly — all of these skills require a language proficiency that sometimes exceeds our expectations. We tend to think of mathematics as a subject that does not require a strong command of language.
For many educators, the challenge of bringing language and math instruction together is a relatively new one. ELL teachers who hadn't taught content areas previously are now being asked to lead or support instruction in the math classroom, and many math teachers who don't see themselves as language instructors are now responsible for providing effective math instruction to ELLs.
High school math teacher Hillary Hansen learned just how big a role language plays in math instruction when she taught her first Basic Math course for ELLs last year. She wanted so much to provide the students with the good foundation they needed, but she felt unable to reach the students or engage them in her lessons, and by the end of the year she was exhausted and frustrated.
That summer she had an opportunity to join a district Sheltered Instruction Observation Protocol SIOP cohort to receive professional development and support to meet the needs of ELLs in content classes. She learned about the importance of language acquisition, building background knowledge, increasing student language production, and explicitly teaching academic language.
She began this school year with a new set of tools and a deeper understanding of the instructional scaffolding ELLs need in order to learn the content while also learning English. I am happy to report that while Hillary still feels challenged and is working very hard, this year has been much more successful for her and her students.
As a result of more effective instruction, her students: Hillary feels that she is providing them with the foundation they need not only to understand the mathematical concepts, but also to successfully interact within a math classroom in order to continue learning more advanced concepts.
Following are some strategies that Hillary and some of the other teachers I spoke with found helpful this year, and that they recommend as best practices when teaching math to ELLs. The importance of teaching academic vocabulary Vocabulary instruction is essential to effective math instruction.
Not only does it include teaching math-specific terms such as "percent" or "decimal," but it also includes understanding the difference between the mathematical definition of a word and other definitions of that word.
The following example, used in a presentation by Dr. Judit Moschkovich of the University of California at Santa Cruz, underscores why vocabulary must be introduced within the context of the content Moschkovich, In this problem, the student is instructed to "find x.
The student even put a note on the page to help the teacher in locating the lost "x". The student understood the meaning of "find" in one context, but not in the appropriate mathematical context.
I recently helped a math teacher create a Sheltered Lesson, and I was surprised to find that there were some vocabulary words that I didn't understand. My lack of familiarity with the words hindered my ability to do the math problem and gave me a deeper empathy for ELLs who struggle in the same way with vocabulary and comprehending math assignments.
Following is a list of tips for explicitly teaching mathematical academic vocabulary: Demonstrate that vocabulary can have multiple meanings. Help students understand the different meanings of words such as "table" and "quarter," as well as how to use them correctly in a mathematical context.
Encourage students to offer bilingual support to each other. Students will understand material better if they explain it to another student, and the new student will benefit from hearing the explanation in their first language. Check the hotlinks for a list of bilingual translations of math vocabulary in multiple languages.
Provide visual cues, graphic representations, gestures, realia, and pictures. Offer students the chance to work with objects and images in order to master vocabulary.
If there aren't enough items for each student, use manipulatives on the overhead or posted throughout the classroom, and demonstrate the vocabulary in front of the students.
For example, Hillary created a Math Word Wall that has three parts: For example, multiplication is portrayed by the following symbols:The easiest way write a complaint letter to a company, is to quickly get to your problem.
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