Helping K-12 Students Communicate Mathematically
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Discussion Overview
NCTM and NCSM Recommendations
The Cycle of Mathematical Understanding
Communication Nuances and Pet Peeves
Strategies for Facilitating Communication
Communication Do’s and Don’ts
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NCTM and NCSM Recommendations
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NCTM and NCSM Recommendations
NCTM calls for instructional programs from prekindergarten through grade 12 to enable each and every student to:�
✦Organize and consolidate their mathematical thinking through � communication
✦Communicate their mathematical thinking coherently and clearly to � peers, teachers, and others
✦Analyze and evaluate the mathematical thinking and strategies of � others
✦Use the language of mathematics to express mathematical ideas � precisely.
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NCTM and NCSM Recommendations
Based on NCTM, one of the components forwarded by NCSM is “communicating mathematical ideas”. Their recommendations in this field includes:
a) students should learn the language and notation of mathematics;�b) students should learn to receive ideas of mathematics through listening, reading, and observing;�c) students should be able to express ideas of mathematics by speaking, writing, drawing pictures and graphics, and demonstrating with concrete models;�d) students should be able to discuss mathematics and put forward questions about mathematics.
In other words, one of the main points of the agreement between the recommendation of NSCM and NCTM is that communicating ideas of mathematics in many kinds of ways (spoken, written, symbolic language, daily language) is very important for the teaching process.
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The Cycle of Mathematical Understanding
Graphical – Numerical/Table – Algebraic/Symbolic – Written/Spoken
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➤ Try to complete the cycle on the handout
➤ Cycle Template (Link pending)
Cycle Practice & Resources
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Strategies for Facilitating Communication
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Always be aware of how much you are talking vs. how much the students are talking/doing.
Facilitating Communication
➤A means by which a teacher can have the students complete vocabulary definitions they have learned or are learning
Using
“Key Concepts”
➤Writing to show understanding of key vocabulary and concepts and to use proper vocabulary
➤Two-Point Problem�➤Examples from Classwork
Writing
Examples
➤The “Four-Step” problem-solving process is not new, but having it organized for the students has them focus on the key vocabulary and processes
Organizing Real-life Problems
Facilitating Communication
➤In teacher-led groups, a teacher can use strategic questioning to get students to talk and write mathematically�(Video1)(Video2)(Video3)
Communicating in Group Work
➤Have students create a math journal
➤Have students make a video explaining how to solve a problem or to teach others to learn a concept�➤Foldables
Journaling
Facilitating Communication
Math/Number/�Algebra Talks
Problem Strings
➤Have students create real-world problems
➤Find projects like the Tech-prep and Long-term projects (see handout)
Using Real-World Math Problem/Projects
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Communication Nuances�and Pet Peeves
Pet Peeves and Nuances
▷Student disorganization
➤Organization Techniques:
∘Using folded notes
∘Using lined paper
∘Using graph paper
▷Students not showing proper organized work
➤See “Showing Algebra Work” in the handout
▷Equations vs. Expressions (and showing work with each)
➤Don’t accept equations for expressions or� expressions for equations
Pet Peeves and Nuances
▷Defining Variables vs. Labels� ➤Students should define their variables concisely in � context (Let: c = “Total cost of renting a scooter”)� ➤Students should not use labels to define variables � (Let: c = $)� ➤Labels give context to a number in a real-world � situation and are paired with the number � ($45 or 3 hours)� ➤The real-world contextual meaning of the x- and � y-intercepts and the slope (rate of change), and � each value of an ordered pair is an essential � communication skill�
Pet Peeves and Nuances
▷Know the difference between solving a system of equations and its ordered pair solution, and solving a univariate equation graphically using a system of equations and its solitary solution
▷Truncating vs. Rounding (and following directions)
▷Mixed numbers vs. Improper (“top heavy”) fractions� ➤Mixed number have their uses but NOT in Algebra
▷Giving math answers (symbolically/numerically) vs. giving real world answers (spoken or written sentences)
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Pet Peeves and Nuances
▷Proofs� ➤Students should be able to give spoken and written � proofs� ➤Students should be proficient with two-column, � algebraic/identity, and paragraph proofs� ➤Proofs should be written so any student at their level � or above can clearly understand each step of the � Proof� ➤“GPD” – Given/Prove/Diagram proofs
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Pet Peeves and Nuances
▷Graphing� ➤Students should be able to create graphs by hand � and using technology� ➤Graphs should not be accepted if they do not have a � title, labels on the axes, or proper scaling numbers
▷Evaluating the Quadratic Formula� ➤Find the value of the discriminant (D) and identify � the number and nature of the roots, then finish � using the formula to find the roots
Pet Peeves and Nuances
▷Students should write mathematics, but also stress proper � English syntax and grammar.
➤Sentences should be complete with noun and a verb.�➤Sentences should begin with a capitalized first word and end with a period.�➤Sentences should never begin with a number in number form. If a sentence starts with a number, then the number should be spelled out. However, when communicating mathematically, the actual number is most typically written, so find a way to not start sentences with a number.�➤Sentences should answer the question at hand with correct vocabulary, proper labels used with the numbers given, and proper context.�➤Sentences should not use “it”, “they”, etc. Don’t assume the reader of your sentence knows the context of the problem. Be specific.
Pet Peeves and Nuances
▷College professor’s expectations
➤Keep all papers neat and well-organized. �➤Papers that are turned in should be professional-looking. They should be clean and free of crinkles and stapled before coming to class. Work should be neat, organized, use proper math symbols, vocabulary, and grammar/structure. Use proper penmanship and sentence structure. Writing should not be too big or too small.�➤Use pencil and erase mistakes. Don’t scribble out mistakes.***�➤Don’t use “switching arrows” if you write answers in the wrong place. Erase and write them in the correct place.
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Communication�Do’s and Don’ts
Do’s and Don’ts
▷Teach students the underlying concepts and let them develop and communicate “shortcuts”.
➤Don’t teach students shortcuts just to get through the lesson/material.
➤Shortcuts without the underlying understanding leads to applying shortcuts inappropriately. (Ex. Only teaching graphing (lines, abs. value, parabolas) the “quickie” way without the underlying patterns/process/concepts. Telling students to “flip the inequality sign when dividing or multiplying an inequality by a negative number without the context of why.)
Do’s and Don’ts
▷Use calculators as a tool, not a crutch.
▷Be ticky-tacky and don’t allow students to communicate in sloppy ways. � ➤Point out student errors in communicating � mathematically (improper vocabulary, � incomplete/poor sentences, poor work shown, � etc.) � ➤Always model proper mathematical communication.
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Thank you!
-Google Slides and electronic version of the handout should be linked on the conference webpage soon.