Information about Problems of the Week

A presentation I made to show how I support students in writing in math.

For many of us, math was just another course we book with a huge book to lug around.

Text But in fact math is an amazing collection of human achievement

So, is math one or the other, or something else completely? Math = Science ? or Math = Religion ?

For you personally, Math is primarily an art, a language, a science, a tool, a way of life, a distant memory, a nightmare, or something else entirely?

How about this idea? Math is an exquisite expression of human culture.

Who is glenn kenyon in 2009?

Who is glenn kenyon in 2009? 48% of my life as a teacher

Who is glenn kenyon in 2009? 48% of my life as a teacher about 4,100 days with students or about 25,000 hours

Who is glenn kenyon in 2009? 48% of my life as a teacher about 4,100 days with students or about 25,000 hours bilingual teacher for 82%

Who is glenn kenyon in 2009? 48% of my life as a teacher Math specialist for 18% about 4,100 days with students or of professional life about 25,000 hours bilingual teacher for 82%

Who is glenn kenyon in 2009? 48% of my life as a teacher Math specialist for 18% about 4,100 days with students or of professional life about 25,000 hours bilingual teacher for 82% 28% in kinder/ﬁrst,

Who is glenn kenyon in 2009? 48% of my life as a teacher Math specialist for 18% about 4,100 days with students or of professional life about 25,000 hours bilingual teacher for 82% 28% in kinder/ﬁrst, 54% in 4/5

Who is glenn kenyon in 2009? 48% of my life as a teacher Math specialist for 18% about 4,100 days with students or of professional life about 25,000 hours bilingual teacher for 82% 28% in kinder/ﬁrst, 54% in 4/5 18% in Middle School

Who is glenn kenyon in 2009? 48% of my life as a teacher Math specialist for 18% about 4,100 days with students or of professional life about 25,000 hours bilingual teacher for 82% 28% in kinder/ﬁrst, 54% in 4/5 18% in Middle School I’ve worked with a little over 1,800 students and their families

Who is glenn kenyon in 2009? 48% of my life as a teacher Math specialist for 18% about 4,100 days with students or of professional life about 25,000 hours bilingual teacher for 82% 28% in kinder/ﬁrst, 54% in 4/5 18% in Middle School I’ve worked with a little over 1,800 students and their families The San Francisco School Teacher. 50% students of color. 50% from public elementary schools. 40% on scholarships.

I think the camel might be a good symbol for me as a teacher. Methodical, self sufficient, agile over difficult terrain, good over the long haul. I trace my own thoughts about how to teach systematic problem solving to my students to this camel. This represents a problem I encountered in my homework while studying for my Masters in Math Education. The each week we were assigned problems to solve and write about. They were called “Problems of the Week”. This process had a profound impact on my practice because I encountered not only fascinating problems, but also was required to write about it and share with my classmates.

Balanced Math Program! Problem! Solving! Math encompasses all these and this is the sweet spot. Technology! Skills! Concepts!

At ﬁrst, I just set my students on a Camila Camel problem with little guidelines. it. There is a freshness when you have to create your own structures, but Camila Camel's harvest, worth its weight there is also a certain fatigue. I also feel it my role to be more a in gold, consists of 3000 bananas. more explicit model for my students. So here is how I The market place where the stash can be simpliﬁed the problem solving strategy. cashed in is 1000 miles away. However, Camila must walk to the market, and can only carry up to 1000 bananas at a time. Furthermore, being a camel, Camila eats one banana during each and every mile she walks (so Camila can never walk anywhere without bananas). How many bananas can Camila get to the market?

Wait, something is missing! I noticed that my 4th and 5th graders really struggled with writing about their thinking in math. They could write the answer in a sentence, and the could tell me a little about how they solved it, but it was not a very satisfying explanation. Other students and their families really did not know how to approach problems without clear algorithms and were clearly stymied. I wanted to persist with these more complex problems, but I also saw that just as I was doing in language arts or social studies, I needed to provide my students with some schemas or outlines to help them organize their thoughts.

Problem of the Week Essay Organizer 1. So I 2. created a Introduction quadrant Strategies/Procedure outline for (reword the problem) them to (how to solve this problem) collect their thoughts in essay form. Relevant Information Dead ends? 4. 3. Solution (s) Reﬂection (What was difﬁcult/surprising)

The problem solving structure and essay come from this amazing book. George Polya Stanford 1944

Original Version The problem 1944 solving structure and essay come from this amazing book. George Polya Stanford 1944

4 Principles of Problem Solving 1st Principle Understand the Problem 2nd Principle Devise a Plan 3rd Principle Carry Out the Plan 4th Principle Review / Extend

4 Principles of Problem Solving 1st Principle Understand the Problem A problem well-stated is a problem half-solved. -John Dewey

4 Principles of Problem Solving 2nd Principle Devise a Plan When you’re prepared, you are more conﬁdent. When you have a strategy, you’re more comfortable.

4 Principles of Problem Solving 3rd Principle Carry Out the Plan I think and think for months and years. 99 times, the conclusion is false. The hundredth time, I am right. -Albert Einstein

4 Principles of Problem Solving 4th Principle Review / Extend Knowledge of self is the mother of all knowledge - Kahlil Gibran

http://mathforum.org/

Ken Johnson Ted Herr Judy Kysh

With your partner, revisit the Camel Problem Focus on Intro, Plan and Reﬂection Camila Camel's harvest, worth its weight in gold, consists of 3000 bananas. The market place where the stash can be cashed in is 1000 miles away. However, Camila must walk to the market, and can only carry up to 1000 bananas at a time. Furthermore, being a camel, Camila eats one banana during each and every mile she walks (so Camila can never walk anywhere without bananas). How many bananas can Camila get to the market?

Introduce

Introduce

Introduce Plan Here is a partial Guess & Check Orderly Lists Solve an equation Draw a diagram Solve a simpler problem Eliminate

Reﬂect What, if anything, did this problem teach you? • What was interesting/intriguing about this problem? • How did this problem challenge you? • How did you manage your time? • What might you do differently next time?

Solve

Solve This problem has many good answers (between 400 and 500 bananas) And one best answer (533 1/3 bananas)

Student Work

comment: comment: comment: comment:

The title of this P.O.W. is Verdania: A Math Oddesy, Chapter 3: Sailing. This P.O.W. is SAMPLE A about two radar stations that both send out a signal at the same time and both hit a target. One station is 90 miles away from the target and the other station is 75 miles away from the target, forming a triangle. I will assume that both stations are on land. The problem is to ﬁnd out where the target is. The key math concepts in this P.O.W. are the Pythagorean theorem and logic. The ﬁrst step I took towards solving this P.O.W. was, I read the problem over about 4 times and still didn’t understand it until I realized that it gave me 2 sides of a possible triangle. One side was 90 miles and the other is 75 miles so, 90 squared is 8,100 and 75 squared is 5,625. 8,100+5,625 is 13,725 and the approximate square root of 13,725 is 117.15 this being how far the two stations are apart. This strategy only works if this is a right triangle that I just created. The next thing I did was, I divided the triangle into two separate triangles by drawing a line from where the 90 degree angle is to about half-way through the hypotenuse. One triangles side lengths were now, 90, 58.6 (half of 117.15), and ?. The other triangles side lengths were now, 75, 58.6 and ?. To ﬁnd the ? of the ﬁrst triangle: 90 squared (8,100) minus 58 squared (3,431) is 4,669. The square root of 4,669 is 68.33, this being how many miles away from shore the target is. My solution to this problem is that the target is 68.33 miles from shore if the two stations are 117.15 miles apart. I know that there are many, many other answers depending on how far apart the two stations are and mine is just one. I think my answer is correct because I revised it everyday for 3 days and I checked it over with my dad, it also makes sense. I found this P.O.W. relatively difficult because I have never been in the situation where I not only need to ﬁnd the answer, I need to ﬁnd the problem. Next time I will manage my time better, and get the essay done sooner. I ranin to a problem while I was writing the essay and that was that I didn’t really understand the question, next time, I’ll make sure that I understand the question before trying to answer it.

Read and evaluate the Sample Essays B and C When ﬁnished, compare your evaluations of one of the essays with your partner Were there major discrepancies in your evaluations? Why?

Issues I consider with student writing:

Issues I consider with student writing: • Student time management.

Issues I consider with student writing: • Student time management. • Student struggles with saliency.

Issues I consider with student writing: • Student time management. • Student struggles with saliency. • Student dependence on peers’ work.

Issues I consider with student writing: • Student time management. • Student struggles with saliency. • Student dependence on peers’ work. • Resistance to describe failed attempts.

Issues I consider with student writing: • Student time management. • Student struggles with saliency. • Student dependence on peers’ work. • Resistance to describe failed attempts. • Issues with logic: “my solution is correct because I checked with “______” and they got the same answer”.

Issues I consider with student writing: • Student time management. • Student struggles with saliency. • Student dependence on peers’ work. • Resistance to describe failed attempts. • Issues with logic: “my solution is correct because I checked with “______” and they got the same answer”. • Superﬁcial reﬂections: “manage time better”.

Issues I consider with my teaching practice:

Issues I consider with my teaching practice: • What are reasonable expectations for written work?

Issues I consider with my teaching practice: • What are reasonable expectations for written work? • How can I use rubric for more POSITIVE feedback?

Issues I consider with my teaching practice: • What are reasonable expectations for written work? • How can I use rubric for more POSITIVE feedback? • How can I provoke more sincere student reﬂection?

Issues I consider with my teaching practice: • What are reasonable expectations for written work? • How can I use rubric for more POSITIVE feedback? • How can I provoke more sincere student reﬂection? • Does the outline allow sufficient creative thought?

Issues I consider with my teaching practice: • What are reasonable expectations for written work? • How can I use rubric for more POSITIVE feedback? • How can I provoke more sincere student reﬂection? • Does the outline allow sufficient creative thought? • How can I promote process over product more effectively?

Issues I consider with my teaching practice: • What are reasonable expectations for written work? • How can I use rubric for more POSITIVE feedback? • How can I provoke more sincere student reﬂection? • Does the outline allow sufficient creative thought? • How can I promote process over product more effectively? Do these questions resonate in your own practice? How do you deal with them?

How this activity can be adapted to other classroom settings: * Being succinct in restating the introduction. Language teacher could give math problems to practice summarizing. * In prompt breakdown: students need the same skills in essay prompt understanding. *Use writing to solidify knowledge of content, metacognitive, sometimes we forget to do this in English class. Reﬂection on how it went. * Unpack the use of quotes in writing: “I picked it because it really shows what I am thinking” Unpacking faulty logic. * Helping kids not to compartmentalize: use math problems in English class. Writing a poem about math. *What do students value, how they value it, what is the most important thing to them: is the camel problem about enjoying the journey or getting the most $$$ or maybe hoard them. * Give ELL students comfortable in math, perhaps this would give them a window into expanding their language use (especially in elementary school) * Notion of critical inquiry: using students’ abilities of making meaning of the problem of their own. Writing is often about document, vs generating ideas, transformative. How human beings in all civilizations use math to make sense of things. Examination of Silk Road. *Make a plan, reﬂect on problem on words, or crosswords. * Talking with partners makes us smarter. Together

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