Conversation With Parents: Math March 1 at 7pm

Conversations with Parents:

Math, The Art of Explanation

We hope you are coming to Mustard Seed School's "Conversations with Parents:  Math".   (Like all these conversations, they are free and open to the public.  Bring a friend.)

If you haven’t yet done it, please register here

The math conversation begins at 7pm, Thursday, March 1 in the Community Room, and the evening is designed to show some of the higher level mathematical thinking we want to develop in students.  For several years, Mustard Seed has been working to build persistence, creativity, collaboration and problem solving skills.  At the same time, we want to build positive attitudes about math to help students see mathematics as an enjoyable and worthwhile pursuit.  During the evening, you will see some of the things students do to achieve these goals.

--Parents will have a chance to try some challenging problems.

--Students will discuss the problems, and present their solutions

--We will have two visiting speakers:  Mr. Thomas Johnson and Mr. Hoyun Cho (See their bio information below.  They have important things to say!)

--There will be a question and answer time

--We'll have cookies afterward, so the conversation can continue.

No homework is required!  Just come to share your thoughts, learn and have fun.

In addition to several Middle School students, we are honored to have two distinguished speakers:

Thomas Johnson graduated from Mustard Seed School in 1998 and received his Bachelor of Science in Mathematics from the University of Scranton in 2006.  He spent a year working in London as a as a secondary school Learning Support Assistant.  In 2008 he completed a Masters in Teaching degree from Seattle University. Since then, he has taught middle and high school mathematics in the Philadelphia public schools.  At present, he teaches Ninth and Tenth Grade geometry in the Friere Charter School. Friere was the first charter school in Pennsylvania to be selected as a National Title 1 Distinguished School. It has won Pennsylvania's Keystone Award for Academic Excellence four times.  

We are asking Mr. Johnson to share his thoughts about the skills students need to develop in their elementary and middle school years to enter high school successfully.  

Thomas is the son of Janet Johnson, Mustard Seed School's Kindergarten teacher from the late 1980's until 2008.  He lives in Philadelphia with his wife, Sarah. Mr. Johnson was an "I Love Math" panelist in February 2009.

Hoyun Cho is a Ph.D. candidate in Mathematics Education at Columbia Teachers College.  His thesis topic is the use of cartoons as a teaching tool in middle school mathematics.  As part of his research, he has been working with Mr. Lawrence and Mustard Seed's Seventh and Eighth Grade students.  He has observed classes, helped students with their Algebra and Pre-Algebra lessons, and designed cartoon-based lessons that can help introduce new topics, review important concepts and clear up misunderstandings.  The goal: that students be engaged with Algebraic concepts in new and interesting ways, that their motivation and enjoyment increase, and that anxiety decrease.

Mr. Cho was an "I Love Math" panelist in February 2011.  He did his undergraduate work at Saginaw Valley State University in Michigan, and his MA in Math Education at Western Michigan University.  Mr. Cho taught middle school and high school mathematics in Korea, before coming to America for further education.  He has been published in several journals, and he has manuscripts in preparation and/or review for several more.  His daughter, Ohanna, is a Kindergarten student at Mustard Seed School. 

Mr. Cho and Mr. Lawrence have been invited to conduct a workshop at the 2012 International Congress of Mathematics Education.  This highly selective conference meets every four years, and will convene in Seoul this July.

The blog posts that follow are some resources you might like to have.  One is list of books by writers who care about children and math education.  Their writings have inspired us at Mustard Seed School.  The other one is a list of ideas to help your child when she is stuck on a problem.  See if you find these ideas helpful.

We hope to see you on the 1^st!  Peace.  Gary

Resources for Parents -- What Do Mathematicians Do?

What Do Mathematicians Do?

2011-2012

This list serves at least two wonderful purposes.  We ask students this question all the time. When they list things mathematicians do, we can always say, "Do you do these things?  When you do them, you are doing the things mathematicians do.  You are a mathematician!"

When a student is stuck, or needs a new strategy, we always ask the question, "What do mathematicians do?"  Somewhere on the list is likely to be a good strategy for any particular problem.

The list is long, gathered from students and adults at Mustard Seed School.  If sometimes a word choice sounds like it came from a child, it probably did.  They own this list!

1.     Solve problems

2.     Create problems

3.     Graph

4.     Predict answers

5.     Make grids

6.     Use coordinates

7.     Discover new things

8.     Create order

9.     Create disorder

10.  Arrange numbers

11.  Rearrange numbers

12.  Support and build technology

13.  Analyze stocks

14.  Find formulas

15.  Explain formulas

16.  Use what they know about small problems to solve big problems

17.  Cook

18.  Ask questions

19.  Experiment with numbers

20.  Count

21.  Estimate

22.  Approximate

23.  Make boundaries

24.  Think, think, think

25.  Prove things

26.  Disprove things

27.  Calculate

28.  Arithmetic

29.  Draw pictures

30.  Make art

31.  Observe patterns

32.  Simplify problems

33.  Make assumptions

34.  Make mistakes

35.  Consider the infinite

36.  Consider the very small

37.  Act out problems

38.  Factor

39.  Speak the language of creation

40.  Speak the language of the universe

41.  Speak the language of God

42.  Collaborate

43.  Get help

44.  Use resources

Resources for Parents -- Books That Inspire Us

Some Books That Inspire Us

Mustard Seed School

February 2012

Enjoyable reading for a general audience (in no particular order):

           1.     What’s Math Got to do With It?            

by Jo Boaler

Viking Penguin, 2008

           2.     Letters to a Young Mathematician            

by Ian Stewart

Basic Books, Perseus Books Group, 2009

                        Good for older students

           3.     Fermat’s Enigma            

by Simon Singh

Anchor Books, 1997

           4.     Mathematics Education for a Changing World            

by Stephen S. Willoughby

Association for Supervision and Curriculum Development, 1990

5.     A Mathematician's Lament                          

by Paul Lockhart

Available on the internet, just Google it

           

6.     Seeing Further:

The Story of Science, Discovery, and the Genius of the Royal Society

edited by Bill Bryson

William Morrow, 2010

7.     The Man Who Counted: A Collection of Mathematical Adventures

By Malba Tahan, Copyright 1972

W. W. Norton and Company, 1993 (English Translation)

Good for students Grades 5 and up 

8.     A Mathematical Mosaic:  Patterns & Problem Solving

By Ravi Vakil

Brendan Kelly Publishing, Inc, 2008

Good for older students

Heavier sledding for the mathematically inclined (no particular order):

9.     The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Richard Courant Lecture in Mathematical Sciences

Delivered at New Yourk University, May 11, 1959

By Eugene P. Wigner, Princeton University

Available on the internet, just Google it

            10.  Fostering Algebraic Thinking            

by Mark Driscoll

Heinemann, 1999

11.  Knowing and Teaching Elementary Mathematics:

Teachers’ Understanding of Fundamental Mathematics

in China and the United States                        

by Liping Ma

            Lawrence Eribaum Associates, Inc, 1999

12.  Is God a Mathematician?

by Mario Livio

Simon & Schuster, 2009

13.  Proofs Without Words:  Exercises in Visual Thinking

             by Roger B. Nelsen

            The Mathematical Association of America, 1993

14.  Proofs Without Words II: More Exercises in Visual Thinking

             by Roger B. Nelsen

The Mathematical Association of America, 2000

The Day Arrives -- February 14, 2012

We All Loved Math (and Still Do)!

  

  

  

It was a great day of Math Love at Mustard Seed School on Tuesday, February 14.  After weeks of hard work, it was time to relax and enjoy it all.  The community room was covered with illustrations of how the math challenges of the past few weeks have been solved.

Students approached the problems with great enthusiasm and creativity.  An Eighth Grade team created puzzle pieces to illustrate a visual proof of the Pythagorean Theorem (above).  Other teams created posters and videos.  One team inserted a digital animation to show how the pieces move on a fixed square to create the proof.  Some ambitious teams found more than one proof.

One of the problems involved Three Brothers and 21 Oak Casks (the problem is reprinted in our older posts).  The solutions created lots of opportunity for students to show their thinking.  This poster from a Seventh Grade team shows how the casks can be divided so each of the three brothers is treated fairly.  Another Seventh Grade team created a 3-D stage set with 21 casks so they could "act out" their answer.  

 

Here is another beautiful demonstration of the Pythagorean Theorem (a Seventh Grade Team.)  

Four panelists from very different backgrounds discussed the impact of math on their lives and careers. 

 

Erin Bellissimo spoke of how she uses math to help people make spending and investment decisions, Steven Orlofsky spoke of how math is used in legal matters to prove or disprove cases, and Gene Schmidt spoke of the deep connectedness between art, math and God.  Then they were joined by none other than Pythagoras, complete with a set of triangles, who was quite interested in watching middle school students prove his theorem.

 

Student teams were celebrated for their astonishing achievements in their mathematical work.

 

Congratulations to all of the Upper School student for their hard work...and for loving math!

February 14th! I Love Math Day

Students are making their last preparations for I Love Math Day.  They are solving problems, creating beautiful presentations to share their solutions, guessing how many chocolate kisses are in a jar, and thinking about the questions they want to ask our distinguished panelists.

This year, Conversations With Parents on Math:  The Art of Explanation will be on March 1.  We are very happy that it follows so closely on the heels of I Love Math Day.  We will have students come to present some of their work and show how problem solving helps achieve some our most important educational goals.

We have a couple of outside guests coming, and they will be able to provide some objective commentary to the proceedings.  They will also be happy to answer questions that parents might have about math education.  We'll be announcing the visiting guests next week, so come back to the blog again then.

Racing Up and Down the Mountain is the latest problems Middle School students are working on.  It was the most challenging of the four, and led to some amazing thinking and presentations.  Send in your questions and solutions to Mr. Lawrence at glawrence@mustardseedschool.org.

Racing Up and Down the Mountain

Racing Up

and Down

the Mountain

Matthew Speedster drove up a mountain road at 50 kilometers per hour.  He turned around instantly at the top and drove back down the same road very fast.  When he returned to his starting place he discovered he had averaged 100 kilometers per hour for the entire trip. 

How fast did he come down the mountain?

When you finish this problem, please share your answer and your thinking with Mr. L.  He may have some follow up questions for you to consider.

February 1 -- Welcome Back!

With only two weeks to go until I Love Math Day, we are happy to announce the panelists for the Middle School audience, and share the next in our series of problems students are working on. Enjoy!

At the heart of I Love Math Day is a panel discussion where students have a chance to meet and ask questions of adults who love math. We are delighted that this year our panelists will be:

Erin Bellissimo: Ms Bellissimo has had a 20 year involvement with Investing, Finance and Consulting. She served as Director of the Heilbrunn Center for Graham & Dodd Investing at Columbia Business School in New York. Her professional experiences have also included investment banking analysis at J.P. Morgan and advisory work at several consultancy firms. Ms Bellissimo was the recipient of the $50,000 Gabelli Prize for her contributions to value investing.

Ms Bellissimo did her undergraduate work at Bucknell University, graduating with a BS in Business Administration and Management. She completed her MBA at Wharton. Three of Erin’s children attend Mustard Seed: Maggie in Kindergarten-Water, Andrew in First, and Katelyn in Third. We are waiting for Charlie to join his MSS siblings soon. Ms Bellissimo serves on the Board of Trustees of Mustard Seed School and chairs the Development Committee. She donated her Gabelli Prize winnings as a gift to the school to acquire laptops and software for student use.

Stephen Orlofsky: Mr. Orlofsky has had a distinguished career as a lawyer and judge for almost 40 years. In his private practice with the firm of Blank Rome in Cherry Hill, New Jersey, Mr. Orlofsky has specialized in complex litigation and alternative dispute resolution. From 1996 to 2003, he served as a United States District Judge for the District of New Jersey.

Prior to leaving the federal bench in 2003, Mr. Orlofsky joined a 13-member Judicial Assessment Team to evaluate the Iraqi judicial system. The team provided advice and counsel to the Coalition Provisional Authority in reconstructing the Iraqi court system.

Mr. Orlofsky did his undergraduate study at City College of New York, and completed his law degree at Rutgers School of Law – Camden. He loves math and reports a sighting of the Pythagorean Theorem in a trial he recently witnessed. He is the grandfather of Vera and Reese in the First Grade, and Ian in the Nest.

Gene Schmidt: Mr. Schmidt is a visual artist living in New York City. He describes the foundation of his artistic practice in two simple sentences: “My father was a preacher and a carpenter. He put thoughts of God in my head and tools in my hands.” Most of Mr. Schmidt’s work over the years has been sculpture, but more recently his sculpture has become an element in larger projects he likes to think of as urban pilgrimages. For Manhattan Measure, the first of these projects, he spent a year measuring the width and length of Manhattan using nearly 30,000 yardsticks, and then stacked them in the shape of a cross on his studio floor.

Mr. Schmidt was born and raised in Quakertown, Pennsylvania. He went to college at Grace University in Omaha, Nebraska, and completed an MFA at the School of Visual Arts in New York. He likes to cook, read, and take walks. He is the father of Justus in the First Grade.

Pythagorean Theorem

Demonstrating the Pythagorean Theorem

You know what a right triangle is.  Right?

It has one right angle. 

(Could it have more than one?  That is worth thinking about for a minute.)

A right triangle always has one side that is longest.

The longest side is always across from the right angle.

It has a name:  hypotenuse.

The other two sides will always be shorter than the hypotenuse.

They always come together to “create” the right angle.

A long time ago, a mathematician named Pythagoras figured out that there is always a relationship between the lengths of the three sides of a right triangle.  We could measure the lengths of the two short sides and call those lengths a and b.  We could measure the length of the hypotenuse and call it c.

Here is the relationship:

a² + b² = c²

Pythagoras liked to picture things geometrically.  So the way he really thought about this formula was like this:  If I take the right triangle, and if I draw a square on each of the three sides, then the areas of the two smaller squares will always add up to equal the area of the big square.

Demonstrate the Pythagorean Theorem using the square and the triangles you were given.

Welcome Back to I Love Math

We hope you enjoyed the Three Brothers and the Oak Casks problem.  Every Middle School team submitted a correct response.  The submissions were exuberant, artful and joyful.

And here is the teaser!  You can see them on display on March 1 at the next Conversation with Parents called Math:  The Art of Explanation.  Some of the submissions were on paper; some on poster board; some on foam board.  A few were digital.  One team provided a Google Doc presentation.  Two teams submitted a video demonstrations.

In case you tried to work it out, the solutions were variations on this theme:
One brother would get 3 full casks, 1 half-full cask, and 3 empty casks.
The other brothers would each get 2 full casks, 3 half-full casks and 2 empty casks.

The best submissions provided this solution, and one other alternative.  Some teams explained why there could be no further solutions.  Very powerful mathematics.  Very powerful thinking.

Below you will find the next problem, The Aunt and Her Nephew.  My brother, the physics professor at University of Chicago, sent me the problem last summer and was having trouble completing the solution.    Before I could figure it out, my daughter had it!

If you have a Middle School student, ask how they did with it.  And see if they will give you any hints.  Enjoy!

The Next I Love Math Problem

The Aunt and Her Nephew

Aunt Annie and her nephew Neal were walking down the street.  In the distance, they saw three well-dressed people disappear down an alley.  Aunt Annie asked Neal if he knew how old they were.  Neal did not.

The three were well-known to Aunt Annie, so she was able to give two helpful clues:

The product of the three ages is 2,450.

The sum of their ages is twice Neal’s age.

Neal was grateful, and went away to think for a few minutes.  Being a clever and creative mathematician, he was able to make good use of the clues.

When Neal returned, he told his Aunt that the clues were very helpful, but they were not enough information.  She agreed.

Aunt Annie added one more clue:

            All three of the ages are less than hers!

Neal was delighted.  He already knew how old Aunt Annie was, and he now had enough information to determine the ages of the three well-dressed people.

How old are the three?

How old is Neal?

How old is Aunt Annie?

Welcome to I Love Math 2012

Welcome!  I Love Math Day is coming up on February 14.   In the preceding weeks, middle school students work in small teams to solve challenging problems.  They are given one week to solve each problem and prepare a solution to submit to the judges.

By way of this blog we encourage all school families to read the problems and join in the fun.  We hope parents and students at all grades will find ways to talk about these problems, puzzle over them, and enjoy the pursuit together.  Click on the Older Posts button below to see the three problems posted so far.

These problems are published for the school community with a one week delay, so middle school students have already submitted their presentations.  However, we love getting solutions from parents and all school families.  Please email Mr. Lawrence with any questions or comments you have, and please send in solutions (glawrence@mustardseedschool.org).

The problem that really counts for the first week is Three Brothers and the Oak Casks. The other two are just for fun. They are great for warming up and thinking outside the box.

I look forward to hearing from you!  Enjoy.  Mr. Lawrence

Ten Soldiers

in Five Rows

Arrange 10 soldiers in five rows in such a way that each row has four soldiers.

Nine Dots

Four Lines

Without lifting your pencil, connect all nine dots using only four straight lines.

•         •         •

•         •         •

•         •         •

Three Brothers And

The Oak Casks

Long, long ago three brothers did a good deed by saving their small town from disaster.  (It had something to do with a fire-breathing dragon, but that’s another story.)

The mayor and all the townspeople rewarded them by presenting 21 beautiful, highly polished oak casks.  They were all sealed tight.  One third of the casks were filled with a rare, precious oil.  One third were half full.  The rest were empty.  The brothers could easily tell which were which by the weight.

Being well-mannered and fair brothers, they wanted to be sure that each one got exactly the same number of beautiful oak casks, and exactly the same amount of oil.  And of course they didn’t want to open the casks just to re-distribute the oil.

They were deep in thought and discussion trying to solve the dilemma.  Just then Beremiz came along and declared, “The division, my friends, can be done without much complication.  Here is the simplest possible solution ………………………….

What do you think Beremiz proposed?