Engaging Fully -- Math At Mustard Seed School

As the 2015-2016 school year began, we shared this perspective on math instruction with all of our Middle School parents.  It says well what we believe about mathematics education:

"One of our hopes for students this year is that they will develop the habit of engaging fully.  The middle school faculty will spend time considering what this means for students.  What does it look like during their school day?  What does it mean as they grow in their academic lives?  How can the faculty foster the habit of engaging fully?

Two of my favorite experts in math education address the issue of engagement:

"There are two versions of math in the lives of many Americans:  the strange and boring subject that they encountered in classrooms and an interesting set of ideas that is the math of the world, and is curiously different and surprisingly engaging.  Our task is to introduce this second version to today's students, get them excited about math, and prepare them for the future."
What's Math Got To Do With It?  
by Jo Boaler, Professor, Stanford Graduate School of Education

"Mathematics is the art of explanation.  If you deny students the opportunity to engage in this activity -- to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs -- you deny them mathematics itself.  So no, I'm not complaining about the presence of facts and formulas in our mathematics classes, I'm complaining about the lack of mathematics in our mathematics classes."
A Mathematician's Lament
Paul Lockhart, Saint Ann's School, Brooklyn, NY  (Ph.D. Columbia University)

Clearly Boaler and Lockhart believe that mathematics -- real mathematics -- is inherently engaging.  Mustard Seed School teachers believe that as well.  We need to be sure that we imbue all of our mathematics instruction with the richness that Lockhart describes.  Our students should be busy posing problems, making conjectures, looking for patterns, making mistakes, critiquing their own work and the work of others, discussing ideas, drawing pictures.  There should be many aha! moments.

Of course, our students need to be fluent and efficient in calculations.  They need to be confident of procedures.  They need speed and accuracy in recall of math facts.  But all of this work should be full of real mathematics as well:  Why does this procedure work?  Is there another algorithm that is easier?  Is it important how I line up the digits?  Why do we write that number that way?  Is the zero important?

We seek this kind of interest and engagement every day.  Beyond this, we seek to provide rich, profound mathematical experiences for children through the I Love Math activities and eighth grade exhibitions.  These parts of the curriculum are specifically designed to help students explore middle school topics in creative and unusual ways.  They are designed to prompt mathematical thought and discourse, and develop habits of mind that will serve students for a lifetime."

March 12 Conversation With Parents

Excitement is building for the upcoming Math Conversation With Parents.  The title this year is:

Better Mathematical Eyesight:  Beauty, Pattern and Form

Mustard Seed School

Community Room, 7pm

In its simplest statement, our plan for the evening is to discuss the connection between these activities in Early Childhood classes:

and this piece of work done by Eight Grade algebra students:

To connect the dots, Mrs. Pargellis and Nathan Johnson are gathering photos and videos that show the role of block building and tessellations in the earliest grades.  These activities are highly quantitative, and help children think about beauty, pattern, form, number, scale, symmetry and perspective.

Jenn Paffenroth and a few of her students will share some work done by Second Graders.  They will demonstrate lessons relating to factors with blocks and manipulatives; other work will be shown using paper and pencil.  What do children "see" through these two different forms?  What patterns emerge? 

Melissa McCallihan has lined up some Thunder and Lightning students to share a classic problem from math history -- Pascal's Triangle.  (see the blog posts of January 27, 2015).  Patterns?  You bet.  Stand back and see what the students have found!

Then three teams of middle school students will share solutions they prepared in response to thought-provoking problems.  Two teams will show solutions to the Bathtub problem (blog post from January 9).  These teams conceived of the problem in very different ways.  The final team will show the solution to the Paper Folding problem (blog post of February 4), which generated the graph above.  The graph is a way to "see" the numerical relationships -- the patterns -- implicit in the problem.  

Come and see what the students are doing!

You can RSVP right now!  Let us know you are coming:  RSVP for March 12

The Big Day Arrives

February 12, 2015 was our day to observe "I Love Math Day" (since Feb. 14 was a Saturday).  It was all that was promised.  Students at every grade level had a chance to enjoy math games, celebrate the important mathematics they have been studying, and have a visit with one or both of our guest mathematicians.

We are grateful to Gene Schmidt and Jay Lawrence who spent the day (the whole day!) visiting with students, listening to their questions and honoring their work.  Our guests were selfless with their time and generous with their mathematical wisdom.

The best way to share the beauty of this day is through pictures.  Here is a glimpse:

Distinguished Mathematicians Coming to Mustard Seed School

We are all mathematicians!  All of our students have been busy doing the things mathematicians do as they prepare for I Love Math Day.  

On February 12, we will be joined by two guests who love math and who use it in their personal and professional lives.  They will be visiting students at every grade level.  They will share some of the ways they use mathematics.  They will hear about some of the problems students have been working on.  They will give math-related books and games to enrich our classrooms and libraries.  

One of our visitors will be Gene Schmidt, father of Justus in the Fourth Grade.  He is a visual artist living in New York City.  Our second guest is physics professor who taught and did research for most of his career at Dartmouth College, Dr. Walter E. Lawrence.  He is the brother of Gary Lawrence.

If you see them at school on February 12, please welcome them!

Gene Schmidt:  Mr. Schmidt is a visual artist living in New York City. In recent years his work has moved out of the studio and onto the streets in a series of projects that combine elements of performance, pilgrimage, and sculpture. The projects are often inspired by, or are meditations on, biblical texts reimagined in a contemporary urban environment. While these projects are done in public for anyone to see and respond to as they wish, for the artist they are spiritual journeys and lengthy prayers with physical weight and dimension.

In Manhattan Measure, nearly 30,000 red yardsticks were used to measure the width and length of Manhattan, and then stacked to form a large cross-shaped sculpture. In Lovetown, PA, a copy of St. Paul’s text on love was made by cutting stencil letters out of 1200 panels of scrap wood. The text was then spelled out, letter by letter, by leaning the panels against buildings, walls, and fences, on a six-mile, twelve-day journey through Philadelphia, the “City of Brotherly Love”.

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 is currently planning his next big project (involving lots of cubes and the word REMEMBER). He is the father of Justus in the Fourth Grade.

Professor Walter E. Lawrence:  Professor Lawrence (Jay) is the eldest of five siblings, the youngest of whom is Gary Lawrence!  He grew up in Cassadaga, NY where winters were long and snow was plentiful.  The family home was near the top of a big hill, which was perfect for tobogganing.   After long toboggan runs, the brothers and sisters had many enjoyable (but exhausting) walks back up the hill together.

Professor Lawrence did not like school very much as a youth, but he did like math problems.  He eventually realized that school could help him do problems better.  He liked math even more when he learned that the natural world works according to rules that are best understood using math.  These include the laws of gravity, centrifugal force, and other mechanical things.  They also include laws governing electricity, magnetism, and light.  They even include very small things like atoms and molecules, and very large things like the orbits of planets in the solar system.

So after graduating from high school in Cassadaga, Jay did his undergraduate work at Carnegie-Mellon and received his Phd from Cornell, where he studied physics, the science of the physical world.  He did postdoctoral work at Stanford.  Then as a grown-up, he taught physics at Dartmouth College.  He is now retired from Dartmouth, but still works on physics problems and has fun using math.  He also likes to ski in the winter.  In summer he likes to swim and to hike with his wife Lenore and their dog, Muku.

The Ultimate Problem for 2015

Middle School students are working on the final I Love Math problem of 2015.  We hope you will take a few minutes to think about it, and we encourage you to discuss it with any of the children who are working on it.

Like all the I Love Math problems, this one is rich in math content appropriate for 6th, 7th and 8th graders.  This problem will have students thinking deeply about things like:

exponential growth
powers of 2
metric conversions
pattern recognition
creating charts and graphs
relationships between numbers
functions
modeling

By the end of the week, their mathematical intuition will be much better informed!  Enjoy!

More Problems for Fourth and Fifth Grade Students

Triangles, triangles.

Students in Thunder and Lightning have been given two more problems to ponder this week.  The first introduced them to the idea of triangular numbers.  This is a simple idea, and helps students grow in the area of pattern recognition.  It helps them see a nice connection between arithmetic and geometry.  It also sets them up nicely for the next problem!

Pascal's triangle is named after the French mathematician Blaise Pascal.  The triangle has been tracked to Persian and Chinese origins as early as the 11th century, well before Pascal's work in the 17th.  However, Pascal used it extensively and identified many of its properties and applications.  

Mustard Seed School students will be looking for patterns and interesting number sequences that emerge from the triangle.  Among other things, they will find triangular numbers, powers of 2, powers of 11, and the Fibonacci sequence.  See what you can find.

A Classic and Tricky Problem

Right now, middle school students are working on a classic and tricky problem.  This is one that PhD's in mathematics have debated publicly, to the embarrassment of those on the wrong side.

What makes it an appropriate challenge for middle school students is this:

•  It encourages students to begin with a guess
•  Then it encourages them to act out the problem
•  The evidence that emerges will improve their mathematical intuition
•  Then they are encouraged to answer the question "Why?"

If you have a middle school student, you can help by playing the game with them (you can do it with three playing cards, representing the three doors).   Keep track of your results.  What do you observe?  How do you explain this result?  

Thunder and Lightning Join the Fun

Fourth and Fifth Graders at Mustard Seed School (known as Thunder and Lightning) received their first I Love Math problems for 2015 today.  Like the middle school students who began their problem-solving a couple of weeks ago, they will be thinking about non-routine problems that demand a high level of creativity and persistence.

If you are new to the I Love Math season, you might want to look at the last three posts.  They show the kinds of problems that students are solving and the spirit in which they are working right now.  The tone is playful and fun.  Yet our goals for the children are as serious as they could possibly be.

We share the problems here so that parents and students at all grade levels can work on them.  Pick any problem that looks interesting to you.  If you have a child in Thunder and Lightning, then we encourage you to ask your child about these problems.  For the matchstick problem, get out some paper and pencil and begin drawing.  Better yet, get some straws or matches and start making triangles.

The Mind Reader problem begs to be acted out.  It takes 5 minutes, and it is lots of fun.  Play along with your student, and prepare to be amazed.

So if you have tried the Mind Reader a few times, will it always work?  Why?  With so many random choices, why do you keep getting an "o" and an "a"?

While these problems are lots of fun, don't miss the significant math content held inside:

divisibility by 3

divisibility by 9

pattern recognition

uniqueness

randomness

concept of proof (when have you proven something?  how do you build an argument?)

Enjoy!

The Second Problem of 2015

Students have turned in some astonishing work on the Bathtub problem.  Almost all teams came up with correct answers, and provided compelling arguments explaining why the answer would be 20 minutes.  A few submissions had unexpectedly creative ways of envisioning the problem.  The problem can become quite simple if we just "see" it correctly.

Time for the second problem:

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

The Sums

of Odd Numbers

Here is a pattern that seems to emerge from observing square numbers.  The sums of odd numbers produce square numbers:

1 = 1  (a square number)

1 + 3 = 4  (another one)

1 + 3 + 5 = 9 (yup!)

1 + 3 + 5 + 7 = 16  (hmmm…….  interesting…….)

1 + 3 + 5 + 7 + 9 = ___________  (well?)

Check a few more.  Will this always be true?

If you don’t think this will always work, come up with a counterexample (in other words, find a case where it does not work).

If you think it will always be true, find a way to show that it will always work (in other words, find a way to convince a skeptic.)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 

This problem comes from Proofs Without Words: Exercises in Visual Thinking by Roger B. Nelsen.  We hope that students will get a visual understanding of what square numbers are, where they come from, and how they grow.  

While students are having fun with this problem, they will hardly notice the serious math content they are learning:

square numbers

square roots

exponents

quadratic growth

the distributive law

binomial expansion

for any given square (81 for example), where does the square root come from?

--There are at least three distinct ways to think about this.  Can you think of three different ways that the square root 9 shows up?  is there a fourth?  (I'll give you one interesting answer:  the number of terms in the addition problem.  The sum of the first 9 odd numbers is 81!  Interesting!  Is this an accident?)

rich connections between arithmetic, algebra and geometry

the importance of seeing numbers and relationships visually

the concept of "proof"

pattern recognition

One hint:  Mathematicians draw pictures.  Think of different ways to draw 1 + 3 + 5.  Is there a way to organize your pictures so that you begin to see squares?  If you are stuck, ask your child to help you think!

Enjoy the problem!

The First Problem for 2015

Middle School students received the first I Love Math problem today.  There is much to say about the problem-solving activities we do each year.  In the coming weeks, we will say a great deal!

For today, however, I want to simply (1) share the problem and let you enjoy it with the students, and (2) let Steven Strogatz explain why he loves this problem so much.  Steven Strogatz is a professor of applied mathematics at Cornell, and the author of "The Elements of Math" series which appeared in the New York Times online edition in 2010.

First, the problem:

Could the problem be any simpler?  Three simple sentences.  Forty-nine simple words.  But plenty to think about.

Steven Strogatz was given this problem by his Uncle Irv when he was a boy.  As an adult, he finds that his thoughts return to it again and again.  In his book The Joy of X, a whole chapter is prompted by his memories of the bathtub problem.  Strogatz says, "There are broader lessons to be learned here -- lessons about how to solve problems approximately when you can't solve them exactly, and how to solve them intuitively, for the pleasure of the Aha! moment."

Dr. Strogatz goes on, "Word problems give us practice in thinking not just about numbers, but about relationships between numbers -- how the flow rates of the faucets affect the time required to fill the tub, for example.  And that is the essential next step in anyone's math education.  Understandably, a lot of us have trouble with it;  relationships are much more abstract than numbers.  But they're also much more powerful.  They express the inner logic of the world around us.  Cause and effect, supply and demand, input and output, dose and response -- all involve pairs of numbers and the relationships between them.  Word problems initiate us into this way of thinking."

We have many hopes, dreams and prayers for Mustard Seed's students in these days of problem solving.  May they grow in this next step of abstraction, and see more of the inner logic of the world around them!

Welcome to I Love Math 2015!

I Love Math Day is coming up on February 12, 2015.

I Love Math is a season in the school year when students, teachers, parents, and the whole community celebrate an everlasting love for math.  It is also a time when we work on some of our most profound hopes for Mustard Seed School students for their growth as mathematicians.  This is a season for:

• helping students learn more of the breadth and beauty of mathematics

• expanding children's understanding of what math is

• playful problem solving

• building persistence

• developing mathematical creativity

Already, middle school students have been considering the question:  "What do Mathematicians Do?"  Each grade level generates its own list.  At the beginning of the year, most classes create a list of 25 or 30 things.  We add to the list as we go through the year, and it often grows to 40 to 60 items.  Who knew?

A list like this serves at least three wonderful purposes.  

• It expands student thinking about what mathematics is.

• It helps children realize that they are mathematicians.  As the list grows, we ask, "Do you do these things?  When you do them, you are doing the things mathematicians do.  You must be a mathematician!"

• It helps students develop a broad array of problem solving strategies.  When a student is stuck, or needs a new strategy, we always ask the question, "What do mathematicians do?"  The list usually contains several helpful strategies for any particular problem.

Please look over the list below to see the kinds of things middle school students think mathematicians do.

Please also return to this blog frequently over the next several weeks.  You will find more postings about the activities of I Love Math season.  Importantly, you will find some of our favorite problems that we can all work on together.  You will also get the news about a distinguished visitor or two who love math.  Stay tuned.

What Do Mathematicians Do?

2014-2015

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