FUNdamentals in Mathematics 2016

A number of mathematicians and mathematics education researchers have recognized the special nature of the mathematical knowledge needed for K-12 teaching and its implications for the mathematical preparation of teachers. In particular, the interviews with Chinese elementary teachers in Liping Ma's 1999 book Knowing and Teaching Elementary Mathematics awakened many mathematicians to this issue and its mathematical substance. The mathematics to which U.S. schoolchildren are exposed from preschool through eighth grade has many aspects. However, at the heart of preschool, elementary school, and middle school mathematics is the set of concepts associated with the term number. Children learn to count, and they learn to keep track of their counting by writing numerals for the natural numbers. They learn to add, subtract, multiply, and divide whole numbers, and later in elementary school they learn to perform these same operations with common fractions and decimal fractions. They use numbers in measuring a variety of quantities, including the lengths, areas, and volumes of geometric figures. From various sources, children collect data that they learn to represent and analyze using numerical methods. The study of algebra begins as they observe how numbers form systems and as they generalize number patterns. Mathematics is often taught in elementary school as a set of algorithms without developing the conceptual understanding needed to move to higher levels. US teachers often have very good procedural understanding of the arithmetic of integers, fractions and decimals, yet a profound conceptual understanding in teachers is essential, as they must provide their students with this needed understanding for reaching algebra and even higher levels of mathematical thinking.

This is the third CTI seminar for me. This website is what remains from the first two seminars. I'll be building the new website for the 2016 seminar as we go. We will be motivated by the Common Core Process Standards, a set of desireable behaviors that teachers and students should emulate. See CCSS for a list of these standards. The first one really stands out, so I'll repeat it here:  Make sense of problems and persevere in solving them.

In addition, this seminar aims to show participants that deep understanding of elementary ideas like place value and fractions is attainable in elementary
classrooms, and that one way to cultivate this understanding is through irresistible problems. The seminar takes the position that learning mathematics can be motivated by interesting problems. The trick is to come up with problems whose solutions either require or strongly motivate the development of the area of mathematics to be learned. One could also take the narrow position that mathematics is about  problem solving.

Fortunately, there are plenty of arithmetic and geometric problems that motivate the need for algebraic thinking. And on top of
that, solving interesting mathematical problems in an appropriate social setting can really be fun. Have a look at the problems
below. You might not be able to solve any of them on the fly. But with two or three partner teachers, you can solve them all. Some of the problems below can be used to build entire lessons. For example, the first problem could motivate the entire section on place-value.

The 2016 Seminar

The first meeting is April 21. Here's the paper on the Area Model  for that meeting.

Our second meeting was May 5. We talked about the critically important topic of Place Value.

Our third and last meeting before summer took place on May 12. We discussed fractions. Please take a good look at the Howe and Wu papers below as you decide on a topic for your unit. I hope you will all read the article by Alan Schoenfeld (starts on page 11 of  this issue of the MAA Focus) called Powerful Classrooms. He lists five key dimensions of a Mathematically Powerful Classroom. 

Our first meeting of the new academic year is due to take place Sept 8 at UNCC. After this all but one meeting will take place at UNCC Center City. Our discussion for this and the next few meeting is about Place Value. In order to see place value in a way that learners of a new language see their native language, we will do our arithemtic in Martian, ie, base 6. My experience is that no matter how hard we work on base 10 problems, we won't see the power of place value until we see it at work in a completely different setting. The first part of the paper is about translating between Earth and Martian numbers and Martian arithemtic. The second part is related to James Tanton's lovely idea of fusing dots.

We looked at cubes for two September meetings. Here's the latest version of the cubes paper.

The last three seminars took place in November, and the topics were Rational Tangles, KenKen, and Magic Figures.

The 2015 Seminar

Before our fist meeting on April 23, please read Roger Howe's very powerful essay on place value. 

The Most Important Thing for Your Child to Learn about Arithmetic  by Roger Howe.

In case you're interested in teaching fractions, here's a paper by Hung-Hsi Wu. We'll talk about this paper at one of the early seminar meetings. Roger Howe has generously agreed to let me post his contribution to the fraction discussion here.

Here's is my set of problems on fractions, and in particular, a small set of problems for Thursday, April 30.

Before the May 14 meeting, please have another look at how Wu does fractions. Note the advantages of using part of a rectangle and of noting that fractions are points on a number line, and their size if precisely their distance from zero.  Also, have a peek at some of Scott Baldridges videos. 

For out meetings in September, here's my paper on Jim Tantons Exploding Dots: Fusing Dots.

Here are some prealgebra problems that some teachers will find useful.

Here is the essay we discuss on Oct. 1 on Modular Arithmetic.

Here is the essay we discuss on Oct. 29 on Divisors

Here is the essay we went through on November 5, Counting.

Here are the two essays to be discussed during the final two seminar meetings,  KenKen and 7-11

Here are the two essays to be discussed during the final  seminar meeting,  CountingWithCubes.

Advice from my friend Dan Teague.

Thankyou to all the CTI fellows for making this a most enjoyable experience for me. I hope we can work together again.

Below is the set of topics from the 2011 CTI seminar.

List of Topics

  1.  Place Value An essay by Roger Howe, Yale University and Susanna Epp, DePaul
    University. Arithmetic, first of whole numbers, then of decimal
    and common fractions, and later of rational expressions and
    functions, is a central theme in school mathematics. This essay
    attempts to point out ways to make the study of arithmetic more
    unified and more conceptual through systematic emphasis of place
    value structure in the decimal number system.
    Ma, Liping (1999). Taken from an online review: `Elementary school
    teachers are expected to teach almost everything: math, reading,
    science, social studies, and writing; along with nurturing,
    soothing, and encouraging. It's not an easy job. It's also hard to
    be an expert in any one piece of the job. But now, many are
    hearing that we're losing the ``math race'' to other countries.
    The drums of ``teacher competency" are booming... and any wise
    teacher knows where the drum sticks will be landing next!''

  2. Exploding Dots
  3. Unit Cube Problems

  4. Place Value Problems

  5. Magic Geograms

  6. Using parity, KenKen, and difference triangles.

  7. Bug in the Plane Problems

  8. Base phi and Fibonacci Representation, dynamic one-pile nim

  9. Perfect Card Trick

  10. Conway's Rational Tangles...Tom Davis' style.

  11. Euler's Formula, using Zome Tools

  12. Julia Robinson Math Festival Prize Problems

  13. Single Pile Nim Games, and Bouton's Nim, Puppies and Kittens

  14. Solving Linear and Quadratic Equations in Z_7

    Other References:

    The Major Topics of School Algebra by Wilfried Schmid and H. Wu  An essay
    listing the topics in high school algebra essential for advanced
    mathematics in college.

    Arithmetic for Parents: A Book for Grownups about Children's Mathematics
    by Ron Aharoni. Online review: `Ron Aharoni writes clearly and
    deeply about the crucial concepts of fundamental maths, how to
    teach them and how not to teach them. He explains the layered and
    subtle structure of elementary maths and how missing a layer can
    lead to frustration and maths anxiety. "There's no royal road the
    maths", an Euclidian quote he emphasizes which summarizes well the
    message in this book. I'm not sure the book is for ``Parents'' as
    its title suggests, but I highly recommend it for both lovers and
    ``haters'' of maths, regardless of their ``parental status.''
    Looking forward to Ron's next book. '

    Here's a collection of units written by New Haven
    mathematics teachers as part of the Yale-New Haven Teachers