Howard Margolis tackles this topic in the following article.

Is Progress Monitoring A Waste of Time?

Teach 4 Mastery materials offer built-in progress monitoring for both teacher and student. This keeps everyone in the loop every day, every lesson. Short-cycle monitoring that gives you the feedback you need to make wise decisions!

Ever wonder why telephone numbers are 3 numbers then 3 more numbers then 4 numbers?

I am not talking about why from a process standpoint. Although it is good to know that the first three numbers are the Area Code, the second three are the City Code, and the last four are the Line Number, I am talking about why is it 3-digits then 3-digits and then 4-digits?

I am asking why not just 10 digits all together?

Well have you ever tried to remember ten things given to you at one time?

Our brains are really complex and amazing, but when things are typically given to us in groups larger than 5 most people start to struggle to remember them. Remembering ten things at once is really a skill few people possess. The working verbal or visual memory is what is used to remember information. When information is grouped into smaller chunks it allows our working memory to remember/recall the numbers much easier.

There are many online resources that offer to test your memory through tests known as Digit Span or Memory Span tests. Currently I can remember around 7 things, 8 is hard. Trying to remember them and then recall them backwards is even more challenging! So, to keep things simple telephone numbers are 3-digits, 3-digits, and 4-digits.

This allows most people to remember telephone numbers, although today's mobile phones make this much easier!

**Students Struggling with Facts:**

Students who struggle in math typically will struggle with being able to remember 3 or 4 digit blocks of numbers. If you have a student who is struggling in math try giving a digit span test and see how they do. If they struggle getting to 5 digits and are 6 years old or older, then they will probably have difficulty memorizing facts or doing word problems with multiple pieces of information.

I'd encourage you to work with them on their visual and verbal memory skills and see if that helps to improve their mathematics abilities! - Dan

Mathematics (specifically arithmetic) is possibly the most important form of communication that you will ever learn.

Math is a language all by itself.

It has a syntax (way in which information must be ordered), specific abstract symbols that have defined meanings, and when used appropriately it can communicate complex ideas and information. Math helps us to communicate value, cost, income, time, measurement, and many other areas critical to our everyday lives.

By learning math w*e learn how to communicate with others around us.*

Understanding arithmetic can open a whole new world related to business, geometry, science and physics. Both time and money are only understood by having a basic knowledge of arithmetic. For example, if I wanted to explain to someone the concept of addition I could use physical (concrete) objects to do so, but if I wanted to make it a language I need to be able to explain it with symbols (abstract).

Therefore learning math is about being able to communicate to someone else without the use of concrete representations. Of course, the best way to help students understand that they have something to communicate is to show them concretely. Then ask them, "How could you communicate that without objects?"

This is when the language of mathematics comes to life. I encourage you to check out the MasterPieces and MasterFractions and see how you can help students move from concrete to abstract with understanding! Until next time...

In August I traveled to Escazu, Costa Rica to do math professional development training with The Country Day School. Read the write-up from Katie Matthias-Fandre, Deputy Director of the school.
About the School:
"The Country Day School was founded by a family in 1963. These pioneers introduced relevant schooling in English for children of expats in Costa Rica. From early days the school's international identity was clear: to provide small classes to students who were interested in an American education and an international pursuit of goals.
Today, with high expectations for engaged learning and a culture where students seek personal excellence in every pursuit, CDS helps each community member become responsible citizens for their communites and the world.
As CDS continues its over 50-year journey, we value each decade and each learner whose personal story betters our own. With 875 students, over 200 faculty and staff, countless parents contributing daily to the school's legacy and the world around us, wecan only be encouraged, humbled and motivated to keep learning together for the sake of our students' borderless future."

At Teach 4 Mastery we believe that students need to understand how to move from concrete to abstract and from abstract to concrete applications. The step in between is sometimes the critical piece that makes the connection for the student! Dr. Pamela Hill recently published this article on the necessity of drawing in education.

Can students have too much tech in the classroom? I certainly think so.

The need for teachers is even more evident when students are given electronic devices. Students, like many adults, would rather use the tech for games or other entertainment purposes than the true purpose it was given, to learn.

A recent research study by Duke University economists Jacob Vigdor and Helen Ladd reports that evidence suggests providing technology and access to high-speed internet would only broaden the gap in math and reading achievement.

Teach 4 Mastery provides hands-on tools to help you and your students to grasp math concepts in a powerful way. The Perceptions program is designed to meet teacher's needs to deliver powerful, explicit instruction in the classroom.

I invite you to take a look and see the difference it can make in your students.

When we consider the word Mastery it has an elusive quality. At what point has someone mastered something? What are the qualifications or criteria for such a title?

In mathematics the area of mastery could be broken down into several subsets that are all important to overall mastery. They may occur at different times during the learning process depending on the emphasis of the instruction given.

Here is a list of the subsets that I find in the learning process:

**Computation** – Having the skills and strategies available to quickly apply the correct process to the problem presented. (*How *to do the problem)

**Concept** – Having the understanding to carefully reason through each step of the process, prioritizing higher function and accuracy. (The reason *Why* each process and step is chosen)

**Application** – Knowing which process to apply to the given problem in each situation. (*When* to use which process)

**Memorization** – Committing to memory the processes and reasoning skills to move from proficiency to fluency to automaticity or unconscious response.

Upon acquiring these subsets, then a student can reflect and demonstrate to others what they have learned with the key being the ability to teach others.

Students need to master skills at each level of arithmetic, including complex applications of those skills, in order to become a true master of their craft. Also, a student teaching back what has been learned is a critical demonstration of the student's ability to:

- Grasp the problem
- Choose a process
- Develop a solution
- Articulate the reasoning behind their choices
- or… Reveal where there are gaps in their understanding

Have you ever pondered that question?

Think about it... what is Math?

Is it formulas, rules, algorithms, computations, concepts, memorization, numbers...?

Recently I decided to so some digging and learn what it is that we do when we are studying math. Here is what I have learned.

Math is the study of:

1. **Numbers** - 0, 1, 2, 3... these are representative symbols of the next part.

2. **Values** - this is the quantity, the amount of things, such as five apples. The quantity is different than the number, even though they both go together. I have found this to be a rather significant concept for students to grasp.

3. **Shapes** - each of the values are represented by shapes. These shapes can be geometrical or non-geometrical.

4.** Space** - each shape takes up a specific amount of space. This space may be physical or abstract, but in early mathematics it is generally very physical.

5. **Interrelationships** - now are aware of those four things then we need to look at how do they all interrelate? How do we combine, compare, or contrast these things together?

6. **Specialized Notations** - these are the specialized symbos (+, -, /, x, =, etc...) that are specific to mathematics only. We need these in order to communicate clearly the interrelationships.

That is what I have found to be the study of math. Now once we realize what those things are, then we can look at computation, concepts, application, memorization as it relates to mastering mathematics.

I encourage you to think about this yourself. Consider sharing it with your students and engage them at a deeper level of conversation.

Developing number sense is a critical skill that students need to learn. When teaching Place Value manipulatives can be very powerful in developing this skill.

Do you see the pattern that is shown using the MasterPieces?

First, look at the unit cubes. You will see that from the top that each cube makes the shape of a square. Then the ten-bar makes the shape of a rectangle and the Hundred-block makes the shape of a square again. What if this pattern continued? What would the shape of 1,000 be?

Well, there are actually two different shapes that 1,000 could take!

Shape 1 is a CUBE. This would be the representation of a 3-dimensional shape with length, width, and height. This would be a good example of 10x10x10 or 10 to the third power. It is a great shape for older students learning more complex applications of mathematics. However, it is not the best example for developing number sense.

Shape 2 would be a RECTANGLE. This shape is a good shape for developing number sense in younger students because it continues the pattern already started with the units (ones), tens, and hundreds places.

Using the MasterPieces we can help students develop number sense of much larger AND smaller numbers!

Each unit cube is 1/2"x 1/2" (Square) Each ten-bar is 1/2" x 5" (Rectangle)

Each hundred-bar is 5" x 5" (Square)

Each thousand-bar would be 5" x 50" (a big long rectangle - explains why we don't make a thousand-bar)

Each ten-thousand-bar would be 50" x 50" (big square)

Each hundred-thousand-bar would be 50" x 500" (HUGE Rectangle) For perspective, 500" is 41'-8" long! That is as high as a four-story building!

Now lets change the perspective a bit to show Place Values lower than 1.

Let's make the 10,000 square (since it is a square) our unit block. So the big 50" x 50" square is now the unit block.

Now follow to the right...

The 5" x 50" rectangle is now one-tenths place. It is one-tenth the size of the unit block.

The next block that is 5" x 5" is now the one-hundredths place. It is 100 times smaller than the unit block.

The next block that is 1/2" x 5" is now the one-thousandths place. It is 1,000 times smaller than the unit block.

And finally the block that is 1/2" x 1/2" is not the ten-thousandths place! It is 10,000 times smaller than the unit block!

Now that is developing number sense!