The problem is magnified when we consider that though students will study algebra, geometry, trigonometry, statistics, and calculus in upper mathematics, they are unlikely to encounter the mathematical sense for these courses in the years preceding them. Indeed number sense provides the foundation upon which we build the subsequent mathematical structure.

Fortunately there are simple strategies that we can implement to help students develop number sense. These strategies can be incorporated into your existing instruction with ease.

This is part one of a series that will explore the components of number sense and help teachers help their students grow mathematically. Though number sense is difficult to define, it is easy to recognize. Five components make up instruction in number sense. They are:

• Number Magnitude
• Estimation
• Mental Computation
• Mathematical Properties
• Effects of Operations

Adults and students who possess these components have a solid grasp on number sense.  We will explore each of these components in upcoming articles, but for now, you can see an example of this instruction in action on this video in which students show their thinking about fractions.

Brad Fulton is a veteran math teacher and nationally recognized educational consultant and keynote speaker. In 2005, he was selected as the C.L.M.S. middle school educator of the year.

Brad and I begin our instruction in solving equations by establishing a solid foundation for understanding how the equal sign functions in mathematics in general and in algebraic equations in particular. You will see us initiate our lesson by asking students to solve equations that are presented in parallel conjunction with a visual model of a balance scale.

As a teacher and an expert in the needs of your specific students, you will decide how much of a foundation already exists in your class and how much still needs to be developed.  Thus you may leave the visual model sooner or later depending upon your students’ conceptual development of equation solving.

This foundation is not only necessary for your students’ understanding of equations, it will also provide invaluable assistance to their teachers in future higher mathematics courses who build upon the foundation we establish. Often students demonstrate some success in mathematics until they reach the secondary level. This can be attributed to the fact that some of them do not really understand mathematics at the elementary level but are simply mimicking our models and processes without conceptual fluency.  As these mathematical processes become more involved in middle and high school, their lack of understanding leaves them with skills that are little more than trying to master algebra by rote procedures. While this worked on simpler or shorter problems in the past, it leaves them wandering without a compass in the rugged algebraic landscape.

One of the components of teaching mathematics using Conceptual Layering is the presentation of concepts using positive whole numbers. Only when students have demonstrated an understanding of the concept are fractions and negative integers introduced. During the course of this training video, we will focus on equations that utilize positive whole numbers to develop the concept. Many of the worksheets in the accompanying PDF handout provide students with opportunities to practice these same concepts with more challenging numbers.” The vocabulary of additive inverses, multiplicative inverses, subtraction, division property of equality, and combining like terms are all covered in this video.

http://www.youtube.com/watch?v=3MkdRAJ1PAA

Brad directs the students’ thinking during the lesson by starting with a simple multiplication problem involving a pair of two-digit numbers; once the students are engaged, he begins to steer them toward much deeper thinking skills that involve concepts that will take them beyond the level of a middle school classroom.

Brad uses specific questioning strategies to elicit this higher order thinking. He asks the students to make “conjectures” and support or reject them using examples.  n using these techniques in his algebra class, Brad is not only focusing on grade-level concepts. He is also laying the foundation for his students’ high school experiences with the postulates, conjectures, and theorems of geometric proofs.

You will also notice a fluid quality to the lesson. Rather than script a predetermined course, Brad is attending to his students’ responses to guide the lesson. This doesn’t mean that Brad enters the activity without a plan–in fact he fully intends the lesson will take his students to the desired destination. At the same time however, Brad is ever cognizant of how his students are reacting with the lesson. He adjusts the scope and the pace of the lesson to ensure that all students are able to remain engaged and to glean the most benefit from this experience.

Today many state standards ask us to focus on specific standards. The purpose of this is to guarantee that all concepts are covered.  However this dissecting of concepts can also segment the students’ thinking at the cost of making important connections. By weaving together many mathematical components, Brad not only covers the algebraic content he intended, he opens the students’ experiences to many other mathematical concepts. Brad covers the number operations of two-digit multiplication, the number sense of understanding the effects of operations, and the ideas of probability, mathematical reasoning, and algebraic thinking.

Brad also wants to begin at the simplest levels of this continuum by asking the students to multiply numbers and ends with the highest level of abstraction in the algebraic application. This incremental teaching technique is called Conceptual Layering. Brad and I have used this revolutionary pedagogical method to achieve incredible data-supported increases in our students’ mathematical progress.  You can learn more about this technique in our DVD’s Leo’s Pattern and Teaching Two-Digit Multiplication Through Conceptual Layering. 

Brad uses questioning during the lesson. There are many opportunities and purposes for the questions we ask students. We can assess their understanding by asking them questions to which we already know the answers. We can ask questions to keep them engaged in the lesson. While these and other goals are fine, Brad also utilizes questioning to help the students formulate and extend their thinking about mathematics.

Brad often asks his students to demonstrate their conjectures by more than a simple raise of their hands. Since mathematics is such a specific content area with definite right answers, students often are reluctant to commit to an opinion or conjecture for fear of being wrong. For this reason, Brad asks them to get up out of their seats or demonstrate their ideas in a physical way. In order to physically respond, students must first think about the issue. This way, Brad ensures that all students are engaged in the mathematical process.

Brad has shown the wealth of mathematics that can be developed from the simplest of mathematical tasks. Students used their knowledge of multiplication, estimation, probability, and mathematical reasoning. By integrating these components, students make deeper and stronger connections.  Interconnection comes about through integration of concepts.  The result is a strong foundation in number sense and algebraic thinking.  By first establishing this solid foundation, students can build a much higher structure of mathematical understanding and fluency. This again is a component of Conceptual Layering.

Much more can be accomplished with this lesson. You could extend the activity to ask students to consider the layout of the digits that would minimize the product of the two-digit numbers.  And of course you can apply this approach to any type of mathematical operation. Younger students can explore addition and subtraction properties while older learners can investigate fractions and exponents. Encased with this DVD is your PDF handout showing templates for not only the activity Brad demonstrated but also for many other operations.

One of the characteristics of the human brain is its fascination with challenge. Brains abhor boredom and tedium. For this reason, a lesson such as this one will engage and empower all levels of students. We’re sure your students will not only benefit from this simple yet mathematically rigorous lesson, they will enjoy the learning process as well.

http://www.youtube.com/watch?v=ofxWAC8awCg

While a textbook is fine for providing skills practice for students, number sense can only be developed through the teaching processes we design for our students.  These number-rich environments are the best soil in which to nurture the higher thinking skills necessary to develop number sense.

Knowing what constitutes number sense in the mind of a student is much like knowing what makes a work of literature great.  We can’t always put it in concrete terms, but we know when it is there.  However, we can list the five factors that make up number sense:

• Estimation

• Mental Computation

• Mathematical Properties

• Effects of Operations

• Number Magnitude

Brad integrates each of these components into the lesson. The questions Brad poses to his class during the activity not only challenge the students, they are what fuels the development of the number sense itself. It is always Brad’s goal to ensure that every student is engaged in the task. Brad easily extends this activity in his algebra class. Thinking accelerates when the numbers are replaced with variables.  Due to the higher level of thinking, far fewer cards are placed on the number line.

This is a powerful yet easy-to-teach activity.  Students incorporate so many of the components of number sense in this lesson.  They used estimation skills, mental computation, and even relied on their previous knowledge of operations with simplifying and comparing fractions.  They reasoned about the effects these operations have, and discussed the magnitudes of the fractions.

This DVD is coupled with a PDF handout to form a powerful tool for effective instruction.  The accompanying handout provides nearly 200 cards of fractions, decimals, percents, and algebraic expressions.  A set of blank cards enables you to design customized cards for your students also.

We are always surprised at how creative and resourceful our students can be when they are given the opportunity to step outside the textbook and journey through the rich world of number sense.  We’re sure your students will enjoy participating in this activity as well, and they’ll be enriching their number sense with fractions, decimals, and percents as well.

We use a technique called “conceptual layering” to gradually transition students from their world of arithmetic on a successful journey to algebra.  We will show you how to build each new concept upon previous ones so that students can navigate this journey smoothly.  Notice that each new skill adds only a slight step to the one previously mastered and will lead smoothly into the subsequent lesson.

We’ll begin by showing how you can establish a solid foundation of understanding of multiplication of numbers and then transition them incrementally into multiplication of binomials and polynomials.

Then we’ll demonstrate how to reverse that process so students can factor polynomials easily.  We’ll cover polynomials in which the coefficient of the lead term is one and also those more challenging examples in which the a term is greater than one.

This DVD is coupled with a pdf handout to form a powerful tool for effective instruction.  The accompanying pdf handout provides comprehensive material for students to practice these strategies.  These can be used for in-class practice or as homework exercises.  Either way they will better prepare your students for the practice material already provided by your textbook. So you are free to use them to supplement the problems in your texts or to replace them entirely.

So often there is a gap between our students’ experiences and what is expected in our standards. You’ll find the strategies presented here provide a crucial instructional link in connecting your students to the material in your text book or on your state test.  Our activities and instructional techniques have been classroom tested, classroom proven with students all over the country.  They are also validated with independent test results.  We’re sure you’ll find they provide an important part of your instructional approach also.

Now we offer our popular workshops in a low-priced DVD format.  These high quality presentations offer four key advantages:

1. Quality: Bill and Brad have been recognized for their high quality and effective staff development at hundreds of presentations throughout the county.  Every single workshop received rave reviews!
2. Affordability: Not only is the cost much lower than hiring a presenter, you own the presentation forever!  When you hire new staff next year, you still own the presentation.
3. Convenience: Can’t get all your teachers out of the classroom for a one-day workshop?  No problem–they can participate in the training at their convenience.  Each DVD comes with a comprehensive printable handout on a CD.  Run as many copies of these classroom-ready and classroom-proven materials as you need.
4. Guidance: With our Moderator’s Guide, you can host your own day of inservice.  Simply download the guide for free from our website and follow the simple instructions for leading your staff through a productive day of staff development.

http://www.youtube.com/watch?v=kht60tE3fhg

Equation Search is a great activity to start the year with, or use it whenever you wish to work on number concepts.

The blank master included as a downloadable PDF file can be used for an overhead transparency or as a student homework master.

The master with some colored “loops” shows some of the ways you can make equations from the given numbers. You may make equations horizontally, vertically, or diagonally.

http://www.youtube.com/watch?v=lrZh6o-qc94

This video gives 4 key steps to success with this method. Learn how to:

1.     Analyze the problem

2.     Use Guess and Check to gain insight into the nature of the solution

3.     Reduce the time taken to arrive at the solution

4.     Become a better problem solver

You can find the PDF file for this activity here.

After you view the video and have a little practice, your number sense will increase and your problem-solving abilities will be stronger. Enjoy!

http://www.youtube.com/watch?v=YymdAQE6l_g

This video gives 4 key steps to success with this method. Learn how to:

1.     Begin the problem solving process

2.     Use your initial guess to gain more clues to the solution

3.     Check your result

4.     Repeat the process until you have the solution

You can find the PDF file for this activity here.

After you view the video and have a little practice, you will be off and running towards being an expert at Guess and Check Problem Solving. Enjoy!

http://www.youtube.com/watch?v=4byKjH6LtxY