Evidence

Fractions- Look at the work of the Rational Number Project.  You can find a compilation of the work from the University of Minnesota, find lesson plans, download fraction manipulatives and read of the success the project has had in public schools. 

In the Multisensory Math Program at ASDEC we use a similar concept based approach.  Students create fraction concept cards for the student notebook.  They choose their manipulatives and assemble card stock graphic organizers which they keep for reference.  The illustrate key vocabulary and concepts and all operations.  Reaction from teachers using them has been extremely enthusiastic.  "The children love them." 

There is evidence to support using circular models first and then transitioning to other models.  Teachers need to solidify concepts before teaching children to simply "push numbers around."  So many teachers tell me that they wish they had been taught this way.  You do not need commercial manipulatives to teach fractions either.  You can print fraction circles on cardstock.  You can use simple construction paper squares.  Folded paper can come later after students fully comprehend the concepts and remember:  fract- is a Latin root that means to break into parts.  I suggest that students need to cut or break something to really understand the linkage. 

In addition, students need to comprehend that the "fractions of one" is a place value concept.  This should be fully understood before decimals are taught.  Students will benefit from using manipulatives to add, subtract and simply fractions and mixed numbers.  Then, students can move on to the abstract level of experience using only numbers. 

I Hate Fractions!

I think I have never met a student who claims to be proficient in fraction concepts and operations.  More often than not, I am met with the title of this post.  Why do you think that so many students feel inadequate when it comes to fractions. 

I met with a student today who loved long division but would not even talk about fractions.  "Could I use a calculator?" she asked.  My students do not generally use calculators in sessions.  I assure success by using student friendly numbers to teach concepts. We use calculators to check answers, to teach concepts such as transformations of functions or in cases where it simply makes sense not to struggle with complex solutions to function applications. We need a more exact answer. 

So why is it that so many students do not feel confident with fractions?  We all need to ask ourselves this question. What is it that we are doing right and what is it that we are not quite conveying?  So much of higher math depends on an understanding of fractions. We need to consider what the research is telling us (circular models are better introduction) and there is a need to move from the concrete to the abstract for many students.  They need to understand the concepts before they begin to push numbers around in operations they do not understand.  In my workshops and courses, I keep hearing the following statement, "I wish I had been taught this way."  Or, "I know how to tell them to get an answer.  I just can't tell them why it works." 

Foundation Concepts

We all look at modern curricula, textbooks, scope and sequence documents etc. and feel a bit overwhelmed.  The Common Core State Standards initiative is one solution.  It doesn't  tell anyone how to teach or insist on specific content.  It merely gives us a common set of skills with which students should be proficient at developmentally appropriate levels, or grades. 

Even the Common Core can seem daunting if you dig too deeply at first.  I would advise checking out the summary document on the website itself.  You will find that the essential skills for each grade level are listed in focus documents that are incredibly simple to comprehend. 

The Common Core Standards for math evolved from the NCTM Focal Points.  For years these focal points formed a terrific guide for what needed to be mastered.  They are based on recent research in how the mind processes math.  This research has, I believe, fundamentally changed the way we teach mathematics. 

We no longer believe that simply having procedural fluency is enough.  Students today must understand what they are doing.  I see this almost everyday in working with students at in algebra.  some of my private clients attend schools which use ancient textbooks, heavy in language and complicated drills but little focus in the underlying concepts.

When we look at the evidence and milestones that students must achieve, consider investing additional time in teaching numeracy, construction and deconstruction of quantity for quantities up to and including ten, and of course, place value.  Place value and the concepts underlying multiplication, division and fractions are essential.  They simply must be done thoroughly.  As important though is the way these concepts are introduced and developed.   They cannot be rushed or taught procedurally for some test.  They must be firmly placed and soundly developed to a level beyond familiarity.  

Manipulatives: Must Be Efficient & Effective For What Is Being Taught

 I continually state that the use of manipulatives is to teach concepts.  Manipulatives are used to give the students a hands-on experience, one that is memorable and helps them interact with different representations.  This is a core principle of UDL (Universal Design for Learning).

The goal of using manipulatives is to illustrate a concept and then get rid of them.  Students should seldom perform calculations with manipulatives unless it is skill building and aids in memory.  For example, using manipulatives to illlustrate/see  calculations of large quantities using craft sticks and a place value mat is extremely useful...for a while.  It reinforces our place value system and allows them to physically experience regrouping and renaming-a concept that is one of our "continuous threads."   Once the student begins to recognize the concept and has formed a mental representation of the procedure involved, we would want to move the student to the representation and abstract levels.

Picturing groups of quantities can certainly explicate the meaning of multiplication and division.  They can help automatize select facts.  They can illustrate the concepts of multiplication and division easily.  They should lead to the use of specific fact families-and for LD students a very few- which are practiced and applied to the automatic level.

An inefficient use of manipulatives would be using counters to solve successive problems beyond the child's fact base knowledge system.  This is where the general education teacher and the special educator may part ways in using a book or set curriculum.  The published curriculum assumes that the child using the textbook has attained certain skill levels.  The special needs student may not have the skills required to use the worksheets and practice pages associated with a specific concept.
This is not to say that a special education student cannot be taught higher level concepts.  It only means that, as the What Works Clearinghouse suggests, struggling students practice math facts daily and as I say, use THOSE facts in their activities.  The teacher may make up a worksheet...yes, in your spare time of course...to fit the needs of the struggling student.  Using a computer program such as Math Type, or the equation editor in MS

Word, the teacher can create a simple worksheet with fewer problems on a page, ample white space, and a restricted set of number facts which can be practiced to complex levels.
Take for example, long division.  The typical text book would ask that the student work with a single digit divisor and two digit dividends until all multiplication facts have been worked through the division algorithm.  Then, as the student approaches multi-digit dividends, the student is expected to have mastered the times table facts.  This would preclude the special needs student from doing the activities.  The special education teacher can easily create a worksheet using one of the tools mentioned above.  The worksheet might use only one times table throughout but include problems with varying levels of complexity.  This would prevent a student from simply using counters to solve problems by hand repeatedly practicing unrelated and isolated facts independently of each other, and leading to frustration without serving to build any mastery of any fact families. 

As always, look at what you are teaching.  Decide what your goal is.  Are you teaching a concept or practicing applications?  Applications do involve complexity, but student should be moved from using know facts to incorporating new ones in ways that build fluency and competence and do not lead to frustration.  

Modifying Materials for differentiation

Students who have special needs are in your classrooms or sitting at dining room tables trying to do homework or worksheets which pose problems.  The font is too small.  There are 30 problems on a page.  There is no space on the worksheet to write calculations, or the space offered is far too small for normal sized digits forcing the student to write in miniscule script.

Keep in mind that students need space.  They do need to conserve paper, trees and the cost of printing; but we do have recycling these days.  Some students are simply unable to recopy all problems, maintain alignment and accommodate their learning needs on prefabricated instructional materials.

If you have special needs students in your class, consider making one document per unit which incorporates specific features to enhance learning.  Create a word processing document with the equation editor or a program such as MathType.   Use 18 point font or larger.  Place fewer problems on a page and provide ample white/work space for computations.  If you are working on a word problems or fractions, provide a space for drawing a representation of the problem.  This is easy to do if you use the "insert table" function and create a template which confines the problem to one quadrant, the representation to another, the calculations to yet one other and a space to write the solution and perhaps a justification of the solution.

In worksheets that  practice long division, integer operations, equation solving, you can also use the "insert shape" function to add lines for alignment, a number line or a space to insert a times table for reference.  Use the "insert text box" to offer students a place to write the number facts they will use.

On the second day of the unit, open the file again and save it under a new name.  This allows you to double click on the equation and change the numbers without changing the format of your document.  You can do this several times to use as warmups or homework.  Thus, you can even create different worksheets for the same class.  Your gifted students receive greater challenges and your students who struggle get a real chance to learn concepts and procedures in a student friendly format.  I am posting two examples for review. 

July 14, 2012
A new summer program.  Once again we are working with a wonderful group of young people.  Students with a variety of gaps in their mathematical tool kits have come to us for a short summer tune up.  In the math section of the middle school program we are working with manipulatives, illustrating concepts and applying them vertically through several operations and at different levels. Based on an intake assessment, we have begun with place value concepts, regrouping in addition and subtraction of whole numbers, and basic fraction concepts.  We began with base ten blocks and asked students to "prove by construction."  They have modeled quantities into the hundreds of thousands using manipulaltive objects.  They have transitioned to the representational level to draw fraction addition and subtraction problems as they regroup from the whole number place value to the fraction place.  Parents report that their children are coming home excited about doing math without pencil and paper.  They are understanding math concepts at an entirely new level and enjoying it more.  Next week we begin earnest work on multiplication, division, exponents, squares and roots.  We will be making conceptual linkages using Unifix cubes, string and beads.   

Thoughts on the Summer Program

Too little time. Too many conceptual gaps. Here we are three weeks into a summer program with students. The young people who appear each day exhibit a huge disparity of skill levels and yet most are woefully unprepared in predictably obvious areas: fractions, decimal operations, estimation, multi-digit multiplication and of course. . .long division. Most of the students have been taught multi-step procedures with few links to the underlying concepts. They demonstrate, individually and collectively, the research supporting an emphasis on numeracy education. They cannot visualize quantity, therefore they cannot apply that knowledge to a useful purpose.

They love games which reinforce patterns. They cut their own fraction manipulatives. They illustrate multi-digit multiplication with place value objects and various representational pictures. They learn a few selected number facts and patterns which aid word retrieval. They apply all of the above to algebra. They simultaneously work on multiplication, division, fractions and solving equations. They perform mental math with decimal fractions as they apply language which supports visualization. "How many pieces the size of one fourth can I make if I start with three and one half?"

We play with dice. We use pictures of pizza drawn on dry erase boards. They chant the perfect squares. They learn the many appearances and applications of the number one. They are a joy to teach.