Post # 29 Diagnostic Prescriptive Teaching

In recent days I have been speaking with numerous individuals who have taken this course and are using the methodology and strategies in practice.  I have sought to elicit from them what their perception of this approach is at its heart.  We have listed core features, strategies and central tenets.

Yes, CRA is central to our approach and it is a good place to begin.  Once one accepts that multisensory is superior to unisensory in reaching the most students in a single lesson, accepting the CRA instructional sequence is a given. At first glimpse the National Math Panel could only endorse the approach in a modest-indirect way, but one must read on to understand why.  Scientists claimed they could not "tease out" from its total, the impact of any one of its components.  This means that it proved difficult to uniquely identify the impact of using manipulatives alone.  We do not use manipulatives without words (auditory input) or a visual component.  That is telling.

Following this trail, one might read into the comments regarding the results from the Rational Number Project.  In it the researchers clearly state that they believe their repeated results in higher test scores was due to the use of manipulatives.   Remember that a key component of the scientific method is "replication of results."

Now to the topic of today,  Diagnostic-Prescriptive teaching.  In listing the components of this approach to teaching math, I was gratified to hear all the standard accommodations; use of manipulatives, transitioning to the representational and abstract, larger font, familiar number facts, ample white space, processing pauses.

What I did not hear often enough was the need to understand following the trail of concepts back to foundation skills.  What are the components of concepts necessary to help students master a task for algebra II or the unit circle in pre-calculus?  It is not so much the need to seek specific strategies for each task.  It is not: "what are the chants or coding strategies" for this particular concept?

Math is a hierarchy of interlinking concepts and skills.  I often make a connection between reading instruction and math, but I often distinguish between them.  One may learn the sounds of letters, the syllable division rules, the morphology of roots and affixes but then still need context to tease out a sophisticated meaning of a polysyllabic word in context in order to comprehend.  The meaning of the word is not always tied to just the sound of a letter or the position of a vowel.  In mathematics, the ties between numeracy and the spatial relationships of quantity create a basis for very complex patterns and applications. 

How then do we reach back?  How does a teacher in an advanced course look back to foundation skills and strategies for teaching them to move students forward and quickly into the topic of the day?  And, how far back does the teacher need to go?  Is it back to "what is two?"  The answer indeed is:  It depends.

Strategies for working with older students are often grounded in truly foundation skills such as place value, regrouping,  multiples and factors, fractions and the part to whole relationship.  These foundation skills, if taught in a purely procedural way, may not be retained by students beyond Friday's test.   Teachers who teach math the way they were taught because they themselves do not know how to illustrate the underlying concepts may only partially instill the skills they hope to teach.  Evidence of this is found in continuing difficulties with fraction concepts and operations beyond the elementary years.  

Today a student said to me, "I can do long division, but not short division."  And indeed, on his pre-assessment he was able to divide a four digit number by nine.  He could not do fourteen divided by three.  Diagnostic.  He did not really understand what division meant.  He was "pushing numbers around."  His pre-assessment score was one of the higher in the class.  He was able to perform complex computations with fractions, but he was stumped by fourteen divided by three.  It took only a few minutes to correct his confusion.  Then, he was ready to roll.   

Yesterday I worked with a fourth grade student who has multiple and varied perceptual difficulties including directionality issues.  He has poor memory for the "words" of arithmetic, meaning his recall of multiplication facts in isolation is extremely poor.  Though he is now quite comfortable with base ten blocks, adding and subtracting with regrouping, his impulsiveness in calling out answers reveals lingering confusions in quantity relationships.  When asked what 100 + 40 would be, he quickly replied, "500."  Only when offered the visual models, could he correct his mistake.  I also believe that for this child, timed drills are fueling his inaccuracies and compounding the problem.  In an effort to get to answers quickly, he makes far more mistakes.  He is not being taught to rationalize or think through his solutions, but to simply generate something as quickly as possible.  This is disastrous for the child with retrieval difficulties and anxiety. 

So, a link from language I will repeat, is to teach the student where she is and not where we want her to be.  And the digression from language that I would make, is that in mathematics we can often bring the student forward in a matter of minutes, in a single lesson or simply a few.  Clarifying a single misconception may move a student a great distance conceptually.  It is the "Oh" that is listen for, that "ah-ha" moment when years of mental fog are blown away as I put something in their hands.