Making Math Real: Long Division Intensive
Online 5-day mini-course via Zoom
LAST OFFERING: Monday-Friday, June 8-12, 2020 | 9:00am to 1:00pm
Registration Fee: $499
MANDATORY PREREQUISITE: All students who enroll in this mini-course must have completed: The 4 Operations & The 400 Math Facts
The focus of the Long Division Intensive will be on:
- Reactivating the concrete connections of the concepts of division to the 9 Lines
- Reviewing the 3 most essential fundamental levels for all of long division: level I: the intro to DMSChB (concrete to abstract), level II: the intro to “Bring Down” (semi-concrete to abstract), and level V: the intro to double-digit divisors (semi-abstract to abstract)
- Extended focus on the 3 most challenging double-digit divisor levels: VI, VII, and VIII: the intro and applications of “When the estimator ‘tricks’ us”: how to identify and “fix the tricks”
- Comprehensive incrementation for estimating and checking quotients for levels VI-VIII
Long Division Intensive Course Description
At approximately 85%, long division with whole numbers has consistently been the highest failure rate nationwide of all elementary grade mathematics content areas. There are a number of reasons this concept-application presents such significant challenges to both students and teachers:
- The cognitive demands on numerous executive processes for activating and sustaining working memory for long division far exceed any of those required for problem solving in other K-5 math concept-applications
- The numerous executive processes for activating and sustaining working memory required for long division typically develop later in life than at the time children learn long division, typically from 3rd through 5th grade. This can make a significant discrepancy between students’ current executive function/working memory development and the degree of executive function/working memory development required to activate and sustain such an extensively long chain linkage of sequential processing necessary for completing long division problems.
- Of all K-5 mathematical concept-applications, long division presents the most complex internal number patterns within single and double-digit divisor-dividend relationships. Consequently, teachers must account for every divisor-dividend relationship spanning multi-step single and double-digit divisor long division with and without remainders, which provides the basis for the eight levels of long division. In the classroom scope and sequence, the eight levels of long division require three years of development, from 3rd through 5th grade.
- In my professional judgement, the demands on the educator to successfully guide students to the full concept-procedure integration of the eight levels of long division spanning single and double-digit divisors is the most challenging of all math content areas to teach across the grades K-12.
- The presentation of long division problems in standard form is derived directly from the fraction form, thereby presenting the divisor before the dividend when perceived in the left to right directionality of reading in English. The presentation of the operator before the operand is an unprecedented relationship for students at this time. Consequently, many students frequently misperceive the correct relationship between the dividend and divisor when they are presented in standard form: 12/4 in fraction form looks like 4÷12 in standard form. The reversed placement of the operator and the operand in long division requires operating in the opposite direction from which students have been taught for the other operations. In addition, subtraction, and multiplication, students operate first in the 1s place, then 10s place, 100s place, etc. However, in long division, the sequence of operation is reversed, requiring operations to begin with the greatest place value and progressing place by place in a decreasing order by place until ending with the 1s place.
- Successful problem solving in long division requires full prior synthesis of multi-digit addition and subtraction with renaming and single and double-digit multipliers in multiplication. It is never indicated to attempt teaching addition, subtraction and multiplication applications while teaching long division. Furthermore, automatic retrieval of the multiplication facts and the direct connection between the multiplication and division facts must precede the introduction to long division.
Required Course Materials: The long division section of The 4 Operations & The 400 Math Facts course binder and all of your notes for long division. (already in participant’s possession)
Optional Supplemental Worksheet Binders: Participants will also benefit from having these spiral worksheet master binders available: Long Division Levels I-IV and Long Division Levels V-VIII (may be purchased if not already in possession)