MATH BCD
THE REVISED MATH LEVELS
Ashwini Poola
80,000
In June 2015, over
Kumon students in
North America were studying in Math levels B through D,
constituting over one-third of KNA’s total math enrollment.
The revision of these materials is an opportunity to improve the study situation of
numerous students. As with previous revisions, the Global Math Team in Japan utilized
student observation, feedback from Instructors, and other research to conclude how
the materials should be changed.
For students studying above grade level to successfully self-learn in Level E and
beyond, they must have the skills and attitude to learn from the worksheets, as well
as a high mastery of the four operations (addition, subtraction, multiplication, and
division). Many students in levels BCD have not yet accumulated enough experience
with self-learning and so do not yet know how to learn from the worksheets. As such,
the introduction of new concepts should be simpler and easier to understand so that
students can successfully self-learn. Levels B, C, and D were revised in order for
students to better develop these skills and therefore experience smoother progress in
the subsequent levels.
The revision was conducted with two policies in mind.
REVISION POLICY 1
Enable further development of student attitude and skills
for learning from worksheets.
From Level E onwards, students are introduced to new concepts through examples,
and these examples occur in the worksheets with increasing frequency.
Refining a student’s
ability to s elf-learn
using examples through
levels BCD will allow a
smooth transition into the
worksheets where this skill
is essential.
In
the
current
worksheets, students found
it challenging to self-learn
some new concepts. As a result, the introduction of these concepts has been revised
so that students can independently learn these concepts from the worksheets. The four
operations of whole numbers, up to D 150, are introduced through guided exercises.
Introduction of new concepts through examples begins with fractions and reduction
on D 151.
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MATH BCD
Many students studying above grade level were confused
by the word problems and addition that introduced vertical
multiplication in the current C 51. The revised C 51 introduces
vertical multiplication through guided exercises that show the
calculation steps to the student more clearly.
current
new
The locations of new and difficult concepts in the current
BCD were also not easily recognized, making it challenging
for Instructors to guide students properly and in a timely
manner. The revised worksheets include thick solid lines on
the first pages of sets where new learning foci are introduced,
as well as thin broken lines on the first pages of sets containing
particularly difficult elements. Both of these lines are followed
by the page and problem number where the new or difficult
concept occurs, serving as an additional reminder to
Instructors that observation or guidance may be necessary.
The current C 61 includes multipliers of 5, 6, and 7, and the
current C 71 multipliers of 8 and 9. This results in students
needing to carry larger numbers and more difficult mental
calculation than is required in previous or later sets.
In the revised C 51-110, multipliers and the frequency of
large carries both increase gradually, so that calculation does
not abruptly become more difficult while students continue to
recall the multiplication table facts.
The number of problems and SCT of some sets have also
been changed in order for completion times to better reflect a
student’s experience completing the worksheets. Basic sections,
such as multiplication tables, have a lower SCT. More difficult
sections have a higher SCT with a wider range between xand y-time to account for the individual differences between
students that are more likely to affect completion of these sets.
Revisions for Level B: Enabling students to…
Review
Revised B 21a
Revised B 91a
1
Addition to
100
11
21
•
31
Addition
of 2-Digit
Numbers
41 51
•
2-3
61
Addition
of 3-Digit
Numbers
71
81
91
•
3-5
REVISION POLICY 2
1-2
2-4
Enable students to study more smoothly
and enthusiastically.
Deepen their understanding of addition with
Smoother study in levels BCD will allow better development
of the calculation skills that will be necessary in later levels.
A student who struggles in completing levels BCD may also
experience a decrease in motivation that persists through the
later levels. Conversely, gradual increases in the difficulty of
the worksheets will enable a student to progress more smoothly
through the levels, strengthening calculation ability without
losing enthusiasm along the way. To accomplish this, the
difficulty of each worksheet section was analyzed, as well as
the factors that caused that particular section to be difficult for
students. Based on these results, certain sets were modified so
that students would not experience a sharp increase in difficulty.
carrying.
In the current worksheets, after vertical addition with
carrying is introduced, there are a few problems that do not
require carrying. Some students have difficulty determining
when carrying is called for and automatically add “1” to the
tens digits, regardless of whether or not it is necessary.
Revision: To counteract this trend, the revised B 21-40
includes more problems that do not require carries. These
problems are distributed throughout the section enabling
students to deepen their understanding of carrying more
smoothly.
FALL 2015 [email protected]
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MATH BCD
Understand 2-digit + 2-digit addition with
consecutive carrying more smoothly.
The current B 41a introduces both consecutive carrying and
sums of 100 and greater at the same time, making it difficult
for students to understand the calculation procedure.
Revision: The revised B 41 introduces sums of 100 and over
through problems that require carrying from only the tens
column on the a-side, before proceeding to carrying from
both the ones column and the tens column on the b-side.
The word problems on the current B 44 and B 49 have been
moved to set B 51-60, and the percentages of problems with
consecutive carrying gradually increase from B 41 to 70,
allowing students to strengthen their understanding of the
calculation procedure.
current
Review
(Subtraction)
101
new
Subtraction
of 2-Digit
Numbers
111
1-2
new
Addition &
Subtraction
of 2-Digit
Numbers
current
Subtraction
of 3-Digit
Numbers
121 131 141 151 161 171 181
•
• •
•
2-3
2-4 2-3
2-4
191
•
3-5
Acquire the skills to calculate 2-digit – 2-digit
subtraction more smoothly.
There are many problems with minuends over 20 in the
current B 101-120, overwhelming some students and making
it challenging for them to recall their subtraction skills from
Level A.
The current B 131-140 introduces 2-digit – 2-digit
subtraction with borrowing, but frequently switches between
problems with borrowing and without before students
understand the procedure of borrowing.
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VOICES FALL 2015
Revision: The subtraction review in B 101-120 will mostly
consist of problems with minuends smaller than 20, allowing
students to more easily recall the calculation procedure from
Level A in B 101-120. More problems with minuends over 20
will be in B 121-130, so that students can master borrowing
from the tens column.
The set introducing 2-digit – 2-digit subtraction with
borrowing includes mostly problems that require borrowing to
give students practice applying the skill. From B 141 onwards,
problems without borrowing are gradually incorporated.
new
The note reminding students to study the multiplication
tables has been moved from B 131 to B 140, and the section title
of B 151-160 has been changed to Addition and Subtraction
of 2-Digit Numbers.
Learn consecutive borrowing more efficiently.
Students study subtraction with borrowing from both the
tens and the hundreds columns from current B 181 onwards.
Revision: Problems with borrowing either from the tens
column or the hundreds column are more frequent in
the revised B 161-180, allowing students to deepen their
understanding of 3-digit subtraction with single borrows
before advancing to consecutive borrows in B 181. Guided
exercises, instead of examples, introduce 3-digit minuends
with a tens digit of zero (such as 102), and the introductory
problem has been revised so that students can learn the
concept more easily.
MATH BCD
Revision: The revised C 51 introduces solving vertical
multiplication through guided exercises with boxes and
tracings, allowing students to independently learn how to
solve the problems from the worksheet. Vertical multiplication
is now introduced on C 53, rather than C 54. The connection
between vertical multiplication and addition is made available
to the student in a footnote, but is no longer the only
explanation in the worksheet.
current
new
Revisions for Level C: Enabling students to…
Review
1
Multiplication
Tables
11
21
31
Multiplication: 2 Digits
x 1 Digit
41
51
•
61
71
81
91
3 or 4
Digits
x1
Digit
101
current
2-3
2-4
3-5
Master 1-digit x 1-digit multiplication more
easily.
To improve the 1-digit x 1-digit multiplication skills,
multipliers that students have studied previously are presented
in a mixed manner on the b-side of the 5 and 10 worksheets
in the current Level C.
Revision: With the aim of further improving students’
1-digit x 1-digit multiplication skills, the number of worksheets
with mixed multipliers will be increased.
Structure of C 11-50
1
2
C 11 Intro
x4
C 21
x6
C 31
x8
C 41
3
4
5
x2
6
7
8
x3
x5
9
10
mixed
mixed
x7
mixed
mixed
mixed
mixed
x9
mixed
Better understand vertical multiplication.
Vertical multiplication is introduced in the current C 51
through a word problem that compares vertical multiplication
and addition. In some cases, students studying beyond grade
level were not able to understand the calculation procedure
on their own.
new
Learn vertical multiplication more smoothly.
The current C 51 intends for students to understand the
calculation procedure of vertical multiplication by comparing
addition and multiplication in word problems; however,
students studying beyond grade level have been observed to
experience difficulty understanding this.
In the current C 51-80, multipliers are fixed in order
for students to continue practicing the 1-digit x 1-digit
multiplication skills strengthened from C 11-50. However, the
problems in C 61-80 have relatively large multipliers, such as
7, 8, and 9, so the completion time becomes longer due to the
more complicated carrying.
Revision: The revised worksheets now introduce vertical
multiplication through calculation problems with hints such
as boxes and tracings. The first problem with carrying now
appears on C 53.
The number of problems and steps for carrying in
C 51-100 have been examined, and the level of difficulty has
been modified so that it increases gradually as students proceed.
The structure with fixed multipliers remains the same.
Revising the learning steps enables students to improve
their skills to calculate carrying while proceeding to the later
sets in a longer span. As a result, the SCT has been extended to
2-4 minutes, allowing Instructors to apply more individualized
study plans.
FALL 2015 [email protected]
I 15
MATH BCD
Acquire the skills for division with remainders
more efficiently.
new
new
Intro to
Division
111
new
Division with
Remainders
121
131
141
•
Problems with quotients over 10 are included throughout
the current C 121-160.
Revision: The revised C 121-150 has only 1-digit quotients,
so that students can develop their division skills more
thoroughly before advancing to C 151-160, which is now the
only set with quotients or divisors over 10.
Division: 2 Digits
÷ 1 Digit
3 Digits
÷ 1 Digit
151
161
171
181
•
•
•
•
2-3
3-4
191
3-5
Better understand division with remainders.
The current C 121-130 introduces division with remainders
through word problems with illustrations, which were
confusing to some students, particularly those studying above
grade level. In other cases, students were able to solve the
problems on C 121 but could not apply the concept to C 122.
Revision: Guided problems with boxes and tracing replace
the word problems, allowing students to learn how to solve
more easily from the worksheets. Division with remainders
is introduced in the revised C 121 through division problems
that more closely resemble those in the preceding set as well
as the rest of the problems in the Division with Remainders
section, allowing students to build upon their knowledge more
easily after solving the introductory problems.
current
new
Revisions for Level D: Enabling students to…
Review
1
Multiplication:
2 Digits x 2 Digits
11
21
3 Digits
x 2 Digits
31
•
2-3
41
•
2-4
3-4
3-5
Master 2-digit x 2-digit multiplication more
easily.
current
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VOICES FALL 2015
new
In some cases, students have trouble understanding the
example on the current D 11 and applying it to the problems
on the same page. Multipliers with units digits of zero are
introduced shortly after the concept of 2-digit x 2-digit
multiplication and in an example without any intermediate
steps.
Revision: The calculation procedure is introduced in the
revised D 11 by tracing, and all the problems on the page have
the same number of columns. The introductory problem on
the revised D 11, more closely resembling the other problems
on the page, allows students to recognize the connection
between the guided problem and the ones to be completed
independently. The revised D 41 has the first appearance
of a multiplier whose units digit is zero, and students can
experience solving these types of problems before being
shown how to omit the intermediate steps on D 45b.
MATH BCD
Master division with 2-digit quotients more
smoothly.
Revision: The introductory problems on the revised D 131
offer more guidance, enabling students to understand the
calculation procedure by themselves more easily. The section
is called Division: Quotients of 2-or-More Digits. The SCT for
D 141-150 has been extended to 4-6 minutes to acknowledge
that the problems with larger digits are included for students
to experience solving them and should not necessarily be
repeated for speed.
current
current
current
new
Fractions
new
Addition Multiplication
and
and
Subtraction
Division
51
61
151
Division by 2-Digit
Numbers
71
81
•
•
•
2-4
3-4
91
new
101 111
3-5
Division:
Quotients of 2
or More Digits
121
131
141
•
•
4-6
Develop the addition, subtraction,
multiplication, and division skills required
for 2-digit division more efficiently.
The a-side of the current D 62-75 combines both horizontal
and vertical multiplication on the same page, leading to some
cases where students solve all of the multiplication vertically
and lose the opportunity to enhance horizontal multiplication
skills.
Revision: Vertical and horizontal multiplication are no
longer combined in the revised D 61-80: the a-side of D 61-70
contains vertical multiplication, and horizontal multiplication
only is on the a-side of D 71-80. Separating the two types
of problems allows students to strengthen each skill more
effectively.
3-5
Reduction
161
171
2-3
181
191
2-4
Start learning fractions more smoothly.
The current D 156a introduces conversion of mixed
numbers to improper fractions by defining the terms and then
comparing the steps to conversion from improper fractions to
mixed numbers.
Revision: The revised D 156a has examples of conversions
both to and from improper fractions, providing more
information for the student to analyze in order to
independently understand how to solve the problems. D 159
includes a footnote demonstrating how to solve using vertical
division which helps students connect to a familiar operation.
FALL 2015 [email protected]
I 17
MATH BCD
Master reduction more easily.
In the current worksheets, students experience reduction
of fractions by 1-digit factors from D 161-175. Some students
have difficulty advancing to the GCF on D 181 with only five
pages of practice reducing by 2-digit factors, D 176-180.
current
new
CURRENT
161
NEW
Factors:
2, 3, 5, 7, 4, 6
Factors:
2, 3, 5, 7
171
Factors: 4, 6, 8, 9
Factors: 4, 6, 8, 9
176
Up to 2-digit
factors
Random 1-digit
factors
181
GCF
Random 1-digit
factors (more 2-digit
denominators)
186
Random factors
Up to 2-digit factors
Random factors
Random factors
(GCF on D 200)
166
191
196
Constant
factors
1-digit
factors
Mixed factors
Constant factors
Up to 2-digit
factors
Mixed factors
Revision: Students practice
reduction of fractions by
1-digit factors from 161-185
in the revised Level D to
strengthen their proficiency
before learning 2-digit
factors on D 186. Rather
than requiring students to
solve using the GCF, the
inst r uc t ions encourage
students to try to reduce in
one step. ■
new
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VOICES FALL 2015
new
new