Student-Program
Alignment and
Teaching
to Mastery
Professor
Siegfried Engelmann
(
Paper
Presented at National Direct Instruction Conference
When
students are taught to mastery, they become
smarter,
acquire information faster, and develop
efficient
strategies for learning. Teachers must have an
understanding
of what mastery is and how to achieve it
in
their students. However, teachers cannot teach to
mastery
without referencing the performance of their
students.
In addition, teachers cannot teach to mastery
without
a program design that supports the approach.
Teaching
to mastery is built upon effective
student/program
alignment. This paper discusses the
features
of a program design that supports mastery,
properties
of mastery, criteria and procedures for
measuring
mastery, procedures for aligning program
placement
with student performance, and the benefits of
mastery.
Features
of a Program Design that Supports
Mastery
A
program design that supports mastery does not
present
great amounts of new information and skill
training
in each lesson. Rather, work is distributed so
new
parts in a lesson account for only 10 - 15 percent of
the
total lesson. The rest of the lesson firms and
reviews
material and skills presented earlier in the
program.
The program assumes that nothing is taught in
one
lesson. Instead, new concepts and skills are
presented
in two or three consecutive lessons to provide
students
with enough exposure to new material that they
are
able to use it in applications. So a lesson presents
material
that is new today; material that is being firmed,
having
been presented in the last two or three lessons;
and
material that was presented even earlier in the
sequence
and is assumed to be thoroughly mastered.
This
material often takes the form of problems or
applications
that require earlier-taught knowledge.
The
amount of new material is relatively small because
most
students are not capable of assimilating more. This
design
provides for some “overlearning”, but having the
program
err in the direction of providing too much
practice
is better than providing too little practice. Work
on
material presented in the preceding few lessons is
needed
to ensure that students are “automatic” with
information
or operations that were previously taught.
The
review of earlier material assures that students use
and
apply what they have learned. Reviews also prompt
students
toward an understanding that they expected to
retain
and use material learned - not just learn it for the
moment.
Basically, most things are taught in the
program
so they can be used in applications or problemsolving
settings.
Therefore, the program is constructed
so
students review and use what they have learned
according
to a systematic schedule. Because reviews
are
a regular feature of every lesson, the program
design
provides daily prompting that material presented
will
appear again. Also, applications that involve earlier taught
skills
provide the kind of practice that students
need
to keep from mixing up different things they are
learning.
If students partially learn things, new learning
is
easily confused with things that are similar. If
students
learn material well, less confusion results.
Mastering
a step at a time. A
program designed with
small
amounts of new material in each lesson is
something
like a stairway. Like a stairway, it needs
strong
support. That support is in the form of the
previously
taught skills and knowledge that are logical
underpinnings
for what is to come next in the program.
Also,
for the stairway to work well, the “steps” in this
series
should be about the same size. Certainly, they
can’t
be fashioned with the accuracy of a physical
stairway,
but they can be designed so they are close to
each
other in size.
If
we conceive of the program as being like a stairway
that
transports students to increasingly complex
performance,
we recognise the supreme importance of
mastery,
what it is, and how it relates to the curriculum.
The
following six points clarify the relationship between
mastery
and the stairway.
1.
The program will function as a stairway if
the
student reaches eve ry stair on
schedule. If students are firmly
on the fifth
stair
(which is analogous to the fifth lesson),
the
new learning that students must achieve
to
reach the sixth stair is manageable. The
students’
position on the fifth stair represents
a
foundation that places the sixth lesson
within
stepping distance. Because the
foundation
is in place, the sixth lesson does
not
overwhelm students with too much new
vocabulary,
unfamiliar or unpracticed
operations,
too much information, or too many
unknown
or unexplained details.
2.
The steps are levelers of individual
differences. Not all students who
stand on
the
fifth stair are the same age, learn at
precisely
the same rate, have equal
intelligence,
or exhibit the same “style” of
learning.
However, every student who is
firmly
on the fifth step is the same with
respect
to the program sequence. Each has
the
skill repertoire and knowledge needed to
take
the next step and reach that step within
30-45
minutes of instruction. Because
students
could not reach the fifth step without
specific
skill and knowledge, the stairway
structure
of a well-designed program serves
as
a leveller. All students with a particular
skill
profile are placed on the same stair.
Certainly,
the program design does not
guarantee
that all students will progress at
exactly
the same rate; however, greatest
individual
differences occur on the very
beginning
levels. On higher levels, after
students
have mastered a battery of skills and
knowledge,
the difference in rate of ascent
for
appropriately placed students is far less
because
all students tend to have enough skill
to
master the new material at around the
same
rate.
3.
The benefits of the design of the program
are
obliterated if a student falls below the
level
of a stair.
This fact holds for students
who
are “smart” as well as those who have a
history
of failure. If a student is below the
fifth
stair and tried to reach the sixth stair
with
one step (which means thoroughly
mastering
the sixth lesson in one period), the
student
must learn substantially more than
students
who are firmly on the fifth stair.
Furthermore,
the student must learn this
material
during the same amount of time
allotted
for students who are firmly on the
fifth
stair. Therefore, the student who is
below
the fifth stair must learn the material at
a
faster rate. The student on the fourth stair
must
learn material at twice the rate of
students
who are correctly placed. The
student
who is on the third stair must learn at
three
times the rate. For the typical student,
a
step that requires three times the amount of
new
learning is too great. Even if the student
is
able to perform acceptably on lesson 6
after
some repetition, the retention rate of the
student
on the subsequent lessons drops
dramatically.
4.
Just as the design of the program
“guarantees”
a successful future for
students
who are firmly based on a stair,
the
design suggests an unsuccessful
future
for a student who is greatly below
that
stair. The
systematic stairway design
does
not provide relief because skills and
knowledge
do not go away. Once introduced,
they
are used throughout the rest of the
program,
either as elements that are used
regularly
(such as a word type that is
learned),
as details that are embedded in
problems
and applications (such as the math
operation
of carrying), or as items that are
frequently
reviewed (such as identifying the
verb
in sentences). Because of this program
design,
once a student falls behind, the
student
will tend not to catch up. If the
student
is initially 3 steps below the lesson,
the
student will probably end up a little more
than
3 steps below the next lesson, a little
further
below the following lesson, and so
forth
until the student is not 3, but 4, steps
below
the level of the lesson, then 5 steps
below,
and so forth.
This
student is not able to benefit from the design of
the
problem, because although the program presents
small
increments of learning, this student must master
large
increments of learning to catch up. For this
student,
the program presents a poorly designed
sequence.
It requires too much new learning and
does
not provide adequate reviews.
5.
Because the program’s design benefits
are
transmitted only to students who are
on
the lesson stairs, student performance
must
match the level of performance
assumed
by each stair. This
goal is
achieved
if teachers teach to mastery.
Mastery
assures that everything that is
supposed
to be taught is taught thoroughly
and
at the time it is introduced in the program
(not
20 or 30 lessons later).
Note,
however, that DI programs are designed with
enough
redundancy that a student who is absent for
two
or three days will not be perfectly lost for the rest
of
the year. Also, if students do not master a new
skill
on the first day it is introduced, the following
lessons
provide at least one - possibly two - reviews
of
the introduction so that students will have sufficient
opportunity
to learn the skill before it is assumed to be
in
their skill repertoire and begins to appear in
applications.
The
problem occurs when students are not brought to
mastery
on skills that will be used later. For instance,
students
in Level 1 of Reading Mastery are
supposed
to be taught to follow the teacher’s
directions
about “touching words” before lesson 30.
The
tasks that the teacher presents require students
to
follow directions to “Touch the first word ... touch
the
next word ... touch the next word ...”
Often
students are not brought to mastery when this
series
of tasks is introduced. These students have
problems
in the lesson range of the 40s because now
they
are expected to first “touch the next word...” and
then
“sound it out.” If they are not firm on touching the
next
word on signal, the activity becomes very sloppy
and
students often become confused about what they
are
supposed to do. If students are taught on time,
however,
they have far less difficulty mastering the
mechanical
steps of touching the next word and then
touching
the individual letters as they sound it out. The
program
design provides for enough practice; however,
that
practice must not be mere exposure or practice with
a
very low standard of performance. The practice must
lead
to mastery.
6.
Most programs do not require teaching
to
mastery. Teaching
to mastery is a
foreign
practice to many experienced
teachers
because most programs do not
require
mastery. Instead of providing
continuous
skill development, these programs
present
topical or thematic units. Students
will
work on a particular unit for a few days
and
then it will be replaced by another unit
that
is not closely related to the first and that
does
not require application of the same skills
and
knowledge. This design, referred to as a
“spiral
curriculum”, is more comfortable for
the
program designers, teacher, and students;
however,
it is inferior for teaching skills and
knowledge.
It
is comfortable for the designers because the design
does
not have to be careful. The designers do not
have
to document that everything that is presented is
“teachable”;
the amount of new learning does not
have
to be carefully measured. The amount of time
required
for a “lesson” does not have to correspond
precisely
to a period, because the design assumes that
different
teachers will take different amounts of time
to
get through a particular “lesson” and “unit”. The
amount
of new material is not controlled. The
expectations
for student performance is low because
teachers
understand that students will not actually
master
the material. They will simply be exposed.
The
accountability of the teacher is therefore more
“comfortable”
because the teacher is not expected to
get
through the material in a specified period of time
or
bring students to mastery. The spiral curriculum is
more
comfortable for students because they are not
required
to learn, use, or apply the skills from one unit
to
the next unit. They quickly learn that even though
they
do not understand the details of a particular unit,
the
unit will soon disappear an be replaced by another
that
does not require application of skills and
knowledge
from the previous unit. The design clearly
reinforces
students for not learning or for learning
often
vague and inappropriate associations of
vocabulary
with a particular topic.
If
the systematic program is like a stairway, the spiral
curriculum
is like a series of random platforms
suspended
on different levels. Students are
mysteriously
transported from one platform to
another,
where they remain for a few days as they
are
exposed to information that is not greatly
prioritized.
Mastery is impractical with a spiral
curriculum
design because many students lack the
background
knowledge they need to stand on a
particular
“platform”. The poor design relieves the
program
designer of assuring that earlier-taught skills
and
knowledge are mastered and used. The poor
design
also relieves students of the responsibility of
learning
to mastery and it relieves the teacher of
teaching
to mastery. It therefore promotes poor
teaching
and poor learning.
In
summary, a program that teaches to mastery is like a
stairway.
Mastery is the guarantee that students are
able
to reach each stair without falling.
Properties
of Mastery
Clearly,
mastery is the handmaiden of a systematic
program.
Mastery is effective for a number of reasons.
The
most important reason is that mastery permits
teachers
to achieve steady reliable progress in student
learning.
When teachers teach to mastery, we can
make
predictions about student performance. We can
very
accurately project where students will be 100
school
days from now or 200 school days from now.
Such
projections are very powerful, but very foreign to
traditional
orientations about learning, which view the
students’
performance as a function of their ability to
learn
and motivation to learn. Therefore, to predict that
student
X will be accurately reading 30 words per
minute
by the end of the kindergarten year would be
something
of a contradiction because it assumes that the
teaching
somehow controls the student’s learning.
The
traditionalist hopes to reach and motivate the
student
and hopes that the student does not have some
type
of mysterious “learning disability” that interferes
with
learning to read. The traditionalist, however, is
unable
to predict who will read and who won’t.
Readiness
tests are tools that are supposed to predict
performance
according to what the student brings to
school.
Because they don’t take into account the kind
of
reading instruction the student will receive, readiness
tests
fail to predict accurately. In fact, the traditional
orientation
to reading has a classification for students
who
are predicted by readiness tests to succeed but who
fail
to learn to read on schedule - specific learning
disabilities.
Note that this label holds fast to the
assumption
that the student’s failure to learn to read has
to
do with a flaw in the student, not a flow in the
instruction.
The school or teacher does not have a
“disability”.
The student does. In other words, for the
traditionalist,
the performance of the student is not
clearly
linked to teaching. The more scientific
orientation
to teaching that by DI espouses assumes that
the
student who meets the entrance requirements for the
program
and who is taught appropriately (to mastery and
on
schedule) will respond in perfectly lawful ways and
will
be reading at a predicted skill level by the end of the
kindergarten
year.
Individualisation must occur from the
beginning.
Projections
are keyed to the performance of a student.
Not
all children entering kindergarten have the same
projections
because not all of them start at the same
place.
Those who enter with more skills have a
headstart and are expected to be farther after
nine
months
of instruction than the child who enters with a
lower
skill level. However, even if children begin as low
performers,
the prediction is that
they will master
beginning
reading skills in kindergarten and will be
reading
by the end of kindergarten. For the child who
enters
with a low skill level, the projected end-of-K-year
performance
may be lesson 120. The projection for the
higher
performer may be double that number.
The
fact that projections are met means that the DI
orientation
to teaching and mastery is correct. Students
will
learn if the teaching is appropriate. If they fail to
learn,
the reason lies not with their inability to learn but
with
the delivery system’s inability to teach.
The
concept of individualisation is closely related to
the
issue
of mastery and to projections about students’
performance.
The teacher cannot teach to mastery
without
referring to the performance of the students
being
taught. The teacher bases decisions about what
to
do next on samples of each student’s behaviour. This
sample
may come from tasks presented to the group,
tasks
presented to individual students, or worksheets and
similar
work samples. DI is designed so students’
thinking
is made overt. The teacher therefore receives
samples
of behaviour at a high rate on everything that
is
being taught. The teacher uses this information to
judge
what rate of presentation is appropriate. If
students
have already learned the skill or concept, the
teacher
is to move on. If the teacher determines that
some
students have not mastered what is being taught,
the
teacher corrects the mistakes and possibly repeats
parts
of the exercise. If quite a few students missed the
item,
the teacher may repeat the entire exercise with the
whole
group, which is more efficient than presenting it
to
some students individually.
In
summary, teaching to mastery is possible only if the
teacher
keys the amount and type of practice students
receive
to the performance of these children.
Criteria
and Procedures for Measuring Mastery
Teaching
to mastery is a difficult procedure for teachers
to
learn. They must learn to reference what to do next
according
to the students’ performance. They must
learn
high, but realistic, expectations for their students.
They
must also learn to coordinate mastery with fast
pacing
so that the lesson is neither a chore for students
nor
busy-work. The teacher uses efficient means of
checking
students’ work, of providing additional practice
and
firm-ups for students who do not achieve mastery
on
skills that were taught, and of providing
reinforcement
for trying hard and for succeeding.
First-time correct procedures. An important key to
teaching
to mastery is the use of first-time correct
procedures.
Procedures for inducing mastery require
the
teacher to interpret students’ performance. The
primary
indicator of mastery is how well students
perform
the first time a particular task or exercise is
presented
in the lesson. Each time a task is presented,
the
group either responds correctly (all students correct)
or
incorrectly (some students giving the wrong response
or
no response). First-time correct means all students
are
correct the first time a task is presented in a lesson.
Also
important is how well students perform on the task
or
exercise if the teacher presents it more than once. If
the
teacher corrects and repeats the task or exercise, it
is
important for students to perform correctly the second
time.
However, for diagnostic purposes, students’
responses
to the first time the task or exercise is
presented
provides the most critical information about
where
students are positioned on the stairway and
whether
they are appropriately placed in the program.
For
instance, the first time the teacher asks a question
such
as, “Do we multiply or divide to solve this
problem?”
or the first time students read a particular
word
list, their responses reveal information about the
mastery
level the students bring to the lesson.
The
students’ pattern of correct responses also provides
important
mastery information. If they are making too
many
mistakes, or if they are not firm on material that
had
been taught earlier and that is assumed to be firm,
they
are placed too far in the program and should be
moved
back. If students give solid indications that they
already
know what the lessons is teaching, the students
may
not be placed as far in the program as they might
be,
and the rate of lesson presentation should increase.
Finally,
the “correct-response” patterns of a group
indicate
whether all students belong in the group or
whether
some should be placed in other groups.
Four
criteria permit precise interpretation of the correct response
performance
for groups and individuals:
Criterion1.
Students should be at least 70% correct on
anything
that is being introduced for the first time.
Criterion
2. Students should be at least 90% correct
on
the parts of the lesson that deal with skills and
information
introduced earlier in the program
sequence.
Criterion
3. At the end of the lesson, all students be
virtually
100% firm on all tasks and activities.
Criterion
4. The rate of student errors should be low
enough
that the teacher is able to complete the lesson
in
the allotted time.
Again,
all the percentages are based on how students
perform
the first time a particular task is presented in
the
lesson. For material that is assumed to be mastered,
the
group should respond perfectly at least 9 out of 10
times.
As
noted above, students’ first-time performance shows
what
they have brought with them to the lesson. That is
the
material that is in their memory and skill repertoire.
The
performance of students after the teacher repeats
the
material indicates only what the students may retain
for
possibly less than 10 minutes. That time span does
not
measure mastery. When students master a skill they
know
it “as well as they know their own name.”
All
four criteria should be considered in evaluating the
mastery
of the group. If students meet the first three
criteria
but can’t seem to get through lessons in the
allotted
time, something is wrong. The following
sections
examine the four criteria in more detail.
Criterion
1. Students should be at least 70 percent
correct
on anything that is being introduced for the
first
time.
This percentage is based on the
understanding
that even the new skills or procedures
that
are being introduced are not composed entirely of
material
that is new. Much of it will be familiar.
Therefore,
the initial rate of correct responses should not
drop
below 70 percent. If students are at mastery on
the
preceding lessons, this outcome will occur in almost
all
cases.
If
students perform much below 70 percent, they are not
learning
the material. If they are only 50 percent
correct,
they may be at a chance level - guessing at the
answers
or the steps in the operation. Their responses
are
not generated by an overall understanding of what
they
are learning. At 70 percent correct, their responses
show
that they are much closer to understanding the
new
material than they are to taking blind stabs at
responding,
and therefore should be able to master the
new
material during the lesson.
Criterion
2. Students should be at least 90 percent
correct
on the parts of the lesson that deal with
skills
and information introduced earlier in the
program
sequence. Criterion
2 is based on the fact
that
students must be completely at mastery on earliertaught
material.
When earlier-taught material occurs in
later
lessons, no reteaching should be required. If
substantial
reteaching is needed, the amount of new
learning
that students must achieve to mater the lesson
becomes
too great. If students are consistently not at
the
90 percent correct level on material that had been
taught
earlier in the program, students need more
extensive
firming and more delayed tests. Possibly, the
teacher
should use a game format in which she asks
students
different questions at the end of the lesson.
Students
who respond correctly receive points. When
virtually
all students consistently earn points, they have
learned
good techniques for learning and retaining
information
presented in the lesson.
Criterion
3. At the end of the lesson, all students
should
be virtually 100 percent firm on all tasks
and
activities.
Criterion
4. The rate of student errors should be
low
enough that the teacher is able to complete the
lesson
in the allotted time. Criteria 3 and 4 go
together.
When the rate of errors for the overall lesson
is
low, the teacher does not need to spend great amounts
of
time firming students, and the teacher should be able
to
complete the lesson in the allotted time. If students
enter
the lesson with skills that permit them to attain 70
percent
correct on new material and 90 percent correct
on
material taught earlier, students should be able to
achieve
virtually 100 percent on all exercises presented
in
the lesson. Achieving this performance level may
require
a little additional firming, but it should not be
necessary
or excessive lesson after lesson. Therefore,
if
Criteria 1 and 2 are met, students should easily
achieve
Criterion 3 and the teacher should be able to
complete
the lesson during the allotted time.
Calculating percentages. Several different
procedures
are
effective for teachers to learn how to “estimate” or
calculate
the percentage of first-time-correct responses.
One
way is to place sticky tabs in the teacher
presentation
book after each task, or affix a sheet of
paper
to the page so the teacher can mark whether the
group
(or individual) correctly responded to each task.
After
the children have responded to ten tasks, the
teacher
simply counts the number of tasks that were
correct.
If seven were correct, the percentage is 70
percent.
(Note: if the teacher repeats a task, she would
not
mark the second-time performance the same way
she
would mark the first-time performance. She could
circle
the second-time performance, note the
performance
in a second column, or use another way to
separate
the first-time performance from performance
on
tasks or exercise that are repeated.)
After
using a procedure of actually counting the
responses
within each exercise, the teacher should try
to
make estimates in her head. One way is to “ball
park”
patterns in terms of whether students are
performing
closer to 50 percent or 100 percent. If they
seem
closer to 50 percent (missing a little less than half
of
what the teacher presents) their first-time percentage