DYNAMIC RESPONSIVE PEDAGOGY: IMPLICATIONS OF MICRO-LEVEL SCAFFOLDING

Abstract

In mathematics education, scaffolding is often viewed as a mechanism to provide temporary aid to learners to enhance mathematical understanding. Micro-level scaffolding is process by which the teacher returns the student(s) to a conceptual point where scaffolding is not needed. Then the teacher creates a series of incrementally more complex tasks leading to the original task. This process is dynamic, as it often requires multiple steps, and it is responsive because involves moment-by-moment assessment, which shapes each increment. In this paper, we present data on how experienced teachers in the Math Minds Initiative employ micro-level scaffolding. Implications of micro-level scaffolding are discussed.

Full text

2015. In Preciado Babb, Takeuchi, and Lock (Eds.). Proceedings of the IDEAS: Designing
Responsive Pedagogy, pp. 198-207. Werklund School of the Education, University of Calgary.
DYNAMIC RESPONSIVE PEDAGOGY: IMPLICATIONS OF
MICRO-LEVEL SCAFFOLDING
Soroush Sabbaghan, Armando Preciado Babb, Martina Metz, Brent Davis
University of Calgary
In mathematics education, scaffolding is often viewed as a mechanism to provide
temporary aid to learners to enhance mathematical understanding. Micro-level
scaffolding is process by which the teacher returns the student(s) to a conceptual
point where scaffolding is not needed. Then the teacher creates a series of
incrementally more complex tasks leading to the original task. This process is
dynamic, as it often requires multiple steps, and it is responsive because involves
moment-by-moment assessment, which shapes each increment. In this paper, we
present data on how experienced teachers in the Math Minds Initiative employ
micro-level scaffolding. Implications of micro-level scaffolding are discussed.
Keywords: Scaffolding; Responsive pedagogy; Mathematics education; Mathematics
for teachers
INTRODUCTION
In this paper, we report on a shift in how elementary school teachers implement scaffolding to
enhance mathematics learning. The data we gathered as researchers in the Math Minds initiative
include video recordings and observations of classrooms that used JUMP Math as their primary
resource. The Math Minds initiative is a five-year partnership that includes the University of
Calgary, JUMP Math, and the Calgary Catholic School District. The project aims to enhance early

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199 IDEAS 2015
numeracy, and our research is framed within the broad goal of understanding what teachers need to
know to effectively teach elementary mathematics.
Our data was collected at a small urban K-6 elementary school in Alberta. The national percentile
ranks (NPR) on the Canadian Test of Basic Skills (Nelson, 2014) was used to track students’
mathematical competencies after one year of participating in the Math Minds initiative, which used
JUMP Math as its primary resource, and included professional development informed by ongoing
research. The results indicated that there was a significant increase in NPRs. Therefore, it would
seem that teachers have had some success in implementing Math Minds principles and in using the
resource effectively.
Drawing from the data we have gathered over the course of two years, we witnessed the evolution
ofteachers’implementationofscaffoldingstrategiesintheMathMindsInitiative.Inthispaper,we
present two types of sample data. First, we describe scaffolding strategies implemented by two
teachers teaching the same lesson, one during their first and one during their second year of
participatingintheinitiative.Second,wereportonteachers’(someusingJUMPMathforabout3
months, and some using JUMP Math for more than a year) scaffolding strategies in response to a
semi-fictional scenario in a professional development session. Finally, we offer some insights on
why micro-level scaffolding strategies are more educative than mainstream strategies.
SCAFFOLDING IN MATHEMATICS EDUCATION
Scaffolding in mathematics education is a structure that has three key components (van Oers, 2014).
First, it is generally considered to be an interactional process between a competent user of
mathematics (teacher or peer) and a student or a group of students. Second, the aim of this process
is to provide appropriate and temporary aid to enhance mathematical understanding, which may

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200 IDEAS 2015
include the learning of mathematical actions and problem solving strategies. Third, scaffolding is
essentially a temporary measure of assistance, and it is supposed to fade away as the learner
becomes more competent.
The idea of supporting a learner through interaction until the learner is able to complete a task
without support is unequivocally connected to the Vygotskian notion of zone of proximal
development (Vygotsky, 1978). Building on this framework, Stone (1993) has suggested that
successful scaffolding does more than allowing the learner to achieve a specific goal in the
immediate context. In other words, Stone asserts that scaffolding is successful when the learner
understands the value of the scaffolding action for future activities. To evaluate the effectiveness of
such scaffolding, it is logical to provide students with opportunities to implement the knowledge
acquired through scaffolding.
Employing scaffolding strategies in mathematics education can be quite a daunting task for the
mathematics teacher. To aid teachers, some scholars have introduced different educative strategies
for implementing scaffolding with different levels of explicitness (see van de Pol 2012). One the
most popular scaffolding strategies is modelling, which is basically showing aspects of task
performance. Giving advice or providing learners with suggestions with the aim of helping them
improve their performance is another strategy. Coaching or giving tailored instructions for
corrective performance is another common scaffolding strategy. Although these strategies are
important in mathematics education, with each serving a different purpose, they are designed to
provide remediation rather than take the students to the edge of their mathematical competence.

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SCAFFOLDING IN THE MATH MINDS INITIATIVE
First year
The video recording data we gathered early in the project seems to indicate that teachers often
employed traditional scaffolding strategies such as modelling, coaching, and giving advice. In one
classroom video recording, we observed the teacher asking students to identify two-digit numbers
on a 100s chart. The procedure was simple. The teacher spoke a number between one and 100, and
the students found the number on their 100s chart. The teacher then randomly asked a student to
come to the Smart Board and highlight the number that was read. In one occasion, the teacher asked
thestudentstofind“43”ontheir100schart. Then the teacher asked a student to come to the Smart
Boardto“find”thenumber43,butthestudenthighlightedthenumber“34,”asshowninFigure1.
Figure 1: Identifying numbers before and after corrective measures.
A transcript of the conversation that followed is shown below:
Teacher: Look at what [name] did. [name] listen to the number. Forty-three. What number
does that end with?
Student: Three

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202 IDEAS 2015
Teacher:Three.Doesthatendinathree?Youhavethecorrectnumber…thedigitsarecorrect,
but where is forty-three?Comebackandseeifyoucouldchangeyourguess…sodo
we look at the 3 column or the 4 column to find forty-three? Where would be go for
43?
Student: here [correctly marks 43].
An analysis of the transcript reveals that the teacher used a coaching strategy by directing the
learner’sattentiontotheonesplacevalue(3or4)inforty-three, then informing the students that
thedigitswerecorrect, and finallyrefocusingthestudent’s attention to theones placevalueby
askingwhichcolumntolookfor“43.”Inthelesson,once“43”wasmarkedbythestudent,the
teachermovedonto“63”andaskedanotherstudenttoidentifyitontheSmartBoard.
Second year
In the second year of the project, a similar incident occurred in the same lesson described above.
The teacher read numbers (from 1 to 100), and the students were asked to put a block on the
numbertheyheardonthe100schart.Theteacherwentaroundtheroomandmonitoredstudents’
performance. In one sequence the teacheraskedstudentstoidentify“47.”However,someofher
studentsplacedtheirblocksonthenumber“74,”asdepictedinFigure2.
Figure2:Student’smisinterpretationof47.

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203 IDEAS 2015
The teacher, who had participated in a series of professional development sessions focusing on
mastery learning (Guskey, 2010) and formative assessment (Wiliam, 2011) as part of the Math
Minds initiative, employed a micro-level scaffolding strategy. An important principle in
micro-level scaffolding is starting with something known. In other words, micro-level scaffolding
is not only a corrective measure per se: It is stepping back and building up in a manner that would
allow the learners to complete the original task correctly and independently. In this instance, the
teacher askedthestudentstoidentifythenumber“40”ontheir100scharts.Thereasonthisnumber
was selected was because it is not possible to confuse the ones digit and the tens digit, as the
number“04”doesnotexistonthe100schart.Weobservedthateveryone in the class correctly
identifiedthenumber40.Next,sheaskedherstudentstoidentifythenumber“41,”andthenshe
monitored the class to make sure that everyone had correctly identified this number. She then asked
her students to identify the following numbers in sequence, each time monitoring to make sure that
everyone had identified the correct number: 42, 43, 44, 45, 46, 47. Her decision to create a task
with this particular sequence was likely informed by variation theory (Marton, 2015; Runesson,
2005; Watson & Mason, 2006), which includes the notion that the development of sequences in a
task should systematically vary in a manner that would allow only one aspect to change while other
aspects remain constant. In this particular sequence, the ones digits vary but the tens digits remain
invariant.Structuringthetaskinthismannerallowedtheteachertofocusthestudents’attentionon
the ones place value, which was meant to rectify any confusion existing between the ones and tens
place values. Next,theteacheraskedthestudentstoidentifythenumber“67.”Thistime,everyone
in the class correctly identified the number on their 100s chart. This allowed the teacher to assess
whether the confusion regarding place values had been alleviated. This objective would not have
beenmethadtheteacheraskedherstudentstoidentifythenumber“55”or“90.”

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204 IDEAS 2015
PERCEPTIONS OF SCAFFOLDING
As part of a professional development session, Figure 2 was presented to a group of elementary
school teachers, who had various degrees of experience using JUMP Math (some a few months,
others more than a year) as their primary resource. They were asked to provide an opinion on a
scenario in which a second-gradeteacheraskedherstudentstoidentifythenumber“47,”andthe
majorityofthestudentsputablockonthenumber“74.”Table1summarizesandcategorizesthe
teachers’suggestionsbasedontheirexperiences.
Less than one year in Math Minds
More than one year in Math Minds
Suggestion
Category
Suggestion
Category
Review skip-counting – ask
them to change their answer
Modelling
Ask to find 10, 20, 30, 40, 41,
42,43,…46
Micro-level
Scaffolding
Ask them to make 47 with
tens and ones blocks
Couching
Ask to find 10, 11, 15, 16, 26,
36, 46
Micro-level
Scaffolding
Ask what comes after 46
Couching
Ask to find 1, 10, 20, 30, 40,
41,42,…,46
Micro-level
Scaffolding
Give hint on which row and
column has the answer
Giving
advice
Ask to find 40, if they can,
ask to find 45, if they can,
ask them to find 46, and 47
Micro-level
Scaffolding
Review 100s chart
Modelling
Table1:Teachers’scaffoldingstrategies.
The classification of suggestions by teacher presented in Table 1 is an indication of a fundamental
difference between teachers, who where participants in the Math Minds initiative for more than one

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205 IDEAS 2015
year, compared to those who were their first year, in how they view appropriate responsive
pedagogical actions. While teacher who were participants in the project for less than one year seem
to be inclined to use more traditional scaffolding strategies such as modelling and coaching,
teacher in the project for more than one year view employing micro-level scaffolding strategies as
responding appropriately. The more experienced Math Minds teachers also mentioned that after
everyoneintheclasswasabletoidentifythenumber“47”ontheir100chart,theywouldasktheir
students to identify a similar number such as 57 or 87 to make sure that the students were skilful
enough to complete a similar task correctly and independently. Furthermore, one of the more
experienced teachers also mentioned that she would extend this exercise by asking her students to
identifythenumberwhich was “one morethan57”in order to createachallengeandkeepher
students engaged.
Implications of dynamic micro-scaffolding
Micro-level scaffolding seems to be deeply embedded in responsive pedagogy. One aspect in
which it differs from mainstream scaffolding strategies is in the frequency of its occurrence. This
type of scaffolding is meant to occur on a moment-by-moment basis. A JUMP Math lesson is
structured in a manner that allows material to be presented in small increments (Mighton, 2007).
After each increment, the teacher assesses whether the students can apply the knowledge
independently and correctly. If for whatever reason the increment suggested by the resource is too
big of a conceptual jump for a particular group of students, the teacher employs micro-level
scaffolding.
Micro-level scaffolding is informed by a series of principles and actions. If students struggle to
correctly implement the knowledge that they have received, the teacher steps back and returns the

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206 IDEAS 2015
students to a point where the teacher is sure that all students can apply their knowledge correctly
and independently. For example, in the scenario presented above, when students could not identify
the number “47” on their 100s chart, the more experienced Math Minds teachers asked their
studentstoidentifyeitherthenumber“1”,“10”,or“40”dependingonwhatthey perceived their
students would be able to do. The teacher would then guide the students by stepping up in small
increments, each time monitoring to make sure that all students responded correctly to the
micro-task, until everyone in the classroom was able to correctly complete the original task (e.g.
identifying the 47 on their 100s chart). Before moving onto the next increment, the teacher needs to
make sure that the students are able to correctly and independently apply this new skill. To do so,
the teacher should ask the students to do a task similar to the one they had just completed, without
providing scaffolding. When all students are successful in their application, the teacher could either
move on to the next increment, or create a more challenging task (but she has be reasonably sure
that his/her students are able to complete this task) to keep students engaged.
Employing micro-level scaffolding ensures that no student gets left behind and that students are
constantly and sufficiently engaged. Furthermore, the lesson structure when using this kind of
scaffolding includes a series of cycles, which contain elements of new content, assessment,
stepping-up/stepping-back, and practice. In this type of lesson structure, students’ responses
inform what the teacher needs to do next, which is in essence a form of responsive pedagogy.
Acknowledgement
We would like to acknowledge Canadian Oil and Sands Limited for their role as sponsor.
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