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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

ofteachers’implementationofscaffoldingstrategiesintheMathMindsInitiative.Inthispaper,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

participatingintheinitiative.Second,wereportonteachers’(someusingJUMPMathforabout3

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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201 IDEAS 2015

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

thestudentstofind“43”ontheir100schart. Then the teacher asked a student to come to the Smart

Boardto“find”thenumber43,butthestudenthighlightedthenumber“34,”asshowninFigure1.

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.Doesthatendinathree?Youhavethecorrectnumber…thedigitsarecorrect,

but where is forty-three?Comebackandseeifyoucouldchangeyourguess…sodo

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’sattentiontotheonesplacevalue(3or4)inforty-three, then informing the students that

thedigitswerecorrect, and finallyrefocusingthestudent’s attention to theones placevalueby

askingwhichcolumntolookfor“43.”Inthelesson,once“43”wasmarkedbythestudent,the

teachermovedonto“63”andaskedanotherstudenttoidentifyitontheSmartBoard.

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

numbertheyheardonthe100schart.Theteacherwentaroundtheroomandmonitoredstudents’

performance. In one sequence the teacheraskedstudentstoidentify“47.”However,someofher

studentsplacedtheirblocksonthenumber“74,”asdepictedinFigure2.

Figure2:Student’smisinterpretationof47.

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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 askedthestudentstoidentifythenumber“40”ontheir100scharts.Thereasonthisnumber

was selected was because it is not possible to confuse the ones digit and the tens digit, as the

number“04”doesnotexistonthe100schart.Weobservedthateveryone in the class correctly

identifiedthenumber40.Next,sheaskedherstudentstoidentifythenumber“41,”andthenshe

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.Structuringthetaskinthismannerallowedtheteachertofocusthestudents’attentionon

the ones place value, which was meant to rectify any confusion existing between the ones and tens

place values. Next,theteacheraskedthestudentstoidentifythenumber“67.”Thistime,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

beenmethadtheteacheraskedherstudentstoidentifythenumber“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-gradeteacheraskedherstudentstoidentifythenumber“47,”andthe

majorityofthestudentsputablockonthenumber“74.”Table1summarizesandcategorizesthe

teachers’suggestionsbasedontheirexperiences.

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

Table1:Teachers’scaffoldingstrategies.

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

everyoneintheclasswasabletoidentifythenumber“47”ontheir100chart,theywouldasktheir

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

identifythenumberwhich was “one morethan57”in order to createachallengeandkeepher

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

studentstoidentifyeitherthenumber“1”,“10”,or“40”dependingonwhatthey 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.

REFERENCES

Guskey, T. (2010). Lessons of mastery learning. Educational Leadership 68(2), 52-57.

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207 IDEAS 2015

JUMP Math (2015). JUMP Math. Retrieved from https://jumpmath.org/jump/en

Marton, F. (2015). Necessary conditions of learning. New York: Routledge.

Mighton, J. (2007). The end of ignorance: Multiplying our human potential. Toronto: Alfred A.

Knopf.

Nelson (2014). Assessment. Retrieved from http://www.assess.nelson.com/default.html

Runesson, U. (2005). Beyond discourse and interaction. Variation: A critical aspect for teaching

and learning mathematics. Cambridge Journal of Education 35(1), 69-87

Stone, CA. (1993). What is missing in the metaphor of scaffolding? In E.A. Forman, M. Minick, &

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Wiliam, D. (2011). Embedded formative assessment. Bloomington, IN: Solution Tree.

Watson, A. & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using

variation to structure sense-making. Mathematical Thinking and Learning 8(2), 91-111.

van de Pol, j. (2012). Scaffolding in teacher-student interaction. Exploring, measuring, promoting

and evaluating scaffolding. Unpublished doctoral dissertation, University of Amsterdam: The

Netherlands.

van Oers, B. (2014). Scaffolding in Mathematics Education. In S. Lerman (Ed.), Encyclopaedia of

Mathematics ducation (pp. 535-538): Springer: The Netherlands.

Vygotsky, L. S. (1978). Mind in society. Cambridge MA: Harvard University Press