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2016. In M. A. Takeuchi, A. P. Preciado Babb, & J. Lock (Eds.). Proceedings of the IDEAS:
Designing for Innovation, pp. 203-212. Calgary, Canada: University of Calgary.
ADDRESSING THE CHALLENGE OF DIFFERENTIATION IN
ELEMENTARY MATHEMATICS CLASSROOMS
Paulino Preciado Babb Martina Metz Soroush Sabbaghan Geoffrey Pinchbeck
Ayman Aljarrah Brent Davis
University of Calgary
Addressing students’ diversity of skills and knowledge for mathematics instruction
has been a common challenge for teachers. This paper reports results from an
innovative partnership of school district, university and curricular material
developers aimed at improving mathematics instruction at elementary level. We
report successful cases of lessons enacting instructional practices that engage all
students in the classroom, ensure they meet expected outcomes, and challenge them
with further bonuses. The cases are analyzed based on mastery of learning, with a
particular focus on continual assessment during class. We also include challenges
we have faced in supporting teachers as they incorporate these practices in their
daily teaching.
Keywords: Mathematics teacher knowledge; student diversity; mastery of
learning; formative assessment.
Academics have emphasised the importance of differentiated instruction for several decades
(Bloom, 1968, Guskey, 2010), and there is now an extensive body of literature on formative
assessment (e.g. Chappuis, 2015; Stiggins, Arter, Chappuis, & Chappuis, 2004; Wiliam, 2011);
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IDEAS 2016 204
however, suggestions for teachers’ immediate responses to in-class assessment are not as
common. This paper addresses the approach to differentiation in mathematics instruction adopted
in the Math Minds initiative, a partnership aimed at improving mathematics instruction at the
elementary level—more details about the initiative and previous research results can be found in
Metz, Sabbaghan, Preciado Babb, & Davis (2015), Preciado Babb, Metz, Sabbaghan, & Davis,
(2015), and Sabbaghan, Preciado Babb, Metz, & Davis, (2015). This approach is informed by
Mastery Learning (Guskey, 2010), the Variation Theory of Learning (Marton, 2015), and
Intrinsic Motivation (Pink, 2011). We provide examples of differentiation based on continuous
assessment from lessons in which all students have engaged in the targeted object of learning in a
way that they met the expectations of the lessons and were further challenged with additional
mathematical explorations and extensions. We also discuss teachers’ difficulties in the
implementation of this approach.
APPROACH TO DIFFERENTIATION
A key principle of Mastery Learning (Guskey, 2010) is that students should move to the next unit
of instruction once they have mastered the concepts or skills required to engage in that unit.
Formative assessment plays an important role in determining when students are ready for the next
step, as well as in identifying students who have mastered the learning outcome and can be
engaged in extensions or enrichment material. One of the principles in the Math Minds initiative
is to assess continually during class with careful attention to critical features of the concepts or
procedures. This assessment is meant to avoid the need for remediation and, at the same time, to
challenge students through bonuses, which are immediate extensions of the work everyone is
doing in the classroom. This approach contrasts with other forms of Mastery Learning in (1) the
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IDEAS 2016 205
frequency of assessment, (2) the focus on pre-empting instead of remediating, and (3) the scope
of the enrichment activities.
In this paper we stress two distinctive features of continuous formative assessment. One feature is
what Wiliam (2011) called “all students’ response system” (p. 87), in which the teacher receives
responses from all students at the same time during class. He suggested, for instance, that
students might write answers to questions on personal mini-boards that are visible to the teacher.
The teacher may respond with what we, in the initiative, call step-backs (Sabbaghan et al., 2015):
a form of scaffolding in which the teacher offers simpler tasks or questions that everyone can
engage in and then continues increasing difficulty—with step-ups—until everyone meets the
immediate learning objective. The Variation Theory of Learning (Marton, 2015) can inform these
step-backs and step-ups, as Metz et al. (2015) explained through an analysis of sequences of
questions that draw attention to features students need to discern to understand particular
mathematical ideas. We acknowledge that this is not a simple input-output process and that the
teacher still has to address the complexities of the classroom; however, this feature provides
specific guidelines for assessment and teacher response.
The other distinctive feature of our use of continuous formative assessment is the way we
conceive bonuses. We do not consider these tasks as ‘enrichment,’ because they are direct
extensions of the work everyone is doing in class. This approach contrasts with Mastery Learning
in which enrichment activities “lie beyond the established curriculum(Guskey, 2010, p. 56).
EXAMPLES OF DIFFERENTIATION
This study involved two elementary schools, each with approximately 150 students. The data
included video recordings from each teacher at both schools, and classroom observations in one
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IDEAS 2016 206
of the schools—more details can be found in Preciado-Babb, et al. (2015), Metz et al., (2015),
and Sabbaghan et al., (2015). We have observed how students who initially struggled with the
content successfully completed the tasks in class, as well as how students continued to engage in
bonus tasks. We show three occurrences stressing different ways in which this form of
differentiation has been enacted in the classrooms.
Example #1: Responsive Improvisation
This example is from a Grade 2 introductory lesson on subtraction. During the lesson, students
showed answers to the teacher’s questions on their mini-boards. The first two questions involved
short stories, such as: “If you had five apples and someone takes three of them, how many would
be left?” Subsequent items were presented as number sentences. Table 1 shows a sequence of
items students had to work with, indicates whether all students showed a correct answer, and
notes the type of response offered by the teacher. The table also indicates whether the teacher
deviated from the original plan, i.e. whether she improvised in response to students’ answers. As
can be seen in the table, not all students provided correct answers even though the teacher
provided individual feedback and re-explained the material. Then she improvised the lesson
sequence by adding easier items, or stepping back, to 7 - 2 and 6 - 4, which involved smaller
numbers than 8 - 5. Once all students showed a correct answer, she provided a harder
question—a step-up—before proceeding with the next part of the lesson—which built, but
differed from, the items in this part.
During class, students also worked independently on a sequence of similar questions, while the
teacher walked through the classroom assisting students who required support. Students who
finished earlier were challenged with variations of the items they had already solved, such as 17 -
4 after 7 - 4.
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Item
In the lesson plan?
All students correct?
Teacher’s responses
7 - 4 =
Yes
No
Feedback and Explanation
8 - 5 =
Yes
No
Feedback, Explanation and Step-back
7 - 2 =
No
No
Feedback and Explanation
6 - 4 =
No
Yes
Step-up
9 - 3 =
No
Yes
Proceed with the next part
Table 1: Task sequence and teacher’s responses: Responsive improvisation
Example #2: Team Work
This lesson involved a combined Grade 4/5 group working on telling time using different
representations. The final activity of the lesson was to indicate on a timeline the times students
get up, eat breakfast, brush teeth, etc., using both digital and analog formats. The teacher assessed
continuously, making sure every student was ready for each next step of the lesson. The class
started by identifying the hour and minute hands on the clock. Then, students were asked to read
the time from an analog clock. First, the focus was only on the hour hand, then on the minute
hand, and finally on both hands (with minutes restricted to multiples of 5). Table 2 shows the
teacher’s responses in a part of the lesson in which she provided easier questions, or step-backs:
8:30 is easier than 12:15 and 8:25 due to the half mark. Once all but three students showed
correct answers, the class began working in pairs: One student said a time and the other had to
indicate it on a mini-clock (with students alternating roles). The teacher walked through the class
checking on all students, but dedicating particular attention to the three students who did not
initially provide correct answers. It is not possible from the video to hypothesize the source of
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IDEAS 2016 208
misconception or difficulty these students experienced; however, teacher’s intervention seemed
effective, as everyone in the class succeeded in providing correct answers.
Item
In the lesson plan?
All students correct?
8:25
Yes
No
12:15
Yes
No
8:30
No
Yes
4:15
Yes
Yes
6:45
Yes
No
Team work
Yes
No
Table 2: Task sequence and teacher’s responses: Team work
The teacher offered a bonus question on the board during the second part: “It is 2:45. I have to
leave in 20 minutes. What time will I have to leave?” Students who already understood how to
indicate time on the analog clock were encouraged to challenge their partners with questions
similar to this one.
Example #3: Individual Assistance
This is an excerpt from a Grade 2 class on the language for comparing two numbers. The teacher
showed an image on the board with a row of spiders and a row of ants and asked how many more
spiders than ants were there. Students wrote their answers on their mini-boards, and the teacher
asked them to show their answers at the same time. The teacher repeated similar questions, but
just with numbers, as shown in Table 3. Most students’ answers were correct; however, one
student struggled with “How many more is 9 than 3?” The teacher supported this student
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IDEAS 2016 209
individually and then asked all students to show their answers: All students showed correct
answers and then the class moved to the next part of the lesson: individual practice.
As in the previous case, the video does not allow hypothesizing regarding the reason for the
student’s struggle, but the teacher’s intervention seemed to be effective for supporting this
student.
Item
In the lesson plan?
All students correct?
Teacher responses
How many more is
9 than 3?
Yes
Yes
Assist one student, Proceed to
the next part of the lesson
Individual practice
Yes
Yes
Walked through the room
while assessing children
Table 3: Task sequence and teacher’s responses: Individual assistance
Original task
Bonus
9 is __ more than 6
6 is __ more than 2
10 is __ more than 7
8 is ___ more than 4
12 is __ more than 6
16 is __ more than 7
19 is __ more than 10
20 is __ more than 10
Table 4: Practice items (first column) and bonus items (Second column): Individual assistance
During individual practice, the teacher supported the student who had struggled, then moved
through the classroom to other tables providing feedback and assistance. She gave a bonus for
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IDEAS 2016 210
students who had finished their practice, as shown in Table 4. The student who initially struggled
completed the bonus successfully.
CONCLUSION
The three examples we have provided show how three teachers used continuous assessment to
respond immediately to all students in their classes. The use of step-backs contrasts with the
common focus on remediation that is evident in different forms of Mastery Learning (Guskey,
2010). These examples also show students engaging in bonus questions that are direct extensions
of the work everyone was doing in the class. This also contrasts with the enrichment and
extension activities typical of Mastery Learning, in which advanced students engage in different
activities that extend beyond the curriculum. Moreover, in our work, ‘non-advanced’ students
could address the same bonus material, as demonstrated by the student who initially struggled
with the questions in Example #3.
This approach to differentiation is not based on selecting different activities for each student, but
on selecting a level of difficulty appropriate to each student within the same activity. The two
stressed features, namely, continuous assessment of all students and the use of bonus tasks, help
to address the diversity of ability and knowledge in the classroom.
We have found, nevertheless, that teachers have difficulties implementing this approach to
assessment (Preciado Babb, et al., 2015). Teachers in the initiative have identified creating
bonuses as particularly challenging. We have also seen teachers focused more on completing the
material than making sure everyone was ready to move on. Finally, teachers tended to rely more
on feedback and explanations in their responses to assessment during class, with less
consideration of step-backs and step-ups. In fact, we believe that teachers’ responses in the
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IDEAS 2016 211
examples provided in this paper could have been even more effective if careful attention to
variation was considered.
We conclude by stressing the need to continue addressing the ways in which teachers respond to
assessment for all students in class. Marton’s (2015) Variation Theory of Learning provides a
framework for both investigating this topic and for informing teacher’s practice, especially for
improvising tasks and questions in response to continuous feedback from students during class.
References
Bloom, B. (1968). Learning for mastery. Evaluation Comment, 1(2), 1–12.
Chappuis, J. (2015). Seven strategies of assessment for learning (2nd Ed.). Hobeken, NJ: Pearson.
Guskey, T. (2010). Lessons of mastery learning. Educational Leadership, 68(2), 52-57.
Marton, F. (2015). Necessary conditions of learning. New York: Routledge.
Metz, M., Sabbaghan, S., Preciado Babb, A. P., & Davis B. (2015). One step back, three forward:
Success through mediated challenge. In A. P. Preciado Babb, M. Takeuchi, and J. Lock (Eds.)
Proceedings of the IDEAS: Designing Responsive Pedagogy Conference, pp. 178 - 186.
Werklund School of the Education, University of Calgary.
Pink, D. (2011). Drive: The surprising truth about what motivates us. New York: Riverhead.
Preciado Babb, A. P., Metz, M., Sabbaghan, S., & Davis, B. (2015). Insights on the relationships
between mathematics knowledge for teachers and curricular material. In T. G. Bartell, K. N.
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Sabbaghan, S., Preciado Babb, A. P., Metz, M., & Davis, B. (2015). Dynamic responsive
pedagogy: Implications of micro-level scaffolding. In A. P. Preciado Babb, M. Takeuchi, and
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ResearchGate has not been able to resolve any citations for this publication.
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