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Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 1
Marzano’s New Taxonomy as a framework for
investigating student affect
Jeff Irvine
Brock University
ABSTRACT
In 1998 Marzano proposed a taxonomy of learning that integrated three domains or
systems: the self system, which involves student motivation; the metacognitive system, involving
goal setting and planning; and the cognitive system, required to complete the task at hand.
Although extant for 20 years, a paucity of studies have utilized this taxonomy, even though
employing Marzano’s taxonomy as a framework is particularly appropriate for studies involving
student affect. This study provides an exemplar of the use of Marzano’s taxonomy as a
framework to investigate the impact of a classroom intervention using active and social strategies
to enhance student participation. Further, this paper provides suggestions for employing
Marzano’s taxonomy in other areas for practising teachers, teacher educators, and educational
researchers.
Keywords: Marzano’s New Taxonomy, engagement, attitude, theorytopractice.
Copyright statement: Authors retain the copyright to the manuscripts published in AABRI
journals. Please see the AABRI Copyright Policy at http://www.aabri.com/copyright.html
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 2
INTRODUCTION
In 1998, Marzano proposed a taxonomy of learning domains that integrated three levels
of processing: self (including motivation), metacognitive, and cognitive (Marzano, 1998;
Marzano & Kendall, 2007). Marzano’s New Taxonomy (MNT) differs from previous taxonomies
in that it comprises three interrelated domains whereas the wellknown Bloom’s (Bloom et al.,
1956) taxonomy addressed only the cognitive domain. Revisions to original Bloom (Anderson &
Krathwohl, 2001) added metacognition, but only as a passive knowledge domain to be acted
upon by the active cognitive domain.
1
Unlike Bloom, MNT is not a strict hierarchy but instead is twodimensional,
encompassing: “(a) flow of processing and information and (b) level of consciousness required to
control execution based on flow of information, and level of consciousness” (Irvine, 2017, p. 2). In
topdown fashion, initially the self system engages, making decisions about whether to engage in a
new task. This is followed by the metacognitive system that sets goals and strategies. Finally, the
cognitive system engages at whatever levels are appropriate to resolve the task. Although Marzano
specifies a hierarchy among the three systems, there is no strict hierarchy within the cognitive
system.
The three active systems of MNT—self (including motivation), metacognitive, and
cognitive—act on three passive knowledge domains: information, mental procedures, and
psychomotor procedures, as shown in Figure 1 (Appendix A). In Marzano’s model, the self system
engages first, making a decision about whether to engage in a new task or continue with the present
task. The metacognitive system then engages to identify goals and select strategies. Once these
goals and strategies are determined, the cognitive system carries out the cognitive activities
required to address the task. While no feedback mechanisms are explicitly included in MNT, the
self system continues to monitor the desirability of continuing with the current task compared to
other alternatives, and the metacognitive system monitors processes to determine efficacy.
The systems of MNT can be further subdivided by strategy, as shown in Figure 2
(Appendix A): Selfsystem strategies examine importance, selfefficacy, emotional response, and
overall motivation; metacognitive system strategies comprise goal specification, process
monitoring, and monitoring for clarity and accuracy; and cognitive system strategies encompass
storage and retrieval, analysis, and knowledge utilization processes.
The flow of processing is illustrated in Figure 3 (Appendix A). Marzano also argues that
his taxonomy is hierarchical based on levels of consciousness, which increase as one proceeds up
the taxonomy. For example, retrieval processes may be automatic, requiring a very low level of
consciousness; however, knowledge utilization requires significantly more conscious thought, as
does goal setting by the metacognitive system, while self system involvement and decision
making requires even more.
Marzano and Kendall (2008) published Designing and Assessing Educational Objectives
to help educators apply the taxonomy, although the work’s instructional strategies are somewhat
basic and need enhancement and augmentation before using them in classroom situations.
Because MNT explicitly addresses self system constructs (such as motivation and
emotions), it is appropriate to investigate whether instructional strategies based on this taxonomy
can positively influence student attitude and engagement, as well as student achievement in
mathematics. Although Marzano and Kendall (2008) outlined ways that MNT could be applied
to learning, specifically in designing and assessing educational objectives, scant empirical
1
For a detailed comparison of MNT and revised Bloom, see Irvine (2017).
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 3
research was found. Indeed, no applications of MNT were found for secondary school education
or secondary school mathematics education. This is surprising because MNT has the potential to
address attitudes and engagement—dimensions of learning that have been identified as critical
for student success and wellbeing (Clarkson, 2013).
REVIEW OF THE LITERATURE
Since Marzano identifies the self system as the first system to engage, followed by the
metacognitive system and then the cognitive system, the discussion below reflects Marzano’s
sequencing in Figure 3 (Appendix A).
Self System: Decision to Engage
Marzano’s self system (see Figure 2, Appendix A) includes four subsystems that involve
examining: importance, efficacy, emotional response, and overall motivation. Marzano considers
motivation to be a superordinate category that combines emotional response, efficacy, and
importance across three dimensions of task engagement: (a) students believe the task is
sufficiently important, (b) students believe they can successfully complete the task, and (c)
students have a positive emotional response in relation to the task (Irvine, 2017).
Marzano’s conception of motivation is based on expectancyvalue theory (Wigfield &
Eccles, 2000), selfefficacy (Bandura, 1997; Pajares, 1997), and, in the case of mathematics,
MWB (Clarkson et al., 2010). The following section examines each subsystem of the self system
in greater detail.
Examining Importance: ExpectancyValue Theory
Expectancyvalue theory suggests that students’ task selection, persistence, and
achievement are predicated on two things: a belief that they will succeed and the value they
assign to the task (Eccles, 1994, 2005, 2009; Eccles & Wigfield, 1995, 2002; Wigfield & Eccles,
2000). In other words, task selection is based on students’ perception of: (a) difficulty with the
task and (b) the ultimate cost of the task (Eccles & Wigfield, 2002; Eccles et al., 1993; Eccles et
al., 1998). The relationship between expectancyvalue theory and selfefficacy therefore is that
students’ perceived ability to complete a task influences their decision to undertake the task.
While Ball et al. (2016) note that selfefficacy and expectancy essentially represent disparate
theoretical constructs, it can be difficult to distinguish them and their associated factors for
research purposes (Irvine, 2018).
The importance component of Marzano’s self system is a central concept of expectancy
value theory. Marzano asks students to respond to questions such as: How important is this to
you? Why do you think it might be important? Can you provide some reasons why it is
important? How logical is your thinking with respect to the importance of this?
Examining Efficacy: SelfEfficacy Theory
The self system’s second subsystem is examining efficacy. Selfefficacy (Bandura, 1997;
Pajares, 1997) involves individuals’ perceptions about their capability to accomplish a task.
Regarding mathematics, Middleton and Spanias (1999) identified a relationship between
perceived mathematical abilities and intrinsic motivation. S. Ross (2008) found that the impact
of selfefficacy was greater than other motivational variables such as goal orientation, intrinsic
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 4
motivation, or an instrumental versus relational view of instruction. Selfefficacy is domain and
task specific (Bandura, 1997). Unfortunately, selfefficacy is very difficult to change, especially
in the short term (J. Ross, 2009). Because of its domain and taskspecificity, students’ self
efficacy will differ for different subjects (e.g., mathematics vs. English) and for different tasks
within each subject. Such factors make selfefficacy a difficult variable to manipulate in the short
or intermediate term.
In relation to selfefficacy, Marzano poses questions such as: How good are you at this?
How well do you think you can do on this? Can you improve at this? How well can you learn
this? How logical is your thinking about your ability to do this?
Examining Emotional Response
The third subsystem of the self system is examining emotional response. This subsystem
identifies affective considerations as being important in the overall decision to engage.
Regarding emotional response, Marzano asks questions such as: What are your feelings about
this? What is the logic underlying these feelings? How reasonable is your thinking? These
questions tend to involve affective dimensions, as well as cognitive questions concerning
reasonableness. A major component of emotional response is interest, which can be construed as
an emotion, as affect, or as a schema (Reeve et al., 2015).
If considered an emotion, “interest exists as a coordinated feelingpurposiveexpressive
bodily reaction to an important life event” (Reeve et al., 2015, p. 80). Interest is activated by the
opportunity for new information or greater understanding. With regards to feeling, interest
involves an alert, positive feeling; in terms of purpose, it creates a motivational urge to explore
and to investigate; as an expression, interest widens the eyelids, parts the lips slightly, and
notably stills the head; and in terms of bodily changes, it decreases heart rate. Collectively, this
coordinated pattern of reactivity facilitates attention, information processing, stimulus
comprehension, and learning (Reeve et al., 2015, p. 80).
A second way of viewing interest is as affect or mood. The two dimensions of affect are
pleasure/displeasure and activation/deactivation. The goal of instruction is to place the student’s
affect/mood in the pleasureactivated quadrant, increasing motivation and stimulating
engagement. The third way of viewing interest is as an emotion schema, which is “an acquired,
processoriented, highly individualized, and developmentally rich construct in which an emotion is
highly intertwined with appraisals, attributions, knowledge, interpretations, and higherorder
cognitions such as the selfconcept” (Reeve et al., 2015, p. 82). This conceptualization of interest is
closely related to identification of value that enables a shift from situational interest to individual
interest (see discussion below). Interest is a predictor of engagement and has been shown to
replenish motivational and cognitive resources when an interested student is engaged in an activity.
Interest is positively and reciprocally correlated with selfefficacy (Bong et al., 2015),
selfconcept (Durik et al., 2015), and selfregulation (Sansone et al., 2015), and is also related to
valuing of content (Kim et al., 2015). The value that students place on particular content is
related to their level of interest for that content. Kim et al. (2015) also demonstrated that interest
and value have an impact on engagement and achievement, with selfefficacy acting as a
moderator variable. For specific content, it has also been shown that value impacts interest. The
greater the value that students place on particular content, the higher the likelihood they will
demonstrate interest in that content (Ainley & Ainley, 2015).
The fourphase model of interest development (Hidi & Renninger, 2006) presents a
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taxonomy of interest development. This model postulates that initial interest is triggered by a
situation or topic (triggered situational interest), which may be fleeting, and may be positive or
negative. If interest in the situation becomes more sustained (maintained situational interest), this
phase is characterized by positive student focus and persistence with the material. If students
develop emerging individual interest, they are likely to independently reengage with the
material or classes and ask curiosity questions, building stored knowledge and stored value about
the material. Finally, at the welldeveloped individual interest stage, students willingly reengage
with the content, selfregulating to reframe questions and seek answers. This level is
characterized by students’ positive feelings towards the material, perseverance through
frustration and challenges, and actively seeking feedback on their learning. The fourphase
model has abundant research evidence supporting it. The present research study focused on the
first two levels of the fourphase model—triggered situational interest and maintained situational
interest—with the hope that some students will become sufficiently engaged in the material to
proceed to the higher two stages of the model.
Examining Overall Motivation
The last subsystem examines overall motivation. Marzano’s concept of overall
motivation is a synthesis of importance (expectancyvalue), selfefficacy, and emotional
response. In this, Marzano is consistent with Hannula’s (2006) model of attitude as well as Di
Martino and Zan’s (2009) three dimensions of attitude. Marzano’s treatment recognizes that
students may be motivated across all three of these dimensions, or some subset of them.
Therefore, the strength of a student’s motivation will vary depending on the number of
dimensions (importance, selfefficacy, emotional response) that are engaged at a specific point in
time. Thus, the level of motivation can and will fluctuate across tasks as well as within tasks.
Students may approach a task with high motivation but become disinterested as the task
progresses. Alternatively, students may approach a task with low initial motivation but become
more motivated while engaging in the task due to increased selfefficacy and confidence that
they can successfully accomplish that task.
Questions posed by Marzano in relation to overall motivation include: How interested are
you in this? How motivated are you to learn this? How would you explain your level of interest
in this? How reasonable is your thinking about your motivation for this?
Instructional strategies that support the self system and motivation include: choice, open
questions, connections to real life, RAFT (role, audience, format, topic), journals, placemat, PMI
(plus, minus, interesting), and explicit questioning about aspects of motivation.
Motivation and Achievement in Mathematics
There is substantial evidence, although not complete agreement, that motivation in
mathematics is positively correlated with mathematics achievement (Hannula, 2006; Koller et al.,
2001; Malmivuori, 2006). This correlation is also bidirectional (Koller et al., 2001; Middleton &
Spanias, 1999), in that such increases in motivation resulted in increases in achievement, which
stimulated further increases in motivation. Further, in a study on streaming students in secondary
schools into applied (nonuniversity track) courses, Maharaj (2014) found that “student
achievement often has more to do with motivation than innate intelligence” (para. 1). Therefore,
when students are unsuccessful in mathematics achievement, the result is decreased motivation,
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Marzano’s New Taxonomy, Page 6
which leads to further low achievement and continued decreases in motivation.
Teachers’ beliefs and practices significantly influence students’ motivation, particularly
in mathematics. For example, Middleton (1995) found that teachers who emphasize content
acquisition instead of considering student motivation tend to decrease student motivation in
mathematics; when the subject of mathematics is “intrinsically motivating” to some but not all
students, “individual differences among students, and the ways in which mathematics education
complements these differences, determine … the degree to which mathematics is perceived as
motivating” (p. 255). Since motivation impacts mathematics achievement, teachers’ attitudes
towards mathematics and their choice of instructional strategies are important dimensions of
influencing student achievement (Middleton & Spanias, 1999). Student motivation typically
decreases over a student’s academic career (Middleton & Spanias, 1999). Cotic and Zuljan
(2009) found that both student cognition and student affect in mathematics were influenced by
instructional strategies that involved problem solving and problem posing.
Because motivation is a superordinate category and therefore very broad, the current
study specifically addressed two subcategories of motivation: student attitudes and engagement.
The study’s duration was approximately 4 weeks. A seminal study by McLeod (1992) found that
engagement can be positively influenced in relatively short time periods, while attitude requires
longer periods of time to be affected. Therefore, the two subdimensions of motivation were
specifically selected as the target of the classroom intervention.
Metacognitive System: Planning and Goal Setting
The second system in MNT is metacognition, defined by Marzano as a separate system,
based on four subsystems: goal specification, process monitoring, monitoring clarity, and
monitoring accuracy. The positioning of metacognition in MNT as the second system to engage
is consistent with earlier work by McCombs and Marzano (1990).
Metacognition has been defined as “the knowledge about and regulation of one’s
cognitive activities in learning processes” (Veenman et al., 2006, p. 3). In a comparison of MNT
and revised Bloom’s taxonomy (RBT), Irvine (2017) contrasts the treatment of metacognition in
the two taxonomies stemming from Flavell’s (1979) division of metacognition into (a)
“declarative knowledge about cognition” and (b) selfregulation, involving “control monitoring
and regulation of cognitive processes” (Irvine, 2017, p. 5). This dualistic treatment is found in
RBT’s approach to metacognition (Anderson & Krathwohl, 2001) in comparison to MNT, as
RBT places metacognition in the domain of knowledge. While Anderson and Krathwohl (2001)
noted some disagreement surrounding metacognition’s categorization under declarative
knowledge, they maintain that metacognition underpins every cognitive process. Still, such
positioning remains inconsistent, as Anderson and Krathwohl label certain aspects of
metacognition as “processes” while RBT assign metacognition to the knowledge domain (Irvine,
2017). The stance in RBT is consistent with researchers who treat metacognition as declarative
knowledge (Veenman et al., 2006). However, Veenman et al. (2006) point out that metacognition
subsumes a number of distinctly different constructs, of which declarative knowledge is only one.
In MNT metacognition is considered separate active system, based on Flavell’s (1979)
second substrate of selfregulation. Jans and Leclercq (1977) defined metacognition as active
judgments that happen throughout learning. Similarly, metacognitive dimensions such as
defining learning goals and monitoring progress towards those goals are dimensions of student
selfregulation (Nunes et al., 2003). The current study used metacognitive strategies to promote
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Marzano’s New Taxonomy, Page 7
student selfregulation and as autonomy supports for students.
A literature review by Veenman et al. (2006) found studies that support the positioning of
metacognition both as domain specific as well as general, and argue such inconsistent positions
may reflect the studies’ respective grain size. For instance, studies assigning metacognition a
“fine grain size” (e.g., for reading strategies) place it in RBT; those involving a “coarser” grain
size (e.g., for problemsolving) adopt Marzano’s position (Irvine, 2017, p. 5).
Such differing interpretations of metacognition thus have different implications. Because
RBT classifies metacognition in the domain of knowledge, metacognition becomes a passive
agent that is acted upon; Marzano, in turn, categorizes metacognition on a higher scale in MNT
(second only to the self system) as a significant, active domain. Overall, metacognition is a key
element in the sequence of processes, bounded by motivation to undertake a task (self system)
and the incitement of cognitive processes needed for the task. RBT offers few examples that
illustrate the appropriateness of metacognition as declarative knowledge (Anderson &
Krathwohl, 2001); MNT, however, recognizes the more active aspects of metacognition, such as
setting goals (Irvine, 2017).
Other research evidence supports the positioning of metacognition as an active rather than
passive system. Hattie (2009), in his synthesis of more than 800 metaanalyses of factors affecting
student achievement, found an effect size of 0.56 for teaching goalsetting strategies, and an effect
size of 0.69 from teaching metacognitive strategies. Meijer et al. (2006), when developing their
metacognitive taxonomy, also considered metacognition to be an active strategy.
Veenman et al. (2006) point to the importance of teaching metacognitive strategies to
enhance student learning, and they identify three researchaffirmed principles for successful
metacognition instruction: embedding metacognitive instruction in the content matter to ensure
connectivity, informing learners about the usefulness of metacognitive activities to make them
exert the initial extra effort, and prolonged training to guarantee the smooth and maintained
application of metacognitive activity. Veenman et al. refer to these principles as the WWW&H
rule: what to do, when, why, and how (p. 9).
Marzano and Kendall (2008) apply a rather simplistic version of these principles in their
text concerning design and assessment of educational objectives, in which they limit
metacognition to goal setting, process monitoring, and monitoring clarity and accuracy. Their
text ignores other metacognitive strategies such as anticipation guides, think aloud, timed retell,
plus/minus/interesting (PMI), and ticket to leave. A number of instructional strategies can be
tailored to address any of the three systems specified in MNT.
Marzano’s dimensions of metacognition (goal specification, process monitoring,
monitoring clarity, and monitoring accuracy) omit some important aspects; namely, planning and
evaluating. Meijer et al. (2006) identify these aspects as components of the highest level of
metacognition. Because metacognition plays an important role in MNT as well as in Marzano’s
theory of behaviour, this study implemented metacognitive instructional strategies throughout
the intervention. Once the metacognitive system has set goals and formulated a plan of action,
the cognitive system engages to analyze and perform the required task.
Cognitive System: Performing the Task
The third system of MNT is the cognitive system, with four sublevels: retrieval,
comprehension, analysis, and knowledge utilization. Cognition is “the mental action or process
of acquiring knowledge and understanding through thought, experience, and the senses”
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 8
(“Cognition,” 2020, para. 1). Cognition has been identified as an important component of all
student learning. Therefore the cognitive system was present in all control and treatment lessons
of the MNT intervention. The MNT intervention involved modifying or adding to base lessons to
explicitly focus on metacognitive and selfsystem attributes, in addition to the cognitive activities
already included in the lessons.
Prior knowledge has been identified as the key cognitive factor in learning mathematics
(Milic et al., 2016). Cognitive competence has been shown to be significantly related to
mathematics achievement as well as students’ selfrating of mathematical ability (Milic et al.,
2016). Of particular note is the notion that “cognition is always for action” (Nathan et al., 2016,
p. 1692) since the instructional intervention in this study took an active stance with respect to
student learning, which may be different than the more passive mathematics lessons that students
had experienced up to this point in their academic careers.
MNT identifies four levels within the cognitive system (lowest to highest): retrieval,
comprehension, analysis, and knowledge utilization. Marzano states that they are ordered based
on the level of processing required. This position is supported by Nokes and Belenky (2011) who
claim that knowledge utilization that supports far transfer requires a significantly higher level of
processing than other cognitive tasks. The two lower levels (retrieval, comprehension) share
similarities with the corresponding levels of RBT. Below is a discussion of the four levels of the
cognitive system, beginning with the lowest level, retrieval.
Cognitive System: Retrieval
Retrieval, the lowest level, involves the activation and transfer of knowledge from
permanent memory to working memory, usually done without conscious thought. This retrieval
may take the form of recognition or recall. Recognition is a simple matching of a prompt or
stimulus with information in permanent memory. Recall involves recognition and production of
related information. Marzano and Kendall (2007) give the example of selecting a synonym for a
word (recognition) contrasted with producing the definition of a word (recall).
Cognitive System: Comprehension
The next level of MNT is comprehension, which consists of two subsystems: integrating
and symbolizing. Integrating involves taking knowledge in a microsystem form and producing a
macrosystem form for that knowledge. This may involve deleting extraneous information,
replacing specific propositions with more generalized ones, or constructing a single proposition
to replace a set of less general propositions. Symbolizing involves creating symbolic
representations of knowledge, in both linguistic form and imagery. The linguistic form is
semantic, while the imagery form involves mental pictures or physical sensations to support
cognition. Thus, teachers may frequently employ graphic organizers, which combine both the
semantic and imagery forms for a specific knowledge set.
Cognitive System: Analysis
The third level of the cognitive system in MNT is analysis, which has several sublevels:
matching, classifying, analyzing errors, generalizing, and specifying (predicting). Matching
involves identification of similarities and differences. Matching has been identified by Atkinson
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 9
et al. (2000) as a critical component of learning from worked examples. Matching is also
important in near transfer (Nokes & Belenky, 2011) and in learning through comparison (Rittle
Johnson & Star, 2011). Classifying requires organizing knowledge into meaningful categories.
Thus, classifying involves identifying defining characteristics, identifying superordinate and
subordinate categories, and justifying these categories. Classifying is used in concept comparison
throughout formal education (RittleJohnson & Star, 2011). Analyzing errors involves the
accuracy, reasonableness, and logic of knowledge. Generalizing is the process of constructing new
generalizations or inferences from knowledge that is already known. RittleJohnson and Star
(2011) point out that generalizing typically involves examination of a range of specific cases in
order to identify commonalities and critical features. Finally, specifying (predicting) extends a
known generalization to other similar situations, and draws conclusions about these new situations.
Cognitive System: Knowledge Utilization
The highest and most complex level of the cognitive system in MNT is knowledge
utilization, which has four sublevels: Decision making, problem solving, experimenting, and
investigating. The knowledge utilization level is unique to MNT, and no similar level exists in
RBT, although Bloom’s synthesis category has elements of some of the subcategories of
knowledge utilization, without specifically addressing knowledge utilization. Decision making
requires selecting among two or more alternatives. This involves thoughtful generation of
alternatives and selecting among them based on sound criteria. Problem solving is a cognitive
process directed at achieving a goal when no solution method is obvious to the problem solver.
Problem solving has also been described as a situation having an initial undesired situation, a
desired end situation, and an obstacle preventing the movement from the initial situation to the
end situation (Irvine, 2015).
Thus, problem solving requires identification of obstacles, generating alternative ways to
accomplish the goal, evaluating the alternatives, and selecting and executing the optimal
alternative. Experimenting requires the generation and testing of hypotheses to understand or
explain a phenomenon, typically from primary data collection. Alternatively, investigating
relates to generating and testing hypotheses based on secondary or historical data.
Instructional strategies that specifically address the cognitive system include concept
attainment, problem posing, timed retell, jigsaw, open questions, explicit questioning, what/so
what double entry, decision trees, and flowcharts. The sublevels of knowledge utilization may
also serve as significant motivational factors since they have a more active stance for students
and involve activities such as investigation and problem solving. All learning involves cognition;
however, cognitive strategies may be used as vehicles to stimulate student engagement and
interest.
MNT AS A FRAMEWORK FOR INVESTIGATING STUDENT AFFECT: AN
EXAMPLE
A mixed methods study (Teddlie & Tashakkori, 2009) examined a set of classroom
activities (“the MNT intervention”) using MNT as the theoretical framework (Figure 4,
Appendix A). This study consisted of student surveys, which were analyzed quantitatively;
student postintervention interviews, analyzed qualitatively; and teacher pre and post interviews,
as well as 20 classroom observations by the researcher. The study involved three classes of
Journal of Instructional Pedagogies Volume 24
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Grade 10 Academic Mathematics at one high school in Ontario, Canada. One class functioned as
a control and did not receive the MNT intervention lessons. The two treatment classes received
lessons that focused on motivation and metacognition while covering the same content as the
control class.
This study was consistent with Veenman et al.’s (2006) three principles in that the
metacognitive instruction is embedded in the mathematics unit involved in the study; students
are made aware of the metacognitive strategies being used; and metacognitive strategies are
embedded throughout the instructional intervention to help foster maintained application of the
strategies.
The MNT intervention utilized activities explicitly linked to an MNT sublevel of the self
system and the metacognitive system (see Appendix B for details of the linkages). Prior to
implementation teachers were given professional learning time to understand the MNT
intervention and make suggestions with regard to its implementation. The intervention was based
on reform mathematics principles (Moyer et al., 2018). Technology was readily available and
utilized where appropriate since the school was a “bring your own device” (BYOD) school.
Method
Teachers delivered all lessons to their own classes. With respect to instruction, treatment
classes received lessons with instructional strategies based on the self and metacognitive
domains, comprising two classes, and the control class received lessons without a focus on
metacognitive and self systems.
Throughout the intervention, the researcher was available as a resource but did not
engage in any classroom teaching. The researcher observed approximately 25% of classes over
the duration of the study, to support implementation fidelity. Observed classes were assessed for
fidelity of implementation against seven criteria identifying the degree to which the lessons
reflected the expectations of the MNT intervention: matching given sequencing of topics;
inclusion of all elements of the MNT intervention; instructional strategies; responses to student
questions; use of manipulatives; use of technology; and responsiveness to student needs. This
method of assessing fidelity of implementation was chosen over selfreport surveys (O’Donnell,
2008) and was reinforced through data obtained from teacher postintervention interviews.
The unit on quadratic functions and quadratic equations was identified by the researcher
as the most appropriate for the study, based on an analysis of the units in the course as well as
comparisons with other secondary mathematics courses. Grade 10 was selected based on the
relative homogeneity of prior knowledge, since all students had completed the Grade 9
Academic Mathematics course. In addition, confounding factors such as the transition from
Grade 8 to Grade 9, and attending a new (and usually larger) school were minimized since the
students had attended the same school in the prior academic year. This unit is one of four units in
the course, with the others being linear systems, analytic geometry, and trigonometry. The
quadratics unit was the second unit taught in the semester, after linear systems.
Before the treatment, all students (both in treatment classes and the control class)
completed surveys on attitude and engagement on computer, smartphone, or tablet. Students
completed weekly reflections, while teachers completed daily reflections, with all reflections
being done online. Summative assessments occurred twice, with one midway through the unit
and the other at the end of the unit, along with a rich assessment task. The summative
assessments were created by the teachers involved in the survey. Both summative assessments
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Marzano’s New Taxonomy, Page 11
consisted of written paperandpencil tests, scored with marking schemes. The researcher
reviewed both assessments prior to their administration. The rich assessment task was designed
by the researcher and assessed with a rubric constructed by the teachers involved in the study,
with researcher input. After the unit was completed, students again completed online surveys on
engagement and attitude.
After completion of the treatment, five student volunteers were identified to participate in
audiotaped interviews. Permission forms were given for parental consent. Five students
volunteered, and all were interviewed after receiving completed permission forms. All students
were assigned pseudonyms when information was reported in the results section. At the conclusion
of the study, both teachers participating in the research were interviewed again, using a separate
targeted interview guide.
In summary, this study sought to examine whether instruction based on MNT that
explicitly targeted dimensions of student metacognition and motivation had positive impacts on
student engagement, attitude, and achievement.
Selected Results: Engagement—Quantitative Findings
Both before and after the intervention, students in both the treatment classes and the
control class completed online surveys from the Dimensions of Student Engagement Survey©
(DSES; Reeve, 2013). The DSES is a 39question Likert scale survey (1=strongly disagree to
5=strongly agree). All surveys were completed online, during class time. The DSES has four
subscales: cognitive, behavioural, emotional, and agentic engagement. For this study, the DSES
had a Cronbach’s α of 0.95.
Pre–Post Comparisons
DSES scores for students in the treatment classes (T
Total
) are shown in Table 1 (Appendix
A). Irvine (2020) reports the intervention’s pre−post results as follows:
Pre and post measures of engagement for T
Total
resulted in a statistically significant
positive effect size of 0.54 (M=0.527, SD=0.694, t(45)=5.29, p<0.001); such an effect
size
is identified as medium (Cohen, 1992) and indicates that the MNT intervention had a
positive impact on student engagement. In addition, all four of the engagement subscales
of the DSES had statistically significant increases. … [Eightyfour percent] of students in
T
Total
showed increases in overall engagement scores (M=0.44, SD=0.816, min=1.46,
max=3.48), selfreported. Overall engagement and all subscales showed statistically
significant; the greatest increase occurred for the agentic engagement subscale (Cohen’s
d=0.73). Agentic engagement is student selfadvocacy subdimension, involving students
selfidentifying interests and preferred learning environments. (p. 19)
Treatment
−
−−
−
Control Comparisons
Irvine (2020) did not find a significant difference in student engagement scores in
pre−post results for the control class, nor for any of the subscales (Table 1, Appendix A):
Prior to the MNT intervention, the control class showed a significant differential
advantage over T
Total
(M= 0.34, SD= 0.158, t(66)=2.140, p=0.036). After the MNT
intervention, no significant differences were found for the control class (M=0.24,
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 12
SD=1.024, t(21)=1.100, p=0.284). Therefore, no change was found for the control class
(not receiving the MNT intervention lessons), while both treatment classes showed
statistically significant increases in engagement. (p. 21)
Selected Results: Attitude—Quantitative Findings
All students in both the treatment classes and the control class completed the Attitude
Towards Mathematics Inventory© (ATMI) both before and after the MNT intervention. The
ATMI (Tapia & Marsh, 2005) is a 40question Likert scale survey (1=strongly disagree to
5=strongly agree) with four subscales: enjoyment, selfconfidence, value, and motivation. For
this study, the ATMI had a Cronbach’s α of 0.978.
Pre–Post Comparisons
ATMI scores for students in the treatment classes (T
Total
) are shown in Table 2(Appendix
A). Irvine (2020) reports the intervention’s pre−post results as follows:
For T
Total
(treatment students) a statistically significant medium effect size of 0.32 was
found (M=0.270, SD=0.0870, t(45)=3.110, p=0.003) .Among the subscales, the only
statistically significant increase was for the selfconfidence subscale. [Seventysix
percent] of students in T
Total
showed a positive increase in their attitudes towards
mathematics (p.23).
TreatmentControl Comparisons
Irvine (2020) found only one significant change for the control class with respect to
attitudes.
For the control class only the selfconfidence subscale (M=3.38, SD=0.660, t(21)=2.608,
p=0.016) was significant, and found a negative change in attitudes toward mathematics
[Table 2, Appendix A]. Prior to the MNT intervention, no statistically significant
differences in attitudes were found between the control class and T
Total
(M=0.007,
SD=0.192, t(66)=0.037, p=0.970). After the MNT intervention, T
Total
had a
statistically significant increase in attitude scores compared to the control class
(M=0.381, SD=0.1372, t(66)=2.781, p=0.007). (p. 26)
Qualitative Results
A convenience sample of five volunteer students were interviewed after completion of
the intervention. The interviews were recorded and the transcribed interviews were analyzed
using content analysis (Krippendorff, 2013) as well as constructivist grounded theory
(Charmaz, 2014). A sample of five is insufficient to formulate theories; however, the student
comments supported the quantitative results. Students indicated that the classroom activities
were enjoyable and interesting, and that the students were more engaged in their own
learning compared to the regular classroom instructional strategies, which typically consisted
of traditional, teachercentred lessons.
Ethical Considerations
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 13
All participants in this study received information letters and returned signed consent
forms prior to the commencement of the study. Students who volunteered to be interviewed
received and returned an additional, separate consent form prior to the interviews taking place.
The university Research Ethics Board approved this study (file #17096).
DISCUSSION
This study used MNT as its framework, which integrates the affective (self) system, the
metacognitive system, and the cognitive system into a coherent whole (Figure 1, Appendix A).
This differs from other taxonomies that typically address only one system. For example, RBT
(Anderson & Krathwohl, 2001) addresses only the cognitive system and relegates metacognition
to a passive information role. Further, Marzano postulates a hierarchical integration of self,
metacognitive, and cognitive systems (Figure 3, Appendix A) that emphasizes the sequential
nature of system engagement, with primacy being given to the self system, which encompasses
student motivation. This is followed by engagement of the metacognitive system, an active
system involving goal setting, planning, and monitoring. Finally, the cognitive system engages to
address and resolve the task. The study described in this paper demonstrates that MNT is a viable
framework for studies involving motivation (self system) and metacognition. While gains in
engagement and attitude were observed, the structure of the intervention did not specifically follow
Marzano’s sequencing of self, metacognitive, and then cognitive systems, since each lesson
included both self and metacognitive dimensions. However, the efficacy of such instructional
features mitigated the potential to modify student affective dimensions in a positive way.
The MNT framework has the potential to enrich practice in a number of areas. One of the
major implications for practice is to raise awareness of the linkages among the three systems of
the MNT framework: self (motivation), metacognition, and cognition.
Schools and Teachers
For current mathematics teachers, the framework provides a template to develop units or
subunits of mathematics content that provide a specific focus on one or more systems, particularly
student motivation and metacognition. Through teachers’ awareness of the importance of these
dimensions over and above the mathematics content, a more studentfocused and studentengaged
classroom climate will develop (see, for example, Irvine, in pressa, in pressb). Inservice
professional learning opportunities need to be provided for practicing teachers to become aware
of the MNT framework and its implications.
An additional constraint is that bridging the theorytopractice gap has frequently been
problematic (e.g., Nuthall, 2004). This can be attributed to a number of factors, including time to
learn and implement the innovation, ease of implementation, and clear and direct relationships
between theory and practice (FarleyRipple et al., 2018). Frequently, workplace socialization and
school culture mitigate against successful implementation (Allen, 2009; Lattimer, 2015). Yet,
“educational research will not have any practical value if it does not affect teaching and learning
in classrooms, no matter how brilliant the design or how magnificent the result” (Wang et al., 2010,
p. 105). By providing teachers with a complete unit instructional intervention, including classroom
activities and lesson plans, and by giving teachers “ondemand” professional learning and support
when requested, this study mitigates these traditional barriers to theorypractice implementation.
Journal of Instructional Pedagogies Volume 24
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As Irvine (2020) points out, this study’s MNT instructional intervention presents a
practical model for educational programs seeking to effect changes in student attitudes or
engagement, as well as a framework to develop comparable learning units and/or activities.
Mathematics teachers may also adopt this study’s instructional intervention to develop similar
approaches for other units in Grade 10 Academic Mathematics (i.e., trigonometry, analytic
geometry, linear systems). The MNT framework could be adopted to plan an entire mathematics
course, on a trial basis, and the outcomes could be investigated further (Irvine, 2020).
Teacher Educators
Teacher educators would benefit from knowledge of the MNT framework and its
relationships to higher order thinking skills (HOTS) and deep learning. With respect to MNT
(Figure 2, Appendix A), HOTS include all the sublevels of the metacognitive system, all
sublevels in the cognitive domain of knowledge utilization, and the sublevels “generalizing” and
“specifying” of the cognitive domain of analysis. The sublevel “specifying” refers to predicting
and may include formulating a hypothesis. Formulating hypotheses will also fall into the
knowledge utilization categories of experimenting and investigating. Lower order thinking skills
would consist of the lower two levels of MNT and the sublevels of analysis not noted above.
As well, MNT makes explicit the roles of student motivation and metacognition in
learning. These concepts could then be included in the curricula for preservice teachers of
mathematics. Since there is now a significant body of research on student attitudes in
mathematics (e.g., Pepin & RoeskenWinter, 2015), the MNT framework provides a structure for
introducing these concepts into preservice courses, as well as a viable framework for lesson
planning with an emphasis on one or more MNT systems.
Educational Researchers
For educational researchers the MNT framework provides a structure for the construction
of studies in one or more of the dimensions of the framework. The study described in this paper
illustrates the utility of the MNT framework for investigating affective dimensions. As a first
step, a complete Grade 10 Academic Mathematics course could be developed and implemented
for a full semester. In doing so, a large body of exemplar materials would be available, and
interactions among variables could be investigated.
The framework would also be useful in structuring studies on student cognition in
mathematics or in other subject areas, as well as multisystem studies linking two or more MNT
systems. The knowledge utilization level of the cognitive system in MNT (problem solving,
investigating, decisionmaking, experimenting) has particularly rich potential for research
studies. Having access to a rich and welldeveloped framework provides researchers with a
structure that is understandable to the participants in a study and may be more easily
communicable to any nonresearchers involved.
CONCLUDING REMARKS
The MNT framework is external to the students’ locus of control, providing a framework
for teachers and educators to develop instructional strategies to positively influence student
behaviours. In 2010, Clarkson et al. proposed the concept of mathematical wellbeing (MWB),
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 15
that provides a fivestage taxonomy based on an internal conception of students’ locus of control:
1. Being aware of and accepting mathematical activity;
2. Responding positively to mathematical activity;
3. Valuing mathematical activity;
4. Having an integrated and conscious value structure for mathematics; and
5. Being independently competent and competent in mathematical activity. (p. 117)
Each level of Clarkson et al.’s taxonomy describes student behaviours and motivation towards
mathematical activity that delineate changes occurring in student beliefs (as indicated by student
behaviours) towards the utility and value of mathematical activities. MWB provides an
enlightening differentiation among the five levels of students’ mathematical beliefs. However,
MWB, in its current form, is not an effective framework for developing instructional strategies to
support students’ progression among the levels. Indeed, Clarkson et al. cite the need for
developing and examining effective instructional techniques in their summary of future research
required to further develop the MWB construct and move it from theory to practice.
The framework provided by MNT is demonstrably useful for structuring research
initiatives. It is also valuable for enriching the knowledge base of inservice teachers, preservice
teachers, and teacher educators. While MNT has been extant for 20 years, the dearth of studies
utilizing this framework is surprising. This is particularly true in the area of student affect, which
is a burgeoning area of research, especially in the field of mathematics (Hannula, 2015; Pepin &
RoeskenWinter, 2015; Schoenfeld, 2015). However, the MNT framework is also a valuable
theoretical framework for studies beyond student motivation and affect, as well as studies
examining the linkages among motivation, metacognition, and cognition.
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APPENDIX A: FIGURES/TABLES
Figure 1
Marzano’s New Taxonomy of Educational Objectives
Note. From Marzano, R., & Kendall, J. (2007). The new taxonomy of educational objectives (2nd
ed.). Corwin Press. Reproduced with publisher’s permission.
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 21
Figure 2
Marzano’s New Taxonomy Showing Sublevels
Note. From Marzano, R., & Kendall, J. (2007). The new taxonomy of educational objectives (2nd
ed.). Corwin Press. Reproduced with publisher’s permission.
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 22
Figure 3
Flow of Processing in Marzano’s New Taxonomy
Note. From Marzano, R., & Kendall, J. (2007). The new taxonomy of educational objectives (2nd
ed.). Corwin Press. Reproduced with publisher’s permission.
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 23
Figure 4
Affective Dimensions Addressed in Marzano’s New Taxonomy Self and Metacognitive Systems
Note. From Irvine, J. (2020). Positively influencing student engagement and attitude in
mathematics through an instructional intervention using reform mathematics principles. Journal
of Education and Learning, 9(2), 48−75. Reproduced with permission.
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 24
Table 1
Dimensions of Student Engagement Scores PostIntervention
Treatment Control
n Sig. Cohen d n Sig.
Engagement (full scale) 46 <0.001*** 0.54 22 0.284
Emotional 46 <0.001*** 0.65 22 0.392
Behavioral 46 0.006** 0.38 22 0.708
Agentic 46 <0.001*** 0.73 22 0.069
Cognitive 46 0.005** 0.31 22 0.329
Note. **significant at p=0.01; ***significant at p=0.001.
Table 2
Attitudes Towards Mathematics Inventory Scores PostIntervention
Treatment Control
Category n Sig. Cohen d n Sig.
Attitude (full scale) 46 0.003** 0.32 22 0.472
Value 46 0.123 − 22 #
Enjoyment 46 0.220 − 22 0.563
Motivation 46 0.358 − 22 0.329
SelfConfidence 46 0.003** 0.33 22 0.016*
Note. **significant at p=0.01; *significant at p=0.05; # t and significance cannot be computed
since mean difference is 0.
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 25
APPENDIX B: LINKING CLASSROOM INTERVENTION TO MARZANO’S NEW
TAXONOMY
Additions to base problems unless indicated as replacement (R)
Expectations
Learning Goals
Metacognition
Focus
Self Focus
–
determine, through
investigation with
and without the use of
technology, that a
quadratic relation of the form
y = ax
2
+ bx + c (a>0) can be
graphically
represented as a parabola, and
that the table
of values yields a constant
second difference
(Sample problem: Graph the
relation
y = x
2
– 4x by developing a
table of
values and plotting points.
Observe the
shape of the graph. Calculate
first and
second differences. Repeat for
different
quadratic relations. Describe
your observations
and make conclusions, using
the appropriate terminology.);
– identify the key features of a
graph of a
parabola (i.e., the equation of
the axis of
symmetry, the coordinates of
the vertex,
the yintercept, the zeros, and
the maximum
or minimum value), and use
the appropriate terminology to
describe them;
*Students will learn
the basic properties
of parabolas and be
able to describe
these properties
using appropriate
mathematical
language
*Students will learn
how to apply
quadratic
regressions to data
sets
*Students will learn
how to use finite
differences to
determine
equations of
quadratic functions
Minds On
Carousel
• crocodile river
• handshake
problem
• pizza cuts
• logpile
• Anticipation
Guide
• Likert scale: interest
• Groups
• Placemat: Tell me
everything you know
about linear relations
Action
Whole class
• Use the method
of finite
differences to find
equations for
each pattern
= +
(linear)
=
+
+
(quadratic)
•
• Think Aloud
• What do we
want to
know; what
do we know;
how can we
connect these
• Likert scale:
importance
Consolidate/D
ebrief
Homework:
Parabolas in
Real Life
• Extend the
pattern to
negative x's using
your equations
• Terminology
(vertex, max/min,
axis of symmetry,
intercepts,
domain, range)
• Journal entry
• How well was
your plan
achieved? Did
it require any
modifications
?
• (R) Connecting Cube
Quadratics
• Homework Crossword
puzzle terminology +
Parabolas in Real Life
–
collect data that can be
represented as a
quadratic relation, from
experiments using
appropriate equipment and
technology (e.g., concrete
materials, scientific probes,
graphing calculators), or from
secondary
sources (e.g., the Internet,
Statistics
Canada); graph the data and
draw a curve
*Students will learn
how to collect and
model data that can
be represented by a
quadratic relation
Minds On
Groups
Use technology to
graph an example
from Curve Fitting and
discuss appropriate
models
• Pairs
• What/So
What plan
solution
method
• Graphic organizer
• Motivation
Action
Groups
Apply quadratic
regressions to obtain
equations for data
given in Curve Fitting
• Groups
• What/So
What revisit
• Journal entry: How
confident are you that
you can solve
problems involving
quadratic relations
• Choice
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 26
of best fit, if appropriate, with
or without
the use of technology (Sample
problem:
Make a 1 m ramp that makes a
15° angle
with the floor. Place a can 30
cm up the
ramp. Record the time it takes
for the can
to roll to the bottom. Repeat
by placing
the can 40 cm, 50 cm, and 60
cm up the
ramp, and so on. Graph the
data and draw
the curve of best fit.);
Consolidate/D
ebrief
Groups
Debrief Parabolas in
Real Life
• Journal entry:
• How well was
your plan
achieved? Did
it require any
modifications
?
• Emoji scales:
• overall motivation
• efficacy
• interest
• importance
–
identify, through
investigation using technology,
the effect on the graph of y = x
2
of transformations (i.e.,
translations, reflections
in the xaxis, vertical stretches
or
compressions) by considering
separately
each parameter a, h, and k
[i.e., investigate
the effect on the graph of y = x
2
of a, h,
and k in y = x
2
+ k, y = (x – h)
2
,
and
y = ax
2
];
*Students will learn
the effect on the
graph of a
quadratic function
of modifying a
parameter in
= ( −
ℎ)
+ k
,
Minds On
Jigsaw
Use technology to
investigate the effect
of various values of
parameters
•
=
•
= −
•
=
+
q
•
=
(
−
)
• What/So
What
• Why does
each
parameter
change result
in the
transformatio
n of the
graph
• Choice
• Choose group for
jigsaw
Action
Whole class
Practice with
various
=
( − )
+
q
• Use strategy
of example,
thinkpair
share
discussion,
worked
questions,
then repeat
• On a scale of 1 to 10,
identify how well you
understand the
impact of changing
parameters
Consolidate/D
ebrief
Whole class
Summarize
transformations
Individual
Journal entry:
summarize the
transformations
of
= ( −
)
+ q
and
the impact of
parameters
• Journal entry
• Given a
specific
= ( −
ℎ)
+ k
describe the steps
you would take to
graph it
• (R) Quadratic Aerobics

explain the roles of a,h, and k
in
=
(
−
ℎ
)
+
k
, using
appropriate terminology to
describe the transformations,
and identify the vertex and axis
of symmetry
sketch, by hand, the graph of
= ( − ℎ)
+ k
by
applying transformations to
the graph of
=
[Sample problem: Sketch the
graph of
=
−
(
−
3
)
+
4
, and verify using technology
*Students will learn
how to sketch and
connect graphs and
equations
= ( −
ℎ)
+ k
,
using appropriate
mathematical
terminology
Minds On
Groups
Matching graphs
and equations
• Groups
• Placemat
• Sketch graphs
from given
equations
and verify
accuracy with
technology
• Snowball PMI
• Role of a, h, k in
=
( − ℎ)
+ k
Action
Individual
Sketch graphs
for various
values of
parameters
Groups
Matching
graphs and
equations
• Verify using
technology
• (R) inside/outside
circle: generate
equation and explain
impact of parameters
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 27

determine the equation in the
form
=
(
−
ℎ
)
+
k
of a
given graph of a parabola
=
(
−
)
+
q
Consolidate/D
ebrief
Pairs
ThinkPairShare to
construct questions
matching graphs,
equations, and
information (domain,
range, intercepts,
vertex, axis of
symmetry)
Inside/Outside Circle
to share with others
• Groups
• What/So
What
• Effect of
various
parameter
changes, how
to recognize
them, how to
verify them
• Likert scale: interest
–
expand and simplify second

degree polynomial
expressions [e.g., (2x + 5)
2
,
(2x – y)(x + 3y)], using a variety
of tools
(e.g., algebra tiles, diagrams,
computer
algebra systems, paper and
pencil) and
strategies (e.g., patterning);
*Students will learn
how to expand and
simplify second
degree expressions,
with and without
manipulatives
Minds On
Groups
Use algebra tiles for
some basic expansions
• Pairs
• Order algebra
tile pieces to
show
expansion
and vice
versa
• Groups
• Discussion
• Why is this/might this
be important to me?
Action
Whole Class
Algebraic expansions
Student practice
•
Groups
• Graffiti
• Step by step
expansion
using algebra
tiles, then
algebraic
expansions
•
Journal entry
• How useful is this to
me?
Consolidate/D
ebrief
Individual
Inside/outside circle:
Student generated
examples of
expansions
Journal entry: Create
an example of each
type of expansion
• Journal entry
• Give
examples of
expansions in
both
directions
with and
without
algebra tiles
• Journal entry: my
favourite expansion
and why
–
factor polynomial
expressions involving
common factors, trinomials,
and differences
of squares [e.g., 2x
2
+ 4x,
2x – 2y + ax – ay, x
2
– x – 6,
2a
2
+ 11a + 5, 4x
2
– 25], using a
variety
of tools (e.g., concrete
materials, computer
algebra systems, paper and
pencil) and
strategies (e.g., patterning);
*Students will learn
how to factor
polynomial
expressions
*Students will learn
how to recognize
and factor special
cases
Minds On
Groups
1)Use algebra tiles for
simple factoring
Using Algebra Tiles
2) Whole class:
Construct a decision
tree for factoring
• 1)Verify
factorizations
by expanding
• 2) Matching
steps for an
example
• 1) Likert scale How fun
is algebraic
manipulation
• 2) Graphic organizer
• Emotions
Action
Whole class
1)Algebraic treatment
of trinomials, perfect
squares, difference of
squares
2)Jigsaw practice
• 1)
Recognition
• What type of
factoring is it
• 2) Pairs
• Timed retell
• Given a card
with a
factorable
expression on
it, explain
how to factor
• 1) Groups
• cartoon placemat
• different groups get
different types of
factoring
• 2) Four corners
• Different types of
factoring at each
corner (multiple
questions on same
type)
Consolidate/D
ebrief
1) Individual practice
Journal Entry: Explain
the relationship
between expanding
and factoring
• 1) Pairs
• One partner
factors, the
other partner
• 1)Emoji scales:
• overall motivation
• efficacy
• interest
• importance
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 28
2) Groups: Write a
script to explain to a
classmate how to
factor (various
expressions)
veri
fies by
expanding
• 2) Game of
Facto
• 2) Journal entry
• How confident are you
that given an
expression to factor,
you can factor it and
verify your answer
–
express
y = ax
2
+ bx + c in the form
y = a(x – h)
2
+ k by completing
the
square in situations involving
no fractions,
using a variety of tools (e.g.
concrete
materials, diagrams, paper and
pencil);
*Students will learn
how to complete
the square, with
and without
manipulatives
Minds On
Groups
Use algebra tiles to
complete Make a
Square
• Think Aloud
• What do we
know, what
do we want
to know, how
are they
related
• Graphic organizer
• Emotions
Action
Whole class
Algebraic complete
the square examples
• Pairs
• Matching
steps for a
numerical
example
• Groups
• Choice apply
completing the square
to various expressions
Consolidate/D
ebrief
Individual
Practice completing
the square
Groups
Think Aloud: What
information can we
obtain by completing
the square
• Ticket to
leave
• Given a
numerical
example,
outline the
steps in
completing
the square
• Ticket to leave
• Choose one of three
expressions and
complete the square
–
determine, through
investigation, and
describe the connection
between the
factors of a quadratic
expression and the
xintercepts (i.e., the zeros) of
the graph
of the corresponding quadratic
relation,
expressed in the form y = a(x –
r)(x – s);
*Students will learn
how to determine
the zeros of a
quadratic relation
and connect them
to xintercepts and
equations
expressed in the
form y = a(x – r)(x –
s);
Minds On
Groups
Matching zeros from
graphs with zeros
from algebra
• Groups
• Outline a plan
to convert to
y = a(x – r)(x –
s)
• Likert scale
• How confident are you
that you can convert
among forms
Action
Whole class
Algebraic intercepts
by factoring
Intercepts using
technology
• Pairs
• Matching
graphs and
equations
• Likert scale
• How interesting do
you find these
conversions
Consolidate/D
ebrief
Individual
Practice finding
intercepts
algebraically and
writing quadratic
functions in the form y
= a(x – r)(x – s); Ticket
to leave: Given values
in
= ( − )
+ q
Rewrite in form
y = a(x – r)(x – s);
•
Journal entry
• How can you
be confident
that you
converted
correctly
• Likert scale
• How important do you
think these
conversions are to you
–
determine the zeros and the
maximum or
minimum value of a quadratic
relation
from its graph (i.e., using
graphing calculators
or graphing software) or from
its
*Students will learn
how to determine
features of a
quadratic relation
(xintercepts,
maximum/minimu
m) from its graph
and from its
equation
Minds On
Groups
Michaela problem
Watch Detroit Airport
video 1
Brainstorm some
questions that you
might ask about the
• Groups
• Graphic
organizer
• Complete the
Polya
organizer
• Graphic organizer
• Right angles for
interest, efficacy,
importance
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 29
defining equation (i.e., by
applying algebraic
techniques);
*Students will learn
how to connect
algebraic and
graphical
techniques to real
life situations and
identify restrictions
fountains, and what
information you
would need to answer
them
Then watch video #2
Action
Groups
Solve real world
problems using a
variety of techniques
• Groups
• Solve,
referring to
Polya
organizer
• Use computer
software or graphing
calculators to solve
problems
Consolidate/D
ebrief
Homework
Whole class
Polya plan
Identify restrictions
based on real life
situation
Watch Detroit Airport
video #3
• Ticket to
leave
• Summarize
plan,
modification,
restrictions,
how to
recognize
• Graphic organizer
• motivation
–
explore the algebraic
development of the
quadratic formula (e.g., given
the algebraic
development, connect the
steps to a
numerical example; follow a
demonstration
of the algebraic development
[student
reproduction of the
development of the
general case is not required]);
*Students will learn
how to develop the
quadratic formula
and apply it ti find
zeros of functions
and xintercepts of
quadratic relations
Minds On
Whole
class
Sample algebraic
solution by factoring
• Groups
• Graph using
technology,
estimate
zeros
• How
confident,
accurate are
zeros
• Groups
• Graph using
technology and
estimate zeros
• Discussion
• How confident are you
that the zeros are
correct and accurate
Action
Whole class
Algebraic
development of
quadratic formula
with values for a,b,c
Algebraic
development of
quadratic formula
with a,b,c
Worked examples
• What/So
what
• Relate
algebraic
steps to
numerical
example
• Likert scale
• importance
Consolidate/D
ebrief
Individual practice
• Timed retell
• Explain the
steps for a
numerical
example
• Graphic organizer
• Emotions
–
solve problems arising from a
realistic situation
represented by a graph or an
equation
of a quadratic relation, with
and
without the use of technology
(e.g., given
the graph or the equation of a
quadratic
relation representing the
height of a ball
over elapsed time, answer
questions such
as the following: What is the
maximum
height of the ball? After what
length of
time will the ball hit the
ground? Over
*Students will learn
how to model real
life situations using
quadratic functions
*Students will learn
how to solve
quadratic models to
answer real life
questions
Minds On
Groups
Dan Meyer basketball
video
• Selfselect
jigsaw
• Plan solution
• Open problems
• Choice
Action
Whole class
Problems worked
examples
• Execute plan
• Gallery walk
• Likert scale
• Confidence
Consolidate/D
ebrief
Groups
getthemath.org
basketball problem
• Ticket to
leave
• Explain plan
and
execution,
restrictions
• Likert scale
• Importance
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 30
what time interval is the height
of the ball
greater than 3 m?).
–
interpret real and non

real
roots of quadratic
equations, through
investigation
using graphing technology, and
relate the
roots to the xintercepts of the
corresponding
relations;
– sketch or graph a quadratic
relation whose
equation is given in the form
y = ax
2
+ bx + c, using a variety
of
methods (e.g., sketching y = x
2
– 2x – 8
using intercepts and symmetry;
sketching
y = 3x
2
– 12x + 1 by completing
the
square and applying
transformations;
graphing h = –4.9t
2
+ 50t + 1.5
using
technology);
*Students will learn
how to interpret
real and nonreal
roots of quadratic
equations
*Students will learn
how to graph
quadratic relations
using a variety of
methods
Minds On
Jigsaw
Graph various
quadratics with real
integer, real decimal,
nonreal roots using
technology
• Anticipation
guide v2
• Emoji scales:
• overall motivation
• efficacy
• interest
• importance
Action
Groups
Find the zeros
algebraically or
explain why this is not
possible
• Groups
• Relate roots
to graphs and
identify
patterns
• Likert scale
• interest
Consolidate/D
ebrief
Groups
Sketch graphs using a
variety of techniques
(complete the square;
factor to find roots;
use technology to
graph to find roots;
table of values
Gallery Walk to share
solutions
• Journal entry
• How can you
tell how
many real
roots a
quadratic
equation will
have?
• Graphic organizer
• motivation
–
solve quadratic equations
that have real roots, using a
variety of methods (i.e.,
factoring, using the quadratic
formula,
graphing) (Sample problem:
Solve
x
2
+ 10x + 16 = 0 by factoring,
and
verify algebraically. Solve x
2
+ x
– 4 = 0
using the quadratic formula,
and verify
graphically using technology.
Solve
–4.9t
2
+ 50t + 1.5 = 0 by
graphing
h = –4.9t
2
+ 50t + 1.5 using
technology.).
*Students will learn
how to solve
quadratic equations
that have real
roots, using a
variety of methods
Minds On
Groups
Build a box
• Four corners
• Choose
solution
method
• Groups
• Discussion:
• Importance
• Efficacy
• Interest
• motivation
Action
Whole Class
Worked examples
• Groups
• Solve a
problem by at
least two
different
methods
• Likert scale
• efficacy
Consolidate/D
ebrief
Groups
Given a problem
solving flowchart,
identify the various
features and then
apply to problems
Problem Solving
Flowchart v2
• Timed retell
• Explain at
least one
method to
partner
• Ticket to leave
• Choose one problem
and present solution
–
compare, through
investigation using technology,
the features of the graph of y =
x
2
and the graph of y = 2
x
, and
determine
the meaning of a negative
exponent and
of zero as an exponent (e.g., by
examining
patterns in a table of values for
y = 2
x
; by
applying the exponent rules for
multiplication
and division).
*Students will learn
to interpret the
meaning of
exponents of 0 and
exponents of a
negative integer
*Students will learn
how to extend the
exponent rules to
exponents of 0 or a
negative integer
Minds On
Groups plac
emat
Compare the graphs
of
=
and
=
2
• domain
• range
• intercepts
• max/min
• Anticipation
guide v3
• (R) Groups
• Money Maker
Action
Whole class
Use Table feature of
graphing calculator to
develop values for
• Matching
• Information
to y = x
2
or y
= 2
x
• Groups
• Exponent Facto
Journal of Instructional Pedagogies Volume 24
Marzano’s New Taxonomy, Page 31
exponents of 0 or
negative integers
Groups
Practice evaluating,
exponent laws
Consolidate/D
ebrief
Individual
Journal entry: Give
several examples of
evaluating powers
with integer
exponents and the
exponent laws
• Inside/outsid
e circle
• Information
to y = x
2
or y
= 2
x
• What/So What
• List some examples
and worked solutions
Review
Minds On
Groups
Construct a summary
page for quadratic
functions
Action
Inside/Outside Circle
Use ThinkPairShare
to each construct and
confirm 3 questions
involving quadratic
functions
Use Inside/Outside
Circle to share with
classmates
• • These periods can be
inserted as needed for
consolidation, skill
building, formative
assessment. They do
not have to be used as
full classes, but a total
of 75x2=150 minutes
may be used in whole
or in part.
Consolidate/D
ebrief
Groups
Solve max/min
problems and
quadratic equation
problems
• •
Consolidate periods (2)
Recommended: use pairs and groups:
gallery walks, jigsaw, inside/outside circle,
carousel, thinkpairshare, create
questions, open questions
• These periods can be inserted as needed for
consolidation, skill building, formative
assessment. They do not have to be used as
full classes, but a total of 75x2=150 minutes
may be used in whole or in part.
RAT
Groups
The painted cube problem
Groups
The painted cube problem v3
Test
Total 26 classes
• Could include a mixed practice day prior to
review/test