CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

Professional Development for the Implementation of High-Level Cognitive Demand

Tasks

A graduate project submitted in partial fulfillment of the requirements

For the degree of Master of Arts in Education, Elementary Education

By

Sarah Gaudet

December 2018

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The graduate project of Sarah Gaudet is approved:

_____________________________________ __________________ Rashanda Zakem Date _____________________________________ __________________ Dr. Minsung Kwon Date _____________________________________ __________________ Dr. Joyce H Burstein, Chair Date

California State University, Northridge

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Table of Contents

Signature Page ii List of Figures iv Abstract v Chapter 1: Introduction 1 Chapter 2: Review of Literature 6 State-Adopted Standards, Assessments, and Tasks 6 Mathematical Task Framework 8 Three Phases of Mathematical Tasks 10 Factors that Contribute to Maintain or Decline Cognitive Demand of Tasks 11 Effect of High-Level Cognitive Demand Tasks on Student Learning 13 Characteristics of Effective Professional Development 14 Chapter 3: Methods 19 Facilitator 19 School Context 21 Mathematical Textbook in Hope Elementary 22 Participants 27 Professional Development 28 Chapter 4: High-Level Cognitive Demand Task: Professional Development 31 Session One: Introduction to High-Level Cognitive Demand Tasks 31 Session Two: Establishing Norms for Implementing High-Level Cognitive

Demand Tasks 36 Session Three: Enriching the Curriculum with High-Level Cognitive Demand Tasks 42 Session Four: Observation of Implementing a High-Level Cognitive Demand Task in a Fifth Grade Classroom and Analysis of Student Work Samples 46 Session Five: Analyzing High-Level Cognitive Demand Task Work Samples 50 Session Six: Creating and Analyzing High-Level Cognitive Demand Tasks 52

Chapter 5: Conclusion 56 Summary 56 What I Learned 58 Limitations 59 Future Research 59 References 60 Appendix A: High-level Cognitive Demand Tasks 62 Appendix B: PowerPoint Presentations 63 Appendix C: Teacher Pre and Post Survey 77 Appendix D: Mathematical Task Analysis Guide 78

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List of Figures

Figure Page

2.1 Shading of Tens and Hundreds 9

2.2 Lunch Menu 9

2.3 Mathematical Tasks Phases 10

2.4 The Decision-Making Process for Presenting Professional Development

Curriculum 16

3.1 High-Level Cognitive Demand Task 24

4.1 Mathematical Practice Standards 33

4.2 Help George 36

4.3 Math Discussion Expectation Anchor Chart 39

4.4 Lion Hunt 47

4.5 Analyzing Student Work Sample Questions 49

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Abstract

Professional Development for the Implementation of High-Level Cognitive Demand

Tasks

By

Sarah Gaudet

Master of Arts in Education, Elementary Education

This graduate project focuses on the importance of providing a professional

development for teachers on how to effectively implement high-level cognitive demand

tasks in K to 5 classrooms. It was found through the review of literature, students are in

need of additional support in achieving the rigorous demand of the Common Core

Standards Mathematics and Mathematical Practice to meet the expectations on the

California Assessment of Student Performance and Progress at the end of the school year.

The results also revealed that the use of high-level cognitive demand tasks had a positive

impact on student learning in mathematics.

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Chapter 1

Introduction

Professional development is vital to improve the quality of teaching and learning.

To increase student learning teachers need to build their mathematical knowledge for

teaching and learn effective teaching practices. In the cross-comparative study between

U.S. and Chinese elementary school teachers, Ma (1999) found that U.S. teachers have

lack of Profound Understanding of Fundamental Mathematics (PUFM). Also, teachers

need to stay up-to-date with current educational policies and initiatives. To meet the

expectations of state-adopted standards and associated assessment systems, teachers,

schools, and districts, have the pressure to improve the quality of teaching (Borko,

Jacobs, Koellner, & Swackhamer, 2015). With the change in the Common Core State

Standards teachers do not feel that they are properly trained in meeting the advancement

and style of teaching to these new standards. This is why teachers need professional

development workshops that provide the materials and tools to help evolve teachers’

instruction in the classroom.

To achieve the ultimate goal of increasing student learning, mathematics

professional development should build teachers’ professional growth in four areas

(National Council of Teachers of Mathematics, 2010): (1) building teachers’

mathematical knowledge and how to use it in practice; (2) build teachers’ understanding

to see, analyze, and answer to students’ thinking; (3) build teachers’ useful practices; and

(4) build relationships and structures that encourage prolonged education. The first

priority in mathematics professional is to add to teachers' knowledge of the content to

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ensure they are providing the students with the best instruction that will then build the

students’ understanding of mathematics. When the teachers have the content knowledge

they can then analyze and better understanding students’ thinking in mathematics.

To meet the demands in evolving education the Los Angeles Unified School

District (LAUSD) provides teachers with a variety of professional development programs

to enhance their teaching skills and to encourage professional growth. My Professional

Learning Network at LAUSD1 provides 67 professional development opportunities for

teachers to voluntarily take. For example, Real and Relevant Mathematics:

Communicating the Purpose of the Lesson gives different strategies on how to effectively

communicate the purpose of the mathematics daily lessons. Another professional

development opportunity is on the California Mathematics Framework to gain a better

understanding for their grade-levels and across grade-levels standards and how to connect

them into their instruction to meet the shift in rigor. In addition, the district is providing

professional development on Cognitively Guided Instruction (CGI). The CGI

professional development helps teachers improve students’ achievement in mathematics

by encouraging teachers to use math problems that allow students to makes sense of the

problems and solve them in meaningful ways.

With all of the professional development programs provided by LAUSD there is

still a need for professional development in high-level cognitive demand tasks.

Henningsen and Stein (1997) argue that “mathematical tasks are central to students

learning” (p. 525). High-level cognitive demand tasks require students to make sense of

the tasks and frequently use several ways to discover the answers to the tasks. In addition,

1 https://lausd.csod.com/GlobalSearch/Search.aspx#q=&s=1&a=

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Henningsen and Stein (1997) suggest that high-level cognitive demand tasks engage

students in learning and allow students to explore different cognitive demands, such as

thinking abstractly and making connections to mathematical concepts, thus allowing the

students to actively participate in longer and more complex activities.

The curriculum adopted by LAUSD, MyMath, is not providing the students with

high-level cognitive demand tasks and therefore the students are not mastering or meeting

standard expectations of the CCSSM and state assessments. In my Masters of Arts in

Education Program at California State University Northridge I took a Seminar in

Elementary School Mathematics Education. In this seminar course I critiqued the

mandated curriculum at my school site. The critique focused on whether or not the

curriculum provided students with high-level cognitive demand tasks and was beneficial

to students learning in mathematics. After performing a professional critique on the

adopted curriculum, MyMath is providing students low-level demand tasks and not

adequately building the students’ understanding of mathematics and this will potentially

affect their connections to mathematics and the real world.

According to the California Department of Education (2018) only 30 to 47

percent of all third, fourth, and fifth grade students in California met or exceeded the

standards on the end of year California Assessment of Student Progress and Performance.

Thus over half of these students have not met or are below the standards, needing

additional support in order to meet and/or exceed the standards. Furthermore, teachers

also need additional assistance on how to supplement the mandated curriculum in a more

creative and relative way so they can provide the students with more rigorous

mathematics instruction that will help students meet grade-level standards and

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expectations on the state testing. Teachers need professional development that provides

specific strategies and implementation tools that will support students’ learning outcomes

in mathematics.

The purpose of my graduate project is to design, facilitate, and implement a

professional development program on high-level cognitive demand tasks. The

professional development will provide opportunities for teachers to teach mathematics

effectively that will support all students in mathematics. The professional development

will be given in the hopes that teachers strengthen their understanding on the

mathematical task and how to assist their students in increasing the students’

understanding and performance in mathematics. The teachers will find different tools,

strategies, and knowledge that will supplement the weakness of the mathematics

curriculum. With the tools, strategies and knowledge teacher will increase students'

problem-solving skills.

This graduate project begins by reviewing literature that supports when high-level

cognitive demand tasks are implemented into mathematics instruction students show an

increase in academic achievement in mathematics. The research and acquired knowledge

on high-level cognitive demand tasks helped influence the creation of a professional

development that will support teachers in implementing high-level cognitive demand

tasks in mathematics instruction to fully prepare elementary school students for the

rigorous demand in the CCSSM and state assessments. In Chapter 3, I provide an

overview of the facilitator and knowledge on high-level cognitive demand tasks, explain

school context information, introduce the mathematic textbook used at the school of

which the professional development will be implemented, and information on the

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participants taking the professional development. The methods chapter also entails how

the professional development was created and the components that make up the

professional development such as the relative research collected on best practices in a

professional development to provide the teachers with an opportunity to build

professional growth and find the material useful for their teaching.

Chapter 4 includes the created professional development. The professional

development includes real student samples, classroom observations, and teacher planning

time to allow the teachers to find different ways to meet the needs of their students.

Chapter 5 provides a conclusion of the research and professional development created to

help improve students' achievement in mathematics through the instruction of high-level

cognitive demand tasks.

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Chapter 2

Review of Literature

This chapter provides a review of literature relevant to the effectiveness of

implementing high-level cognitive demand tasks in K to 5 classrooms. There is a large

need for professional development and support for implementing high-level cognitive

demand task in mathematics instruction and curriculum. I will start by providing

empirical evidence that supports the need for supplemental instruction tools that assist

students in meeting the rigorous demands of the state-adopted standards and end-of-year

testing. The mathematical task framework will be introduced and the features of different

levels of cognitive demand tasks will be defined. The previous research on implementing

high-level cognitive demand tasks guides the design of content-focused, sustained,

collective, and effective professional development.

State-Adopted Standards, Assessments, and Tasks

In August 2010, the California State Board of Education adopted the California

Common Core State Standards: Mathematics (CA CCSSM), published in January 2013.

This document is intended to prepare all students to graduate from high school and

prepare them academically for college and career endeavors (p. 2). The CA CCSSM

includes two categories of standards: Eight Mathematical Practice Standards and

Mathematical Content Standards. The handbook states, “the CA CCSSM call for

mathematical practices and mathematical content to be connected as students engage in

mathematical tasks” (p. 3). With the California Assessment of Student Progress and

Performance, which is a rigorous and computer adaptive state achievement test for grades

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3 to 8 and 11 and produced by the Smarter Balanced Assessment Consortium (SBAC),

teachers are now seeking different approaches in teaching mathematics that will meet the

needs of all students.

To meet the expectations of CA CCSSM, students need additional support in

high-level thinking, problem solving and communicating in mathematics. In order to

enhance students’ learning, Henningsen and Stein (1997) argue that “mathematical tasks

are central to students learning” (p. 525). Mathematical task is defined as a classroom

activity that is devoted to a mathematical idea and can involve several related problems

or extended work on a single complex problem (Stein & Smith, 1998). The task engages

students by using multiple techniques that allows the student to think about the math task

in a variety of ways. These types of tasks are called high-level cognitive demand tasks.

High-level cognitive demand tasks require students to make sense of the tasks and

frequently use several ways to discover the answers to the tasks. In addition, Henningsen

and Stein (1997) suggest that high-level cognitive demand tasks engage students in

learning and allow students to explore different cognitive demands, such as thinking

abstractly and making connections to mathematical concepts, thus allowing the students

to actively participate in longer and more complex activities. Teachers are searching for

ways to meet the rigorous demands of the new standards; Stein and Smith’s (1998)

Framework for High-Level Cognitive Demand Tasks provide this opportunity for

teachers.

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Mathematical Task Framework

The Mathematical Task Framework consists of two levels with two different tasks

in each level: “memorization” and “procedures without connections” as lower-level

demand tasks; “procedures with connections” and “doing mathematics” as high-level

demand tasks. Smith and Stein (1998) created a mathematical task analysis guide that

identifies features of each cognitive demand level.

The two low-level cognitive demand tasks in the Mathematical Task Framework

are “memorization” and “procedures without connections” tasks. Memorization tasks

include reproducing learned facts, rules, formulas, and definitions. The tasks are not

ambiguous and cannot be solved using procedures. The tasks have no connection to

concepts or meanings and are focused on success of the correct answer with no

explanation. A sample memorization task would be: What is the digit value of the

underlined digit in the number 0.98? The students are expected to respond, nine-tenths.

“Procedures without connections” use algorithms. The tasks have little ambiguity

and require limited cognitive demand for the competition. The tasks also have no

connection to the concepts or meaning to the procedure being used. Here is an example of

what a procedure without connections task looks like: How many times greater is 46 than

0.46? The students are expected to respond with 46/10= 4.6 4.6/10=0.46, it is 100 times

greater.

The two high-level cognitive demand tasks in the mathematical task framework

are “procedures with connections” and “doing mathematics” tasks. “Procedures with

connections” tasks require cognitive effort and understanding on the procedures used to

solve the problem. These tasks are usually represented in multiple ways, such as visuals,

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manipulatives, symbols, and problem situations. A sample procedure with connections

tasks would be: Write the decimal and fraction for each model shown in Figure 2.1. Are

the two models equal? Explain to a friend why. The students expected response would be

4/10= 0.4 and 40/100=0.40. Yes, the models are equal because each model has the

equivalent shaded area. Therefore, four tenths is equal to forty hundredths.

Figure 2.1 Shading of Tens and Hundreds (Stein & Smith 2011, p. 10)  

The last high-level cognitive demand task, doing mathematics, requires the

students to think complexly, explore, and understand the nature of the math concepts.

The students need to use their prior knowledge and experiences to analyze the tasks and

have rigor through problem solving. As shown in Figure 2.2, an example for doing

mathematics would be: You are going to the mall. At the mall, you will buy lunch but

you cannot buy more than one of the same item, and you cannot spend more than

$10.00. Using the menu to the left, find out how many different lunches you could buy

without receiving any change back.

Task Hot dog $1.75

Hamburger $2.25 Cheeseburger $2.75

Chips $0.75 Nachos $1.50

French Fries $0.75 Soda $1.50 Water $1.25

Figure 2.2 Lunch Menu (Orrill, 2015)

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Three Phases of Mathematical Tasks

Henningsen and Stein (1997) describe mathematical tasks in a framework that

passes through three phases as seen in Figure 2.3.    

Figure 2.3 Mathematical Task Phases (Stein & Smith, 2011, p.11)

It starts with the developing of the tasks as seen in the math curriculum and

instructional material. The next phase is how the teacher sets up the tasks. This phase is

the development of the task and how it will enhance students “mathematical

understanding, reasoning, and problem solving” (p. 528). The teacher must determine the

goals for the tasks and how the teacher will encourage the students throughout the solving

of the tasks. During this phase the teacher has set up the “classroom norms, task

conditions, and teacher and student dispositions” (p. 529). The final phase is student-

based. The students engage in the tasks and the cognitive process that leads to learning

outcomes. The students’ learning outcomes will reflect how the tasks are implemented, so

it is important that the teacher has selected an appropriate task and has introduced the

task to the students with the use of the established classroom norms to result in high

levels of student success.

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Factors that Contribute to Maintain or Decline Cognitive Demand of Tasks

Henningsen and Stein (1997) analyzed the data from 144 tasks used in

Quantitative Understanding: Amplifying Student Achievement and Reasoning

(QUASAR) projects in classrooms. Fifty-eight of the classrooms were recognized as

giving tasks that encourage high-level cognitive demand tasks in mathematics. The data

showed that the students were engaged in the tasks because the tasks built on their prior

knowledge. Moreover, the teacher scaffolded, modeled, and sustained the pressure for

explanation and meaning, as well as providing an appropriate amount of time to complete

the task.

There are many factors when implementing High-level Cognitive Demand Tasks.

Henningsen and Stein also found that classrooms which had lower-level cognitive

demand tasks was due to inappropriate tasks, lack of classroom management, or too

much or too little time given on the tasks, yielding results that focused more on finding

the correct answer. They found that there are many factors that can lead students to have

more engagement in high-level cognitive demand tasks. The factors include making the

tasks appropriate for the students so the students can make a connection to the tasks,

enough support from the teacher with scaffolding and pressing students for meaningful

explanations, as well as establishing effective classroom norms.

Henningsen and Stein (1997) explain that high-level cognitive demand tasks can

be too challenging for students and result in students eagerly awaiting the teacher’s

explanation of the task. When students feel that the task must be completed quickly, with

a correct answer, the high-level cognitive demand of the task can be diminished. Another

dynamic that can lead to unproductive task implementation is the absence of “students’

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prior knowledge, interest, and motivation” (p. 526). To encourage and motivate students,

classroom norms can be set up as guidelines for students when participating in these

tasks. Some of these norms include, “the physical organization of the classroom, the

allotted time on the tasks, transitions from various activities, establishment of

accountability, and how disciplines are handled” (p. 526).

Other norms in the classroom that can impact student learning is the physical

organization of the classroom and how it will promote student engagement. If the

students can communicate with their peers and feel comfortable in their environment

there will be a positive impact on student learning (Henningsen & Stein, 1997). When

the norms are established, the students know what is expected of them within the

academic setting. The students and the teachers are held accountable during the tasks

because the students understand how the tasks should be completed and to what degree of

quality. The teacher is held accountable by allowing the students to struggle during the

difficult tasks and provide support and guidance to help the students persevere through

the task. The allotted time on a task should be appropriate for that particular group of

students. The time given should not be rushed, allowing the students to wrestle with the

tasks and deepen their cognitive ability. The tasks should also not go too long. Too much

time on a task can decline the high-level cognitive ability of the students because the

students become discouraged when the task is too difficult. One sign that the task is too

difficult may be that the students cannot get past the struggle and want more support from

the teacher. The teacher needs to establish clear norms when it comes to expectations in

the classroom. Additionally, the teacher sets up rules and expectations when it comes to

transitions from tasks, subjects, recess, and lunch. The teacher also needs to have set

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classroom rules and expectations to hold the students accountable for their behaviors. The

students need to know there are consequences for inappropriate behaviors, allowing them

to keep behavior management to a minimum.

Moreover, Henningsen and Stein (1997) suggest that a positive correlation exists

between having a mathematical understanding and knowledge of the activities in which

students participate. This is essential to supporting the implementation of the high-level

cognitive demand tasks. With the implementation of the classroom norms and selection

of appropriate tasks that the students can engage in, the cognitive process during the tasks

can remain at a higher level (Henningsen & Stein, 1997). These researchers also state that

students cannot be expected to engage in this active process if they are not being

supported in their classroom environment or encouraged to think, reason, and problem

solve with high-level cognitive demand tasks.

In order to maintain a high-level of engagement, Smith and Stein (1998) found

that when students are introduced to tasks it is important that the tasks are grade-level

appropriate. Low-level tasks do not produce high-levels of engagement. Smith and Stein

also believed that high-level cognitive demand tasks have a positive effect on the day-to-

day development of students’ understanding on what it means to do mathematics.

Effect of High-Cognitive Demand Tasks on Student Learning

Ni, Zhou, Cai, Li, Li, and Sun (2017) examined the relationship amongst three

cognitive features of mathematical instruction tasks (high-level cognitive demand,

multiple representation, and multiple solution methods) and students learning outcomes

among 1,779 students from thirty Chinese fifth grade classrooms. One of the hypotheses

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of the study was to find out if “high-level cognitive demand tasks would have an impact

on computation fluency, routine problem solving, conceptual understanding, and complex

problem solving” (p. 5). Another hypothesis was that “instructional tasks of multiple

representation or those of multiple solution methods would positively predict routine

problem solving and complex problem solving” (p. 5). Thus, the study was designed to

observe both the cognitive and affective learning outcomes in relation to the three

cognitive qualities of instructional tasks.

As a result of their study, the “students’ achievement on the three cognitive

measures improved significantly from the pre-assessment to the post assessment” (p. 8).

The results supported the hypothesis that tasks of high-level cognitive demand positively

impacted student learning and interest level in mathematics. Therefore, students were

participating and collaborating with their peers on the tasks at hand.

In this first section of the review of literature I defined what high-level cognitive

demand tasks are, how to properly implement the tasks, and reviewed the research on

implementing high-level cognitive demand tasks and the effect of high-level cognitive

demand tasks on student learning outcomes.

Characteristics of Effective Professional Development

In this section, I will provide current research on effective professional

development and non-effective professional development modalities. Professional

development for teachers is essential to improving mathematics instruction (Desimone,

Garet, Birman, Porter, & Yoon, 2003). Guskey (2003) states effective professional

development is supporting teachers in having a deeper understanding of the content

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taught and how students learn from the content. He also explains that professional

development involves clear and well-organized time along with having the professional

development on-site. When teachers attend professional development, the teachers are

looking for an effective facilitator that delivers material in a variety of ways that includes

visuals, activities, conversations, tasks, and a variety of modalities.

The facilitator of the professional development needs to be engaging and involved

in the professional development. The teachers need to be in a comfortable environment

because most of the teachers will be attending the professional development after a long

teaching day. Another way to give a positive professional development to teachers is to

have the professional development focus on what teachers need in the classrooms

(Lemlech, 1995). Teachers want to feel as though they are being treated as professionals

and that their time is being honored. Lemlech (1995) also believes the activities in the

professional development need to be at an adult level. Teachers do not appreciate or

engage in activities that feel childlike.

  When receiving a professional development, teachers need to feel as though the

facilitator has empathy towards their feelings and situations. In an effort to be

collaborative, Lemlech (1995) found over the years that teachers will admit to the areas

of need, so it is important to converse with teachers to gain insights on what they are

looking for in a professional development. Teachers want the facilitators to be

knowledgeable in the subject area and be able to present the material in an efficient way

using best pedagogical practices. When the content introduces a new methodology, the

material should be presented through a presentation that is interactive. Lemlech (1995)

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displays a model that suggests what an active presentation should look like as seen in

Figure 2.4.    

Figure 2.4 The decision-making process for presenting professional development curriculum. (Lemlech, 1995, p.182)

The teacher should start with a verbal and experiential presentation. From there the

presentation should display actual student material and use adult level material with

student methodology. Moreover, it is important for the facilitator to start with a verbal

and experiential presentation.

The National Council of Teachers of Mathematics (2010) states that teachers can

acquire their mathematical content understanding in a variety of ways. Some of the ways

teachers can develop their content knowledge is by solving and discussing mathematics

problems, examining students’ mathematical thinking, collaborating with other teachers,

and analyzing the implementation in the classrooms during professional development

sessions.

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When teachers solve and discuss mathematic problems with other teachers, the

teachers can discuss solutions and methods which, in return, will deepen their

understanding of the mathematical content. Teachers can also deepen their understanding

by noticing and analyzing student thinking during mathematics and problem solving.

Professional development can assist teachers in understanding how students think in

mathematics. Teachers can use the knowledge from the professional development to

focus more on the students’ thinking than evaluating the students’ work. For example, a

group of elementary teachers were given a professional development on Cognitively

Guided Instruction and it was noted that the teachers learned how to select engaging tasks

that would support students at their own mathematical levels in order for the students to

demonstrate their mathematical thinking strengths and advanced strategies on how to get

the correct answers in a tasks (NCTM, 2010).

NCTM also expressed that collaboration between teachers can allow them to talk

about their best practices and learning outcomes inside the classroom. Professional

conversations can encourage and support teachers to start experimenting with new

approaches being discussed in professional development. When teachers share their

strategies and efforts about implementing new methodologies in the classroom, this

collaboration becomes another teaching tool during the professional development to

which the teachers respect and respond. In conclusion, teachers need the opportunity to

analyze students’ work and engage in mathematical content to better serve their students.

In addition, teachers acquire new and collaborative ideas when they have the time to

work with colleagues and the opportunity to attend professional development.

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In the review of literature, I learned how the existing research on high-level

cognitive demand tasks directly correlates with the increase in students’ achievement in

mathematics. In addition, the review of literature explained how to effectively implement

high-level cognitive demand tasks into the classroom as a supplemental tool to the

mandated curriculum. Furthermore, the research showed the importance of professional

development as an aid in expanding teachers’ knowledge and understanding in teaching

mathematics. When developing a professional development, it is important to incorporate

the best practice strategies that I learned about in the review of literature.

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Chapter 3. Methods

In the review of literature, I explained what high-level cognitive demand tasks

are, talked about the different levels of high and low-level cognitive demand tasks,

discussed how the high-level cognitive demand tasks should be implemented, and gave

research and data that supports the effectiveness of high-level cognitive demands tasks.

In this section I will provide background knowledge on the facilitator and expertise the

facilitator has gained from her schooling and attendance of professional development

workshops. The school context will be provided for where the professional development

will be implemented and to provide a better understanding of the demographics and

population at this school site. The curriculum at Hope Elementary School was critiqued

by the facilitator in a seminar course at CSUN which provided data that high-level

cognitive demand tasks were needed to increase student achievement in mathematics. A

brief overview of the participants that will be in attendance of the professional

development will be given and the overall arrangement and content in the professional

development will be discussed below.

Facilitator

I have been a credentialed multiple subject teacher in the Los Angeles Unified

School District (LAUSD) for four years. Within those four years, I have taught first and

fifth grades. In the past four years I have attended many professional development

workshops. Some being at my school site and others offered throughout the district.

Particular professional development workshops I have taken are: Meeting the Needs of

Gifted Learners, Cognitively Guided Instruction, Working with Collaborative Groups,

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MyMath Curriculum Training, and Differentiating Instruction for All Level Learners. I

have found a lot of the professional development workshops to be effective and enhance

my teaching skills in order to provide my students with enriched instruction in all

different subject areas. Some of the strategies or activities I found to be engaging or

effective were when the facilitator was respectful of our time, treated us like

professionals, or when I would engage in activities that directly connected to my

teaching. I want to provide teachers with an effective professional develop that they feel

they could use in their classrooms to help increase student achievement in mathematics.

In my second year of teaching, I started a Masters of Arts in Education Program

with a specialization in Elementary Education at California State University Northridge

(CSUN). Among many other courses, the Seminar in Elementary School Mathematics

Education inspired me to design a high-quality professional development program for

teachers. As part of the final project for the course, I created an hour-long professional

development on high-level cognitive demand tasks and implemented the professional

development program for my grade-level teachers at my school site. Based on my

experience of designing, facilitating, and evaluating my own professional development, I

decided to further expand the ideas of high-level cognitive demand tasks and design a

more content-focused, sustained, and collaborative professional development on high

level cognitive demand tasks. The goal is to provide this professional development

program for all the teachers at my school site and potentially to other teachers, especially

for whom mainly use low-level cognitive demands tasks from mathematics textbooks.

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School Context

The professional development will be provided to teachers and staff at Hope

Elementary School2. Hope Elementary is a public affiliated charter school in the LAUSD.

Hope is located in a suburban neighborhood with a diverse population and the majority of

the people living in the community are from low to middle socioeconomic status. Hope

Elementary is a Title One school with 450 students and 17 classrooms. Sixty-six percent

of students at this school are receiving free or reduced-lunch plans. The students at Hope

are mainly Hispanic, with a small percentage of African American, Middle Eastern,

Asian, and Caucasian students. Sixty-six percent of the student population come from

homes with a single parent income or two working parents contributing to the household.

Moreover, there is minimal parent involvement due to parents’ commitment to their work

or the lack of ability to communicate due to a language barrier. The small percentage of

about 20% of parent involvement is from parents that do not work full time or parents

that try to find time to participate in school functions on their days off.

In fifth grade the students take the Smarter Balanced Assessment Consortium

(SBAC) that assess the students’ achievement on the fifth-grade Common Core State

Standards. After the first year of teaching fifth grade, I noticed that my students had

difficulties with solving SBAC tasks and they were not well-prepared to take a high-stake

test at the end of the year. The test includes several problem-solving tasks that require

students to solve a problem using multiple strategies and explain their reasoning for solving

the problem. After analyzing the mathematics textbooks, I found that MyMath curriculum

2 Hope Elementary School is a pseudonym.

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did not provide rich opportunities for students to solve high-level cognitive demand tasks

that would prepare them for the SBAC test.

At the beginning of the school year our staff had the opportunity to get together and

review the third, fourth, and fifth grade SBAC scores. The mathematics test results showed

that 49% of third grade students nearly met or did not meet grade-level standards, 61% of

fourth grade students nearly met or did not meet grade-level standards and 75% of fifth

grade students nearly met or did not meet grade-level standards. These results were

alarming to myself and our staff. To help the students meet grade-level standards in

mathematics, we need to collaborate on strategies that can provide a high-quality learning

opportunity for students. The administrator and teachers agreed that it was important to

have an opportunity to attend professional development in mathematics in order to support

our students when it came to learning mathematics content and prepare them for taking the

end of the year SBAC.

Mathematical Textbook in Hope Elementary

Hope Elementary adopted MyMath curriculum to replace an outdated math

curriculum in hopes to increase student achievement in mathematics. In my Seminar

Elementary School Mathematics course I had to critique the mathematics curriculum

provided at Hope Elementary. After a critique of MyMath, I discovered that it is does not

give the students many tasks that are challenging or provide them a chance to become

critical thinkers in a way that will display their knowledge and understanding in

mathematics. MyMath uses a traditional perspective that is substantially lectured-based

with a focus on the basic mastery skills of literacy computation. Many of the tasks in the

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MyMath are memorization and procedural leaving little room for connecting ideas to

procedures.

In my analysis of the MyMath curriculum, I found that the teacher performs more

tasks than the students. It is noticeable how there is not enough student actions happening

in the MyMath curriculum. A weakness found in MyMath is that the students do not have

a lot of control when it comes to their learning. The expectations MyMath has for the

students closely matches the traditional theoretical perspective. It has all of the traditional

components. Ultimately, students are expected to learn through the teacher’s traditional

layout of direct instruction, progress monitoring and assessments. Students learn the basic

facts needed to attain success in the real world and are given the chance to build from

what is taught. These expectations closely align with society’s expectations of direct

instruction.

Henningsen and Stein (1997) believe students need to have more time, more

discussions and hands-on learning by using models, working with peers, investigating,

and using their own reasoning and problem-solving skills. For example, high-level

cognitive demand tasks allow the students to use different strategies within a problem,

collaborate with their peers, use models, explain and reason why they solved a problem

the way they did.

In addition to MyMath curriculum, I implemented high-level cognitive demand

tasks to provide my students the opportunity to develop high-cognitive abilities in

mathematics. The students also needed to form connections on why we do math, rather

than just doing math for math’s sake. The students needed to build their understanding of

how to explain their thinking and justify their reasoning to better support and prepare the

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students for the California Smarter Balanced Assessment Consortium (SBAC) and meet

the fifth grade CA CCSSM along with the eight mathematic practices in which the

students need to make sense of problems, persevere in solving them, reason abstractly

and quantitatively.

In 2017, I implemented high-level cognitive demand tasks into my fifth grade

classroom. With the implementation of high-level cognitive demand tasks, the data

collected from student work sample showed a progression in the students’ skills and

abilities to engage in problem-solving tasks with their peers. Figure 3.1 displays two

different examples of tasks used in my research.

Figure 3.1 High-level Cognitive Demand Tasks (NC Department of Public Instruction: Formative Instruction and Assessment Tasks)

The first task implemented shows that all students were able to successfully complete the

problem, but only 63% of the students were able to explain their reasoning, and 38% of

the students needed more support on expressing their reasoning behind their problem

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solving. The second task shows more improvement in the students’ explanation

responses, with 42% of the students able to give an explanation of their reasoning and

successfully complete the tasks, and 39% of the students were able complete the tasks

and give a more advanced explanation than the first. Some of the students were still

struggling with the tasks and giving reasoning for their problem solving.

At the end of data collection, I had implemented five different tasks. After

implementing the five tasks, the students had shown more success with problem solving

and giving explanations for their reasoning throughout the problem-solving process. The

data collected resulted in 75% of the students being able to solve the problems, engage in

the student mathematic conversations, and give an explanation to their answers; 12% of

the students were able to solve the problem and engage in the discussions, but struggled

with giving an efficient explanation; and 12% of the students were still struggling with

solving the problem, engaging in the conversation, and giving an explanation for their

problem solving strategies.

To prepare the students for the high-level cognitive demand tasks, I developed

clear expectations and norms in my classroom not only during mathematics, but

throughout the school day. The students knew what was expected of them during an

academic setting and the more informal settings, like recess and lunch. Moreover, the

students knew that a productive struggle during math was the norm and that making

mistakes usually leads to other learning possibilities. I taught the students how to engage

in math discussions by creating math talk sentence starters and putting the math talk

expectations on the board in a visual manner so all students could check the board for

sentences needed for explanations. The students were put into seating arrangements of

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four to six to facilitate peer discussions. In all, I prepared the students for high-level

cognitive demand tasks by preparing them to make sense of problems and persevere in

solving them, along with providing reasoning and an explanation on how they solved the

problems.

I provided scaffolding tools to support my students throughout the process by

gradually increasing the difficulty of tasks. I began the implementation of tasks by using

lower-level cognitive demand tasks for all students to participate comfortably and feel at

ease learning how to give explanations in class. The first tasks implemented were

memorization tasks and then I moved the task up a level to the procedures without

connections. These tasks were problem solving questions given in the MyMath

curriculum. I would use these problems to build the students’ confidence and

understanding on how to focus on one task during math instruction instead of doing

multiple problems on a page under time pressure.

As soon as the students were used to the new math routine, I increased the tasks to

high-level cognitive demand tasks. The class started with procedures with connections

tasks. An example of the tasks that were completed in my class can be found in Appendix

A. The students, at first, wanted more guidance and pressed me for approval, but I

encouraged the students to take advantage of the struggle and opportunity to problem

solve. Over time the students understood that I was not going to tell them how to solve

the problem or give the answer. I was pleased with the outcome in the change of my

pedagogy because many of the students were able to meet their full potential in

successfully completing high-level cognitive demand tasks, while other students made

progress in meeting the demands of the tasks. With all of the data collected and with my

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professional observation, I needed to supplement MyMath curriculum with high-level

cognitive tasks in order to provide support for my students in meeting the mathematical

practices.

Participants

Because Hope Elementary is a small school with low enrollment, we have a

relatively small number of staff. Currently we have one kindergarten and transitional

kindergarten-spilt teacher, two kindergarten teachers, two first grade teachers, one first

and second split teacher, two second grade teachers, one second and third split teacher,

two third grade teachers, two fourth grade teachers, two fifth grade teachers, one K to 2

special education autistic teacher and one 3 to 5 special education autistic teacher. In total

there are 17 teachers along with one coordinator, one instructional coach, and one

administrator, our principal, that will be participating and receiving the professional

development. Thirteen of the teachers have been teaching for 10 or more years and four

teachers have less than five years of teaching experience.

Teachers in Hope Elementary have received minimal professional development

in mathematics. Two years ago, the local district personnel came to our school site to

give a presentation on MyMath curriculum. The presentation was not helpful when it

came to enriching the content for high achieving students. In the meantime, we are

currently receiving ongoing training in Cognitively Guided Instruction (CGI) and will

have a total of ten training sessions throughout the year. When the CGI professional

development is completed I plan on implementing the professional development I created

on high-level cognitive demand tasks. The teachers need further professional

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development on mathematics in order to strengthen their mathematic instruction to

provide the students with opportunities in meeting the rigorous demands of the California

Common Core Standards Mathematical Practices such as, MP1: make sense of problem

solving and persevere in solving them.

Professional Development

The professional development includes strategies that were discussed in the

literature review such as, engaging activities, collaboration, research and reflection. The

professional development consists of six three-hour sessions, one session a week for six

weeks. Each session includes activities for the teachers to engage in and provide research

on high-level cognitive demand tasks. The whole group and teacher discussions will be

collaborative in nature and allow for reflection.

The professional development will be spread out into six different sessions over a

six-week period to allow teachers time to implement high-level cognitive demand tasks

into their classrooms. The teachers will be able to come back to each session with

feedback and ask any questions they may still have to improve their teaching and

understanding of high-level cognitive demand tasks. A clear and well-organized timeline

will help lead to an effective professional development.

In each of the six sessions I will be delivering material in a variety of ways. Just

like in the classroom with students, teachers need multiple modalities when learning new

content. Throughout the professional development I will be presenting information using

visuals with charts, video and teacher-lead lessons, as well as kinesthetic activities and

engaging discussions.

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As previously discussed in the literature review, teachers are deepening their own

understanding in the content if they have sufficient time to discuss, share, and engage in

mathematics. This is why I have provided engaging activities for the teachers to work

collaboratively in every session. I also provide time for the teachers to feel as though

their own professional judgment matters and that they, too, have their own knowledge on

how to teach math. I will also be giving the teachers time to work in their grade-level

clusters to allow them the opportunity to create appropriate tasks and to gather data that

will help give more insight on whether or not students are meeting grade-level standards.

The opportunity to work with their grade-levels helps to honor and respects the teachers’

time and allows them the opportunity to collaborate with their colleagues.

The activities given will be tasks that students can do, but the teachers will be

tackling the tasks as adults. I am not asking the teachers to take on the roles of students

because research on professional development shows that teachers do not respond well to

childlike activities. The teachers will analyze tasks to determine if the tasks are low-level

or high-level cognitive demand tasks, they will analyze their own curricula material and

will engage in high-level cognitive demand tasks that their students will be working on

when implemented in the classroom. Along with analyzing and engaging in the tasks, the

teachers will also create their own high-level cognitive demand tasks, appropriate for the

level of their students. The teachers will be able to discuss student solutions and thinking,

which will also lead to teachers developing a deeper understanding in the mathematical

content.

In one of the sessions I will be demonstrating the implementation of a high-level

cognitive demand task in my own classroom with fifth grade students. This will provide

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the teachers an opportunity to see how high-level demand tasks can be used in a real

classroom environment. The teachers can observe the lesson and ask any unanswered

questions they may still have before they too implement it in their own classrooms. This

will also allow me to bring actual student work to our next session so that teachers can

analyze and discuss student thinking and problem-solving strategies.

By the end of the six sessions the teachers will have the chance to create and

implement two different high-level cognitive demand tasks in their classrooms. We will

all meet and discuss their implementation by talking about things that went well and areas

the teachers feel they still need support in. The teachers will be able to analyze their own

student’s work to see what their next steps are in the process of high-level cognitive

demand task implementation. As the professional with the knowledge on this content, I

will make myself available to all teachers and staff members after the six sessions if they

still need guidance and support with implementing high-level demand tasks.

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Chapter 4

High-Level Cognitive Demand Task: Professional Development

This chapter focuses on the content of a professional development to help teachers

learn how to effectively implement high-level cognitive demand tasks. The professional

development consists of six different professional development seminars that will be

conducted at the school site. The PowerPoint Presentations for all six sessions can be

found in Appendix B.

Session One: Introduction to High-Level Cognitive Demand Tasks

The first session will provide the teachers with a brief overview of the six

professional development sessions. Each session will occur over a three-hour period, with

all teachers at the school site in attendance. The teachers will be asked to sit at the tables

with other teachers of the same grade-level, so the teachers can collaborate with their

grade-level teachers during the professional development. All supplies, including paper,

pens, and pencils, will be offered. There will be one ten-minute break halfway through

the session.

The introduction to the session will include a video that will engage the teachers.

After watching the video, I will lead a discussion on the importance for the professional

development and why Hope Elementary School needs more professional development on

mathematics. At this time, I will give interesting findings about high-level cognitive

demand tasks found in the review of literature. The research from Stein and Smith

(1998), Smith and Stein (1998), and Henningsen and Stein (1997) will be presented to

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support high-level cognitive demand tasks. The session will then end with a discussion on

what the next steps are for the future sessions.

Introduction (40 minutes)

This session will begin with the facilitator introducing herself and offering

refreshments in the back of the room. All participants will have received an agenda at

sign-in, so the facilitator will briefly go over the agenda at the start of the session. All of

the teachers will then partake in a short survey to get a better understanding of how the

teachers feel about math and teaching math. The same survey will be given at the end of

the professional development to evaluate if the teachers’ attitudes have changed about

math and teaching mathematics. The survey questions can be found in Appendix C.

The facilitator will then engage the teachers with a TED Talk video3 presented by

Dan Meyer (2010) called “Math Class Needs a Makeover.” Meyer talks about how our

students are not prepared to work through word problems in a way that involves

initiative, perseverance, and retention. Meyer then gives examples of word problems in

his math textbook and how he rewrites some of the problems to get the students to engage

in the task with initiative, perseverance, and retention.

After watching the TED Talk video, I will then make a transition into a whole

group discussion on how the teachers think MyMath curriculum in terms of supporting

our students in meeting the California Common Core State Standards: Mathematics

Content and Mathematical Practices. The facilitator will then display a chart of the eight

Mathematical Practices as seen in Figure 4.1.

3 https://www.ted.com/talks/dan_meyer_math_curriculum_makeover

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Figure 4.1 Mathematical Practice Standards Rubric (Dooms, 2013)

The facilitator and the teachers will discuss what the Mathematical Practices are

assessing and if we as a staff are giving our students the opportunity to engage in math

tasks that allow them to meet the rigorous demand of these practices. For example,

mathematical practice three: construct viable arguments and critique the reasons of

others. Are the students prepared to have mathematical conversations that explain their

understanding of problem-solving?

Purpose and Warm Up: (60 Minutes)

The discussion on the Mathematical Practices will leave teachers wondering how

we can help our students meet the rigor in these standards. At this time the facilitator will

introduce what high-level cognitive demand tasks are based on the research provided in

the review of literature. The teachers will engage in the activity “High or Low” where

they have to separate six different tasks into low-level demand tasks and high-level

demand tasks. Once the teachers have completed the task, the facilitator will then review

the four different levels of cognitive demand tasks explained by Smith and Stein (1998)

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in the Mathematical Task Analysis Guide that can be found in Appendix D. The teachers

will have to go back to the task cards and determine which task fits into the four different

levels. The teachers will have to share out their reasoning for sorting the task card with

that particular level. The teachers should attempt to complete as many of the tasks

possible with the remaining time.

Break (10 Minutes)

After a long work day and 100 minutes of professional development, it is

important to let the teachers know their time is appreciated and that the facilitator is being

empathetic towards their feelings and that is why there should a break in the middle of

the session. The break will give teachers time to get up and move around, use the

restroom, or get more refreshments. The break can also boost the morale of the

professional development and make the second portion of the session more enjoyable and

meaningful.

Connection to Research (15 minutes)

Now that the teachers have had a chance to listen to a TED Talk about revamping

math instruction, discussed the features of high-level cognitive demand tasks, and sorted

mathematical tasks into “High or Low,” the facilitator will present the results that

implementing high-level cognitive demand tasks contributed to an increase in students’

mathematics achievement (e.g., Henningsen & Stein, 1997; Ni, Zhou, Cai, Li, Li, & Sun,

2017).

Start to Implementation (25 minutes)

Now the facilitator will hear from the teachers about what classroom norms they

already have in place. The teachers will share with their group the norms they have in

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their classroom, while one of the participants writes a list of all the norms discussed.

Once the groups are finished they will share out to the whole group. This allows the

teachers to feel that their professional judgement on classroom norms matters in this

professional development. The facilitator will compile the lists from the group and

display them on a parking lot chart paper. Some of the norms that may be shared out

would be raising your hand before speaking, respectful to other students’ ideas, respect

others materials, be an active listener, and remain in seat unless told otherwise. The

teachers will have these as a reference if they need more positive norms to use in their

classrooms.

Math Task (20 Minutes)

After giving the teachers the foundational understanding of what high-level

cognitive demand tasks are and empirical-evidence that supports student achievement

with these tasks, it is only right for the teacher to have an opportunity to engaging in a

high-level cognitive demand task. The teachers will work independently on the task

“Help George.” Once the teachers are finished with the task they will have a chance to

share with their groups. After the groups share out a few teachers will be asked to come

up and share their work. As the teachers are working on the task the facilitator will be

circling the room asking guiding questions or encouraging positive work habits. The

facilitator is modeling some of the classroom norms the teachers earlier discussed. This is

the final activity of the first session. Figure 4.2 shows an example of the math task given

to the teachers.

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Figure 4.2 Help George (NC Department of Public Instruction: Formative Instruction and Assessment Tasks)

Take Away and Next Steps (15 Minutes)

The facilitator will ask the teachers to share out any take-away they may have

from this session. This will be an opportunity for the facilitator to get feedback on the

presentation and gauge teacher learning outcomes. The facilitator will then give a brief

explanation on the next session and what the teachers should bring to the second session.

In addition, the facilitator will encourage the teachers to establish the classroom norms or

implement a new norm learned today and to encourage their students to start having

conversation about the math instruction. This will guide the teachers for the next session

on the implementation of high-level cognitive demand mathematical tasks.

Session Two: Establishing Norms for Implementing High-Level Cognitive Demand

Tasks

The second session of the professional development will cover more information,

research, and data that support the implementation of high-level cognitive demand tasks.

The goal for this session is for all teachers to understand the purpose of and how to start

implementing high-level cognitive demand tasks. The presentation will also include

videos and tasks from Graham Fletcher, a classroom teacher and math specialist that

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advocates for best practice in elementary mathematics. The teachers will also have an

opportunity to analyze their own MyMath curriculum for their grade-level.

Introduction (15 Minutes)

To start the second session of the professional development, the facilitator will

briefly review the first session and ask if there are any burning questions the teachers

may have about high-level cognitive demand tasks. The facilitator can write the questions

the teachers ask on the Parking Lot displayed in the room and then the facilitator will

preview the agenda for today’s session. After the whole group discussion, the facilitator

will play a video by Fletcher4 that shows a progression of learning from grade-level to

grade-level. Fletcher has made multiple progression videos on different math concepts, so

the facilitator will choose one of the videos that the whole group would like to see. The

video runs for about seven minutes. The progression video by Fletcher is important for

the teachers because using students’ prior knowledge is the key factor that contribute to

maintaining high-level cognitive demand tasks. The video will provide the teachers a

better understanding of how the mathematical standards build from one grade-level to the

next in order to successfully create tasks that will continue to build on students’

understanding and knowledge of mathematics.

Purpose and Warm Up (30 Minutes)

When teachers are presented with a professional development they want to know

why they are receiving this professional development. A question the teachers will be

asking in this professional development is, why do they need to know what high-level

4 https://gfletchy.com/progression-videos/

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cognitive demand tasks are? The facilitator will explain the why in this session. The

facilitator will explain that high-level cognitive demand tasks will help prepare the

students with the Smarter Balanced Assessment Consortium, the rigorous expectations of

the Mathematical Practices, and mastering the California Common Core State Standards:

Mathematics. Students need additional support in high-level thinking, problem solving,

and communication in math instruction. Teachers need to provide the students with the

skills to get the students to persevere in challenging mathematical tasks not only in the

classroom, but in order to make the connections with math and the real world.

As a warm up the teachers will be asked to get into K to 5 groups. The teachers

will go to Fletcher’s website and select an intriguing 3-Act Task. The facilitator will

provide guided questions for the teachers to answer as they go through the three acts.

Examples of the guided questions are: Using the Mathematics Task framework provided

by Stein and Smith (1998), what is the cognitive demand of the 3-Act task? How does

this task look like in your classroom? What factors influence on maintaining or declining

the cognitive demands of the 3-Act task? The teachers will engage in the task and then

have a discussion on how to implement the 3-Act task in their classrooms. This task and

discussion are preparing the teachers for the presentation on implementation in this

session. The facilitator will also provide an article written by Lomax, Alfonzo, Dietz

Kleyman, and Kazemi (2017) on Three-Act Tasks. This article provides an idea on how

and why using these tasks are beneficial for student learning, how to have mathematical

conversations, identify key information needed to solve a problem, and make connections

or revise their thinking, all of which are part of high-level cognitive demand tasks.

Implementation Content (60 Minutes)

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The second session will dive deeper into the implementation of the high-level

cognitive demand tasks. The group will review classroom norms and the facilitator will

show the norms, rules, and expectations currently implemented and displayed in their

classroom as seen in Figure 4.3.

Figure 4.3 Math Discussion Expectation Anchor Chart

Other norms, rules, and expectations that can be implemented in the classroom are

Talk Moves: Re-voicing what others have said, provide your own reasoning, add on to

others thinking, and giving others enough wait time to think. Establish rules such as being

respectful, be safe, be responsible, be kind, and be honest. Hold the students accountable

for being ready to learn by teaching them to be active listeners, enter and exit prepared,

always try their best, and respect yourself and others. The facilitator will encourage the

teachers to also have visual displays in their classroom for the students to refer to when

needed. It is important to have the norms, rules, and expectations in place because the

students need to know that they are held accountable for their behaviors and that there are

40

consequences for their actions. Having classroom norms, rules, and expectations will

keep the classroom management minimal and that will leave more time for mathematics.

Another aspect into successful implementation is the physical arrangement of the

classroom. Henningsen and Stein (1997) believe the physical arrangement of the

classroom should be in a way that promotes student engagement that allows them to

communicate with their peers in a comfortable environment. The facilitator will present

ways to physically organize the classroom that will encourage and allow students to

collaborate and engage in mathematical tasks. The set ups do not need to be permanent

arrangements, but should be used during high-level cognitive demand tasks. The

facilitator will give two visuals in the PowerPoint of a classroom set up that display a

positive physical arrangement that will be beneficial when working on high-level

cognitive demand tasks. The physical display of the classroom is important because it

encourages the students to engage and collaborate during the math tasks and to have math

related conversations.

Time is a big factor when implementing high-level cognitive demand tasks. The

timing needs to be appropriate for the task and for the students performing the tasks.

Henningsen and Stein (1997) and Smith and Stein (1998) encourage teachers to allow the

students to struggle through the tasks, but know when the students are reaching a level of

frustration that will decrease the level of cognitive ability. They also state that the time

given on the task should be appropriate for that particular group of students. The

facilitator will give the analogy of Goldie Locks and the Three Bears for time. Teachers

need to use their professional judgement and find what the appropriate time is for their

students. The importance of time is that it should not be rushed, so the students can work

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through the struggle as the teacher asks guiding questions. Another important factor on

time, is that there should not be too much time given. Too much time can decrease the

high-level cognitive demand because the students become discouraged and frustrated

when the task is too difficult. The facilitator will make it clear that the teachers need to

use their professional judgement and knowledge about their own students to know the

warning signs of a task that is too rushed or given too much time.

In addition to implementation, the students will need encouragement from their

peers and teacher because these tasks will be challenging for the students. Smith and

Stein (1998) state that the tasks will need to be appropriate to the grade-level and the

students learning abilities to keep the high-level of engagement. The three phrases of

high-level cognitive demand tasks are developing the tasks, establishing the goals for the

tasks, and getting the students to successfully engage in the tasks. At the end of the

implementation section, the facilitator will give credit to the authors: Henningsen and

Stein (1997), Smith and Stein (1998), Stein and Smith (1998), and Stein, Smith and

Henningsen (2000) for the provided research.

Break (10 Minutes)

MyMath Curriculum Analysis (60 Minutes)

Teachers need time to analyze their own curriculum to formulate their own

professional judgement if the MyMath curriculum provided by LAUSD is enough to

support students in meeting the required expectations. The facilitator will give the grade-

level groups instructions for analyzing their MyMath curriculum. The groups will choose

one chapter to focus on. The teachers will look for the problem-solving tasks and

determine if the tasks are high or low-level cognitive demand tasks. The teachers will

42

need to determine if the students are being challenged, asked to collaborate, give

explanations and reasoning for their problem solving. The teachers will share out their

findings at the end of the activity. As the groups are digging into the curriculum the

facilitator will walk the room going from group to group asking guiding questions, or

answering any questions the groups may have. Some questions the facilitator will ask are:

are the students being challenged? Are the students asked to collaborate with their peers?

Are the students asked to explain their reasoning or give explanation to their problem-

solving? This task will better prepare the teachers for the upcoming session when the

teachers will need to create their own high-level cognitive demand tasks from the

MyMath curriculum.

Next Steps (10 Minutes)

At the end of session, the facilitator will give the teachers time to ask any

questions they may still have. After the questions are answered the facilitator will let the

teachers know what materials they will need for the upcoming session and what next

steps they should take in their classrooms. The facilitator will encourage the teacher to try

a Gfletchy’s 3-Act Task and to report back to the group at the next session. As the

sessions progress the teachers should be trying new strategies in their classroom to better

prepare them for the implementation of high-level cognitive demand tasks.

Session Three: Enriching the Curriculum with High-Level Cognitive Demand Tasks

The third session focuses on creating high-level cognitive demand tasks to help

supplement the MyMath curriculum. The teachers will work in their grade-level teams to

select a task from MyMath and redesign the task to make it a high-level cognitive

demand task. The task the teachers created will be solved and critiqued by another grade-

43

level team. This session will allow teachers to work with their colleagues to enhance the

curriculum and to advance their thinking on how to develop a low-level task into a high-

level cognitive demand task.

Introduction (15 Minutes)

At the beginning of the session the facilitator will go over the agenda for the third

session. After the introduction of all the items on the agenda for that day, the teachers

will be presented with a task that will require them to reflect on their week in the

classroom. They will be asked to write down two highs and one low for teaching and

learning. Teachers need a time to share out positive learning moments they are

experiencing in the classroom and also be able to use this time to vent every day

frustrations. The teachers will also provide feedback on how the implementation of high-

level cognitive demand tasks are going in an oral discussion. With a whole group

discussion, the facilitator will be able to gauge how the teachers are feeling about the new

mathematics content (high-level cognitive demand tasks) and what mood they are in

during this session and thus give more time or less time to the topics planned.

Warm Up (30 Minutes)

The warm up task, “Multiply It,” that the teachers will engage in is a high-level

cognitive demand task that will require the teachers to use visuals, different methods of

problem solving, reasoning, and communicate with their peers. The facilitator will use a

task from MyMath curriculum and will revise it to a high-level cognitive demand task.

Teachers will work on this task individually to allow them an opportunity to engage in

the task without any distractions. The teachers will be given enough time to work through

the presented task. When they finish the problem-solving on their own, they will share

44

their work with their grade-level teams and have a group discussion on how they solved

the problem and share out their reasoning on why they used the strategy they used. This

warm up will get the teachers ready for creating their own high-level cognitive demand

task with using MyMath.

Creating a Task (75 Minutes)

To give the teachers an idea of how high-level tasks are implemented in a

classroom the facilitator will show the teachers a YouTube video5 called “Best Practices:

Math Effective Tasks,” by Fairfax Network-Fairfax County Publish Schools. The video

shows a classroom working on a task and how the teacher facilitates the tasks for the

students. The video will display questions the teacher asks the students to maintain the

high-level cognitive demand of the task. For the next engagement task the teachers will

work with their grade-level teams to modify a math problem in the MyMath curriculum

to implement in their classrooms before the next professional development session. The

teachers will also compose guiding questions and list tools students will need to complete

the task.

Giving the teachers an opportunity to work together to create a task may produce

more accountability to their grade-level and using tasks in the classroom. Giving the gift

of time to collaborate, most teachers will use this professional development time wisely

to produce high-level cognitive demand tasks that could become a grade-level common

assessment.

Break (10 Minutes)

5 https://www.youtube.com/watch?v=XI3-52B0V6s

45

Critiques (40 Minutes)

After the grade-levels are done creating a high-level cognitive demand task, the

facilitator will distribute the tasks to other groups. The teachers will then work in their

grade-levels teams to critique tasks using the questions the facilitator provided: What is

the cognitive demand level of the task? How can you maintain the high-level cognitive

demand of the task? What prior knowledge will the students need to have to perform the

task? What Mathematical Practice is aligned with the task? The teachers can also critique

the level of the task by using the Mathematical Task Analysis Guide that was given out in

session one. The facilitator will be strategic in distributing the tasks. For example, a first-

grade task will be given to kindergarten teachers, so that the progression of mathematical

learning can be seen. The progress of the tasks needs to be seen by the different grade-

levels to make the teachers aware of the content being taught in the upcoming grade so

the students will come to that grade-level prepared to perform high-level cognitive

demand tasks appropriate for that grade-level. The teachers will also be using learning

progressions, the common core standards, and mathematical practices to critique the new

high-level cognitive demand tasks.

Next Steps (10 Minutes)

Wrapping up the third session will include an exit ticket for the teachers to

complete. The prompt will encourage the teachers to try the newly created high-level

tasks in their classrooms. It will also give the facilitator a sneak-peek into what the

teachers are thinking and feeling about the presentation on high-level cognitive demand

tasks. The facilitator will preview session four and encourage teachers to use the tasks

before the next professional session. Teachers will be expected to implement one high-

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level cognitive demand task, gather baseline data, and student work samples to be

analyzed in the fourth session.

Session Four: Observation of Implementing a High-level Cognitive Demand Task in a

Fifth Grade Classroom and Analysis of Student Work Samples

The fourth session will provide the teachers an opportunity to observe a fifth-

grade classroom perform a high-level cognitive demand task and analyze their own

student work samples to formulate a baseline for what their students need when

performing high-level cognitive demand tasks.

Introduction (20 Minutes)

At the beginning of the session the facilitator introduces the agenda for the fourth

session. After the introduction of all the items on the agenda for that day, the teachers

will be asked to share out about how the high-level cognitive demand task went in their

classroom. The teachers will be given a few guided questions they can answer, so the

facilitator can have a better understanding on the teacher’s attitudes after implementing

high-level cognitive demand tasks in their classroom. The key focus for this session is the

observation of a classroom. The facilitator will give an explanation about the observation.

The whole group will go into a fifth-grade class and observe them working on a high-

level cognitive demand task. The teachers will be told they should not participate in the

classroom activity. They can take notes during the observation to use for their

professional practice.

Observation (60 Minutes)

The teachers will watch a fifth-grade teacher instruct her class through a high-

level cognitive demand task. The observers will sit around the back of the room and

47

observe how the teacher introduces the task to the students. The teacher will read the

problem with the students and go over the provided tools the students can use when

working on the provided task. The task introduced to the students is shown in Figure 4.4.

Figure 4.4 Lion Hunt (NC Department of Public Instruction: Formative Instruction and Assessment Tasks)

The students will have been grouped heterogeneously and in a physical

arrangement that encourages mathematic conversations and collaboration. The teacher

will circulate around the classroom, monitor students’ progress in the task, and aske

guiding questions when the students seek extra support. For example, the teacher may

ask, “tell me about your drawing?” or “So why did you break into 4 parts?”. The

observers will observe how the teacher and the students work together to work through a

challenging task without decreasing the cognitive demand of the mathematical task. After

the observation, the teachers will be given a ten-minute break.

Whole Group Discussion (20 Minutes)

After the observation, the teacher and facilitator will participate in a whole group

discussion about what they observed in the fifth-grade classroom. The teachers will

provide an explanation on what they noticed or liked that the teacher or students did in

the math class. They will share out some of the strategies for teaching methods the

teacher used while the students worked on the task. The teachers will share out whether

they though the task was high-level and appropriate for the students and if the level of

48

demand increased or decreased student achievement. The will also reflect on the norms

and arrangement of the classroom and if it encouraged student communication. The

teachers can provide feedback on what they would do differently or how they would add

to the lesson. The discussion will help teachers add to their professional understanding on

how to effectively implement high-level cognitive demand tasks in their math class.

Analysis of Data and Student Work Samples (60 Minutes)

Allowing teachers the opportunity to work in grade-levels to analyze student work

samples and data collected from the high-level cognitive demand task is an effective

strategy to promote professional growth in the teacher’s ability in implementing high-

level cognitive demand tasks. The teachers will work in their grade-level teams to discuss

the students’ performance on the task. The teachers can get a baseline understanding of

students’ strengths and weaknesses in high-level cognitive demand tasks. The teachers

can analyze the student work samples and select students that understand or need

additional support on the use of models, give an explanation for their problem solving, or

use a variety of strategies to solve a task. The facilitator will display questions the

teachers should be answering in order to properly analyze the student work sample. The

questions can be seen in Figure 4.5.

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Figure 4.5 Analyzing Student Work Sample Questions

The teachers will be able to create heterogenous groups that can potentially

support the students when working on the next implemented task. In addition, the

teachers can formulate what next steps will be needed in order to meet the needs of all the

students in the classroom.

Once all of the teachers have had time to analyze student work and data, each

grade-level team will be asked to share out what they found was effective with their

instruction, what the students struggled with, and what next steps they will be taking for

the next implementation. This discussion will provide the teachers an opportunity to hear

what was effective in other classrooms, and what other students were struggling with.

Knowing common areas of need in a school can help targeted intervention. The teacher

will get to know what other grade-levels are doing in their classrooms and how the

instruction is going, so in the following school year there is a better understanding on

how and what students learned in mathematics.

Wrap Up (15 Minutes)

At the end of the fourth session, the facilitator will go over what the teachers

should do in their classroom before the session five and what materials will be needed in

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the next session. Before the teachers leave they will be asked to fill out an exit ticket that

will provide feedback on how the teachers felt about the classroom observation and if it

was effective for this professional development.

Session Five: Analyzing High-Level Cognitive Demand Task Work Samples

The fifth session of the professional development focuses more on analyzing

student work samples and giving the teachers grade-level planning time to accumulate

high-level cognitive demand tasks to use in their classrooms. As a whole group, they will

watch a video of a first-grade class working on a task and how the teacher questions the

students during the process.

Introduction and Warm Up (30 Minutes)

In the beginning of the session the facilitator will introduce the agenda for the

fifth session. The warm up activity for this session will include a video6 of a first-grade

classroom performing a math task. The facilitator will ask the teachers to pay attention to

what questions the teacher is asking the students, how the students are explaining their

problem solving, how the students came up with different strategies for one task, and

what academic language was used in the students’ responses. This video connects to

high-level cognitive demand tasks because the teacher asks quidding questions to keep

the students engaged and prompts them in giving reasoning and details for their problem-

solving. The students came up with different strategies for solving the task and were able

to share with the class why they solved the task the way they did. This video shows how

first grade students are able to complete a high-level cognitive demand task and maintain

6 https://www.youtube.com/watch?v=o9gybhyRPHw&t=197s

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engagement to communicate their thinking during problem-solving. After watching the

video, the whole group will engage in a discussion about what they noticed in the video.

The facilitator will specifically want to hear from the first-grade teachers about what they

noticed and thought about the video. The facilitator will ask if the students were

performing a task aligned with the mathematical practices and how would they be graded

based on their understanding of the content and performance on a high-level cognitive

demand task.

Creating More Rigorous Tasks (60 Minutes)

Before the teachers start working on creating more tasks, the facilitator will

provide the teachers with a handout on how to modify tasks (see Appendix D). The

handout will help the teachers with the next activity when they are creating, redesigning,

or researching more high-level cognitive demand tasks. The goal for this activity is for

the teachers to work in their grade-level teams to accumulate as many high-level

cognitive demand tasks possible, so they can implement these tasks in their classroom

throughout the year to practice the skills needed to meet the rigorous expectation in the

CA CCSS Mathematical Practices, and SBAC. The teachers can also collaborate with

other grade-levels to share strategies, tools, or ask questions they have.

Break (10 Minutes)

Student Work Samples (45 Minutes)

The facilitator will display some tasks that they have had students work on in the

past. After reviewing the tasks, the facilitator and the whole group will then look over

student work samples from the displayed task list in Appendix A. The work samples will

be displayed in the PowerPoint for session five. The facilitator will lead the conversation,

52

asking the teacher how they would grade these students and if they can rationalize what

the students were thinking when working on the task, did the students display a high-

level of understanding for the task at hand? The facilitator will then ask the teachers how

they would help support or challenge these particular students for the next task they

would complete.

Next Steps (10 Minutes)

The facilitator will encourage the teachers to continue implementing high-level

cognitive demand tasks in their math classes and will ask the teachers to bring a task and

student work samples to the next session to share with the group. The facilitator will go

over what the sixth session will cover, so the teachers can be prepared for the last session.

Session Six: Creating and Analyzing High-level Cognitive Demand Tasks

The last session of the professional development is going to focus on student

work samples, and creating tasks that show the progression through the CCSS-M. The

teachers will work in heterogeneous groups that include K to 5 teachers. The teachers

will have an opportunity to share out their experiences in the classroom and how they feel

implementing high-level cognitive demand tasks.

Introduction and Warm Up (30 Minutes)

The facilitator will go over the agenda for the last session and explain how today

will be focusing on student work and how we have grown as professionals. The warm up

activity for the last session will include story telling. The teachers will be asked to share

out how the high-level cognitive demand tasks are going in their classroom. They will

share out how the students are responding to the tasks and if the teachers are noticing

improvement in the achievement levels in mathematics. The facilitator will also ask the

53

teachers if there are any areas the teachers are still struggling in and how he/she can

provide additional support to the teachers.

Gallery Walk (30 Minutes)

The teachers were asked to bring a sample high-level cognitive demand task that

was implemented in their classroom and provide student work samples for that particular

task. The teachers will set up their task and student work samples on the tables and the

whole group will walk around the room for a gallery walk. The teachers will read and

analyze other teacher’s tasks and student work samples. The teachers will examine the

student work and analyze what they were thinking when solving the task. They will be

looking to see if the students displayed understanding for the content being taught and

used different strategies to solve the task. At the end of the gallery walk the teachers will

share out what they noticed about all of the tasks and student work samples. The teachers

will share out if the high-level cognitive demand tasks viewed were effective with

improving student achievement in mathematics.

Progression Tasks (60 Minutes)

It is important for the teachers to understand the progression through the CA

Common Core Standards and work together in order to provide the students with proper

instruction that will help the student learn the content as they advance to the next grade-

level. As discussed in Chapter 2, high-level cognitive demand tasks require students to

use their prior knowledge in order to maintain the high-level of demand in the task. The

teachers will be grouped in K to 5 groups. The groups will select a CCSS Mathematical

Practice, such as MP1: make sense of problem solving or MP3: construct viable

arguments to create a task that will show the progression of the standard in each grade-

54

level. The teachers will need to start with a kindergarten standard to start and then build

on the task all the way to meet a fifth-grade standard. With this activity the teachers will

clearly see how tasks need to be implemented in all grade-levels in order to continue

building the high-level cognitive demand in mathematics. The teachers will also see the

importance of implementing the tasks in all grade-level so the students become familiar

with having mathematical conversations and the use of academic vocabulary to maintain

high-level cognitive demand. Smith and Stein (1998) state that the student needs to be

familiar with the use of vocabulary and how to effectively have structured mathematical

conversations to sustain high-level cognitive demand in mathematics. The facilitator will

provide Gfletchy’s Progression Videos as a resource to the teacher to assist with the task

building.

Break (10 Minutes)

Share Out (30 Minutes)

The teachers will have the opportunity to share their progression tasks with the

rest of the group. These tasks can be used in the classrooms as well, and can be used to

help differentiate instruction in the classrooms. The idea of creating progression tasks

will help the teachers in assisting the students based on their levels of understanding in

order to close the achievement gap in mathematics.

End of Session (10 Minutes)

At the end of the session, the facilitator will thank the teachers for their

participation and provide them with the facilitator’s contact information in the event that

the teachers need additional support or have questions about high-level cognitive demand

tasks. The facilitator will display a link to a post-survey that the teachers will be asked to

55

take. The survey has the same questions the teachers answered in the beginning of the

professional development. The responses will inform the facilitator is the teachers have

acquired more knowledge and confidence in teach high-level cognitive demand tasks.

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Chapter 5

Conclusion

Summary

After analyzing MyMath curriculum and students’ mathematics performance on

the state-adopted standards (California Common Core State Standards) and associated

assessment (Smarter Balanced Assessment Consortium) at Hope Elementary School, I

found that teachers need additional support for implementing mathematical tasks in the

mandated curriculum. Considering the needs of teachers, I have explored the design,

structure, content, activities, and outcomes of effective mathematics professional

development that ultimately increases students’ learning in mathematics. Among many

other professional development opportunities, this project focuses on the design of

professional development on implementing high-level cognitive demand tasks. The

MyMath curriculum was not sufficient enough in meeting the rigorous demand in the

state-adopted standards and associated assessment and that is why teachers need

supplemental resources with high-level cognitive demand tasks.

The central guiding research question in this graduate project was: How can we

create a professional development program that supports teachers to implement high-

level cognitive demand tasks? What challenges or affordances exist in the professional

development program that focuses on implementing high-level cognitive demand tasks in

the context of using the mandated low-level cognitive demands tasks mathematical

curriculum?

In this project, I introduced the mathematical task framework, identified features

of each cognitive demand level task, provided examples of each level, and factors

57

contribute to maintain or decline the high-level cognitive demands of task. High-level

cognitive demand tasks consist of four different levels. Each level brings different

expectations and rigor to the task. Stein and Smith (1998) explain the four different levels

(“memorization” and “procedures without connections” as lower-level demand tasks;

“procedures with connections” and “doing mathematics” as high-level demand tasks).

The order of the levels advances from the lowest to the highest level of demand. To meet

the high expectation and rigorous demand of state standards and assessments, the goal is

to have the students performing tasks at level three or four. High-level cognitive demand

tasks allow students to explore different cognitive demands and to actively participate in

longer and more complex activities. The empirical evidence shows that high-level

cognitive demand tasks increase student achievement in mathematics (Henningsen &

Stein, 1997; Ni et al., 2017).

Also, I identified characteristics of effective professional development which

allowed me to design an effective professional development. Research (NCTM, 2010;

Lemlech1995; Guskey, 2003) provided effective strategies to create my own professional

development on high-level cognitive demand tasks. It includes being respectful and

empathetic of the teacher’s time, making the content relevant and useful, as well as

providing the teachers time to digest and utilize the materials given.

Overall this graduate project examined the importance of implementing high-level

cognitive demand tasks in K through 5 classrooms. In order to meet the high expectations

in the CA CCSS-M and SBAC, students need more rigorous and challenging

mathematical tasks. Teachers need to provide students with the opportunity to persevere

through challenging problems in order to get the students to make connections between

58

mathematical instruction and using mathematics in the real world. High-level cognitive

demand tasks allow the students to think critically, build connections, collaborate with

their peers and support reasoning through the problem-solving process.

What I Learned

As a professional practitioner conducting research on high-level cognitive

demand tasks, I have advanced my knowledge on effective teaching practices to

implement into the classroom. I have found that students can benefit from high-level

cognitive demand tasks. Students can build their confidence in solving challenging

mathematical tasks, have meaningful mathematical conversations with their peers and

give an explanation and reasoning that supports their thinking and problem solving. Since

I have implemented high-level cognitive demand tasks in my own classroom, I have

noticed growth in my students’ mathematical understanding. The students are able to

make connections between the real world and what they are learning in the math lesson.

The students also see the importance of mathematical tasks and enjoy working through

them.

The research on high-level cognitive demand tasks has not only improved my

instruction in mathematics, but it has also provided me with the knowledge to be able to

support my students in creating a positive learning environment. After establishing norms

and expectations from the start of school year, my students are comfortable in the

classroom and know it is a safe place to try and learn new things.

Not only have I advanced my skills as a teacher, but I have also developed the

skills needed to be an instructional leader at my school site. The literature review on

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characteristics of effective professional development has equipped me with the

confidence, knowledge and understanding on how to provide meaningful and enriching

professional developments on high-level cognitive demand tasks to teachers at my school

site and potentially district-wide.

Limitations

This graduate project focused on research that supports the implementation of

high-level cognitive demand tasks and best practices of effective professional

development. Therefore, the limitation to this graduate project was that another

mathematical professional development was already in place at the school site, so all of

our professional development time was already booked for the year. LAUSD mandated

that selected schools were required to receive professional development on Cognitively

Guided Instruction (CGI). The professional development on CGI could provide

foundational knowledge for a future professional development on high-level cognitive

demand tasks.

Future Research

My goals for future research will be to keep track of the effectiveness of the

implementing high-level cognitive demand tasks on students’ mathematics performance

on California Common Core State Standards Mathematical Practices Smarter Balanced

Assessment Consortium Mathematics. Collecting data from the high-stake assessment

and mastery of the grade-level standards will allow me the opportunity to assist my staff

with this professional development in the upcoming school year.

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References

Borko, H., Jacobs, J., Koellner, K., & Swackhamer, L. E. (2015). Mathematics professional development: Improving teaching using the problem-solving cycle and leadership preparation models. New York: Teachers College Press.

California Department of Education. (n.d.). Retrieved October 28, 2018, from

https://www.cde.ca.gov/ Desimone, L. M., Porter, A. C., Garet, M. S., Yoon, K. S., & Birman, B. F. (2002).

Effects of Professional Development on Teachers’ Instruction: Results from a Three-year Longitudinal Study. Educational Evaluation and Policy Analysis, 24(2), 81-112. doi:10.3102/01623737024002081

Fairfax Network - Fairfax County Public Schools. (2013, December 11). Retrieved

October 28, 2018, from https://www.youtube.com/watch?v=XI3-52B0V6s Fletcher, G. (n.d.). Questioning My Metacognition. Retrieved October 20, 2018, from

https://gfletchy.com/ Guskey, T. R. (2003). What Makes Professional Development Effective? Phi Delta

Kappan, 84(10), 748-750. doi:10.1177/003172170308401007 Henningsen, M., & Stein, M. K. (1997). Mathematical Tasks and Student Cognition:

Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning. Journal for Research in Mathematics Education, 28(5), 524-5. doi:10.2307/749690

Lemlech, J. K. (1995). Becoming a professional leader. New York: Scholastic. Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers understanding

of fundamental mathematics in China and the United States. New York: Routledge.

McGraw-Hill Education (Firm). (2014). My math. Columbus, OH: McGraw-Hill

Education. Meyer, D. (2010, March). Retrieved October 15, 2018, from

https://www.ted.com/talks/dan_meyer_math_curriculum_makeover N. (2013). Common Core State Standards Mathematics. Retrieved September 9, 2018,

from https://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf National Council of Teachers of Mathematics. (2010). Professional Development

Research Brief: Mathematics Professional Development. Pp. 1 – 7. Reston, VA: NCTM.

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Ni, Y., Zhou, D. R., Cai, J., Li, X., Li, Q., & Sun, I. X. (2017). Improving cognitive and

affective learning outcomes of students through mathematics instructional tasks of high cognitive demand. The Journal of Educational Research,1-16. doi:10.1080/00220671.2017.1402748

Orrill, C. Hawley (2015, May 11). Tasks, Questions, and Practices. Retrieved December

8, 2018, from https://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/Tasks,-Questions,-and-Practices/

Schools, P. C. (2016, April 22). Retrieved October 28, 2018, from

https://www.youtube.com/watch?v=o9gybhyRPHw&t=197s Smith, M. S., & Stein, M. K. (1998). Reflections on Practice: Selecting and Creating

Mathematical Tasks: From Research to Practice. National Council of Teachers of Mathematics, 3(5), 344-350. Retrieved September 9, 2018, from www.jstor.org/stable/41180423

Stein, M. K., & Smith, M. S. (1998). Mathematical Tasks as a Framework for Reflection:

From Research to Practice. National Council of Teachers of Mathematics, 3(4), 268-275. Retrieved September 9, 2018.

Stein, M. K., Smith, M. S., & Henningsen, M. A. (2000). Implementation Standards-

Based Mathematics Instruction A Casebook for Professional Development (E. A. Silver, Ed.). New York, NY: Teachers College Press.

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Appendix A: High-Level Cognitive Demand Tasks

Task 1: A box of cereal contains 340 grams of carbohydrates. If there are 12 servings, about how many grams are there in one serving? Show how you estimated. Explain why your answer is reasonable. (MyMath, 2014, p. 254)

Task 2: Cheryl has a collection on sports cards and has 8 pages in the album. Each page of the album holds 14 sports cards. If she completely fills the album, how many sports cards does she have? Explain how you got your answer. (MyMath, 2014, p. 260)

Task 3: George is having a hard time solving division problems, and he has asked you for his help. Here is George’s strategy:

485 ÷ 4 = ? 4 ÷ 4 = 1 8 ÷ 4 = 2

5 ÷ 4 = 1 remainder 1 1 + 2 + 1 = 4

484 ÷ 4 = 4 r 1 What is George doing wrong? Explain how George can fix his strategy so that it works. (Don’t teach him a new strategy!!! Help him fix this one!) Why does this strategy work? In what contexts would this be a good strategy to use? When would this not be a good strategy to use? Explain your reasoning.

Task 4: An adult lion can eat a lot of meat in one sitting. If a pride of lions eats a water buffalo that has 1,182 pounds of meat, and each adult lion eats 66 pounds of meat, how many adult lions will the water buffalo feed? Will there be enough food left over to feed 4 cubs, if each cub needs 13 pounds of meat? Solve this problem using 2 different strategies. For each strategy, write a sentence to explain why your strategy works. (NC Department of Public Instruction: Formative Instruction and Assessment Tasks)

Task 5: Part 1: It’s the beginning of the school year and time to get supplies for the classroom. Mr. Burton has ordered some supplies for his students and is wondering if he has enough for his 32 students.

1.   Mr. Burton ordered 512 pencils. How many pencils will each student receive? Show your work by writing an equation or drawing an array or an area model.

2.   Mr. Burton ordered 12 boxes of ball point pens and each box contains 12 pens. He would like each student to have 5 pens. Did Mr. Burton order enough boxes? Explain why or why not.

(NC Department of Public Instruction: Formative Instruction and Assessment Tasks)

(NC Department of Public Instruction: Formative Instruction and Assessment Tasks)

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Appendix B: PowerPoint Presentations

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68

69

70

71

72

73

74

75

76

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Appendix C: Teacher Pre and Post Survey

1. I personally enjoy math.

o! Strongly agree o! Agree o! Neither agree nor disagree o! Disagree o! Strongly disagree

2. I feel confident teaching math.

o! Strongly agree o! Agree o! Neither agree nor disagree o! Disagree o! Strongly disagree

3. MyMath is all I need to teach math.

o! Strongly agree o! Agree o! Neither agree nor disagree o! Disagree o! Strongly disagree

4. My students are well prepared for SBAC.

o! Strongly agree o! Agree o! Neither agree nor disagree o! Disagree o! Strongly disagree

5. I have heard of high-level cognitive demand tasks.

o! Yes o! No

6. I am ready to go home!! o! Strongly agree o! Agree o! Neither agree nor disagree o! Disagree o! Strongly disagree

7. I am confident with implementing high-level cognitive demand tasks?

o! Strongly agree o! Agree o! Neither agree nor disagree o! Disagree o! Strongly disagree

8. I found this professional development to be effective.

o! Strongly agree o! Agree o! Neither agree nor disagree o! Disagree o! Strongly disagree

9. I will continue to use high-level cognitive demand tasks in my classroom.

o! Very likely o! Likely o! Neither likely nor unlikely o! Unlikely o! Very unlikely

10. I could use additional support for high-level cognitive demand tasks.

o! Yes o! No

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Appendix D: Mathematical Task Analysis Guide

!

!

Strategies for Modifying Tasks

Increasing the cognitive demands of tasks. • Ask students to create real-world stories for “naked number” problems. • Include a prompt that asks students to represent the information another way (with a picture, in a table, a graph, an equation, with a context). • Use a task “out of sequence” before students have memorized a rule or have practiced a procedure that can be routinely applied. • Eliminate components of the task that confine student thinking or provide too much scaffolding. • Create opportunities for repeated reasoning or pattern finding • Create a prompt that asks students to write about the meaning of the mathematics concept. • Add a prompt that asks students to make note of a pattern or to make a mathematical conjecture and to test their conjecture. • Include a prompt that requires students to make a generalization. • Include a prompt that requires students to compare solution paths or mathematical relationships and write about the relationship between strategies or concepts. • Select numbers carefully so students are more inclined to note relationships between quantities (e.g., two tables can be used to think about the solutions to the four, six, or eight tables).