## Mathematics Assessment Project

Assessing 21 st century math, welcome to the mathematics assessment project, materials from the math assessment project.

The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students’ mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress. The tools are relevant to any curriculum that seeks to deepen students' understanding of mathematical concepts and develop their ability to apply that knowledge to non-routine problems.

More about the Math Assessment Project

## Formative Assessment Lessons: Classroom Challenges

100 lessons for formative assessment, some focused on developing math concepts, others on solving non-routine problems. A Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

## Summative Assessment Tasks

A set of 94 exemplar summative assessment tasks to illustrate the range of performance goals required by CCSSM. The tasks come with scoring rubrics and examples of scored student work.

## Prototype Tests

Complete summative test forms and rubrics designed to help teachers and students monitor their progress using a range of task types similar to the 'Tasks' section.

## Professional Development Modules

5 Prototype modules that encourage groups of teachers to explore the practical and pedagogical concepts behind the materials, such as formative assessment, collaborative learning and the use of unstructured problems.

## The TRU Math Tools Suite

The Teaching for Robust Understanding of Mathematics (TRU Math) suite is a set of tools with applications in Professional Development and research based around a framework for characterizing powerful learning environments.

State, district and CCSSI standards appear courtesy of their respective authors. All other material Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham.

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## Designing Assignments for Learning

The rapid shift to remote teaching and learning meant that many instructors reimagined their assessment practices. Whether adapting existing assignments or creatively designing new opportunities for their students to learn, instructors focused on helping students make meaning and demonstrate their learning outside of the traditional, face-to-face classroom setting. This resource distills the elements of assignment design that are important to carry forward as we continue to seek better ways of assessing learning and build on our innovative assignment designs.

## On this page:

Rethinking traditional tests, quizzes, and exams.

- Examples from the Columbia University Classroom
- Tips for Designing Assignments for Learning

## Reflect On Your Assignment Design

Connect with the ctl.

- Resources and References

Cite this resource: Columbia Center for Teaching and Learning (2021). Designing Assignments for Learning. Columbia University. Retrieved [today’s date] from https://ctl.columbia.edu/resources-and-technology/teaching-with-technology/teaching-online/designing-assignments/

Traditional assessments tend to reveal whether students can recognize, recall, or replicate what was learned out of context, and tend to focus on students providing correct responses (Wiggins, 1990). In contrast, authentic assignments, which are course assessments, engage students in higher order thinking, as they grapple with real or simulated challenges that help them prepare for their professional lives, and draw on the course knowledge learned and the skills acquired to create justifiable answers, performances or products (Wiggins, 1990). An authentic assessment provides opportunities for students to practice, consult resources, learn from feedback, and refine their performances and products accordingly (Wiggins 1990, 1998, 2014).

Authentic assignments ask students to “do” the subject with an audience in mind and apply their learning in a new situation. Examples of authentic assignments include asking students to:

- Write for a real audience (e.g., a memo, a policy brief, letter to the editor, a grant proposal, reports, building a website) and/or publication;
- Solve problem sets that have real world application;
- Design projects that address a real world problem;
- Engage in a community-partnered research project;
- Create an exhibit, performance, or conference presentation ;
- Compile and reflect on their work through a portfolio/e-portfolio.

Noteworthy elements of authentic designs are that instructors scaffold the assignment, and play an active role in preparing students for the tasks assigned, while students are intentionally asked to reflect on the process and product of their work thus building their metacognitive skills (Herrington and Oliver, 2000; Ashford-Rowe, Herrington and Brown, 2013; Frey, Schmitt, and Allen, 2012).

It’s worth noting here that authentic assessments can initially be time consuming to design, implement, and grade. They are critiqued for being challenging to use across course contexts and for grading reliability issues (Maclellan, 2004). Despite these challenges, authentic assessments are recognized as beneficial to student learning (Svinicki, 2004) as they are learner-centered (Weimer, 2013), promote academic integrity (McLaughlin, L. and Ricevuto, 2021; Sotiriadou et al., 2019; Schroeder, 2021) and motivate students to learn (Ambrose et al., 2010). The Columbia Center for Teaching and Learning is always available to consult with faculty who are considering authentic assessment designs and to discuss challenges and affordances.

## Examples from the Columbia University Classroom

Columbia instructors have experimented with alternative ways of assessing student learning from oral exams to technology-enhanced assignments. Below are a few examples of authentic assignments in various teaching contexts across Columbia University.

- E-portfolios: Statia Cook shares her experiences with an ePorfolio assignment in her co-taught Frontiers of Science course (a submission to the Voices of Hybrid and Online Teaching and Learning initiative); CUIMC use of ePortfolios ;
- Case studies: Columbia instructors have engaged their students in authentic ways through case studies drawing on the Case Consortium at Columbia University. Read and watch a faculty spotlight to learn how Professor Mary Ann Price uses the case method to place pre-med students in real-life scenarios;
- Simulations: students at CUIMC engage in simulations to develop their professional skills in The Mary & Michael Jaharis Simulation Center in the Vagelos College of Physicians and Surgeons and the Helene Fuld Health Trust Simulation Center in the Columbia School of Nursing;
- Experiential learning: instructors have drawn on New York City as a learning laboratory such as Barnard’s NYC as Lab webpage which highlights courses that engage students in NYC;
- Design projects that address real world problems: Yevgeniy Yesilevskiy on the Engineering design projects completed using lab kits during remote learning. Watch Dr. Yesilevskiy talk about his teaching and read the Columbia News article .
- Writing assignments: Lia Marshall and her teaching associate Aparna Balasundaram reflect on their “non-disposable or renewable assignments” to prepare social work students for their professional lives as they write for a real audience; and Hannah Weaver spoke about a sandbox assignment used in her Core Literature Humanities course at the 2021 Celebration of Teaching and Learning Symposium . Watch Dr. Weaver share her experiences.

## Tips for Designing Assignments for Learning

While designing an effective authentic assignment may seem like a daunting task, the following tips can be used as a starting point. See the Resources section for frameworks and tools that may be useful in this effort.

## Align the assignment with your course learning objectives

Identify the kind of thinking that is important in your course, the knowledge students will apply, and the skills they will practice using through the assignment. What kind of thinking will students be asked to do for the assignment? What will students learn by completing this assignment? How will the assignment help students achieve the desired course learning outcomes? For more information on course learning objectives, see the CTL’s Course Design Essentials self-paced course and watch the video on Articulating Learning Objectives .

## Identify an authentic meaning-making task

For meaning-making to occur, students need to understand the relevance of the assignment to the course and beyond (Ambrose et al., 2010). To Bean (2011) a “meaning-making” or “meaning-constructing” task has two dimensions: 1) it presents students with an authentic disciplinary problem or asks students to formulate their own problems, both of which engage them in active critical thinking, and 2) the problem is placed in “a context that gives students a role or purpose, a targeted audience, and a genre.” (Bean, 2011: 97-98).

An authentic task gives students a realistic challenge to grapple with, a role to take on that allows them to “rehearse for the complex ambiguities” of life, provides resources and supports to draw on, and requires students to justify their work and the process they used to inform their solution (Wiggins, 1990). Note that if students find an assignment interesting or relevant, they will see value in completing it.

Consider the kind of activities in the real world that use the knowledge and skills that are the focus of your course. How is this knowledge and these skills applied to answer real-world questions to solve real-world problems? (Herrington et al., 2010: 22). What do professionals or academics in your discipline do on a regular basis? What does it mean to think like a biologist, statistician, historian, social scientist? How might your assignment ask students to draw on current events, issues, or problems that relate to the course and are of interest to them? How might your assignment tap into student motivation and engage them in the kinds of thinking they can apply to better understand the world around them? (Ambrose et al., 2010).

## Determine the evaluation criteria and create a rubric

To ensure equitable and consistent grading of assignments across students, make transparent the criteria you will use to evaluate student work. The criteria should focus on the knowledge and skills that are central to the assignment. Build on the criteria identified, create a rubric that makes explicit the expectations of deliverables and share this rubric with your students so they can use it as they work on the assignment. For more information on rubrics, see the CTL’s resource Incorporating Rubrics into Your Grading and Feedback Practices , and explore the Association of American Colleges & Universities VALUE Rubrics (Valid Assessment of Learning in Undergraduate Education).

## Build in metacognition

Ask students to reflect on what and how they learned from the assignment. Help students uncover personal relevance of the assignment, find intrinsic value in their work, and deepen their motivation by asking them to reflect on their process and their assignment deliverable. Sample prompts might include: what did you learn from this assignment? How might you draw on the knowledge and skills you used on this assignment in the future? See Ambrose et al., 2010 for more strategies that support motivation and the CTL’s resource on Metacognition ).

## Provide students with opportunities to practice

Design your assignment to be a learning experience and prepare students for success on the assignment. If students can reasonably expect to be successful on an assignment when they put in the required effort ,with the support and guidance of the instructor, they are more likely to engage in the behaviors necessary for learning (Ambrose et al., 2010). Ensure student success by actively teaching the knowledge and skills of the course (e.g., how to problem solve, how to write for a particular audience), modeling the desired thinking, and creating learning activities that build up to a graded assignment. Provide opportunities for students to practice using the knowledge and skills they will need for the assignment, whether through low-stakes in-class activities or homework activities that include opportunities to receive and incorporate formative feedback. For more information on providing feedback, see the CTL resource Feedback for Learning .

## Communicate about the assignment

Share the purpose, task, audience, expectations, and criteria for the assignment. Students may have expectations about assessments and how they will be graded that is informed by their prior experiences completing high-stakes assessments, so be transparent. Tell your students why you are asking them to do this assignment, what skills they will be using, how it aligns with the course learning outcomes, and why it is relevant to their learning and their professional lives (i.e., how practitioners / professionals use the knowledge and skills in your course in real world contexts and for what purposes). Finally, verify that students understand what they need to do to complete the assignment. This can be done by asking students to respond to poll questions about different parts of the assignment, a “scavenger hunt” of the assignment instructions–giving students questions to answer about the assignment and having them work in small groups to answer the questions, or by having students share back what they think is expected of them.

## Plan to iterate and to keep the focus on learning

Draw on multiple sources of data to help make decisions about what changes are needed to the assignment, the assignment instructions, and/or rubric to ensure that it contributes to student learning. Explore assignment performance data. As Deandra Little reminds us: “a really good assignment, which is a really good assessment, also teaches you something or tells the instructor something. As much as it tells you what students are learning, it’s also telling you what they aren’t learning.” ( Teaching in Higher Ed podcast episode 337 ). Assignment bottlenecks–where students get stuck or struggle–can be good indicators that students need further support or opportunities to practice prior to completing an assignment. This awareness can inform teaching decisions.

Triangulate the performance data by collecting student feedback, and noting your own reflections about what worked well and what did not. Revise the assignment instructions, rubric, and teaching practices accordingly. Consider how you might better align your assignment with your course objectives and/or provide more opportunities for students to practice using the knowledge and skills that they will rely on for the assignment. Additionally, keep in mind societal, disciplinary, and technological changes as you tweak your assignments for future use.

Now is a great time to reflect on your practices and experiences with assignment design and think critically about your approach. Take a closer look at an existing assignment. Questions to consider include: What is this assignment meant to do? What purpose does it serve? Why do you ask students to do this assignment? How are they prepared to complete the assignment? Does the assignment assess the kind of learning that you really want? What would help students learn from this assignment?

Using the tips in the previous section: How can the assignment be tweaked to be more authentic and meaningful to students?

As you plan forward for post-pandemic teaching and reflect on your practices and reimagine your course design, you may find the following CTL resources helpful: Reflecting On Your Experiences with Remote Teaching , Transition to In-Person Teaching , and Course Design Support .

The Columbia Center for Teaching and Learning (CTL) is here to help!

For assistance with assignment design, rubric design, or any other teaching and learning need, please request a consultation by emailing [email protected] .

Transparency in Learning and Teaching (TILT) framework for assignments. The TILT Examples and Resources page ( https://tilthighered.com/tiltexamplesandresources ) includes example assignments from across disciplines, as well as a transparent assignment template and a checklist for designing transparent assignments . Each emphasizes the importance of articulating to students the purpose of the assignment or activity, the what and how of the task, and specifying the criteria that will be used to assess students.

Association of American Colleges & Universities (AAC&U) offers VALUE ADD (Assignment Design and Diagnostic) tools ( https://www.aacu.org/value-add-tools ) to help with the creation of clear and effective assignments that align with the desired learning outcomes and associated VALUE rubrics (Valid Assessment of Learning in Undergraduate Education). VALUE ADD encourages instructors to explicitly state assignment information such as the purpose of the assignment, what skills students will be using, how it aligns with course learning outcomes, the assignment type, the audience and context for the assignment, clear evaluation criteria, desired formatting, and expectations for completion whether individual or in a group.

Villarroel et al. (2017) propose a blueprint for building authentic assessments which includes four steps: 1) consider the workplace context, 2) design the authentic assessment; 3) learn and apply standards for judgement; and 4) give feedback.

## References

Ambrose, S. A., Bridges, M. W., & DiPietro, M. (2010). Chapter 3: What Factors Motivate Students to Learn? In How Learning Works: Seven Research-Based Principles for Smart Teaching . Jossey-Bass.

Ashford-Rowe, K., Herrington, J., and Brown, C. (2013). Establishing the critical elements that determine authentic assessment. Assessment & Evaluation in Higher Education. 39(2), 205-222, http://dx.doi.org/10.1080/02602938.2013.819566 .

Bean, J.C. (2011). Engaging Ideas: The Professor’s Guide to Integrating Writing, Critical Thinking, and Active Learning in the Classroom . Second Edition. Jossey-Bass.

Frey, B. B, Schmitt, V. L., and Allen, J. P. (2012). Defining Authentic Classroom Assessment. Practical Assessment, Research, and Evaluation. 17(2). DOI: https://doi.org/10.7275/sxbs-0829

Herrington, J., Reeves, T. C., and Oliver, R. (2010). A Guide to Authentic e-Learning . Routledge.

Herrington, J. and Oliver, R. (2000). An instructional design framework for authentic learning environments. Educational Technology Research and Development, 48(3), 23-48.

Litchfield, B. C. and Dempsey, J. V. (2015). Authentic Assessment of Knowledge, Skills, and Attitudes. New Directions for Teaching and Learning. 142 (Summer 2015), 65-80.

Maclellan, E. (2004). How convincing is alternative assessment for use in higher education. Assessment & Evaluation in Higher Education. 29(3), June 2004. DOI: 10.1080/0260293042000188267

McLaughlin, L. and Ricevuto, J. (2021). Assessments in a Virtual Environment: You Won’t Need that Lockdown Browser! Faculty Focus. June 2, 2021.

Mueller, J. (2005). The Authentic Assessment Toolbox: Enhancing Student Learning through Online Faculty Development . MERLOT Journal of Online Learning and Teaching. 1(1). July 2005. Mueller’s Authentic Assessment Toolbox is available online.

Schroeder, R. (2021). Vaccinate Against Cheating With Authentic Assessment . Inside Higher Ed. (February 26, 2021).

Sotiriadou, P., Logan, D., Daly, A., and Guest, R. (2019). The role of authentic assessment to preserve academic integrity and promote skills development and employability. Studies in Higher Education. 45(111), 2132-2148. https://doi.org/10.1080/03075079.2019.1582015

Stachowiak, B. (Host). (November 25, 2020). Authentic Assignments with Deandra Little. (Episode 337). In Teaching in Higher Ed . https://teachinginhighered.com/podcast/authentic-assignments/

Svinicki, M. D. (2004). Authentic Assessment: Testing in Reality. New Directions for Teaching and Learning. 100 (Winter 2004): 23-29.

Villarroel, V., Bloxham, S, Bruna, D., Bruna, C., and Herrera-Seda, C. (2017). Authentic assessment: creating a blueprint for course design. Assessment & Evaluation in Higher Education. 43(5), 840-854. https://doi.org/10.1080/02602938.2017.1412396

Weimer, M. (2013). Learner-Centered Teaching: Five Key Changes to Practice . Second Edition. San Francisco: Jossey-Bass.

Wiggins, G. (2014). Authenticity in assessment, (re-)defined and explained. Retrieved from https://grantwiggins.wordpress.com/2014/01/26/authenticity-in-assessment-re-defined-and-explained/

Wiggins, G. (1998). Teaching to the (Authentic) Test. Educational Leadership . April 1989. 41-47.

Wiggins, Grant (1990). The Case for Authentic Assessment . Practical Assessment, Research & Evaluation , 2(2).

## Wondering how AI tools might play a role in your course assignments?

See the CTL’s resource “Considerations for AI Tools in the Classroom.”

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## Mathematics Classroom Assessment: A Framework for Designing Assessment Tasks and Interpreting Students’ Responses

Classroom assessment could contribute substantially to improving students’ mathematics learning. The process of classroom assessment involves decisions about how to elicit evidence, how to interpret it, and how to use it for teaching and learning. However, the field still needs to further explore how assessment tasks could guide forthcoming instructional adjustments in the mathematics classroom. Towards the endeavor of unpacking the classroom assessment, we present a framework that provides a lens to capture the interplay between the design of mathematics assessment tasks and the analysis of students’ responses. To do so, we relied on existing frameworks of mathematics assessment tasks, and on issues that pertain to the design of tasks. The proposed framework consists of three types of mathematics assessment tasks, their respective competencies, and the characterization of students’ responses. The framework is exemplified with students’ responses from a fourth-grade classroom, and is also used to sketch different students’ profiles. Issues regarding the interpretation of students’ responses and the planning of instructional adjustments are discussed.

## 1. Introduction

1.1. scope of the paper.

Classroom assessment serves as a process for gathering and interpreting evidence to identify students’ progress in order to make decisions about forthcoming actions in day-to-day teaching [ 1 , 2 ]. It would be particularly useful to have lenses that support the interpretation of the evidence elicited from students’ responses in a systematic manner, in order to better understand where students stand on the learning continuum, and in what ways students’ learning could be enhanced. In this paper, we focus on a planned process—that of written assessment tasks. Mathematics tasks differ based on the expected cognitive demand, the mathematical competencies, the thinking processes, the solution strategies, the level of students’ understanding that determine the ways students respond to the tasks, and the kind of information elicited. However, there are limitations in existing frameworks, due to placing less emphasis on the interplay between mathematics assessment tasks and ways of interpreting the evidence elicited from the tasks that could lead to the decision making regarding forthcoming instructional adjustments. Limitations also result from not attending to the bounded classroom context and the descriptive features of students’ work.

Herein, we present a two-dimensional framework that attempts to align the design of assessment tasks with the ways students’ responses could be analyzed. The design of the tasks is intended to elicit information about students’ competencies in mathematics tasks with various levels of contextual familiarity. We also investigate the ways in which students’ responses could be analyzed. The contribution of this framework lies in the identification of a selected competency for each type of mathematics task, which is then used to characterize students’ work. In this way, the framework attempts to characterize descriptive features of students’ work, which are more likely to provide information for effective feedback [ 3 ]. We also use empirical data from fourth-grade students to sketch students’ profiles, and then turn to discuss how it would be possible to align these profiles with forthcoming instructional adjustments.

## 1.2. Literature Review

We focus on classroom assessment for formative purposes, using tasks of everyday mathematics to elicit evidence of students’ learning. We then review existing frameworks to identify important components of mathematics assessment tasks. We discuss issues that pertain to the design of assessment tasks and, finally, elaborate on the interpretation of students’ responses and teachers’ actions.

## 1.2.1. Classroom Assessment

Research suggests that classroom assessment practices for formative purposes have the potential to increase student achievement, and to lead to long-term gains in mathematics performance [ 1 , 4 ]. Particularly, the use of assessment data to individualize instruction has been associated with significant increases in students’ achievement [ 5 ]. It has also been found that the speed of learning can be doubled, and the gap between high and low achievers can be reduced [ 1 , 6 ]. Assessment techniques that are embedded within the classroom instruction have also been shown to support teachers in developing better understanding of students’ thinking and misconceptions [ 7 ].

Classroom assessment for formative purposes consists of eliciting evidence using instruments that are aligned with instruction and the specific domain, identifying patterns in students’ learning, combining the evidence with general principles to provide meaningful feedback, embedding the assessment into the instructional practice, and designing instructional adjustments [ 8 , 9 , 10 , 11 ]. However, less emphasis has been placed on the aspect of instructional adjustments [ 12 ].

Classroom assessment for formative purposes could range from formative assessment lessons [ 13 ] to cognitive diagnostic items [ 14 ]. Formative assessment lessons present a rather integrated ongoing approach of assessment [ 13 ], while cognitive diagnostic items are intended to provide fine-grained analysis of students’ cognitive knowledge [ 15 ]. Classroom assessment practices also include journal reflection, questioning techniques, challenging assignments, assessment tasks, and open-ended performance tasks [ 16 , 17 ]. The various approaches tend to capture students’ learning in order to connect assessment with instruction [ 18 ]. The interpretation could be based on the identification of misconceptions, the categorization of students’ strategies, and the quality of students’ arguments.

Among the various practices for eliciting evidence of students’ current learning, the focus in this paper is on a planned process—that of written assessment tasks. Tasks could provide information about where students stand in terms of learning progression, as well as their levels of understanding [ 19 ]. Mathematics tasks for assessment purposes could be used in everyday mathematics teaching, depending on what was taught and what the teacher intends to assess. Tasks should not necessarily form a test, but independent tasks have the potential to provide chunks of information regarding students’ learning in terms of the teaching and learning processes. Empirical results suggest that, within the context of one unit in primary school, it is possible to employ rather extensive formative assessment practices [ 20 ].

## 1.2.2. Frameworks for Mathematics Assessment Tasks

We review and analyze frameworks that are relevant to the design and analysis of assessment tasks ( Table 1 ). The first framework, “classroom challenges”, presents four genres of tasks, and was designed to assess and enhance students’ ability to solve multistep, non-routine problems [ 13 ]. The second framework presents three levels of thinking [ 21 ]. The first framework aligns with a rather radical approach to the classroom culture—that of designing whole lessons of formative assessment—while the second seems to focus more on the design and selection of independent tasks. In this paper, our approach to formative assessment aligns more closely with the design of tasks instead of lessons, as we regard this as an intermediary step along the endeavor of integrating formative assessment in school classrooms.

Existing frameworks for assessment tasks.

We also review frameworks of assessment tasks that are widely used—even in large-scale studies, mainly for summative purposes—to identify important components that need to be taken into consideration. We agree with Thompson et al. that “a given assessment task can be either formative or summative, depending on how the information gathered from that task is used” [ 22 ] (p. 4). Harlen also suggests that assessment information could be used “for both summative and formative purposes, without the use for one purpose endangering the effectiveness of use for the other” [ 23 ] (p. 215).

Most of these frameworks, including the two aforementioned ones, seem to place emphasis on important mathematical processes, and on procedural and conceptual aspects of mathematical ideas. The MATH taxonomy is a modification of Bloom’s taxonomy for structuring mathematics assessment tasks, and describes the skills that a particular task assesses [ 24 ]. Bloom et al. developed a taxonomy for the design and assessment of classroom activities that consist of knowledge, comprehension, application, analysis, synthesis, and evaluation [ 25 ]. The TIMSS framework was developed for the purpose of large-scale assessments to compare students’ mathematics achievement in different cognitive domains and content areas, identify trends in students’ performance, and inform evidence-based decisions for improving educational policy and practice across countries over more than 25 years [ 26 ]. The MATH taxonomy, TIMSS framework, and de Lange levels seem to have been influenced by Bloom’s taxonomy. Furthermore, the QCAI framework was designed to assess students’ understanding, reasoning, problem solving, and communication in different content areas in order to measure growth in mathematics over time [ 27 ]. Finally, the SPUR framework suggests that teachers need to assess understanding of the mathematical content that they teach from four dimensions in order to ensure a balanced perspective in teaching and assessment: algorithms and procedures (skills), underlying principles (properties), applications (uses), and diagrams, pictures, or other visual representations (representation) [ 28 ].

The first five frameworks attempt to highlight the nature of mathematics by incorporating important mathematical processes such as problem solving, reasoning and proof, communication, connections, and representation. The last five frameworks refer to the application of procedures or skills in various ways to mathematical concepts or relationships. The second, third, and fourth frameworks more clearly incorporate the idea of assessing students from reproduction to application, and then to mathematical reasoning.

The categorization of tasks in the frameworks above is informative about the kinds of processes that students would need to engage with throughout the assessment tasks. However, the mere categorization into the types of knowledge or processes presents limitations when it comes to how a classroom teacher could be informed about their students’ learning during a series of lessons on a mathematical idea. Assessment for formative purposes is administered according to students’ needs, and is closely associated with the curriculum [ 29 ]. Sociocognitive and sociocultural theories also seem more suitable for classroom assessment—particularly for achieving alignment between the curriculum and the classroom instruction [ 29 ]. Hence, the proposed framework in this paper relies on these existing frameworks, but also attempts to move a step further by making links between the processes and the interpretation of students’ responses. We aimed for a framework that sheds light on students’ emergent, robust, or even fragmented understanding as they engage with mathematical ideas within a classroom community.

## 1.2.3. Design of Assessment Tasks

The assessment tasks should be meaningful and worthwhile opportunities to learn, as well as being accessible to students [ 30 ]. They should drive classroom learning activities and indicate what kinds of instruction should be encouraged [ 28 , 30 , 31 ]. Tasks that are intended to elicit students’ thinking are usually longer than typical tasks—such as multiple-choice tasks—and take more time to complete, since they engage students with a higher cognitive load [ 31 ]. It is inevitable that different types of tasks provide different types of evidence regarding students’ understanding. Shorter tasks could be used to provide instant feedback to the teacher about students’ understanding, while longer tasks could provide insight into students’ thinking, and opportunities for classroom discussions.

Students’ previous experiences and familiarity with the mathematical idea(s) being assessed could change the expected student processes [ 32 ]. Students tend to solve tasks that share critical properties with textbooks’ tasks by recalling facts and procedures, while they use creative reasoning for those tasks that do not share those critical properties [ 33 ]. The structure, with respect to the level of openness of the task, is another element to be taken into consideration [ 34 ]. Structuring the task into successive parts lowers its intended demand [ 31 ]. The amount and complexity of textual and visual information, such as the use of terminology and complex sentences, increase students’ reading load [ 35 ]. The complexity of the task could also be determined by the number of steps and variables [ 31 ]. Overall, the way the language is used, as well as the forms of the questions in the tasks, relate to how students engage with them [ 34 ].

Another issue is the context in which the tasks are framed. On the one hand, the context could make the task accessible to students, and give them latitude to display what they know [ 30 ]; on the other hand, the context creates challenges in students’ engagement and in the decision making of the task design (e.g., whether the context plays critical role in the mathematization process) [ 21 ]. These design issues also moderate the feedback that the teacher receives based on students’ engagement with the task. However, the way in which this happens is poorly understood [ 6 ].

## 1.2.4. Interpretation of Students’ Responses and Teachers’ Actions

The analysis of students’ responses in mathematics assessment tasks needs further study in order to lead to meaningful insight that informs teachers about forthcoming instructional adjustments. Pellegrino, Chudowsky, and Glaser mention that “cognition, observation, and interpretation must be explicitly connected and designed as a coordinated whole. If not, the meaningfulness of inferences drawn from the assessment will be compromised” [ 36 ] (p. 2). Indeed, “good teaching decisions are based on high-quality information” [ 37 ] (p. 100).

Analytic rubrics could be used to interpret students’ responses in tasks, which result in identifying elements that should be included in the response [ 37 ]. Another approach is the use of holistic rubrics, in which the overall quality of students’ work is assigned to predetermined categories [ 37 ]. For example, rubrics have been used to support teachers and students to provide feedback for students’ competencies, and to help both understand the competencies required [ 38 ]. Rubrics support the feedback process which, in turn, seems to have a major impact on students’ learning [ 39 ].

Teachers who have a better understanding of the learning goals might design richer learning experiences, be more prepared to provide effective formative feedback, and plan remediation instruction. Teachers’ forthcoming adjustments based on elicited evidence could include immediate modification of instructional decisions, planning instructional activities, diagnosing learning difficulties, placing students into learning sequences, recording for later use, and even eliciting further evidence [ 37 ]. Teachers need to know how to ensure that the inferences made from assessment tasks are of sufficient quality to understand where the learner is along the learning continuum, and to inform decisions about the next instructional steps to be taken [ 40 ].

Instructional actions that are effective in supporting students’ learning of procedures and skills would differ from those that are appropriate for developing students’ understanding and sense-making [ 41 ]. Tasks with a lower level of challenge may help students to engage easily with classroom activities, as may tasks with multiple representations or solving processes [ 42 ]. Moreover, too many challenging tasks in a limited time may demotivate students, even if such tasks promote mathematical reasoning [ 42 ]. The types of tasks, the variation in challenge level, and the timing are issues to be considered when planning instructional adjustments to support the learning of mathematics.

## 1.2.5. Aims of the Paper

The purpose of this paper is twofold: first, it aims to present a framework, and second, to examine its application for classroom assessment. The framework provides a tool and the relevant language for designing mathematics assessment tasks and analyzing students’ responses to them. The framework is exemplified with students’ actual responses in assessment tasks in order to develop insight into how the framework could be employed to explore students’ learning of the mathematical idea(s) under study. To do so, we sketch students’ profiles, and then use the framework to set the grounds for making hypotheses for further instructional adjustments.

## 2.1. Proposed Framework

The framework is presented in Table 2 ; it aligns the design of mathematics assessment tasks with the analysis of students’ responses. The first column presents the names of the three types of tasks—reproduction, application, and generation and reflection tasks. In the second column, we refer to the mathematical processes that students are expected to engage with. These processes are partially determined by the contextual familiarity, which is presented in the third column of the table. The contextual familiarity relies on the previous teaching and learning experiences in the classroom, which are known to the classroom teacher. Students’ responses in each type of task are analyzed through a selected competency shown in the fourth column. Then, students’ responses are characterized using the descriptions presented in the fifth column.

Framework for classroom assessment tasks.

## 2.1.1. Expected Processes

Several of the frameworks presented in Table 1 appear to agree in assessing students from reproduction tasks to higher level thinking tasks. We relied on the categorization of the task processes in these frameworks to define the expected processes for the three types of tasks, and then further refined these processes with reference to the contextual familiarity. Herein, each process is described according to how students are expected to engage with mathematical ideas. Hence, the processes are operationalized with consideration of the previous teaching and learning experiences in the classroom. Mathematical ideas include facts, rules, definitions, and procedures.

In reproduction tasks, students are expected to rely on recalling mathematical ideas. The minimum requirement is reliance on memory, since the contextual familiarity is that students have had extensive practice with these mathematical ideas (e.g., repeating the same definition in classroom, practicing the multiplication tables). Students may respond not only by reproducing, but also by reconstructing mathematical ideas. Such tasks are part of everyday mathematics teaching, and could inform the teacher whether students are able to respond to tasks that they have practiced extensively.

In application tasks, students are expected to apply mathematical ideas. It does not suffice for students to reproduce taught ideas; they need to decide which mathematical ideas to use, and in what way to use them, according to the format of the task. In detail, the variation in the format of the task creates the need to make inferences and adjust the taught mathematical ideas accordingly.

In generation and reflection tasks, students are expected to reflect on mathematical ideas and generate arguments, justifications, strategies, and models. In such a task, “it requires a process of stepping back and reflection on the information being connected” [ 43 ] (p. 5). Students need to decide not only how to adjust the mathematical ideas to the format of the task, but also how to make sense of the structure of mathematics.

Identifying tasks that correspond to these three types of processes relies on the expected formulation of the tasks. For example, a reproduction task for second-grade students might be an application task for first-grade students. An application task might also engage students in reproducing a known algorithm. Hence, we relied on identifying the expected processes by modifying the approach of the “expected formulation of tasks” for the case of assessment tasks. “The expected formulation of a task represents the path the students in a particular classroom community are anticipated to follow if their community engaged with the task in the ways designed in the curricular resource from which the task was derived” [ 44 ] (p. 70). For the case of assessment tasks, the expected formulation of a task relies on the path that students are anticipated to follow based on what preceded in the lesson plan, and the curriculum materials used in the classroom.

## 2.1.2. Contextual Familiarity

Mathematical knowledge is developed through the personal (mental) and the institutional (contextual) dimensions [ 45 ]. Hence, the assessment needs to be relevant to the context in which the student participates [ 46 ]. We delineate the adaptation of the “expectation formulation” of assessment tasks by focusing on the contextual familiarity, and in particular determining how familiar the format of the task is, as well as the work procedure to complete the task. We relied only on students’ prior experiences in the classroom, which are known to the teacher, and acknowledge the limitation that students have further experiences from prior grades and the home environment. Since the framework also aims to become a tool for classroom teachers, we focused on a rather simple categorization of the format and the work procedure as “familiar” or “unfamiliar”.

The format of the task refers to how the request of the task is presented, and how the information is given. The format could change due to variation in representations, scenarios, the number of steps, or examples of numbers/shapes. The work procedure refers to the steps for completing the task. In reproduction tasks, both the format and the work procedure are expected to be familiar. The familiarity results from extensive opportunities for practice. In application tasks, the format is expected to be unfamiliar, while the work procedure is expected to be familiar. Thus, students need to identify how to use the taught mathematical idea(s) in an unfamiliar format, but afterwards, the procedure to complete the task is expected to be familiar. The unfamiliar format needs to be substantially different, often in a nuanced way, depending on the mathematical idea(s) under study (e.g., relying on students’ common misconceptions). In generation and reflections tasks, both the format and the work procedure are expected to be unfamiliar. Hence, students not only need to interpret and identify what kind of taught mathematical idea is relevant to the task, but they also need to construct a series of steps to reach a conclusion.

## 2.1.3. Competency

In the context of this study, we defined as a competency per type of task the mechanism that acts as a lens to analyze students’ responses. In mathematics education, there is great consensus that students need to engage in representation, reasoning and proving, communication, problem solving, generalization, making connections, and modelling [ 27 , 47 ]. These are called processes, practices, or competencies [ 48 ], and also appear in the majority of the frameworks presented in Table 1 . However, for the purpose of classroom assessment, we identified constraints in identifying, for example, at which point of the learning continuum a student is at problem solving for a taught mathematical idea (e.g., addition of fractions). Another constraint was that communication and representation, for example, could be seen as media that convey students’ thinking as identified in different types of tasks. Furthermore, we aimed to identify competencies that could be applicable to a range of mathematical topics for primary mathematics, and could also be used for the characterization of students’ responses. The selected competencies for this framework are fluency, flexibility, and reasoning (fourth column in Table 2 ).

For reproduction tasks, the teacher would intend to explore how fluent the student is in recalling taught mathematical ideas, considering their extent of practice and familiarity with the task. For application tasks, the focus is on students’ flexibility, as the teacher would intend to elicit how students’ mathematical ideas are adapted, related, kept coherent, and “freed from specific contexts” [ 43 ] (p. 3) in various task formats. Generation and reflection tasks turn the focus to students’ reasoning. Reasoning is a common term in mathematics education, often having a meaning close to thinking. Here, reasoning is the production of assertions and justified inferences to reach conclusions using, for example, deductive, inductive, and abductive processes [ 49 ].

## 2.1.4. Characterization of Students’ Responses

Based on the selected competency for each type of task, the framework presents characterizations of students’ responses. The characterization relies on snapshots of aspects of students’ learning being assessed in the assessment tasks ( Table 3 ). The evidence from students’ responses to a reproduction task could indicate developed fluency, developing fluency, or limited fluency. In the same way, the evidence from an application task indicates developed flexibility, developing flexibility, or limited flexibility, while evidence from a generation and reflection task could suggest developed reasoning, developing reasoning, or limited reasoning.

Characterization of students’ responses.

The framework could be viewed horizontally and vertically in a dynamic fashion. Students’ responses could be compared along the continuum in order to identify how students respond to the same tasks (vertical interpretation). Hence, the teacher could decide on how the whole class performs to the processes of different tasks. Students’ responses could also be used to describe their profiles (horizontal interpretation). Hence, the teacher could decide on what instructional adjustments are most appropriate for each student.

## 2.2. Development of the Framework

The development of the framework started with the analysis of existing frameworks, and the mathematical ideas under study, by examining the mathematical standards in the curriculum, the terminology, the expected representations, and students’ common misconceptions. Students’ familiarity with the tasks was determined by exploring the types of tasks found in textbooks—since teachers and students rely extensively on the unique textbook series used in all state schools in the educational context under study—as well as teachers’ lesson plans when these were available [ 50 ]. We also explored the content quality by looking at whether the content was sufficiently consistent with the current priorities of the field of mathematics education in order for the tasks to be worthwhile [ 51 ]. We also discussed with mathematics education experts what kinds of evidence each assessment task was meant to elicit. Two mathematics educators, who are experienced in the design of tasks for primary mathematics, advised us on the design and analysis of tasks. Then, we turned our attention to the task features and the specification of the tasks by considering issues that pertain to the design of tasks for classroom assessment. Further on, we implemented the assessment tasks and piloted the characterization of students’ responses. We administered 161 tasks to 5 classrooms from grade 4 to grade 6 over the course of a whole school year. The assessment tasks were administered in collaboration with the classroom teachers when the mathematical ideas assessed in the tasks were taught in the respective classrooms. Students solved the tasks independently. We then analyzed students’ responses to explore whether their responses revealed the expected processes [ 51 ]. Herein, we present the final version of the framework, and empirical data from one classroom, to illustrate the application of the framework for classroom assessment.

## 2.3. Design of Assessment Tasks

We exemplify the framework with assessment tasks on multidigit multiplication, and discuss the analysis of students’ responses. The origin of multiplication is based on repeated addition and the schema of correspondences [ 52 ]; it is a binary operation with two distinctive inputs, and students need to coordinate the multiplicand (number of elements in each set) and the multiplier (number of such sets), along with the procedure to find the product [ 53 ].

Multidigit multiplication includes a series of steps for finding the product, and relies on extending single-digit multiplication [ 54 ]. Students need to achieve two coordinations: the first coordination is between the magnitudes of factors and the magnitudes of products, while the second coordination is between the expanded forms of factors and the distributive property [ 54 ]. Multiplication methods rely on multiplying digits—either manipulating the digits in their expanded form (e.g., 3 in 36 as 3 tens, or 30) or manipulating them as single digits. Particularly, students’ understanding of the distributive property prepares them for finding the product in multidigit multiplication [ 55 ] in fractions and algebra [ 54 ]. The different types of situations that involve multiplication are equal groups, multiplicative comparison, Cartesian product, and rectangular area [ 56 ].

The assessment tasks for multidigit multiplication were designed and selected based on the contextual familiarity. The decisions were based primarily on the examination of textbooks, since the teaching approach depicted in textbooks is anticipated to be the dominant one in classrooms since teachers, in the educational context in this study, rely heavily on textbooks for planning and implementing their lessons [ 50 ]. In fourth-grade textbooks, the lessons begin with how to use single-digit multiplications to find multidigit multiplications in which one of the factors is a multiple of 10, using the commutative and associative properties. Then, attention turns towards strategies for estimating the product. Afterwards, rectangular arrays are used to find the product of two- and three-digit numbers with one-digit numbers. This approach is then linked with the distributive property of multiplication over addition and subtraction. Finally, the lessons probe students to explore different forms of vertical algorithms (e.g., expanded forms and shorter forms), before reaching the standard algorithm. Below, we present four assessment tasks.

The “reproduction task” (RT) is shown in Figure 1 ; it explores whether students could reproduce two different methods to find the product of a two-digit number by one-digit numbers. Both the standard algorithm and the use of the distributive property are expected to have been taught and practiced beforehand in the classroom. Hence, students are anticipated to have extensive familiarity with the format of the task and the work procedure.

Reproduction task.

The first “application task” (AT1) is shown in Figure 2 . The task intends to engage students in comparing mathematical expressions in different forms (e.g., varying the place of addition and multiplication symbols, the place of digits). The first set of expressions intends to explore whether students understand the distributive property, and whether they would consider the expression (5 + 54) × (1 + 54) as equivalent to 6 × 54. The second set of expressions intends to investigate whether students would inappropriately apply the commutative property by ignoring the place value of numbers. Students are asked to explain their rationale in order to provide further insight into their thinking.

Application task 1.

The second “application task” (AT2) is shown in Figure 3 . The task asks students to use the given information (i.e., 34 × 9 = 306) to find the product in the other expressions, where either the multiplier or the multiplicand differs. The task intends to engage students in adjusting the procedure of the distributive property, since they are asked not to analyze one of the factors in tens and units, but to analyze them according to the given information. In the two application tasks, the work procedure is familiar, but the format of the tasks is unfamiliar, since they have to interpret the given information carefully and adapt the known algorithms.

Application task 2.

The “generation and reflection task” (GRT) is shown in Figure 4 . Students are asked to form an argument to justify whether they agree or disagree with the given statement by exploring how the numerical structure of the factors relates to the product. They are expected to reflect on the structure of the numbers, and to find a counterexample. They also need to verbalize their argument. The format of task is unfamiliar, as is the work procedure, since students need to decide how to work in order to justify an answer.

Generation and reflection task.

We present the analysis of students’ responses from one fourth-grade classroom with 21 students in order to exemplify the application of the framework for classroom assessment. We elaborate on the process of analysis, as well as the vertical and horizontal perspectives of the framework.

## 3.1. Process of Analysis

The analysis of students’ responses followed two stages: (1) one researcher used the characterizations to code the students’ work; (2) the other two researchers independently coded a sample of students’ responses. Any discrepancies were discussed with the whole group of researchers until consensus was reached.

## 3.2. Vertical Perspective

The vertical perspective of the framework provides an overall picture of the classroom ( Table 4 ). The analysis suggests that the majority of the students have developed fluency in using the procedure for finding the product. However, the class needs to work further on adapting the procedure to different formats, since 16 students showed limited or developing flexibility. The majority of students also showed limited reasoning. Hence, the results suggest that the teaching needs to focus on instructional actions to enhance students’ flexibility and reasoning.

Vertical perspective.

## 3.3. Horizontal Perspective

We also elaborate on the horizontal perspective of the framework by presenting selected students’ profiles—namely, the profiles of Lina, Manolis, Eleonora, Evita, and Makis ( Figure 5 ). The selection aimed to (1) illustrate all of the different characterizations (presented in Table 4 ) by relying on students’ responses (i.e., developing and developed fluency; limited, developing, and developed flexibility; and limited, developing, and developed reasoning), and (2) reveal different profiles of students according to how they responded across the tasks. For example, Lina showed developing fluency, and limited flexibility and reasoning, while Makis also showed developing fluency, but developing flexibility and developed reasoning. In this way, it is then possible to compare different students’ profiles, and to use the profiles as cases for discussing instructional adjustments.

Students’ profiles.

Lina. Lina used the taught procedures in the RT, but made computational mistakes when using the distributive property of multiplication over addition and subtraction ( Figure 6 ). It was not possible for her to apply the procedure of multiplication in the ATs. Lina mentioned that 6 × 54 was greater than (5 + 54) × (1 + 54), without converting the second expression into a comparable form (e.g., 59 × 55) to the first one. Similarly, Lina did not use the procedure flexibly to compare the expressions 42 × 9 and 49 × 2. In the AT2, Lina applied the distributive property by splitting the number into tens and digits, without adapting the procedure flexibly based on the given product. In the GRT, Lina mentioned that the product of a two-digit number by 2 is a two-digit number. Lina is developing fluency, but limited flexibility and reasoning are evident in these four tasks for the concept of multiplication under study. In total, seven students had the same profile as Lina.

Lina’s responses.

Manolis. Manolis showed a systematic method of work in the RT ( Figure 7 ). In the AT1, he compared the multiplicands in the first set of expressions, while he tried to find the product in the latter set to compare the expressions. He relied more on the taught procedure than on the magnitude of the numbers (i.e., 4 tens times 9 compared to 4 tens times 2), and made computational mistakes. In the AT2, he adjusted the taught procedure to the context of the task by analyzing the multiplicands based on the given information. In the GRT, Manolis agreed with the given statement, and gave an example to justify his answer. He did not explore the whole spectrum of two-digit numbers to refute the statement. Manolis showed developed fluency and flexibility, while his mathematical reasoning was limited.

Manolis’ responses.

Eleonora. Eleonora also used the taught procedure fluently in the RT ( Figure 8 ). In the AT1, she decided that 59 × 55 is bigger than 6 × 54, without finding the product. In the latter set of expressions, Eleonora decided that the change in the place of numbers does not matter, and said that 42 × 9 is greater than 49 × 2. Even though she adapted the procedure flexibly to respond to the AT1, this was not the case in the AT2, in which she found the product by analyzing the number in tens and units, without considering the given information (i.e., students were anticipated to split 11 into 9 + 2). She either did not consider the given statement, or she faced difficulties in extending her current method of using the procedure of distributive property to find the product. In the GRT, Eleonora found that the double of 50 is a three-digit number, thus presenting a counterexample to refute the argument. Hence, Eleonora showed developed fluency and reasoning, and developing flexibility in the four tasks. In total, four students had the same profile as Eleonora.

Eleonora’s responses.

Evita. Evita used the procedure fluently to find the product in the RT ( Figure 9 ). In the AT1, she was looking for the “right place” of the addition and multiplication signs in order to decide whether the expressions are equivalent. She decided that since 2 × 9 = 9 × 2, the expression 42 × 9 must be equal to 49 × 2. In the AT2, she did not consider the given information to find the products, and instead used the procedure she knew (i.e., analyzing the bigger number in tens and units). Her fluency in reproducing the procedure was noticeable, but she did not show any flexibility in adapting the mathematical ideas to other formats. In the GRT, she identified that there is a set of numbers for which this statement would not be true. However, her reasoning was not presented in a coherent manner. Overall, Evita’s responses in the four tasks indicate developed fluency, limited flexibility, and developing reasoning.

Evita’s responses.

Makis. Makis showed developing fluency in the RT due to computational mistakes in the distributive property over subtraction ( Figure 10 ). In the AT1, he decided that 6 × 54 is greater than (5 + 54) × (1 + 54), because he identified a different sign than the expected one. In the latter expression, he seems to have made an estimation for the product. In the AT2, he used the distributive property similarly to the way in which it was used in the RT, without adapting it to the given information. Lastly, in the GRT, he provided a counterexample to refute the statement. Hence, his responses indicate developing fluency and flexibility, and developed reasoning.

Makis’ responses.

## 4. Discussion

4.1. vertical view of the framework.

Using the framework as a lens to design tasks and interpret students’ responses is intended to give the teacher an overall idea of the level of the class before delving into further analysis of each student’s level of competency. The framework aims to present an approach that is integrated along the continuum of teaching, learning, and assessment. Specifically, during or after the introduction of a new mathematical idea in a mathematics lesson or in a series of lessons, the teacher could use assessment tasks based on the framework to elicit evidence and interpret students’ responses. The interpretation of students’ responses could then guide the preparation and implementation of the next lesson(s). It is not necessary to offer several tasks within each of the three types of task. The number of tasks per type depends on what aspects of the taught mathematical ideas the teacher aims to assess, and in what ways these aspects are entailed in the designed tasks. In this paper, we focus on students’ written work. However, the framework is not incapable of being used during informal observations of students’ work, or during talk in the classroom.

## 4.2. Horizontal View of the Framework

Central to the framework is the role of the teacher and the previous learning opportunities in classroom. The expected processes (i.e., recall from memory, application, and generation) are framed by the contextual familiarity (i.e., students’ familiarity with the format of the task and the work procedure). A drawback of large-scale assessment is that it misses the qualitative insights on which classroom assessment could rely in order to characterize students’ emergent ideas, and the ways in which students could improve [ 29 ].

The evidence herein from students’ written responses could be viewed from different perspectives and for different purposes. For example, some could focus more on cognitive difficulties, while others focus more on the level of engagement with important mathematical processes (e.g., representing, modelling, connecting). We do not suggest that different perspectives are contradictory—at times they are complementary. To address this concern, the proposed framework presents a selected competency for each type of task, which is aligned with the expected processes and contextual familiarity. The analysis of students’ responses and the language to characterize the responses are aligned with the selected competencies. The characterizations indicate whether the student has reached a satisfying level (e.g., “developed” flexibility), whether the student is still developing the competency (e.g., “developing” flexibility), or whether the response does not provide evidence that the student is developing the competency (e.g., “limited” fluency). The framework is operationalized for the classroom teachers to inform them about students’ learning in a timely manner, in order that they might use the evidence to plan instructional adjustments.

## 4.3. Instructional Adjustments

The field of classroom assessment should focus more on how to move directly from the evidence about students’ understanding to the description of appropriate teaching actions [ 36 ]. The framework aligns the design of tasks with the analysis of students’ responses in order to set the grounds for developing hypotheses about the respective alignment with instructional adjustments.

Characterizing the level of students’ fluency could suggest possible actions for the classroom teacher. The case of “developing fluency” (Lina and Makis) indicates that further opportunities for practice could be provided in order to attend more to the series of steps, and to precision in calculations. Since students are expected to recall the mathematical idea from memory, opportunities to enhance this recall could be valued. Further study is needed in order to explore how much practice should be provided, and in what intervals. These answers might vary according to the mathematical idea(s) under study. The case of “limited fluency” is rather puzzling. If a student does not reproduce the taught mathematical idea, then student characteristics and teaching approaches should be studied further.

The cases of “developing flexibility” (Manolis and Eleonora) and “limited flexibility” (Evita and Lina) suggest that features of the tasks were not taken into consideration by the students in order to adapt the taught mathematical idea. Hence, timely feedback and focusing on the features of the tasks could enhance their learning [ 42 ]. In addition, the teaching opportunities could be infused with a variety of formats across the mathematical ideas.

For “developing reasoning” (Evita) and “limited reasoning” (Lina and Manolis), a useful approach might be the development of classroom discussions in which students are asked to persuade their classmates about their line of thought. Scholars suggest the use of prompted self-explanation and accountable talk for the learning processes of understanding and sense-making [ 41 ]. Nevertheless, further research is needed in order to provide insight into effective instructional adjustments. We agree that different processes would require different instructional adjustments. It is more likely that using the same examples with different numbers would make students better at reproducing than applying or reasoning.

## 4.4. Limitations

The framework is a starting point to discuss and elaborate further on the interplay between the design of assessment tasks and the analysis of students’ responses. Statistical analysis from various classroom settings could provide further insight. Additionally, the framework could be used and be adapted to other educational contexts and grade levels. It would be interesting to explore whether the identified processes, the contextual familiarity, the competencies, and the characterization of students’ responses are applicable and meaningful to other content areas. We anticipate that the proposed framework may have much greater validity for primary teachers, since we relied on several topics of primary mathematics for its development. The three types of processes are widely used in mathematics education and beyond. However, further research is needed in order to explore in what ways the three competencies and the characterizations of students’ responses are perceived and applied by the classroom teachers.

Moreover, it would be useful to explore the extent to which the proposed framework might be relevant in settings that use a different textbook than the one on which the framework was developed, or in settings that rely on a varied set of instructional resources rather than a textbook. We relied on a textbook series that is organized per mathematical topic. Hence, an adaptation of the framework would be needed in order to use it alongside a textbook series that is organized per mathematical process. Regarding the use of varied instructional resources, it is anticipated that the adaptation of the “expected formulation of tasks” for assessment tasks would support the application of the framework to such settings. The design of tasks and the interpretation of students’ responses rely extensively on the anticipated path based on what preceded in the classroom context, which determines the contextual familiarity (i.e., task format and work procedure), irrespective of the number of instructional resources used. Furthermore, it would be purposeful to explore the instructional adjustments in real classroom settings based on the hypotheses drawn from the framework, and how these adjustments relate to students’ learning.

## 5. Conclusions

There is evidence that classroom assessment for formative purposes has the potential to improve students’ learning [ 4 , 7 ]. This is a timely issue that needs to be further explored by relying on empirical evidence and systematic research. However, its effective implementation in classrooms is still in the early stages. In this paper, we move a step forward by presenting a framework that captures the interplay of the design of mathematics assessment tasks and the analysis of students’ responses along the continuum of teaching, learning, and assessment. The proposed framework provides an operational tool for the purpose of classroom assessment; it aims to provoke research that would develop insight into meaningful evidence for enhancing students’ learning of mathematics, and to set the grounds for systematically exploring instructional adjustments.

## Author Contributions

Conceptualization: E.D., C.C., D.P.-P.; Methodology: E.D., C.C., D.P.-P.; Analysis: E.D., C.C., D.P.-P.; Writing—Reviewing: E.D., C.C., D.P.-P. All authors have read and agreed to the published version of the manuscript.

This research received no external funding.

## Institutional Review Board Statement

Not applicable.

## Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

## Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## IM 6–12 Math: Grading and Homework Policies and Practices

By Jennifer Willson, Director, 6–12 Professional Learning Design

In my role at IM, working with teachers and administrators, I am asked to help with the challenges of implementing an IM curriculum. One of the most common challenges is: how can we best align these materials to our homework and grading practices? This question is a bit different from “How should we assess student learning?” or “How should we use assessment to inform instruction?”

When we created the curriculum, we chose not to prescribe homework assignments or decide which student work should count as a graded event. This was deliberate—homework policies and grading practices are highly variable, localized, and values-driven shared understandings that evolve over time. For example, the curriculum needed to work for schools where nightly, graded assignments are expected; schools where no work done outside of class is graded; and schools who take a feedback-only approach for any formative work.

IM 6–8 Math was released in 2017, and IM Algebra 1, Geometry, and Algebra 2 in 2019. In that time, I’ve been able to observe some patterns in the ways schools and teachers align the materials to their local practices. So, while we’re still not going to tell you what to do, we’re now in a position to describe some trends and common ways in which schools and districts make use of the materials to meet their local constraints. Over the past four years, I have heard ideas from teachers, administrators, and IM certified facilitators. In December, I invited our IM community to respond to a survey to share grading and homework policies and practices. In this post I am sharing a compilation of results from the 31 teachers who responded to the survey, as well as ideas from conversations with teachers and IMCFs. We hope that you find some ideas here to inform and inspire your classroom.

## How do teachers collect student responses?

Most teachers who responded to the survey collect student work for assessments in a digital platform such as LearnZillion, McGraw-Hill, ASSISTments, Edulastic, Desmos, etc. Others have students upload their work (photo, PDF, etc.) to a learning management system such as Canvas or Google classroom. Even fewer ask students to respond digitally to questions in their learning management system.

## How do teachers tend to score each type of assessment, and how is feedback given?

The table shows a summary of how teachers who responded to the survey most often provide feedback for the types of assessments included in the curriculum.

## How are practice problems used?

Every lesson in the curriculum (with a very small number of exceptions) includes a short set of cumulative practice problems. Each set could be used as an assignment done in class after the lesson or worked on outside of class, but teachers make use of these items in a variety of ways to meet their students’ learning needs.

While some teachers use the practice problems that are attached to each lesson as homework, others do not assign work outside of class. Here are some other purposes for which teachers use the practice problems:

- extra practice
- student reflection
- as examples to discuss in class or use for a mini-lesson
- as a warm-up question to begin class
- as group work during class

## How do teachers structure time and communication to “go over” practice problems?

It’s common practice to assemble practice problems into assignments that are worked on outside of class meeting time. Figuring out what works best for students to get feedback on practice problems while continuing to move students forward in their learning and work through the next lesson can be challenging.

Here are some ways teachers describe how they approach this need:

- We don’t have time to go over homework every day, but I do build in one class period per section to pause and look at some common errors in cool-downs and invite students to do some revisions where necessary, then I also invite students to look at select practice problems. I collect some practice problems along with cool-downs and use that data to inform what, if anything, I address with the whole class or with a small group.
- Students vote for one practice problem that they thought was challenging, and we spend less than five minutes to get them started. We don’t necessarily work through the whole problem.
- I post solutions to practice problems, sometimes with a video of my solution strategy, so that students can check their work.
- I assign practice problems, post answers, invite students to ask questions (they email me or let me know during the warm-up), and then give section quizzes that are closely aligned to the practice problems, which is teaching my students that asking questions is important.
- I invite students to vote on the most challenging problem and then rather than go over the practice problem I weave it into the current day’s lesson so that students recognize “that’s just like that practice problem!” What I find important is moving students to take responsibility to evaluate their own understanding of the practice problems and not depend on me (the teacher) or someone else to check them. Because my district requires evidence of a quiz and grade each week and I preferred to use my cool-downs formatively, I placed the four most highly requested class practice problems from the previous week on the quiz which I substituted for that day’s cool-down. That saved me quiz design time, there were no surprises for the students, and after about four weeks of consistency with this norm, the students quickly learned that they should not pass up their opportunity to study for the quiz by not only completing the 4–5 practice problems nightly during the week, but again, by reflecting on their own depth of understanding and being ready to give me focused feedback about their greatest struggle on a daily basis.
- I see the practice problems as an opportunity to allow students to go at different paces. It’s more work, but I include extension problems and answers to each practice problem with different strategies and misconceptions underneath. When students are in-person for class, they work independently or in pairs moving to the printed answer keys posted around the room for each problem. They initial under different prompts on the answer key (tried more than one strategy, used a DNL, used a table, made a mistake, used accurate units, used a strategy that’s not on here…) It gives the students and I more feedback when I collect the responses later and allows me to be more present with smaller groups while students take responsibility for checking their work. It also gets students up and moving around the room and normalizes multiple approaches as well as making mistakes as part of the problem solving process.

## Quizzes—How often, and how are they made?

Most of the teachers give quizzes—a short graded assessment completed individually under more controlled conditions than other assignments. How often is as varied as the number of teachers who responded: one per unit, twice per unit, once a week, two times per week, 2–3 times per quarter.

If teachers don’t write quiz items themselves or with their team, the quiz items come from practice problems, activities, and adapted cool-downs.

## When and how do students revise their work?

Policies for revising work are also as varied as the number of teachers who responded.

Here are some examples:

- Students are given feedback as they complete activities and revise based on their feedback.
- Students revise cool-downs and practice problems.
- Students can revise end-of-unit assessments and cool-downs.
- Students can meet with me at any time to increase a score on previous work.
- Students revise cool-downs if incorrect, and they are encouraged to ask for help if they can’t figure out their own error.
- Students can revise graded assignments during office hours to ensure successful completion of learning goals.
- Students are given a chance to redo assignments after I work with them individually.
- Students can review and revise their Desmos activities until they are graded.
- We make our own retake versions of the assessments.
- Students can do error logs and retakes on summative assessments.
- We complete the student facing tasks together as a whole class on Zoom in ASSISTments. If a student needs to revise the answers they notify me during the session.

## Other advice and words of wisdom

I also asked survey participants for any other strategies that both have and haven’t worked in their classrooms. Here are some responses.

What have you tried that has not worked?

- Going over practice problems with the whole class every day. The ones who need it most often don’t benefit from the whole-class instruction, and the ones who don’t need it distract those who do.
- Grading work on the tasks within the lessons for accuracy
- Leaving assignments open for the length of the semester so that students can always see unfinished work
- Going through problems on the board with the whole class does not correct student errors
- Most students don’t check feedback comments unless you look at them together
- Grading images of student work on the classroom activity tasks uploaded by students in our learning management systems
- Providing individual feedback on google classroom assignments was time consuming and inefficient
- Allowing students to submit late and missing work with no penalty
- Trying to grade everything
- Below grade 9, homework really does not work.
- Going over every practice problem communicates that students do not really think about the practice problems on their own.

What else have you tried that has worked well?

- My students do best when I consistently assign practice problems. I have tried giving them an assignment once a week but most students lose track. It is better to give 2–3 problems or reflective prompts after every class, which also helps me get ahead of misconceptions.
- I don’t grade homework since I am unsure who completes it with or for the students.
- A minimum score of 50% on assignments, which allows students the opportunity to recover, in terms of their grade in the class
- Time constraints imposed during remote learning impact the amount and type of homework I give as well as what I grade
- Give fewer problems than normal on second chance assignments
- I have used platforms such as Kahoot to engage students in IM material. I also build Google Forms to administer the Check Your Readiness pre-assessment and End-of-Unit assessments, but I may start using ASSISTments for this in the future.
- The value of homework in high school is okay, but personally I skip good for great.
- Students are able to go back and revise their independent practice work upon recognizing their mistakes and learning further about how to solve the problems.
- Sometimes I select only one or two slides to grade instead of the whole set when I use Desmos activities.
- Allow for flexibility in timing. Give students opportunities for revision.
- Frequent short assessments are better than longer tests, and they allow students to focus on specific skills and get feedback more frequently.
- Especially during the pandemic, many of my students are overwhelmed and underachieving. I am focusing on the core content.
- Homework assignments consist of completing Desmos activities students began in class. Additional slides contain IM practice problems.
- I am only grading the summative assessment for accuracy and all else for completion. I want the students to know that they have the room to learn, try new strategies and be wrong while working on formative assessments.

What grading and homework policies have worked for you and your students that aren’t listed? Share your ideas in the comments so that we can all learn from your experience.

What would you like to learn more about? Let us know in the comments, and it will help us design future efforts like this one so that we can all learn more in a future blog post.

We are grateful to the teachers and facilitators who took the time to share their learning with us.

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- Designing a Frictional Roller Coaster With Math and Physics!

## Hands-on Activity Designing a Frictional Roller Coaster With Math and Physics!

Grade Level: 12 (11-12)

(seven 50-minute sessions)

This activity also uses some non-expendable (usable) items; see the Materials List for details.

Group Size: 3

Activity Dependency: A Tale of Friction

Subject Areas: Algebra, Measurement, Physics, Problem Solving

NGSS Performance Expectations:

## Activities Associated with this Lesson Units serve as guides to a particular content or subject area. Nested under units are lessons (in purple) and hands-on activities (in blue). Note that not all lessons and activities will exist under a unit, and instead may exist as "standalone" curriculum.

Te newsletter, engineering connection, learning objectives, materials list, worksheets and attachments, more curriculum like this, pre-req knowledge, introduction/motivation, vocabulary/definitions, investigating questions, troubleshooting tips, activity scaling, additional multimedia support, user comments & tips.

With their breathtaking elevation changes and speeds, spine-chilling roller coasters rides are the star attractions of amusement parks. All the various up, down and around designs all work because of gravity, inertia and friction. In this activity, by designing a simple roller coaster, students consider the same forces that professional engineers do when designing rides. Students apply mathematics, energy conservation principles and computation software to completely define the path and optimal dimensions for their designs. Then they use simple materials to build and test functional roller coaster models, which gives them the opportunity to address prototype construction problems and the unanticipated details that impact the expected design performance. Also like professional engineers, students find solutions to these problems.

After this activity, students should be able to:

- Create a design given certain requirements and constraints.
- Create, using parabolas, a piecewise differentiable function given specific vertices and determining tangency points.
- Estimate the velocity of a rolling body along a curved path, considering friction forces.
- Use computational software (such as Excel®) to evaluate and graph functions.
- Build a functional model from a mathematically generated design.
- Determine the possible causes why physical models might not behave exactly as expected in the theoretical, mathematically derived, “on-paper” design.

## Educational Standards Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN) , a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g. , by state; within source by type; e.g. , science or mathematics; within type by subtype, then by grade, etc .

Ngss: next generation science standards - science, common core state standards - math.

View aligned curriculum

Do you agree with this alignment? Thanks for your feedback!

## International Technology and Engineering Educators Association - Technology

State standards, texas - math, texas - science.

Each group needs:

- polyethylene pipe wrap insulation, such as Armacell Tubolit half-inch x 6-ft length for ~$1 from Home Depot ; although foam pipe insulation material that is larger than the half-inch size can be used as the coaster rails/channel, the half-inch size is more flexible
- vinyl bullnose archway corner bead, such as Clark Western ¾-in x 10-ft length from Home Depot for ~$4, which will be cut into approximately one-inch segments; alternatively, use some other material/way to provide L-bracket-type support to attach the curving half-pipe insulation rail to a cardboard backing
- 1 corrugated cardboard or plastic sheet, 3 x 4-ft size works well; note; large pieces of cardboard can be scavenged from large bicycle and appliance boxes; alternatively, have students tape together smaller cardboard pieces (see Figure 13)
- a few glass marbles, half-inch diameter; make sure the marble diameter is less than the internal diameter of the pipe insulation so the rolling marbles have only one contact point with the insulation surface
- masking tape (1-inch thick) and super glue, for adhering pipe insulation to itself and to cardboard/plastic backing
- scissors and/or utility knife, for cutting pipe insulation and vinyl corner bead
- yardstick or 1-meter ruler with inch-scale markings
- computer with Microsoft® Excel® and PowerPoint®, or similar software applications
- Project Rubric , one per group
- A Frictional Roller Coaster Pre-Quiz , one per person
- (optional) 2 x 4-foot pegboard with one-inch-apart drilled holes, used as a mat to make it easier to graph/draw points on the cardboard backing

To share with the entire class:

- projector and computer to show the class the A Frictional Roller Coaster Presentation , a PowerPoint® file, including some online videos
- kitchen scale, or other small scale capable of measuring the mass of a marble in grams
- (optional) capability to take photographs and/or videos of the roller coaster models in use
- Conic sections, parabolas
- Microsoft® Excel® formulas and graphs
- Microsoft® PowerPoint®
- Solving non-linear systems of equations
- Piecewise functions
- Continuous functions
- Differential calculus and derivatives
- Differentiable piecewise functions
- Concepts and equations presented in the associated lesson: A Tale of Friction

(Be ready to show the 17-slide A Frictional Roller Coaster Presentation , a PowerPoint® file, including some online videos. The slides are animated, so a mouse or keyboard click brings up the next text, image and/or slide.)

(Slides 1-2) Does anyone have any idea of the cost of roller coaster projects? (Play some of the three short videos linked in slide 2 about roller coaster construction and computer modeling.)

(Slides 3-5) Do you have any idea of the cost of projects like these? Guess the cost for each of these three roller coasters (also shown in Figure 2). (Write students’ guesses on the board. Then tell them the answers.) El Toro cost ~$25 million, Thunder Dolphin cost ~$37 million, and Millennium Force cost ~$25 million. (How close were they in their cost estimations?) Disney’s Expedition Everest ride at Walt Disney World in Florida is the most expensive roller coaster, costing $100 million and taking more than six years to create.

For projects that cost so much—like these world-class roller coasters—the designs and construction are complex and challenging tasks for engineers.

(Slide 7) Have you ever worked on a roller coaster project? (Expect students to answer yes. If so, ask them to briefly describe what they did. Expect that some students may have used a marble as a cart and a flexible material to construct the roller coaster path. They placed an initial point very high and then created a path with hills, valleys and the most loops possible. Then they tested the path and if the marble stopped or flew away, they modified the path until the marble was able to complete the entire path. Next, if desired, play some of the nine videos linked on slide 7 to show some student roller coaster school projects.)

Great, I’m sure you really had fun working on those projects. Do you think engineers do the same? I mean, do you think they build a complex structure, then test and modify the path sections where the cart gets stuck or flies off? (Expect students to say something like, “no, that does not sound like a reasonable process since they would have to demolish parts of the built structure and build again.”)

You are right. That approach would be terribly inefficient. To make real-world structures, engineers first create an efficient design that takes into consideration all the conditions and factors that may affect performance. They often create virtual simulations of the coaster in action (as you saw in two of the videos). One of the many factors engineers must consider is friction. In your school roller coaster projects, you were aware that friction was an important factor. You saw how the marbles never reached a height equal to their initial drop heights and every time the marble achieved a lower height. While you understood that this happened because of friction, you were unable to estimate its effect.

In today’s activity, you will work in teams like professional engineers who work on those expensive multi-million-dollar projects from the design to the final product . This is your design challenge :

- Design a simple roller coaster , that is, one with two dimensions—only hills and valleys—but consider friction forces from the beginning of the design process. Define the path by a function so that every point on the path is precisely defined with an (x, y) pair. Precisely determine the maximum possible height of every hill such that the marble is able to complete the entire path, moved only by its potential energy at the initial point of the path. The initial velocity is zero and the final velocity must be zero, or practically zero.
- Build and test a prototype using a marble as the cart, pipe insulation for the coaster’s path and a large cardboard sheet to hold the model track. Your prototype dimensions must match those in your design, and like the roller coasters created by professional engineers, your prototype must behave as expected in your design. No major changes in the path or dimensions are permitted. If you run into problems, you will determine why your prototype is not performing as expected.

Typical middle and high school roller coaster projects do not quantify the friction force present. For students in these grade bands, it is explained that roller coasters work because potential energy transforms into kinetic energy and vice versa, and that the initial potential energy builds up the kinetic energy to propel a cart through the remainder of the ride. It is also explained that the work done by friction forces is what eventually stops the cart, but no quantification is provided for this work along the coaster’s path. At most, the overall work done by friction is estimated by computing the kinetic-potential energy losses using the work-energy theorem (R1).

In the high school AP Physics C course, students solve some problems involving friction, using the work-energy theorem. They may find the velocity of a body sliding on a flat surface while considering the friction present. So high school students are able to understand the solution of a spherical body rolling along a curved path, with friction being considered.

In the associated lesson, A Tale of Friction , it is explained in detail using high school differential calculus and physics how to find a formula to determine the velocity of a sphere rolling along a simple roller coaster path considering friction (1):

Equation (1) is the magic key to successfully design a functional simple roller coaster with its path defined by a function f (x) . Because it is not easy to find a single function describing a roller coaster’s path, f (x) must be defined as a differentiable piecewise function (see Figure 1). Depending on the time available for this activity, and your students’ math and physics background, decide what portions of the lesson to teach to your students, so that at the end they understand equation (1), and how to use it to test whether their designs will work or not.

In this activity, students design a Russian Mountain path using upward-opening and downward-opening parabolas. With parabolas, it is relatively simple to create a differentiable piecewise function. They are easy to define, given the vertex and a point on the graph.

As in any engineering project, constraints and conditions must be considered in the design. The roller coasters will be fabricated from polyethylene pipe wrap insulation, use a rolling marble as the cart, and be attached to a flat surface. Our roller coaster curves must resemble those of real Russian Mountains (Figure 2) and the slopes must not be very steep because the marbles may slide instead of roll.

A roller coaster’s initial height and slope determines its length. The higher the initial point, the longer the path. It is best that the initial roller coaster slope not be steeper than ±2.5, otherwise the marble may slide instead of roll. This guideline is an outcome from testing different initial slopes.

It is recommended that inches be used for the measurements and calculations because all construction and auxiliary materials are measured in inches. Following are the details for the design and construction of a roller coaster.

The design begins by defining the initial (highest) roller coaster point (x 0 , y 0 ) and the slope of the parabola at that spot. This highest point is on the left branch of the first upward-opening parabola (see Figure 3). To simplify calculations, place the initial point on the y -axis, and the vertices of the upward-opening parabolas on the x -axis. This example describes an initial point 25 inches above the x -axis: (x 0 , y 0 ) = (0, 25), and a slope of ±2.5 at this point. With this initial data, the vertex of the first upward-opening parabola (h, k) = ( h 1 , 0) can be determined.

Because the parabola vertex is involved in these calculations, the vertex form equation is used:

Solving this system, we find that:

The equation for the first parabola is then:

The next two upward-opening parabolas may be placed more or less arbitrarily. Remember to place their vertices at the same level as the parabola 1 vertex, but not too close to each other since the closer the vertices, the steeper the path of the parabolas joining them will be, and the marble is more inclined to slide instead of roll if the path is too steep. Because a parabola is symmetric with respect to its vertical axis, it is convenient if all three upward parabolas have the same width (see Figure 3); this choice simplifies the involved calculations very much.

In this example (Figure 3), the next vertices are ( h 3 , k 3 ) = (56, 0) and ( h 5 , k 5 ) = (84, 0). These three upward-opening parabolas join smoothly with three downward-opening parabolas. The heights of the vertices of the downward-opening parabolas must be such that the marble passes that hill with a very low velocity but enough to continue rolling to the next hill and past it and so on. It is expected that the marble stops (or almost stops) at the end of the designed path. These heights cannot be set up arbitrarily; the friction along the path must be considered. If we roll a marble along a single U-path, we know that friction stops the marble before it reaches a height equal to the initial height. Therefore, equation (1) must be applied in every upward-opening parabola to estimate the height the marble may reach at the end of that portion of the path. Once that height is determined, the vertex for the downward-opening parabolas may be defined, and the next step is to join them smoothly with the upward-opening parabolas.

For the parabola with vertex ( h 3 , k 3 ) = (56, 0) and same width as (3), the corresponding equation is:

and for the parabola with vertex ( h 5 , k 5 ) = (84, 0), the corresponding equation is:

The x -coordinate of the vertices of the downward-opening parabolas is at the middle of the x -coordinates of the vertices of the upward-opening parabolas (see Figure 5). Determine the height of these vertices using equation (1):

v i = v (0) = 0

Now it is necessary to evaluate equation (1) along the parabolic path (3). For this purpose, the path is divided into little straight sections and on every section a final velocity is computed, taking as initial velocity the final velocity computed for the previous section. These calculations can be easily performed in Excel® (see Figure 4). (Note: Expect students to create their own spreadsheets, guided by the teacher.)

The previous computations show that the marble will stop at a point with horizontal distance from the origin between 34.907 and 34.908 inches, reaching a height of no more than 13.89 inches. It is reasonable to place the vertex of the second parabola at about 13 inches high.

Placing the vertex of parabola 2 (see Figure 5) at the middle of the parabolas 1 and 3 vertices, ( h 2 , k 2 ) = (38, 13), the equation for parabola 2 is:

To find the value of coefficient a , the condition that parabolas 1 and 2 must be tangent to each other (Figure 5), must be solved. This condition is expressed as two equations:

From (7), substituting in (6):

Grouping and solving for x :

Substituting this value in (7), and solving for a :

The equation for parabola 2 is then:

Using equation (8), we find the y -coordinate of the tangency point:

Because of symmetry, (8) is also tangent to parabola 3. This second tangency point is at the same horizontal distance from the parabola 2 vertex, but to the right, and of course, at the same height (Figure 5):

The first two branches of the roller coaster path are completely defined:

To find the equation of the second downward-opening parabola (Figure 6), repeat the above procedure. Equation (5) must be used beginning at x 0 = 0 again to estimate the height the marble may reach at the end of parabola 3. Once this height is determined, the parabola 4 vertex can be defined, and its equation determined solving the tangency condition with parabola 3.

Using Excel® again (Figure 7):

From these computations, it is reasonable to place the vertex of the fourth parabola at about 8-inches high. Again, placing this vertex between the parabolas 3 and 5 vertices, ( h 4 , k 4 ) = (70, 8), the equation for parabola 4 is:

Again applying the tangency condition to parabolas 3 and 4:

and solving equations (9) and (10), the value of the constant a and the tangency point coordinates is determined:

The next two branches of the roller coaster path are now completely defined:

The last two parabolas need to be determined in the same way. Analyzing the velocities along the parabolas 1 to 5 (Figure 8):

The maximum height predicted for parabola 5 is 4.5555 inches. The vertex of parabola 6 is placed at ( h 6 , k 6 ) = (96, 4.5) where the marble is expected to stop (or almost stop). The proposed equation for this parabola is then:

Applying again the tangency condition to parabolas 5 and 6:

The equation for parabola 6 is then:

Figure 10 shows the graph of the predicted velocities along the path of this design:

At this point, a natural question is: How did friction behave along the path? Friction was considered in order to predict the above velocities and to design the roller coaster path, but until now, friction has not been explicit. To answer this question, it is necessary to evaluate the equations obtained in the associated lesson that correspond to the static friction coefficient and the friction force for the rolling marble:

To express the friction force in Newtons (N), it is necessary to change the units used until now, inches, into meters (1 m = 39.37 inches). So, the derivative of our piecewise function in meters is:

Taking g = 9.81 m/s, and a marble mass m = 0.004 kg, again use Excel® (Figure 11) to evaluate the coefficient of friction μ s and the friction force f s .

By definition, the friction force is proportional to the normal component of the body’s weight, and the constant of proportionality is the coefficient of friction. But for a curved roller coaster path, the normal component changes at every point on the path, the same as the slope of the tangent line (Figure 12).

Notice also that the friction coefficient and the friction force are always considered positive because the friction force always opposes the body’s movement (Figure 12). In the work-energy equation obtained in the associated lesson, A Tale of Friction :

Before the Activity

- This activity and its associated lesson is suitable for the end of the first semester of the school year for high school AP Calculus courses, serving as a major grade for the last six-week period with the results presentation-report taken as the first-semester finals test.
- Before beginning this activity with students, practice every one of the steps described in the Background section, especially the calculations and graphs in the Rolling with Friction Calculations Example (an Excel® file). Notice that a Rolling with Friction Calculations No Calculus Example is also provided. Feel free to contact the author with any questions or suggestions.
- It is recommended that you create a working roller coaster model of your own to show to students.
- Arrange for an open work space where students have plenty of room to work.
- Make sure every team has access to a computer with Excel® and PowerPoint® (or similar/equivalent) software applications.
- Define which of the necessary construction materials you will provide and which you will ask students to provide.
- Teach the associated lesson, A Tale of Friction , making sure that students understand the equations they need to use to conduct this activity.
- Make copies of A Frictional Roller Coaster Pre-Quiz (one per student) and A Frictional Roller Coaster Project Rubric (one per team). Decide on a presentation-report deadline date and put that into the rubric before making copies of it.
- Before beginning the activity, administer the pre-quiz to students to provide practice and review in creating differentiable piecewise functions.
- Be ready with a computer, software and projector to show students the A Frictional Roller Coaster Presentation , a PowerPoint® file.

With the Students—Day 1: Activity Introduction (50 minutes)

- Present the Introduction/Motivation content to the class, using the accompanying slide presentation.
- Explain the overarching engineering design challenge and the specifics of what students need to do.
- Show students an example working roller coaster track prototype so they have some idea of what they might construct.
- Make it very clear that this is not another elementary-middle school roller coaster project like they may have done in the past or like the ones they saw in the linked videos. In this project, they cannot undo-redo the model to make it work. The entire path must be defined mathematically through a function, and their functionality supported with calculations before teams construct the roller coaster models.
- If some students ask if a ripple may be included in the roller coaster paths their teams design, answer as follows: A ripple requires more complex math expressions for the path (remind them that loops are not functions) and the computations involving velocity and friction require the incorporation of additional physics concepts like centripetal forces and accelerations. It is important to first understand perfectly well all the concepts and calculations in this project, before introducing loops into roller coaster paths. So we are not permitting loops in the designs for this project.
- Let students organize themselves in teams of three or four students each.
- Tell students what materials are provided and what materials they are to bring from home.
- Hand out the rubric.

With the Students—Day 2: Path Design (50 minutes)

- Ask students to define the initial heights and slopes of their coaster paths, and then use those numbers to find the equation of the first upward-opening parabola. They can also define the equations for parabolas 3 and 5, placing them at reasonable distances from parabola 1, depending on the planned model span (Figure 5).
- Then, using formula (1) and Excel®, direct the groups to each determine the maximum height the marble is able to reach after being dropped from the initial height. This height gives them what they need to define the height of the vertex of the second parabola (the first downward-opening parabola).
- Next, student find the equation of parabola 2, which is mid and tangent to parabolas 1 and 3.

With the Students—Day 3: Path Design (50 minutes)

- Students, again using equation (1), determine the maximum height the marble may reach when dropped from highest point and rolled through the parabolas 1-2-3 path.
- Have groups define the vertex height of parabola 4 (the second downward-opening parabola) and find the corresponding equation.
- Have teams determine the maximum height the marble may reach at the end of parabolas 1-2-3-4-5. Students may decide to either end the coaster’s path at that point or define a sixth parabola. Encourage them to include a sixth parabola. Instruct students to generate graphs for the designed path and for the expected velocities along the path (Figure 9).
- Note: Depending on students’ physics background and available time, ask them to generate graphs for the friction forces along the path.

With the Students—Days 4-6: Model Construction (50 minutes)

- Have students begin model construction using the final design dimensions they calculated during the previous work days.
- Use the pipe insulation material to build the physical models. This material is inexpensive and easy to work with. Remind students that the prototype model must match the design dimensions.
- To hold together and stabilize the physical model, have teams use cardboard sheets, masking tape, and vinyl L-bracket (corner bead) supports (see Figure 17).
- Permit teams to use different kinds of materials, but require students to first inform the teacher about any intended changes/additions.
- Suggest that students take photographs or videos of the various construction stages to include in their final presentations-reports (refer to the rubric).
- Require groups to test their models at least 10 times, recording the results. Direct the teams to draw conclusions about their designs and their physical models. Ask: Is the model behaving as expected? If not, why not and what conclusions can be made? What were the failures? What might explain the discrepancies/failures? How did you solve them?
- Model construction tips:
- Use a large cardboard (or plastic) sheet as the backing for each two-dimensional coaster path. If you only have smaller cardboard pieces, it works well to make a 3 x 4-ft backing by taping them together, edge-to-edge, rather than overlapping them. To aid in storage and transport, tape the sheets together such that they can be folded at the joints (see Figure 13).

- Split the pipe insulation in half lengthwise and use glue and tape to join together the split segments (see Figure 14) to make long coaster rails. Be careful to not let glue or tape get on the interior channel of the pipe insulation because that interferes with the rolling marble.

- To sketch the designed coaster path on the joined-together cardboard (or plastic) sheets requires some precise points. One easy and accurate way to draw the points is to use pegboard with a grid of one-inch-apart drilled holes (see Figure 15). Start by drawing a coordinate axis on the cardboard backing (15A). Then align two perpendicular lines of pegboard holes with the xy -axes drawn on the cardboard (15B, 15C). Use the pegboard holes to help graph points on the cardboard for the integer coordinates (or almost-integer coordinates; consider 14.05 or 14.1 to be “14”) of the designed coaster path (15D). Move along the horizontal axis of the pegboard to continue drawing points. Then sketch the coaster path by drawing lines to join the points.

- Cut the vinyl corner bead into one-inch pieces. These 90°-angle pieces (or any other system you devise) serve as L-brackets to support the pipe insulation channel—the coaster’s rail. Tape them at intervals along and parallel to the sketched path (Figure 16).

- Then use small pieces of rolled tape to adhere the pipe insulation channel onto the L-bracket supports made of corner bead pieces (Figure 17). Doing this completes the mathematically designed Russian Mountain roller coaster path prototype so it is ready for testing.

- To test the design functionality, use glass marbles as the roller coaster carts. It is important to use marbles with a diameter that is smaller than the pipe insulation’s inner diameter because all the calculations are based on using marbles that have only one point of contact with the coaster rail surface so the friction effect is as described in formula (1). A bigger marble will have many contact points and consequently more friction than what was considered in the calculations.

Day 7: Prepare and Present Project Report (50 minutes)

After school/on their own time, have each group create a slideshow or video to present to the rest of the class the different steps of its “professional engineering” work. Let students select and organize the information to include in the presentation-report, guided by the rubric. See the Assessment section for more information.

acceleration: The change in the velocity of an object. The average acceleration is defined as the change in velocity divided by time. The instantaneous acceleration is defined as the acceleration at any particular time period. In calculus, given the velocity of a body as a function of the time, the instantaneous acceleration is the derivative if the velocity with respect to the time.

continuous function: In a very basic approach, a function whose graph does not have any void and/or it is not broken, that is, can be drawn without lifting the pencil from the paper. More formally, a function is continuous at a point in its domain if a sufficiently small change in the input results in an arbitrarily small change in the output. In calculus, a function is continuous at a point x = c if and only if all the next three conditions are met: 1) the function is defined at x = c, 2) the limit of the function at x = c exists, and 3) the limit and the value of the function at x = c are equal.

derivative: The limit of the ratio of the change in a function to the corresponding change in its independent variable as the latter change approaches zero. Geometrically, this rate of change gives the slope of the tangent line at a point on the function’s graph. The graph of the function at that point must be continuous and smooth, that is, the function cannot have a peak at this point.

differentiable function : A function whose derivative exists at each point in its domain. Geometrically, the function whose graph is continuous and smooth such that a tangent line exists for every point on the graph.

energy conservation: A principle that states that in a system that does not undergo any force from outside the system, the amount of energy is constant, irrespective of its changes in form.

friction: The resistance to motion of one object moving relative to another. The surface resistance to relative motion, as of a body sliding or rolling. The force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other.

gravitational force: The force of attraction between masses; the attraction of the earth’s mass for bodies near its surface. The gravitational force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them.

inertial force: A force that resists a change in velocity of an object. It is equal to the applied force, but with opposite direction. Because of this force, every object in a state of uniform motion, or rest, tends to remain in that state unless an external force is applied to it.

kinetic energy: The energy possessed by an object due to its motion or movement. It is proportional to its mass and to the square of its velocity. The magnitude of this energy arises from the net work done on the object, to accelerate it from rest to its final velocity. Kinetic energy can be transformed again in work.

kinetic friction: A force that acts between moving surfaces. It is also known as sliding friction or moving friction, and it is the amount of retarding force between two objects that are moving relative to each other.

maximum of a function: The largest value of a function, either within a given range or on the entire domain. Formally, a function f (x) has a maximum at x = c, a < c < b, [a, b] a subinterval of the function’s domain, if f I > f (x), for a ≤ x ≤ b.

mechanical energy: The capacity of a physical system to change an object’s state of motion.

mechanical work: The amount of energy transferred by a force that moves an object. The work is calculated by multiplying the applied force by the amount of movement of the object.

minimum of a function: The smallest value of a function, either within a given range or on the entire domain. Formally, a function f (x) has a minimum at x = c, a < c < b, [a, b] a subinterval of the function’s domain, if f I< f (x), for a ≤ x ≤ b.

parabola: A conic section formed by the intersection of a vertical cone by a plane parallel to the cone’s side. A curve where any point is at an equal distance from a fixed point, the focus, and a fixed straight line, the directrix.

parabola vertex: The point where a parabola crosses its axis of symmetry. It is the maximum point when the parabola opens downwards, or the minimum point if the parabola opens upwards.

piecewise function: A function that is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function’s domain. Also known as a hybrid function.

potential energy: The energy possessed by a body by virtue of its position relative to others. In a gravitational field, potential energy is the energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the earth for the object.

roller coaster: An amusement park ride that consists on an elevated railroad track designed with sharp curves and steep slopes on which people move in small, fast and open rolling cars.

Russian Mountains: The oldest roller coaster design and a predecessor to the modern day roller coaster. Descended from Russian winter sled rides on hills of ice, these early roller coasters were open wheeled carts or open train carts on tracks of elevated up-hills and down-hills supported by wooden or steel structures.

secant line: The straight line joining two points on a curve.

static friction: The friction that exists between a stationary object and the surface on which it rests—the force that keeps an object at rest and that must be overcome to start moving an object. Pushing horizontally with a small force, static friction establishes an equal and opposite force that keeps the object at rest.

system of equations: A set of equations working together at once. Systems of equations are called linear if all the equations in the system are linear (their graphs are all straight lines). Systems of equations are called non-linear if at least one of the equations in the system is not linear. Also called simultaneous equations.

tangent line: The straight line that touches a curve at a point without crossing over.

velocity: The time rate of change of position of a body in a specified direction. The average velocity is defined as the change in position divided by the time of travel. The instantaneous velocity is simply the average velocity at a specific instant in time. In calculus, given the position of a body as a function of the time, the instantaneous velocity is the derivative of the position with respect to the time

work-energy theorem: A principle that states that the work done by all forces acting on a particle equals the change in the particle’s kinetic energy.

Pre-Activity Assessment

Pre-Quiz: Before beginning the activity, administer A Frictional Roller Coaster Pre-Quiz to students as practice in creating differentiable piecewise functions. Before moving on to conduct the activity, verify that every student is able to create a differentiable piecewise function joining two parabolas.

Activity Embedded Assessment

Checking In: During path design days, verify that students understand how to apply formula (1) to the path created with parabolas. Check their computations in their Excel® files.

Post-Activity Assessment

Professional Results Presentations: Along with their working roller coaster models, have teams prepare (after school and on their own) and present slide or video presentations to the class that summarize the process from design to calculations to prototype construction, including an overall analysis of the work done, problems encountered/resolved, and conclusions. This includes groups answering questions about why their models might not have performed exactly as expected; see the Investigating Questions section. Refer to the Project Rubric for expectations and grading suggestions.

Making Sense : Have students reflect about the science phenomena they explored and/or the science and engineering skills they used by completing the Making Sense Assessment .

During the final presentations, engage each team to reflect on why their models might not have performed exactly as expected. Many of these issues stem from the construction process and the difficulty in fabricating models that have the exact calculated dimensions (theory vs. reality). Suggested questions are provided below, along with example/possible student answers.

Q: What construction problems did you encounter?

A: It was difficult to evenly cut the pipe insulation in half lengthwise.

A: It was a challenge to place the half-pipe channel in the exact calculated positions. We had to estimate where to tape the vinyl holders/brackets and their inclinations so that the pipe insulation would pass through all the desired points with the required slope. A slightly misplaced holder affects the path’s shape.

A: It is difficult to place the pipe insulation channel with precision of less than one-quarter inch.

Q: What surprise, unanticipated (not considered) details did you encounter during construction?

A: A way to keep vertical and stable by itself the cardboard backing used to hold the coaster path.

A: The cardboard has a bit of curvature so the path is “not exactly” two dimensional.

A: If the pipe insulation was slightly twisted, it could cause a marble to fly away.

Q: Did your prototype model behave as expected?

A: In general, it behaved as expected—the marble passed all the hills—but its velocity at the end of the path was higher than the expected “almost zero” velocity.

Q: If not behaving as expected, why? What conclusions can be made?

A: The path hill heights were set lower than the maximum calculated heights, so the velocity of the marble along the path was always greater than the calculated velocity.

Q: What were some failures and their solutions?

A: The model failed if we were not careful with the way the marble was dropped at the beginning of the path. If not dropped smoothly, the marble may bounce past the first inches of the path, not attain the calculated velocity and not pass the first hill. So we needed to be careful to place and release the marble gently.

A: The model failed when we used a marble of diameter equal to the pipe insulation internal diameter. In this case, the friction on the marble is more because many points of contact exist, instead of just one, as estimated in the calculations, and then the marble stops sooner. So we made a point to always use marbles with diameters less than the pipe insulation internal diameter.

You are welcome to contact the author, Miguel R. Ramirez, at [email protected] or 832-386-2922 to ask any questions related to this activity.

For lower grades, make simplifications in the path design process, but not in the physics or calculations to test the models. Instructor guidance for this approach:

- Have students with no calculus background draft by hand the roller coaster path, while keeping the same restrictions: no loops and a simple Russian Mountain path composed of parabola-like curves.
- For the calculations, use the Rolling with Friction Calculations No Calculus Example so that students do not have to perform the complex calculations by themselves. Graphs of the path, values, the rolling marble’s velocities along the path, and approximated friction values, are automatically displayed. Formula (1) is included in this worksheet to test design functionality.
- Have students draft their roller coaster paths by hand on graph paper and determine the x-y coordinates of points on the path (Figure 18A). Remind them that the drawings they make also serve as scaled blueprints of the models they will create, and suggest they use a 1-unit: 1-inch scale. Then direct them to enter these sketch coordinates into the no calculus spreadsheet in the x-y -columns in worksheet Path . Then, the designed path appears as a graph next to these columns (18B). The path is still a sequence of straight segments but it is softened in the graph.

In addition, the velocities along the path are automatically calculated and displayed. The velocities are graphed below the path’s graph, and the values are displayed in a column to the right of the graphs (Figure 18B).

Changes made in the x-y column values are immediately displayed on the path’s graph, enabling students to see whether or not their changes produce an odd path. Expect students to use the process of set points > check > modify > visualize, until a functional path is obtained.

Any #NUM! error that shows up for an entry in the velocities column indicates a point corresponding to that entry where the body/marble stops rolling, indicating that the design requires modification for that portion.

The no-calculus spreadsheet also estimates the friction force along every segment of the path using a non-calculus alternative to formula (15):

- Depending on your students’ math and physics background, decide how much of the associated lesson is convenient to teach. Even if the students are not ready for the entire lesson, you must present formula (1) to them because it is the main tool to check whether the designs work or not. To understand this formula and why the spreadsheet works, students must understand that:
- Friction is a non-conservative or dissipative force. It always opposes movement and its effect can be measured by evaluating the work it does. For a solid sphere rolling on an inclined straight surface, the work done by friction is proportional to the weight of the object and the surface steepness (19).
- When friction is present, the mechanical energy (ME) of a system: kinetic energy (K) + potential energy (U), is not constant. The work-energy theorem (18) states that the ME losses are equivalent to the work done by the friction force.
- The velocity of a solid sphere rolling on an inclined surface (1), can be determined using the work-energy theorem (18)
- A final recommendation: Use of the provided spreadsheet requires basic Excel® knowledge. The teacher must be familiar with this spreadsheet before the activity. The spreadsheet is designed to hold up to 40 points but can be modified as needed.

Project support AP Calculus First Semester Project online tutorial for students at Sophia (by the author): https://www.sophia.org/playlists/ap-calculus-first-semester-project , on the following topics:

- Friction concepts and solutions for the problem of a body rolling on a surface with friction
- Rotational kinematics and dynamics
- Piecewise differentiable functions

Project support for Excel® and PowerPoint®:

Excel: Get Started with Formulas and Functions . Microsoft Office Support. https://support.office.com/en-us/article/Get-Started-with-Formulas-and-Functions-e0b10c56-700c-4961-a7b2-a0fc5866735e

Tips for Making Effective PowerPoint Presentations . May 28, 2009. National Conference of State Legislatures. http://www.ncsl.org/legislators-staff/legislative-staff/legislative-staff-coordinating-committee/tips-for-making-effective-powerpoint-presentations.aspx

Microsoft PowerPoint Tutorials. Electric Teacher. http://www.electricteacher.com/tutorial3.htm

High school students learn how engineers mathematically design roller coaster paths using the approach that a curved path can be approximated by a sequence of many short inclines. They apply basic calculus and the work-energy theorem for non-conservative forces to quantify the friction along a curve...

Students explore the physics exploited by engineers in designing today's roller coasters, including potential and kinetic energy, friction and gravity. During the associated activity, students design, build and analyze model roller coasters they make using foam tubing and marbles (as the cars).

Students build their own small-scale model roller coasters using pipe insulation and marbles, and then analyze them using physics principles learned in the associated lesson. They examine conversions between kinetic and potential energy and frictional effects to design roller coasters that are compl...

Students act as mechanical, civil, and structural engineers as they design and build a roller coaster with their teammates that allows a table tennis ball to roll through the entire model unassisted. As students design and build their roller coaster, they will learn about kinetic and potential energ...

Alonso, Marcelo, & Edward, Finn. Fundamental University Physics, Volume 1 Mechanics . Addison-Wesley, 1966 (Spanish version: Fisica Volumen 1, Mecanica . Fondo Educativo Interamericano, 1976).

Briggs, William L., Cochran, Lyle, and Gillett, Bernard. Calculus AP Edition . Upper Saddle River, NJ: Pearson Education, 2014.

Brooks, Meade. Physics Concepts in Action | Physics Roller Coaster . 2014. Collin College, Frisco, TX. http://iws.collin.edu/mbrooks/student%20research/projects/Rollercoaster%20Simulation/index.html

Demana, Franklin, et al. Precalculus, Graphical, Numerical, Algebraic . Second Edition. Pearson, 2016.

Design a Roller Coaster (step by step applet). 2016. Amusement Park Physics: What are the forces behind the fun? Annenberg Foundation. https://www.learner.org/exhibits/parkphysics/coaster/

Giancoli, Douglas C. “Energy dissipation (friction) in a roller coaster.” Physics for Scientists and Engineers with Modern Physics. Google Book. Pp. 198. (R1) https://books.google.com/books?id=xz-UEdtRmzkC&pg=PA198&lpg=PA198&dq=energy+dissipation+in+a+roller+coaster&source=bl&ots=3N9n_vS6HA&sig=yKGlmZOWOrKKzfgw31vSMT_Bq6s&hl=en&sa=X&ved=0ahUKEwibroKfv7jSAhVETCYKHaalCl04KBDoAQgjMAI#v=onepage&q=energy%20dissipation%20in%20a%20roller%20coaster&f=false

Halliday, David, & Resnick, Robert. Physics, Parts 1 and 2 Combined. Second Edition . John Wiley & Sons, Inc., 1966 (Spanish version: Fisica, Edicion Combinada Partes I y II, CECSA, Tercera Edicion, 1976).

Harris, Benson. University Physics . Second Edition, John Wiley & Sons, Inc., 1995.

How Do Roller Coasters Work? Wonder of the Day #1239. Physical Science, Science, Wonderopolis.org. National Center for Families Learning. http://wonderopolis.org/wonder/how-do-roller-coasters-work

Larson, Ron, Edwards, Bruce, and Hostetler, Robert P. Calculus of a Single Variable. Eighth Edition. Boston, MA: Houghton-Mifflin, 2006.

Liddle, Scott. Physics of Roller Coasters (lesson). 2007. TeachEngineering Digital Library. https://www.teachengineering.org/lessons/view/duk_rollercoaster_music_less

Liddle, Scott: Building Roller Coasters (activity). 2007. TeachEngineering Digital Library. https://www.teachengineering.org/activities/view/duk_rollercoaster_music_act

Morrow, Mandy. The 10 Most Expensive Roller Coasters in the World . The Richest. June 11, 2014. http://www.therichest.com/luxury/most-expensive/the-10-most-expensive-roller-coasters-in-the-world/

Neumann, Erik. Roller Coaster (physics, equations, graphs; great interactive applet to make custom roller coasters). Last revised June 5, 2017. MyPhysicsLab.com. http://www.myphysicslab.com/roller/roller-single/roller-single-en.html

Pownal, Malcom. Functions and Graphs, Calculus Preparatory Mathematics , Prentice-Hall, 1983.

Roller Coaster Physics . Real-World Physics Problems.com. http://www.real-world-physics-problems.com/roller-coaster-physics.html

Sastamoinen, Shawna. Roller Coaster Physics: The Science Behind the Thrills . 2002. Physics 211X, University of Alaska Fairbanks. http://ffden-2.phys.uaf.edu/211_fall2002.web.dir/shawna_sastamoinen/Roller_Coasters.htm

Young, Hugh D., & Freedman, Roger. University Physics with Modern Physics .14th Edition, Pearson, 2016.

## Contributors

Supporting program, acknowledgements.

The author expresses his thanks to Tony Gardea, Galena Park High School Principal; Seretha Augustine, Galena Park High School Associate Principal for Curriculum; and Gerard Kwaitkowski, Galena Park High School Math Specialist.

Last modified: February 11, 2022

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## MS Word Cover Page Templates

Download, personalize & print, mathematics project front page designs.

Posted By: admin 26/10/2022

Your project needs a front page if you want it to make an impression on the reader. The front page is a clear sign that you have worked on the content of the project as well as its appearance.

It is the first page of the assignment of a math student who wants people to see the project file and recognize that it is a math project at a first glance. This front page describes the mathematics project the student has worked on.

## Information to be shown on the front page of the project?

The main information that you are allowed to showcase on the front page of your math project includes:

- The name of the institute in which the student has been enrolled
- Title of the project
- Name and roll number of the student

Depending on the personal choice of a student, a front page can be filled with lots of other details that are relevant to the math project.

## Attributes of a great front-page design:

Know the following before you start designing a cover page for a mathematics project:

## Know the purpose of the front page:

When you are clear about the objective of designing a front page for your mathematics front page, you will not do anything wrong. In general, a front page is always designed to reflect the math project that has been kept inside it. In other words, the front page always introduces the project it covers. Therefore, it should front only introductory details.

## Know which design will be best:

When it comes to designing the front page of the math project, students often feel overwhelmed because they don’t know what type of design will go with the project itself. Due to this confusion, they often choose the wrong design. However, if you are aware of the project that you have chosen, you can easily incorporate its reflection on the front page.

## Align the content with the design:

Many students make the mistake that they add the content to the front page wherever they find space. However, this is not the right strategy. If you are working on the front page and you have known all the intricacies of the design, you will know where the content can be added and where it will look reasonable. Sometimes, taking care of margins and orientation makes the job easy.

## Use the template:

Students of mathematics often struggle with designing the front page as they are not generally well-versed in the artistic aspects of the front page. For such students, different websites provide different types of front pages. Some front pages of mathematics can go with any type of project.

So, if you are not well aware of the design aspects, you can choose the generic front page of the subject of mathematics and use it as your project’s first page after certain types of modifications.

The template, no doubt, is very helpful for students who want to focus on their project instead of the front page.

MS Word File 3MB

MS Word File 2MB

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This paper presents ideas for constructing mathematics homework assignments based on lessons from the cognitive science of learning and memory. In particular, we focus on two popular techniques from the field: spaced practice and mixed practice.

Tools for School and District Leaders. The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers ...

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75. The Moore Method: A Pathway to Learner-Centered Instruction, Charles A. Coppin, W. Ted Mahavier, E. Lee May, and G. Edgar Parker. 76. The Beauty of Fractals: Six Different Views, Denny Gulick and Jon Scott, Editors. 77. Mathematical Time Capsules: Historical Modules for the Mathematics Classroom, Dick Jardine and Amy Shell-Gel - lasch, Editors. 78.

Structure of a PSMT for Maths Methods Tips for Writing a PSMT Breaking Down the Instrument-Specific Marking Guide (ISMG) What is a Problem-Solving and Modelling Task (PSMT)? A PSMT is an assessment task designed to evaluate your ability to respond to an investigative mathematical scenario or stimulus.

Incorporating Lessons from Cognitive Science: Spacing and Mixing in Mathematics Homework Assignment Design Lori Carmack ABSTRACT This paper presents ideas for constructing mathematics...

The College Mathematics Journal. 23, 5 (1992): 362 - 372. Winkelmes, Mary-Ann. "Building Assignments that Teach." Essays on Teaching Excellence. Vol. 19, no. 5 POD Network in Higher Education, (2008). _____. Transparency in Teaching: Faculty Share Data and Improve Students' ... assignment design at their own discretion, ...

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This study used a randomized pretest-posttest clustered design to examine the effect of 3 homework purposes (i.e., practice, preparation, and extension) on 6th graders' mathematics achievement and how this relationship was modulated by the amount of completed homework.A total of 27 mathematics teachers and their 638 students participated in this study.

For example, the curriculum needed to work for schools where nightly, graded assignments are expected; schools where no work done outside of class is graded; and schools who take a feedback-only approach for any formative work. IM 6-8 Math was released in 2017, and IM Algebra 1, Geometry, and Algebra 2 in 2019.

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Noticing individual differences in the classroom was an important element of educational discourse related to mathematics education. Mathematical creativity was an aspect of the classroom that was getting deep attention at this time because it paid attention to differences, and lateral thinking was one of the approaches related to creativity. However, the design of mathematical assignments ...

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Students apply high school-level differential calculus and physics to the design of two-dimensional roller coasters in which the friction force is considered, as explained in the associated lesson. In a challenge the mirrors real-world engineering, the designed roller coaster paths must be made from at least five differentiable functions that are put together such that the resulting piecewise ...

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Math Worksheet Generator Create and Print Your Own Custom Math Worksheet s for Various Levels and Topics! Choose a Main Topic to Create Custom Math Worksheets Addition Algebra Decimals Division Fractions Factors and Multiples Mixed Operations Multiplication Number Charts and Graphs Pre-Algebra Subtraction Time Word Problems Factors, Multiples..

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